Alternative Analysis of the Michaelis-Menten Equations

ERIC Educational Resources Information Center

Krogstad, Harald E.; Dawed, Mohammed Yiha; Tegegne, Tadele Tesfa

2011-01-01

Courses in mathematical modelling are always in need of simple, illustrative examples. The Michaelis-Menten reaction kinetics equations have been considered to be a basic example of scaling and singular perturbation. However, the leading order approximations do not easily show the expected behaviour, and this note proposes a different perturbation…

Modelling a Simple Mechanical System.

ERIC Educational Resources Information Center

Morland, Tim

1999-01-01

Provides an example of the modeling power of Mathematics, demonstrated in a piece of A-Level student coursework which was undertaken as part of the MEI Structured Mathematics scheme. A system of two masses and two springs oscillating in one dimension is found to be accurately modeled by a system of linear differential equations. (Author/ASK)

A Simple Interactive Software Package for Plotting, Animating, and Calculating

ERIC Educational Resources Information Center

Engelhardt, Larry

2012-01-01

We introduce a new open source (free) software package that provides a simple, highly interactive interface for carrying out certain mathematical tasks that are commonly encountered in physics. These tasks include plotting and animating functions, solving systems of coupled algebraic equations, and basic calculus (differentiating and integrating…

Ultrasensitive dual phosphorylation dephosphorylation cycle kinetics exhibits canonical competition behavior

NASA Astrophysics Data System (ADS)

Huang, Qingdao; Qian, Hong

2009-09-01

We establish a mathematical model for a cellular biochemical signaling module in terms of a planar differential equation system. The signaling process is carried out by two phosphorylation-dephosphorylation reaction steps that share common kinase and phosphatase with saturated enzyme kinetics. The pair of equations is particularly simple in the present mathematical formulation, but they are singular. A complete mathematical analysis is developed based on an elementary perturbation theory. The dynamics exhibits the canonical competition behavior in addition to bistability. Although widely understood in ecological context, we are not aware of a full range of biochemical competition in a simple signaling network. The competition dynamics has broad implications to cellular processes such as cell differentiation and cancer immunoediting. The concepts of homogeneous and heterogeneous multisite phosphorylation are introduced and their corresponding dynamics are compared: there is no bistability in a heterogeneous dual phosphorylation system. A stochastic interpretation is also provided that further gives intuitive understanding of the bistable behavior inside the cells.

The Identities Hidden in the Matching Laws, and Their Uses

ERIC Educational Resources Information Center

Thorne, David R.

2010-01-01

Various theoretical equations have been proposed to predict response rate as a function of the rate of reinforcement. If both the rate and probability of reinforcement are considered, a simple identity, defining equation, or "law" holds. This identity places algebraic constraints on the allowable forms of our mathematical models and can help…

Matrix Solution of Coupled Differential Equations and Looped Car Following Models

ERIC Educational Resources Information Center

McCartney, Mark

2008-01-01

A simple mathematical model for the behaviour of how vehicles follow each other along a looped stretch of road is described. The resulting coupled first order differential equations are solved using appropriate matrix techniques and the physical significance of the model is discussed. A number possible classroom exercises are suggested to help…

Getting to the Bottom of a Ladder Problem

ERIC Educational Resources Information Center

McCartney, Mark

2002-01-01

In this paper, the author introduces a simple problem relating to a pair of ladders. A mathematical model of the problem produces an equation which can be solved in a number of ways using mathematics appropriate to "A" level students or first year undergraduates. The author concludes that the ladder problem can be used in class to develop and…

Simple Examples of the Interpretation of Changes in Kinetic and Potential Energy under Galilean Transformations

ERIC Educational Resources Information Center

Ginsberg, Edw S.

2018-01-01

The compatibility of the Newtonian formulation of mechanical energy and the transformation equations of Galilean relativity is demonstrated for three simple examples of motion treated in most introductory physics courses (free fall, a frictionless inclined plane, and a mass/spring system). Only elementary concepts and mathematics, accessible to…

Solving ordinary differential equations by electrical analogy: a multidisciplinary teaching tool

NASA Astrophysics Data System (ADS)

Sanchez Perez, J. F.; Conesa, M.; Alhama, I.

2016-11-01

Ordinary differential equations are the mathematical formulation for a great variety of problems in science and engineering, and frequently, two different problems are equivalent from a mathematical point of view when they are formulated by the same equations. Students acquire the knowledge of how to solve these equations (at least some types of them) using protocols and strict algorithms of mathematical calculation without thinking about the meaning of the equation. The aim of this work is that students learn to design network models or circuits in this way; with simple knowledge of them, students can establish the association of electric circuits and differential equations and their equivalences, from a formal point of view, that allows them to associate knowledge of two disciplines and promote the use of this interdisciplinary approach to address complex problems. Therefore, they learn to use a multidisciplinary tool that allows them to solve these kinds of equations, even students of first course of engineering, whatever the order, grade or type of non-linearity. This methodology has been implemented in numerous final degree projects in engineering and science, e.g., chemical engineering, building engineering, industrial engineering, mechanical engineering, architecture, etc. Applications are presented to illustrate the subject of this manuscript.

Keep Your Distance! Using Second-Order Ordinary Differential Equations to Model Traffic Flow

ERIC Educational Resources Information Center

McCartney, Mark

2004-01-01

A simple mathematical model for how vehicles follow each other along a stretch of road is presented. The resulting linear second-order differential equation with constant coefficients is solved and interpreted. The model can be used as an application of solution techniques taught at first-year undergraduate level and as a motivator to encourage…

Introducing Algebraic Structures through Solving Equations: Vertical Content Knowledge for K-12 Mathematics Teachers

ERIC Educational Resources Information Center

Wasserman, Nicholas H.

2014-01-01

Algebraic structures are a necessary aspect of algebraic thinking for K-12 students and teachers. An approach for introducing the algebraic structure of groups and fields through the arithmetic properties required for solving simple equations is summarized; the collective (not individual) importance of these axioms as a foundation for algebraic…

Taguchi method for partial differential equations with application in tumor growth.

PubMed

Ilea, M; Turnea, M; Rotariu, M; Arotăriţei, D; Popescu, Marilena

2014-01-01

The growth of tumors is a highly complex process. To describe this process, mathematical models are needed. A variety of partial differential mathematical models for tumor growth have been developed and studied. Most of those models are based on the reaction-diffusion equations and mass conservation law. A variety of modeling strategies have been developed, each focusing on tumor growth. Systems of time-dependent partial differential equations occur in many branches of applied mathematics. The vast majority of mathematical models in tumor growth are formulated in terms of partial differential equations. We propose a mathematical model for the interactions between these three cancer cell populations. The Taguchi methods are widely used by quality engineering scientists to compare the effects of multiple variables, together with their interactions, with a simple and manageable experimental design. In Taguchi's design of experiments, variation is more interesting to study than the average. First, Taguchi methods are utilized to search for the significant factors and the optimal level combination of parameters. Except the three parameters levels, other factors levels other factors levels would not be considered. Second, cutting parameters namely, cutting speed, depth of cut, and feed rate are designed using the Taguchi method. Finally, the adequacy of the developed mathematical model is proved by ANOVA. According to the results of ANOVA, since the percentage contribution of the combined error is as small. Many mathematical models can be quantitatively characterized by partial differential equations. The use of MATLAB and Taguchi method in this article illustrates the important role of informatics in research in mathematical modeling. The study of tumor growth cells is an exciting and important topic in cancer research and will profit considerably from theoretical input. Interpret these results to be a permanent collaboration between math's and medical oncologists.

A new mathematical solution for predicting char activation reactions

USGS Publications Warehouse

Rafsanjani, H.H.; Jamshidi, E.; Rostam-Abadi, M.

2002-01-01

The differential conservation equations that describe typical gas-solid reactions, such as activation of coal chars, yield a set of coupled second-order partial differential equations. The solution of these coupled equations by exact analytical methods is impossible. In addition, an approximate or exact solution only provides predictions for either reaction- or diffusion-controlling cases. A new mathematical solution, the quantize method (QM), was applied to predict the gasification rates of coal char when both chemical reaction and diffusion through the porous char are present. Carbon conversion rates predicted by the QM were in closer agreement with the experimental data than those predicted by the random pore model and the simple particle model. ?? 2002 Elsevier Science Ltd. All rights reserved.

A new solution procedure for a nonlinear infinite beam equation of motion

NASA Astrophysics Data System (ADS)

Jang, T. S.

2016-10-01

Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth-order nonlinear partial differential equation. To answer the question, a pseudo-parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively,therefore, that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here.

An accessible four-dimensional treatment of Maxwell's equations in terms of differential forms

NASA Astrophysics Data System (ADS)

Sá, Lucas

2017-03-01

Maxwell’s equations are derived in terms of differential forms in the four-dimensional Minkowski representation, starting from the three-dimensional vector calculus differential version of these equations. Introducing all the mathematical and physical concepts needed (including the tool of differential forms), using only knowledge of elementary vector calculus and the local vector version of Maxwell’s equations, the equations are reduced to a simple and elegant set of two equations for a unified quantity, the electromagnetic field. The treatment should be accessible for students taking a first course on electromagnetism.

Simple linear and multivariate regression models.

PubMed

Rodríguez del Águila, M M; Benítez-Parejo, N

2011-01-01

In biomedical research it is common to find problems in which we wish to relate a response variable to one or more variables capable of describing the behaviour of the former variable by means of mathematical models. Regression techniques are used to this effect, in which an equation is determined relating the two variables. While such equations can have different forms, linear equations are the most widely used form and are easy to interpret. The present article describes simple and multiple linear regression models, how they are calculated, and how their applicability assumptions are checked. Illustrative examples are provided, based on the use of the freely accessible R program. Copyright © 2011 SEICAP. Published by Elsevier Espana. All rights reserved.

Equation-free modeling unravels the behavior of complex ecological systems

USGS Publications Warehouse

DeAngelis, Donald L.; Yurek, Simeon

2015-01-01

Ye et al. (1) address a critical problem confronting the management of natural ecosystems: How can we make forecasts of possible future changes in populations to help guide management actions? This problem is especially acute for marine and anadromous fisheries, where the large interannual fluctuations of populations, arising from complex nonlinear interactions among species and with varying environmental factors, have defied prediction over even short time scales. The empirical dynamic modeling (EDM) described in Ye et al.’s report, the latest in a series of papers by Sugihara and his colleagues, offers a promising quantitative approach to building models using time series to successfully project dynamics into the future. With the term “equation-free” in the article title, Ye et al. (1) are suggesting broader implications of their approach, considering the centrality of equations in modern science. From the 1700s on, nature has been increasingly described by mathematical equations, with differential or difference equations forming the basic framework for describing dynamics. The use of mathematical equations for ecological systems came much later, pioneered by Lotka and Volterra, who showed that population cycles might be described in terms of simple coupled nonlinear differential equations. It took decades for Lotka–Volterra-type models to become established, but the development of appropriate differential equations is now routine in modeling ecological dynamics. There is no question that the injection of mathematical equations, by forcing “clarity and precision into conjecture” (2), has led to increased understanding of population and community dynamics. As in science in general, in ecology equations are a key method of communication and of framing hypotheses. These equations serve as compact representations of an enormous amount of empirical data and can be analyzed by the powerful methods of mathematics.

Effective algorithm for ray-tracing simulations of lobster eye and similar reflective optical systems

NASA Astrophysics Data System (ADS)

Tichý, Vladimír; Hudec, René; Němcová, Šárka

2016-06-01

The algorithm presented is intended mainly for lobster eye optics. This type of optics (and some similar types) allows for a simplification of the classical ray-tracing procedure that requires great many rays to simulate. The method presented performs the simulation of a only few rays; therefore it is extremely effective. Moreover, to simplify the equations, a specific mathematical formalism is used. Only a few simple equations are used, therefore the program code can be simple as well. The paper also outlines how to apply the method to some other reflective optical systems.

Mathematical Analysis and Optimization of Infiltration Processes

NASA Technical Reports Server (NTRS)

Chang, H.-C.; Gottlieb, D.; Marion, M.; Sheldon, B. W.

1997-01-01

A variety of infiltration techniques can be used to fabricate solid materials, particularly composites. In general these processes can be described with at least one time dependent partial differential equation describing the evolution of the solid phase, coupled to one or more partial differential equations describing mass transport through a porous structure. This paper presents a detailed mathematical analysis of a relatively simple set of equations which is used to describe chemical vapor infiltration. The results demonstrate that the process is controlled by only two parameters, alpha and beta. The optimization problem associated with minimizing the infiltration time is also considered. Allowing alpha and beta to vary with time leads to significant reductions in the infiltration time, compared with the conventional case where alpha and beta are treated as constants.

A new mathematical approach for shock-wave solution in a dusty plasma

DOE Office of Scientific and Technical Information (OSTI.GOV)

Das, G.C.; Dwivedi, C.B.; Talukdar, M.

1997-12-01

The problem of nonlinear Burger equation in a plasma contaminated with heavy dust grains has been revisited. As discussed earlier [C. B. Dwivedi and B. P. Pandey, Phys. Plasmas {bold 2}, 9 (1995)], the Burger equation originates due to dust charge fluctuation dynamics. A new alternate mathematical approach based on a simple traveling wave formalism has been applied to find out the solution of the derived Burger equation, and the method recovers the known shock-wave solution. This technique, although having its own limitation, predicts successfully the salient features of the weak shock-wave structure in a dusty plasma with dust chargemore » fluctuation dynamics. It is emphasized that this approach of the traveling wave formalism is being applied for the first time to solve the nonlinear wave equation in plasmas. {copyright} {ital 1997 American Institute of Physics.}« less

Searching fundamental information in ordinary differential equations. Nondimensionalization technique.

PubMed

Sánchez Pérez, J F; Conesa, M; Alhama, I; Alhama, F; Cánovas, M

2017-01-01

Classical dimensional analysis and nondimensionalization are assumed to be two similar approaches in the search for dimensionless groups. Both techniques, simplify the study of many problems. The first approach does not need to know the mathematical model, being sufficient a deep understanding of the physical phenomenon involved, while the second one begins with the governing equations and reduces them to their dimensionless form by simple mathematical manipulations. In this work, a formal protocol is proposed for applying the nondimensionalization process to ordinary differential equations, linear or not, leading to dimensionless normalized equations from which the resulting dimensionless groups have two inherent properties: In one hand, they are physically interpreted as balances between counteracting quantities in the problem, and on the other hand, they are of the order of magnitude unity. The solutions provided by nondimensionalization are more precise in every case than those from dimensional analysis, as it is illustrated by the applications studied in this work.

Mathematical Metaphors: Problem Reformulation and Analysis Strategies

NASA Technical Reports Server (NTRS)

Thompson, David E.

2005-01-01

This paper addresses the critical need for the development of intelligent or assisting software tools for the scientist who is working in the initial problem formulation and mathematical model representation stage of research. In particular, examples of that representation in fluid dynamics and instability theory are discussed. The creation of a mathematical model that is ready for application of certain solution strategies requires extensive symbolic manipulation of the original mathematical model. These manipulations can be as simple as term reordering or as complicated as discovery of various symmetry groups embodied in the equations, whereby Backlund-type transformations create new determining equations and integrability conditions or create differential Grobner bases that are then solved in place of the original nonlinear PDEs. Several examples are presented of the kinds of problem formulations and transforms that can be frequently encountered in model representation for fluids problems. The capability of intelligently automating these types of transforms, available prior to actual mathematical solution, is advocated. Physical meaning and assumption-understanding can then be propagated through the mathematical transformations, allowing for explicit strategy development.

A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion.

PubMed

Dipierro, Serena; Valdinoci, Enrico

2018-07-01

Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurring in the Purkinje cells, whose malfunctioning is known to be related to the lack of voluntary coordination and the appearance of tremors. Also, a classical mathematical feature is that (fractional) parabolic equations possess smoothing effects, in contrast with the case of hyperbolic equations, which typically exhibit shocks and discontinuities. In this paper, we show how a simple toy-model of a highly ramified structure, somehow inspired by that of the Purkinje cells, may produce a fractional diffusion via the superposition of travelling waves that solve a hyperbolic equation. This could suggest that the high ramification of the Purkinje cells might have provided an evolutionary advantage of "smoothing" the transmission of signals and avoiding shock propagations (at the price of slowing a bit such transmission). Although an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that it is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media. The case of a travelling concave parabola with sufficiently small curvature is explicitly computed. The new link that we propose between time fractional diffusion and hyperbolic equation also provides a novelty with respect to the usual paradigm relating time fractional diffusion with parabolic equations in the limit. This paper is written in such a way as to be of interest to both biologists and mathematician alike. In order to accomplish this aim, both complete explanations of the objects considered and detailed lists of references are provided.

Gastric residual volume (GRV) and gastric contents measurement by refractometry.

PubMed

Chang, Wei-Kuo; McClave, Stephen A; Hsieh, Chung-Bao; Chao, You-Chen

2007-01-01

Traditional use of gastric residual volumes (GRVs), obtained by aspiration from a nasogastric tube, is inaccurate and cannot differentiate components of the gastric contents (gastric secretion vs delivered formula). The use of refractometry and 3 mathematical equations has been proposed as a method to calculate the formula concentration, GRV, and formula volume. In this paper, we have validated these mathematical equations so that they can be implemented in clinical practice. Each of 16 patients receiving a nasogastric tube had 50 mL of water followed by 100 mL of dietary formula (Osmolite HN, Abbott Laboratories, Columbus, OH) infused into the stomach. After mixing, gastric content was aspirated for the first Brix value (BV) measurement by refractometry. Then, 50 mL of water was infused into the stomach and a second BV was measured. The procedure of infusion of dietary formula (100 mL) and then water (50 mL) was repeated and followed by subsequent BV measurement. The same procedure was performed in an in vitro experiment. Formula concentration, GRV, and formula volume were calculated from the derived mathematical equations. The formula concentrations, GRVs, and formula volumes calculated by using refractometry and the mathematical equations were close to the true values obtained from both in vivo and in vitro validation experiments. Using this method, measurement of the BV of gastric contents is simple, reproducible, and inexpensive. Refractometry and the derived mathematical equations may be used to measure formula concentration, GRV, and formula volume, and also to serve as a tool for monitoring the gastric contents of patients receiving nasogastric feeding.

Is the Reaction Equilibrium Composition in Non-ideal Mixtures Uniquely Determined by the Initial Composition?

NASA Astrophysics Data System (ADS)

Sefcik, Jan

1998-05-01

Reaction equilibrium can be mathematically described by the equilibrium equation and the reaction equilibrium composition can be calculated by solving this equation. It can be proved by non-elementary thermodynamic arguments that for a generic system with given initial composition, temperature and pressure there is a unique stable equilibrium state corresponding to the global minimum of the Gibbs free energy function. However, when the concept of equilibrium is introduced in undergraduate chemistry and chemical engineering courses, such arguments are generally not accessible. When there is a single reaction equilibrium among mixture components and the components form an ideal mixture, it has been demonstrated by a simple, elegant mathematical argument that there is a unique composition satisfying the equilibrium equation. It has been also suggested that this particular argument extends to non-ideal mixtures by simply incorporating activity coefficients. We show that the argument extension to non-ideal systems is not generally valid. Increasing non-ideality can result in non-monotonicity of the function crucial for the simple uniqueness argument, and only later it leads to non-uniqueness and hence phase separation. The main feature responsible for this is a composition dependence of activity coefficients in non-ideal mixtures.

Antioxidant Capacity: Experimental Determination by EPR Spectroscopy and Mathematical Modeling.

PubMed

Polak, Justyna; Bartoszek, Mariola; Chorążewski, Mirosław

2015-07-22

A new method of determining antioxidant capacity based on a mathematical model is presented in this paper. The model was fitted to 1000 data points of electron paramagnetic resonance (EPR) spectroscopy measurements of various food product samples such as tea, wine, juice, and herbs with Trolox equivalent antioxidant capacity (TEAC) values from 20 to 2000 μmol TE/100 mL. The proposed mathematical equation allows for a determination of TEAC of food products based on a single EPR spectroscopy measurement. The model was tested on the basis of 80 EPR spectroscopy measurements of herbs, tea, coffee, and juice samples. The proposed model works for both strong and weak antioxidants (TEAC values from 21 to 2347 μmol TE/100 mL). The determination coefficient between TEAC values obtained experimentally and TEAC values calculated with proposed mathematical equation was found to be R(2) = 0.98. Therefore, the proposed new method of TEAC determination based on a mathematical model is a good alternative to the standard EPR method due to its being fast, accurate, inexpensive, and simple to perform.

Simulating Bone Loss in Microgravity Using Mathematical Formulations of Bone Remodeling

NASA Technical Reports Server (NTRS)

Pennline, James A.

2009-01-01

Most mathematical models of bone remodeling are used to simulate a specific bone disease, by disrupting the steady state or balance in the normal remodeling process, and to simulate a therapeutic strategy. In this work, the ability of a mathematical model of bone remodeling to simulate bone loss as a function of time under the conditions of microgravity is investigated. The model is formed by combining a previously developed set of biochemical, cellular dynamics, and mechanical stimulus equations in the literature with two newly proposed equations; one governing the rate of change of the area of cortical bone tissue in a cross section of a cylindrical section of bone and one governing the rate of change of calcium in the bone fluid. The mechanical stimulus comes from a simple model of stress due to a compressive force on a cylindrical section of bone which can be reduced to zero to mimic the effects of skeletal unloading in microgravity. The complete set of equations formed is a system of first order ordinary differential equations. The results of selected simulations are displayed and discussed. Limitations and deficiencies of the model are also discussed as well as suggestions for further research.

Simple Examples of the Interpretation of Changes in Kinetic and Potential Energy Under Galilean Transformations

NASA Astrophysics Data System (ADS)

Ginsberg, Edw. S.

2018-02-01

The compatibility of the Newtonian formulation of mechanical energy and the transformation equations of Galilean relativity is demonstrated for three simple examples of motion treated in most introductory physics courses (free fall, a frictionless inclined plane, and a mass/spring system). Only elementary concepts and mathematics, accessible to students at that level, are used. Emphasis is on pedagogy and concepts related to the transformation properties of potential energy.

Inattentive Drivers: Making the Solution Method the Model

ERIC Educational Resources Information Center

McCartney, Mark

2003-01-01

A simple car following model based on the solution of coupled ordinary differential equations is considered. The model is solved using Euler's method and this method of solution is itself interpreted as a mathematical model for car following. Examples of possible classroom use are given. (Contains 6 figures.)

As Simple as Possible, But No Simpler: A Gentle Introduction to Simulation Modeling

DTIC Science & Technology

2006-12-01

cultures, people waiting for a bus mimic the concept by standing in a row. However, there are some cultures where no line forms but it is considered...mathematical equations such as the equations of motion Report Documentation Page Form ApprovedOMB No. 0704-0188 Public reporting burden for the...PERSON a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

From Heuristic to Mathematical Modeling of Drugs Dissolution Profiles: Application of Artificial Neural Networks and Genetic Programming

PubMed Central

Mendyk, Aleksander; Güres, Sinan; Szlęk, Jakub; Wiśniowska, Barbara; Kleinebudde, Peter

2015-01-01

The purpose of this work was to develop a mathematical model of the drug dissolution (Q) from the solid lipid extrudates based on the empirical approach. Artificial neural networks (ANNs) and genetic programming (GP) tools were used. Sensitivity analysis of ANNs provided reduction of the original input vector. GP allowed creation of the mathematical equation in two major approaches: (1) direct modeling of Q versus extrudate diameter (d) and the time variable (t) and (2) indirect modeling through Weibull equation. ANNs provided also information about minimum achievable generalization error and the way to enhance the original dataset used for adjustment of the equations' parameters. Two inputs were found important for the drug dissolution: d and t. The extrudates length (L) was found not important. Both GP modeling approaches allowed creation of relatively simple equations with their predictive performance comparable to the ANNs (root mean squared error (RMSE) from 2.19 to 2.33). The direct mode of GP modeling of Q versus d and t resulted in the most robust model. The idea of how to combine ANNs and GP in order to escape ANNs' black-box drawback without losing their superior predictive performance was demonstrated. Open Source software was used to deliver the state-of-the-art models and modeling strategies. PMID:26101544

From Heuristic to Mathematical Modeling of Drugs Dissolution Profiles: Application of Artificial Neural Networks and Genetic Programming.

PubMed

Mendyk, Aleksander; Güres, Sinan; Jachowicz, Renata; Szlęk, Jakub; Polak, Sebastian; Wiśniowska, Barbara; Kleinebudde, Peter

2015-01-01

The purpose of this work was to develop a mathematical model of the drug dissolution (Q) from the solid lipid extrudates based on the empirical approach. Artificial neural networks (ANNs) and genetic programming (GP) tools were used. Sensitivity analysis of ANNs provided reduction of the original input vector. GP allowed creation of the mathematical equation in two major approaches: (1) direct modeling of Q versus extrudate diameter (d) and the time variable (t) and (2) indirect modeling through Weibull equation. ANNs provided also information about minimum achievable generalization error and the way to enhance the original dataset used for adjustment of the equations' parameters. Two inputs were found important for the drug dissolution: d and t. The extrudates length (L) was found not important. Both GP modeling approaches allowed creation of relatively simple equations with their predictive performance comparable to the ANNs (root mean squared error (RMSE) from 2.19 to 2.33). The direct mode of GP modeling of Q versus d and t resulted in the most robust model. The idea of how to combine ANNs and GP in order to escape ANNs' black-box drawback without losing their superior predictive performance was demonstrated. Open Source software was used to deliver the state-of-the-art models and modeling strategies.

Multiple-generator errors are unavoidable under model misspecification.

PubMed

Jewett, D L; Zhang, Z

1995-08-01

Model misspecification poses a major problem for dipole source localization (DSL) because it causes insidious multiple-generator errors (MulGenErrs) to occur in the fitted dipole parameters. This paper describes how and why this occurs, based upon simple algebraic considerations. MulGenErrs must occur, to some degree, in any DSL analysis of real data because there is model misspecification and mathematically the equations used for the simultaneously active generators must be of a different form than the equations for each generator active alone.

Distribution of thermal neutrons in a temperature gradient

NASA Astrophysics Data System (ADS)

Molinari, V. G.; Pollachini, L.

A method to determine the spatial distribution of the thermal spectrum of neutrons in heterogeneous systems is presented. The method is based on diffusion concepts and has a simple mathematical structure which increases computing efficiency. The application of this theory to the neutron thermal diffusion induced by a temperature gradient, as found in nuclear reactors, is described. After introducing approximations, a nonlinear equation system representing the neutron temperature is given. Values of the equation parameters and its dependence on geometrical factors and media characteristics are discussed.

Thrust and Propulsive Efficiency from an Instructive Viewpoint

ERIC Educational Resources Information Center

Kaufman, Richard D.

2010-01-01

In a typical engineering or physics curriculum, the momentum equation is used for the determination of jet engine thrust. Even a simple thrust analysis requires a heavy emphasis on mathematics that can cause students and engineers to lose a physical perspective on thrust. This article provides for this physical understanding using only static…

Undergraduate paramedic students cannot do drug calculations.

PubMed

Eastwood, Kathryn; Boyle, Malcolm J; Williams, Brett

2012-01-01

Previous investigation of drug calculation skills of qualified paramedics has highlighted poor mathematical ability with no published studies having been undertaken on undergraduate paramedics. There are three major error classifications. Conceptual errors involve an inability to formulate an equation from information given, arithmetical errors involve an inability to operate a given equation, and finally computation errors are simple errors of addition, subtraction, division and multiplication. The objective of this study was to determine if undergraduate paramedics at a large Australia university could accurately perform common drug calculations and basic mathematical equations normally required in the workplace. A cross-sectional study methodology using a paper-based questionnaire was administered to undergraduate paramedic students to collect demographical data, student attitudes regarding their drug calculation performance, and answers to a series of basic mathematical and drug calculation questions. Ethics approval was granted. The mean score of correct answers was 39.5% with one student scoring 100%, 3.3% of students (n=3) scoring greater than 90%, and 63% (n=58) scoring 50% or less, despite 62% (n=57) of the students stating they 'did not have any drug calculations issues'. On average those who completed a minimum of year 12 Specialist Maths achieved scores over 50%. Conceptual errors made up 48.5%, arithmetical 31.1% and computational 17.4%. This study suggests undergraduate paramedics have deficiencies in performing accurate calculations, with conceptual errors indicating a fundamental lack of mathematical understanding. The results suggest an unacceptable level of mathematical competence to practice safely in the unpredictable prehospital environment.

Development of abstract mathematical reasoning: the case of algebra

PubMed Central

Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja

2014-01-01

Algebra typically represents the students’ first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students’ ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16–17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15–16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students’ transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition. PMID:25228874

Development of abstract mathematical reasoning: the case of algebra.

PubMed

Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja

2014-01-01

Algebra typically represents the students' first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students' ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16-17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15-16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students' transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.

A new adaptive estimation method of spacecraft thermal mathematical model with an ensemble Kalman filter

NASA Astrophysics Data System (ADS)

Akita, T.; Takaki, R.; Shima, E.

2012-04-01

An adaptive estimation method of spacecraft thermal mathematical model is presented. The method is based on the ensemble Kalman filter, which can effectively handle the nonlinearities contained in the thermal model. The state space equations of the thermal mathematical model is derived, where both temperature and uncertain thermal characteristic parameters are considered as the state variables. In the method, the thermal characteristic parameters are automatically estimated as the outputs of the filtered state variables, whereas, in the usual thermal model correlation, they are manually identified by experienced engineers using trial-and-error approach. A numerical experiment of a simple small satellite is provided to verify the effectiveness of the presented method.

Deriving the Regression Equation without Using Calculus

ERIC Educational Resources Information Center

Gordon, Sheldon P.; Gordon, Florence S.

2004-01-01

Probably the one "new" mathematical topic that is most responsible for modernizing courses in college algebra and precalculus over the last few years is the idea of fitting a function to a set of data in the sense of a least squares fit. Whether it be simple linear regression or nonlinear regression, this topic opens the door to applying the…

The Pendulum: A Paradigm for the Linear Oscillator

ERIC Educational Resources Information Center

Newburgh, Ronald

2004-01-01

The simple pendulum is a model for the linear oscillator. The usual mathematical treatment of the problem begins with a differential equation that one solves with the techniques of the differential calculus, a formal process that tends to obscure the physics. In this paper we begin with a kinematic description of the motion obtained by experiment…

Real-Time Exponential Curve Fits Using Discrete Calculus

NASA Technical Reports Server (NTRS)

Rowe, Geoffrey

2010-01-01

An improved solution for curve fitting data to an exponential equation (y = Ae(exp Bt) + C) has been developed. This improvement is in four areas -- speed, stability, determinant processing time, and the removal of limits. The solution presented avoids iterative techniques and their stability errors by using three mathematical ideas: discrete calculus, a special relationship (be tween exponential curves and the Mean Value Theorem for Derivatives), and a simple linear curve fit algorithm. This method can also be applied to fitting data to the general power law equation y = Ax(exp B) + C and the general geometric growth equation y = Ak(exp Bt) + C.

The diffusion approximation. An application to radiative transfer in clouds

NASA Technical Reports Server (NTRS)

Arduini, R. F.; Barkstrom, B. R.

1976-01-01

It is shown how the radiative transfer equation reduces to the diffusion equation. To keep the mathematics as simple as possible, the approximation is applied to a cylindrical cloud of radius R and height h. The diffusion equation separates in cylindrical coordinates and, in a sample calculation, the solution is evaluated for a range of cloud radii with cloud heights of 0.5 km and 1.0 km. The simplicity of the method and the speed with which solutions are obtained give it potential as a tool with which to study the effects of finite-sized clouds on the albedo of the earth-atmosphere system.

Theoretically Investigating the Nature of Spacetime- A grand definition of what clocks measure

NASA Astrophysics Data System (ADS)

Egie, Meru

Einstein's special theory of relativity established time as a dimension of reality, explaining physically the mathematical stipulations of Lorentz transformation equations that are required to keep the validity of Maxwell's equations of light and explain the null result of Michelson-Morley experiment. Our current understanding of time is relativistic, that is time is not absolute but runs differently depending on the frame of reference, yet this description uncovers so little about the fundamental reality of time. Using mathematical arguments derived from a simple thought experiment, both Lorentz transformation equations and Einstein's far reaching conclusions of his 1905 paper on the electrodynamics of moving bodies are obtained with arguments that suggest no prior knowledge of both Einstein and Lorentz works. This work attempts uncovering the fundamental nature of what clocks measure and a major implication of this is that the fourth dimension could just be a persistent illusion caused by the existence of space. Gratitude to Mr. Jon Egie for his support and Aghogo Rita for her listening ears.

Inclusion of unsteady aerodynamics in longitudinal parameter estimation from flight data. [use of vortices and mathematical models for parameterization from flight characteristics

NASA Technical Reports Server (NTRS)

Queijo, M. J.; Wells, W. R.; Keskar, D. A.

1979-01-01

A simple vortex system, used to model unsteady aerodynamic effects into the rigid body longitudinal equations of motion of an aircraft, is described. The equations are used in the development of a parameter extraction algorithm. Use of the two parameter-estimation modes, one including and the other omitting unsteady aerodynamic modeling, is discussed as a means of estimating some acceleration derivatives. Computer generated data and flight data, used to demonstrate the use of the parameter-extraction algorithm are studied.

The Parker-Sochacki Method--A Powerful New Method for Solving Systems of Differential Equations

NASA Astrophysics Data System (ADS)

Rudmin, Joseph W.

2001-04-01

The Parker-Sochacki Method--A Powerful New Method for Solving Systems of Differential Equations Joseph W. Rudmin (Physics Dept, James Madison University) A new system of solving systems of differential equations will be presented, which has been developed by J. Edgar Parker and James Sochacki, of the James Madison University Mathematics Department. The method produces MacClaurin Series solutions to systems of differential equations, with the coefficients in either algebraic or numerical form. The method yields high-degree solutions: 20th degree is easily obtainable. It is conceptually simple, fast, and extremely general. It has been applied to over a hundred systems of differential equations, some of which were previously unsolved, and has yet to fail to solve any system for which the MacClaurin series converges. The method is non-recursive: each coefficient in the series is calculated just once, in closed form, and its accuracy is limited only by the digital accuracy of the computer. Although the original differential equations may include any mathematical functions, the computational method includes ONLY the operations of addition, subtraction, and multiplication. Furthermore, it is perfectly suited to parallel -processing computer languages. Those who learn this system will never use Runge-Kutta or predictor-corrector methods again. Examples will be presented, including the classical many-body problem.

Perceptual learning modules in mathematics: enhancing students' pattern recognition, structure extraction, and fluency.

PubMed

Kellman, Philip J; Massey, Christine M; Son, Ji Y

2010-04-01

Learning in educational settings emphasizes declarative and procedural knowledge. Studies of expertise, however, point to other crucial components of learning, especially improvements produced by experience in the extraction of information: perceptual learning (PL). We suggest that such improvements characterize both simple sensory and complex cognitive, even symbolic, tasks through common processes of discovery and selection. We apply these ideas in the form of perceptual learning modules (PLMs) to mathematics learning. We tested three PLMs, each emphasizing different aspects of complex task performance, in middle and high school mathematics. In the MultiRep PLM, practice in matching function information across multiple representations improved students' abilities to generate correct graphs and equations from word problems. In the Algebraic Transformations PLM, practice in seeing equation structure across transformations (but not solving equations) led to dramatic improvements in the speed of equation solving. In the Linear Measurement PLM, interactive trials involving extraction of information about units and lengths produced successful transfer to novel measurement problems and fraction problem solving. Taken together, these results suggest (a) that PL techniques have the potential to address crucial, neglected dimensions of learning, including discovery and fluent processing of relations; (b) PL effects apply even to complex tasks that involve symbolic processing; and (c) appropriately designed PL technology can produce rapid and enduring advances in learning. Copyright © 2009 Cognitive Science Society, Inc.

GENASIS Mathematics : Object-oriented manifolds, operations, and solvers for large-scale physics simulations

NASA Astrophysics Data System (ADS)

Cardall, Christian Y.; Budiardja, Reuben D.

2018-01-01

The large-scale computer simulation of a system of physical fields governed by partial differential equations requires some means of approximating the mathematical limit of continuity. For example, conservation laws are often treated with a 'finite-volume' approach in which space is partitioned into a large number of small 'cells,' with fluxes through cell faces providing an intuitive discretization modeled on the mathematical definition of the divergence operator. Here we describe and make available Fortran 2003 classes furnishing extensible object-oriented implementations of simple meshes and the evolution of generic conserved currents thereon, along with individual 'unit test' programs and larger example problems demonstrating their use. These classes inaugurate the Mathematics division of our developing astrophysics simulation code GENASIS (Gen eral A strophysical Si mulation S ystem), which will be expanded over time to include additional meshing options, mathematical operations, solver types, and solver variations appropriate for many multiphysics applications.

Bayesian parameter estimation for nonlinear modelling of biological pathways.

PubMed

Ghasemi, Omid; Lindsey, Merry L; Yang, Tianyi; Nguyen, Nguyen; Huang, Yufei; Jin, Yu-Fang

2011-01-01

The availability of temporal measurements on biological experiments has significantly promoted research areas in systems biology. To gain insight into the interaction and regulation of biological systems, mathematical frameworks such as ordinary differential equations have been widely applied to model biological pathways and interpret the temporal data. Hill equations are the preferred formats to represent the reaction rate in differential equation frameworks, due to their simple structures and their capabilities for easy fitting to saturated experimental measurements. However, Hill equations are highly nonlinearly parameterized functions, and parameters in these functions cannot be measured easily. Additionally, because of its high nonlinearity, adaptive parameter estimation algorithms developed for linear parameterized differential equations cannot be applied. Therefore, parameter estimation in nonlinearly parameterized differential equation models for biological pathways is both challenging and rewarding. In this study, we propose a Bayesian parameter estimation algorithm to estimate parameters in nonlinear mathematical models for biological pathways using time series data. We used the Runge-Kutta method to transform differential equations to difference equations assuming a known structure of the differential equations. This transformation allowed us to generate predictions dependent on previous states and to apply a Bayesian approach, namely, the Markov chain Monte Carlo (MCMC) method. We applied this approach to the biological pathways involved in the left ventricle (LV) response to myocardial infarction (MI) and verified our algorithm by estimating two parameters in a Hill equation embedded in the nonlinear model. We further evaluated our estimation performance with different parameter settings and signal to noise ratios. Our results demonstrated the effectiveness of the algorithm for both linearly and nonlinearly parameterized dynamic systems. Our proposed Bayesian algorithm successfully estimated parameters in nonlinear mathematical models for biological pathways. This method can be further extended to high order systems and thus provides a useful tool to analyze biological dynamics and extract information using temporal data.

Modeling and control of flexible space platforms with articulated payloads

NASA Technical Reports Server (NTRS)

Graves, Philip C.; Joshi, Suresh M.

1989-01-01

The first steps in developing a methodology for spacecraft control-structure interaction (CSI) optimization are identification and classification of anticipated missions, and the development of tractable mathematical models in each mission class. A mathematical model of a generic large flexible space platform (LFSP) with multiple independently pointed rigid payloads is considered. The objective is not to develop a general purpose numerical simulation, but rather to develop an analytically tractable mathematical model of such composite systems. The equations of motion for a single payload case are derived, and are linearized about zero steady-state. The resulting model is then extended to include multiple rigid payloads, yielding the desired analytical form. The mathematical models developed clearly show the internal inertial/elastic couplings, and are therefore suitable for analytical and numerical studies. A simple decentralized control law is proposed for fine pointing the payloads and LFSP attitude control, and simulation results are presented for an example problem. The decentralized controller is shown to be adequate for the example problem chosen, but does not, in general, guarantee stability. A centralized dissipative controller is then proposed, requiring a symmetric form of the composite system equations. Such a controller guarantees robust closed loop stability despite unmodeled elastic dynamics and parameter uncertainties.

Simple Chaotic Flow with Circle and Square Equilibrium

NASA Astrophysics Data System (ADS)

Gotthans, Tomas; Sprott, Julien Clinton; Petrzela, Jiri

Simple systems of third-order autonomous nonlinear differential equations can exhibit chaotic behavior. In this paper, we present a new class of chaotic flow with a square-shaped equilibrium. This unique property has apparently not yet been described. Such a system belongs to a newly introduced category of chaotic systems with hidden attractors that are interesting and important in engineering applications. The mathematical model is accompanied by an electrical circuit implementation, demonstrating structural stability of the strange attractor. The circuit is simulated with PSpice, constructed, and analyzed (measured).

Tutorial on Fourier space coverage for scattering experiments, with application to SAR

NASA Astrophysics Data System (ADS)

Deming, Ross W.

2010-04-01

The Fourier Diffraction Theorem relates the data measured during electromagnetic, optical, or acoustic scattering experiments to the spatial Fourier transform of the object under test. The theorem is well-known, but since it is based on integral equations and complicated mathematical expansions, the typical derivation may be difficult for the non-specialist. In this paper, the theorem is derived and presented using simple geometry, plus undergraduatelevel physics and mathematics. For practitioners of synthetic aperture radar (SAR) imaging, the theorem is important to understand because it leads to a simple geometric and graphical understanding of image resolution and sampling requirements, and how they are affected by radar system parameters and experimental geometry. Also, the theorem can be used as a starting point for imaging algorithms and motion compensation methods. Several examples are given in this paper for realistic scenarios.

Undergraduate paramedic students cannot do drug calculations

PubMed Central

Eastwood, Kathryn; Boyle, Malcolm J; Williams, Brett

2012-01-01

BACKGROUND: Previous investigation of drug calculation skills of qualified paramedics has highlighted poor mathematical ability with no published studies having been undertaken on undergraduate paramedics. There are three major error classifications. Conceptual errors involve an inability to formulate an equation from information given, arithmetical errors involve an inability to operate a given equation, and finally computation errors are simple errors of addition, subtraction, division and multiplication. The objective of this study was to determine if undergraduate paramedics at a large Australia university could accurately perform common drug calculations and basic mathematical equations normally required in the workplace. METHODS: A cross-sectional study methodology using a paper-based questionnaire was administered to undergraduate paramedic students to collect demographical data, student attitudes regarding their drug calculation performance, and answers to a series of basic mathematical and drug calculation questions. Ethics approval was granted. RESULTS: The mean score of correct answers was 39.5% with one student scoring 100%, 3.3% of students (n=3) scoring greater than 90%, and 63% (n=58) scoring 50% or less, despite 62% (n=57) of the students stating they ‘did not have any drug calculations issues’. On average those who completed a minimum of year 12 Specialist Maths achieved scores over 50%. Conceptual errors made up 48.5%, arithmetical 31.1% and computational 17.4%. CONCLUSIONS: This study suggests undergraduate paramedics have deficiencies in performing accurate calculations, with conceptual errors indicating a fundamental lack of mathematical understanding. The results suggest an unacceptable level of mathematical competence to practice safely in the unpredictable prehospital environment. PMID:25215067

"In Situ" Observation of a Soap-Film Catenoid--A Simple Educational Physics Experiment

ERIC Educational Resources Information Center

Ito, Masato; Sato, Taku

2010-01-01

The solution to the Euler-Lagrange equation is an extremal functional. To understand that the functional is stationary at local extrema (maxima or minima), we propose a physics experiment that involves using a soap film to form a catenoid. A catenoid is a surface that is formed between two coaxial circular rings and is classified mathematically as…

A Theoretical Analysis of the Influence of Electroosmosis on the Effective Ionic Mobility in Capillary Zone Electrophoresis

ERIC Educational Resources Information Center

Hijnen, Hens

2009-01-01

A theoretical description of the influence of electroosmosis on the effective mobility of simple ions in capillary zone electrophoresis is presented. The mathematical equations derived from the space-charge model contain the pK[subscript a] value and the density of the weak acid surface groups as parameters characterizing the capillary. It is…

Rosenzweig instability in a thin layer of a magnetic fluid

NASA Astrophysics Data System (ADS)

Korovin, V. M.

2013-12-01

A simple mathematical model of the initial stage of nonlinear evolution of the Rosenzweig instability in a thin layer of a nonlinearly magnetized viscous ferrofluid coating a horizontal nonmagnetizable plate is constructed on the basis of the system of equations and boundary conditions of ferrofluid dynamics. A dispersion relation is derived and analyzed using the linearized equations of this model. The critical magnetization of the initial layer with a flat free surface, the threshold wavenumber, and the characteristic time of evolution of the most rapidly growing mode are determined. The equation for the neutral stability curve, which is applicable for any physically admissible law of magnetization of a ferrofluid, is derived analytically.

A mathematical model of nutrient influence on fungal competition.

PubMed

Jabed A Choudhury, M; M J Trevelyan, Philip; P Boswell, Graeme

2018-02-07

Fungi have a well-established role in nutrient cycling and are widely used as agents in biological control and in the remediation of polluted landscapes. Competition for resources between different fungal communities is common in these contexts and its outcome impacts on the success of such biotechnological applications. In this investigation a mathematical model is constructed to represent competition between two fungal colonies that have access to different resources. It is shown that the model equations display a multitude of travelling wave solutions and that the outcome of competition between two fungal biomasses can be controlled through the simple manipulation of the nutrient resources available to each. The model equations are also numerically integrated to illustrate the range of outcomes arising from fungal competition and these results are placed in context of established experimental observations. Copyright © 2017 Elsevier Ltd. All rights reserved.

Mathematical modeling of spinning elastic bodies for modal analysis.

NASA Technical Reports Server (NTRS)

Likins, P. W.; Barbera, F. J.; Baddeley, V.

1973-01-01

The problem of modal analysis of an elastic appendage on a rotating base is examined to establish the relative advantages of various mathematical models of elastic structures and to extract general inferences concerning the magnitude and character of the influence of spin on the natural frequencies and mode shapes of rotating structures. In realization of the first objective, it is concluded that except for a small class of very special cases the elastic continuum model is devoid of useful results, while for constant nominal spin rate the distributed-mass finite-element model is quite generally tractable, since in the latter case the governing equations are always linear, constant-coefficient, ordinary differential equations. Although with both of these alternatives the details of the formulation generally obscure the essence of the problem and permit very little engineering insight to be gained without extensive computation, this difficulty is not encountered when dealing with simple concentrated mass models.

Reduced modeling of signal transduction – a modular approach

PubMed Central

Koschorreck, Markus; Conzelmann, Holger; Ebert, Sybille; Ederer, Michael; Gilles, Ernst Dieter

2007-01-01

Background Combinatorial complexity is a challenging problem in detailed and mechanistic mathematical modeling of signal transduction. This subject has been discussed intensively and a lot of progress has been made within the last few years. A software tool (BioNetGen) was developed which allows an automatic rule-based set-up of mechanistic model equations. In many cases these models can be reduced by an exact domain-oriented lumping technique. However, the resulting models can still consist of a very large number of differential equations. Results We introduce a new reduction technique, which allows building modularized and highly reduced models. Compared to existing approaches further reduction of signal transduction networks is possible. The method also provides a new modularization criterion, which allows to dissect the model into smaller modules that are called layers and can be modeled independently. Hallmarks of the approach are conservation relations within each layer and connection of layers by signal flows instead of mass flows. The reduced model can be formulated directly without previous generation of detailed model equations. It can be understood and interpreted intuitively, as model variables are macroscopic quantities that are converted by rates following simple kinetics. The proposed technique is applicable without using complex mathematical tools and even without detailed knowledge of the mathematical background. However, we provide a detailed mathematical analysis to show performance and limitations of the method. For physiologically relevant parameter domains the transient as well as the stationary errors caused by the reduction are negligible. Conclusion The new layer based reduced modeling method allows building modularized and strongly reduced models of signal transduction networks. Reduced model equations can be directly formulated and are intuitively interpretable. Additionally, the method provides very good approximations especially for macroscopic variables. It can be combined with existing reduction methods without any difficulties. PMID:17854494

A Fast Method of Deriving the Kirchhoff Formula for Moving Surfaces

NASA Technical Reports Server (NTRS)

Farassat, F.; Posey, Joe W.

2007-01-01

The Kirchhoff formula for a moving surface is very useful in many wave propagation problems, particularly in the prediction of noise from rotating machinery. Several publications in the last two decades have presented derivations of the Kirchhoff formula for moving surfaces in both time and frequency domains. Here we present a method originally developed by Farassat and Myers in time domain that is both simple and direct. It is based on generalized function theory and the useful concept of imbedding the problem in the unbounded three-dimensional space. We derive an inhomogeneous wave equation with the source terms that involve Dirac delta functions with their supports on the moving data surface. This wave equation is then solved using the simple free space Green's function of the wave equation resulting in the Kirchhoff formula. The algebraic manipulations are minimal and simple. We do not need the Green's theorem in four dimensions and there is no ambiguity in the interpretation of any terms in the final formulas. Furthermore, this method also gives the simplest derivation of the classical Kirchhoff formula which has a fairly lengthy derivation in physics and applied mathematics books. The Farassat-Myers method can be used easily in frequency domain.

A Numerical Study on Microwave Coagulation Therapy

DTIC Science & Technology

2013-01-01

hepatocellular carcinoma (small size liver tumor). Through extensive numerical simulations, we reveal the mathematical relationships between some critical parameters in the therapy, including input power, frequency, temperature, and regions of impact. It is shown that these relationships can be approximated using simple polynomial functions. Compared to solutions of partial differential equations, these functions are significantly easier to compute and simpler to analyze for engineering design and clinical

How students process equations in solving quantitative synthesis problems? Role of mathematical complexity in students' mathematical performance

NASA Astrophysics Data System (ADS)

Ibrahim, Bashirah; Ding, Lin; Heckler, Andrew F.; White, Daniel R.; Badeau, Ryan

2017-12-01

We examine students' mathematical performance on quantitative "synthesis problems" with varying mathematical complexity. Synthesis problems are tasks comprising multiple concepts typically taught in different chapters. Mathematical performance refers to the formulation, combination, and simplification of equations. Generally speaking, formulation and combination of equations require conceptual reasoning; simplification of equations requires manipulation of equations as computational tools. Mathematical complexity is operationally defined by the number and the type of equations to be manipulated concurrently due to the number of unknowns in each equation. We use two types of synthesis problems, namely, sequential and simultaneous tasks. Sequential synthesis tasks require a chronological application of pertinent concepts, and simultaneous synthesis tasks require a concurrent application of the pertinent concepts. A total of 179 physics major students from a second year mechanics course participated in the study. Data were collected from written tasks and individual interviews. Results show that mathematical complexity negatively influences the students' mathematical performance on both types of synthesis problems. However, for the sequential synthesis tasks, it interferes only with the students' simplification of equations. For the simultaneous synthesis tasks, mathematical complexity additionally impedes the students' formulation and combination of equations. Several reasons may explain this difference, including the students' different approaches to the two types of synthesis problems, cognitive load, and the variation of mathematical complexity within each synthesis type.

Synesthesia affects verification of simple arithmetic equations.

PubMed

Ghirardelli, Thomas G; Mills, Carol Bergfeld; Zilioli, Monica K C; Bailey, Leah P; Kretschmar, Paige K

2010-01-01

To investigate the effects of color-digit synesthesia on numerical representation, we presented a synesthete, called SE, in the present study, and controls with mathematical equations for verification. In Experiment 1, SE verified addition equations made up of digits that either matched or mismatched her color-digit photisms or were in black. In Experiment 2A, the addends were presented in the different color conditions and the solution was presented in black, whereas in Experiment 2B the addends were presented in black and the solutions were presented in the different color conditions. In Experiment 3, multiplication and division equations were presented in the same color conditions as in Experiment 1. SE responded significantly faster to equations that matched her photisms than to those that did not; controls did not show this effect. These results suggest that photisms influence the processing of digits in arithmetic verification, replicating and extending previous findings.

The influence of wind-tunnel walls on discrete frequency noise

NASA Technical Reports Server (NTRS)

Mosher, M.

1984-01-01

This paper describes an analytical model that can be used to examine the effects of wind-tunnel walls on discrete frequency noise. First, a complete physical model of an acoustic source in a wind tunnel is described, and a simplified version is then developed. This simplified model retains the important physical processes involved, yet it is more amenable to analysis. Second, the simplified physical model is formulated as a mathematical problem. An inhomogeneous partial differential equation with mixed boundary conditions is set up and then transformed into an integral equation. The integral equation has been solved with a panel program on a computer. Preliminary results from a simple model problem will be shown and compared with the approximate analytic solution.

Collective instabilities

DOE Office of Scientific and Technical Information (OSTI.GOV)

K.Y. Ng

2003-08-25

The lecture covers mainly Sections 2.VIII and 3.VII of the book ''Accelerator Physics'' by S.Y. Lee, plus mode-coupling instabilities and chromaticity-driven head-tail instability. Besides giving more detailed derivation of many equations, simple interpretations of many collective instabilities are included with the intention that the phenomena can be understood more easily without going into too much mathematics. The notations of Lee's book as well as the e{sup jwt} convention are followed.

Modelling biochemical reaction systems by stochastic differential equations with reflection.

PubMed

Niu, Yuanling; Burrage, Kevin; Chen, Luonan

2016-05-07

In this paper, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain D, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain D is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain D, and if not perform an orthogonal projection. It is found that the projection onto the closure D¯ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach. Copyright © 2016 Elsevier Ltd. All rights reserved.

Consistent three-equation model for thin films

NASA Astrophysics Data System (ADS)

Richard, Gael; Gisclon, Marguerite; Ruyer-Quil, Christian; Vila, Jean-Paul

2017-11-01

Numerical simulations of thin films of newtonian fluids down an inclined plane use reduced models for computational cost reasons. These models are usually derived by averaging over the fluid depth the physical equations of fluid mechanics with an asymptotic method in the long-wave limit. Two-equation models are based on the mass conservation equation and either on the momentum balance equation or on the work-energy theorem. We show that there is no two-equation model that is both consistent and theoretically coherent and that a third variable and a three-equation model are required to solve all theoretical contradictions. The linear and nonlinear properties of two and three-equation models are tested on various practical problems. We present a new consistent three-equation model with a simple mathematical structure which allows an easy and reliable numerical resolution. The numerical calculations agree fairly well with experimental measurements or with direct numerical resolutions for neutral stability curves, speed of kinematic waves and of solitary waves and depth profiles of wavy films. The model can also predict the flow reversal at the first capillary trough ahead of the main wave hump.

Equations of Interdoublet Separation during Flagella Motion Reveal Mechanisms of Wave Propagation and Instability

PubMed Central

Bayly, Philip V.; Wilson, Kate S.

2014-01-01

The motion of flagella and cilia arises from the coordinated activity of dynein motor protein molecules arrayed along microtubule doublets that span the length of axoneme (the flagellar cytoskeleton). Dynein activity causes relative sliding between the doublets, which generates propulsive bending of the flagellum. The mechanism of dynein coordination remains incompletely understood, although it has been the focus of many studies, both theoretical and experimental. In one leading hypothesis, known as the geometric clutch (GC) model, local dynein activity is thought to be controlled by interdoublet separation. The GC model has been implemented as a numerical simulation in which the behavior of a discrete set of rigid links in viscous fluid, driven by active elements, was approximated using a simplified time-marching scheme. A continuum mechanical model and associated partial differential equations of the GC model have remained lacking. Such equations would provide insight into the underlying biophysics, enable mathematical analysis of the behavior, and facilitate rigorous comparison to other models. In this article, the equations of motion for the flagellum and its doublets are derived from mechanical equilibrium principles and simple constitutive models. These equations are analyzed to reveal mechanisms of wave propagation and instability in the GC model. With parameter values in the range expected for Chlamydomonas flagella, solutions to the fully nonlinear equations closely resemble observed waveforms. These results support the ability of the GC hypothesis to explain dynein coordination in flagella and provide a mathematical foundation for comparison to other leading models. PMID:25296329

Automatic Semantic Generation and Arabic Translation of Mathematical Expressions on the Web

ERIC Educational Resources Information Center

Doush, Iyad Abu; Al-Bdarneh, Sondos

2013-01-01

Automatic processing of mathematical information on the web imposes some difficulties. This paper presents a novel technique for automatic generation of mathematical equations semantic and Arabic translation on the web. The proposed system facilitates unambiguous representation of mathematical equations by correlating equations to their known…

Exploring the relationship between math anxiety and gender through implicit measurement

PubMed Central

Rubinsten, Orly; Bialik, Noam; Solar, Yael

2012-01-01

Math anxiety, defined as a negative affective response to mathematics, is suggested as a strong antecedent for the low visibility of women in the science and engineering workforce. However, the assumption of gender differences in math anxiety is still being studied and results are inconclusive, probably due to the use of explicit measures such as direct questionnaires. Thus, our primary objective was to investigate the effects of math anxiety on numerical processing in males and females by using a novel affective priming task as an indirect measure. Specifically, university students (23 males and 30 females) completed a priming task in which an arithmetic equation was preceded by one of four types of priming words (positive, neutral, negative, or related to mathematics). Participants were required to indicate whether the equation (simple math facts based on addition, subtraction, multiplication, or division) was true or false. People are typically found to respond to target stimuli more rapidly after presentation of an affectively related prime than after an affectively unrelated one. In the current study, shorter response latencies for positive as compared to negative affective primes were found in the male group. An affective priming effect was found in the female group as well, but with a reversed pattern. That is, significantly shorter response latencies were observed in the female group for negative as compared to positive targets. That is, for females, negative affective primes act as affectively related to simple arithmetic problems. In contrast, males associated positive affect with simple arithmetic. In addition, only females with lower or insignificant negative affect toward arithmetic study at faculties of mathematics and science. We discuss the advantages of examining pure anxiety factors with implicit measures which are free of response factors. In addition it is suggested that environmental factors may enhance the association between math achievements and math anxiety in females. PMID:23087633

Exploring the relationship between math anxiety and gender through implicit measurement.

PubMed

Rubinsten, Orly; Bialik, Noam; Solar, Yael

2012-01-01

Math anxiety, defined as a negative affective response to mathematics, is suggested as a strong antecedent for the low visibility of women in the science and engineering workforce. However, the assumption of gender differences in math anxiety is still being studied and results are inconclusive, probably due to the use of explicit measures such as direct questionnaires. Thus, our primary objective was to investigate the effects of math anxiety on numerical processing in males and females by using a novel affective priming task as an indirect measure. Specifically, university students (23 males and 30 females) completed a priming task in which an arithmetic equation was preceded by one of four types of priming words (positive, neutral, negative, or related to mathematics). Participants were required to indicate whether the equation (simple math facts based on addition, subtraction, multiplication, or division) was true or false. People are typically found to respond to target stimuli more rapidly after presentation of an affectively related prime than after an affectively unrelated one. In the current study, shorter response latencies for positive as compared to negative affective primes were found in the male group. An affective priming effect was found in the female group as well, but with a reversed pattern. That is, significantly shorter response latencies were observed in the female group for negative as compared to positive targets. That is, for females, negative affective primes act as affectively related to simple arithmetic problems. In contrast, males associated positive affect with simple arithmetic. In addition, only females with lower or insignificant negative affect toward arithmetic study at faculties of mathematics and science. We discuss the advantages of examining pure anxiety factors with implicit measures which are free of response factors. In addition it is suggested that environmental factors may enhance the association between math achievements and math anxiety in females.

A tutorial solution to scattering of radiation in a thin atmosphere bounded below by a diffusely reflecting, absorbing surface

NASA Technical Reports Server (NTRS)

Buglia, J. J.

1982-01-01

A simple tutorial method, based on a photon tracking procedure, is described to determine the spherical albedo for a thin atmosphere overlying a reflecting surface. This procedure is used to provide a physical structure with which to interpret the more detailed but highly mathematical analyses presented. The final equations are shown to be in good numerical agreement with more exact solutions for thin atmospheres.

Application of Mathematical Modeling in Potentially Survivable Blast Threats in Military Vehicles

DTIC Science & Technology

2008-12-01

elastic – compression and tension of body under loading if elastic tolerances are exceeded, (b) viscous – when fluid matter is involved in the...lumbar spine biomechanical response. The model is a simple spring and damper system and its equation of motion is represented as: 2...dynamic motion. The seat structural management system was represented using Kelvin spring damper element provided in MADYMO. In the actual seat system

Evaluation of massless-spring modeling of suspension-line elasticity during the parachute unfurling process

NASA Technical Reports Server (NTRS)

Poole, L. R.; Huckins, E. K., III

1972-01-01

A general theory on mathematical modeling of elastic parachute suspension lines during the unfurling process was developed. Massless-spring modeling of suspension-line elasticity was evaluated in detail. For this simple model, equations which govern the motion were developed and numerically integrated. The results were compared with flight test data. In most regions, agreement was satisfactory. However, poor agreement was obtained during periods of rapid fluctuations in line tension.

Matrix De Rham Complex and Quantum A-infinity algebras

NASA Astrophysics Data System (ADS)

Barannikov, S.

2014-04-01

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A ∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin-Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006-48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein-Leites simple associative algebras with odd trace q( N), and gl( N| N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin-Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.

Equating TIMSS Mathematics Subtests with Nonlinear Equating Methods Using NEAT Design: Circle-Arc Equating Approaches

ERIC Educational Resources Information Center

Ozdemir, Burhanettin

2017-01-01

The purpose of this study is to equate Trends in International Mathematics and Science Study (TIMSS) mathematics subtest scores obtained from TIMSS 2011 to scores obtained from TIMSS 2007 form with different nonlinear observed score equating methods under Non-Equivalent Anchor Test (NEAT) design where common items are used to link two or more test…

Mathematical and computational model for the analysis of micro hybrid rocket motor

NASA Astrophysics Data System (ADS)

Stoia-Djeska, Marius; Mingireanu, Florin

2012-11-01

The hybrid rockets use a two-phase propellant system. In the present work we first develop a simplified model of the coupling of the hybrid combustion process with the complete unsteady flow, starting from the combustion port and ending with the nozzle. The physical and mathematical model are adapted to the simulations of micro hybrid rocket motors. The flow model is based on the one-dimensional Euler equations with source terms. The flow equations and the fuel regression rate law are solved in a coupled manner. The platform of the numerical simulations is an implicit fourth-order Runge-Kutta second order cell-centred finite volume method. The numerical results obtained with this model show a good agreement with published experimental and numerical results. The computational model developed in this work is simple, computationally efficient and offers the advantage of taking into account a large number of functional and constructive parameters that are used by the engineers.

Microwave heating and joining of ceramic cylinders: A mathematical model

NASA Technical Reports Server (NTRS)

Booty, Michael R.; Kriegsmann, Gregory A.

1994-01-01

A thin cylindrical ceramic sample is placed in a single mode microwave applicator in such a way that the electric field strength is allowed to vary along its axis. The sample can either be a single rod or two rods butted together. We present a simple mathematical model which describes the microwave heating process. It is built on the assumption that the Biot number of the material is small, and that the electric field is known and uniform throughout the cylinder's cross-section. The model takes the form of a nonlinear parabolic equation of reaction-diffusion type, with a spatially varying reaction term that corresponds to the spatial variation of the electromagnetic field strength in the waveguide. The equation is analyzed and a solution is found which develops a hot spot near the center of the cylindrical sample and which then propagates outwards until it stabilizes. The propagation and stabilization phenomenon concentrates the microwave energy in a localized region about the center where elevated temperatures may be desirable.

Impact of the Equation of State in Models for Surfactant Spreading Experiments

NASA Astrophysics Data System (ADS)

Levy, Rachel

2014-11-01

Pulmonary surfactant spreading models often rely on an equation of state relating surfactant concentration to surface tension. Mathematically, these models have been analyzed with simple functional relationships. However, to model an experiment with a given fluid and surfactant, a physically meaningful equation of state can be derived from experimentally obtained isotherms. We discuss the comparison between model and experiment for NBD-PC lipid (surfactant) spreading on glycerol for an empirically-determined equation of state, and compare those results to simulations with traditionally employed functional forms. In particular we compare the timescales by tracking the leading edge of surfactant, the central fluid height and dynamics of the Marangoni ridge. We consider both outward spreading of a disk-shaped region of surfactant and the hole-closure problem in which a disk-shaped surfactant-free region self-heals. Support from NSF-DMS-FRG 0968154, RCSA-CCS-19788, and HHMI.

A Mathematics Software Database Update.

ERIC Educational Resources Information Center

Cunningham, R. S.; Smith, David A.

1987-01-01

Contains an update of an earlier listing of software for mathematics instruction at the college level. Topics are: advanced mathematics, algebra, calculus, differential equations, discrete mathematics, equation solving, general mathematics, geometry, linear and matrix algebra, logic, statistics and probability, and trigonometry. (PK)

Optimization and Control of Agent-Based Models in Biology: A Perspective.

PubMed

An, G; Fitzpatrick, B G; Christley, S; Federico, P; Kanarek, A; Neilan, R Miller; Oremland, M; Salinas, R; Laubenbacher, R; Lenhart, S

2017-01-01

Agent-based models (ABMs) have become an increasingly important mode of inquiry for the life sciences. They are particularly valuable for systems that are not understood well enough to build an equation-based model. These advantages, however, are counterbalanced by the difficulty of analyzing and using ABMs, due to the lack of the type of mathematical tools available for more traditional models, which leaves simulation as the primary approach. As models become large, simulation becomes challenging. This paper proposes a novel approach to two mathematical aspects of ABMs, optimization and control, and it presents a few first steps outlining how one might carry out this approach. Rather than viewing the ABM as a model, it is to be viewed as a surrogate for the actual system. For a given optimization or control problem (which may change over time), the surrogate system is modeled instead, using data from the ABM and a modeling framework for which ready-made mathematical tools exist, such as differential equations, or for which control strategies can explored more easily. Once the optimization problem is solved for the model of the surrogate, it is then lifted to the surrogate and tested. The final step is to lift the optimization solution from the surrogate system to the actual system. This program is illustrated with published work, using two relatively simple ABMs as a demonstration, Sugarscape and a consumer-resource ABM. Specific techniques discussed include dimension reduction and approximation of an ABM by difference equations as well systems of PDEs, related to certain specific control objectives. This demonstration illustrates the very challenging mathematical problems that need to be solved before this approach can be realistically applied to complex and large ABMs, current and future. The paper outlines a research program to address them.

A Gauge-generalized Solution for Non-Keplerian Motion in the Frenet-Serret Frame

NASA Astrophysics Data System (ADS)

Garber, Darren D.

2009-05-01

The customary modeling of perturbed planetary and spacecraft motion as a continuous sequence of unperturbed two-body orbits (instantaneous ellipses) is conveniently assigned a physical interpretation through the Keplerian and Delaunay elements and complemented mathematically by the Lagrange-type equations which describe the evolution of these variables. If however the actual motion is very non-Keplerian (i.e. the perturbed orbit varies greatly from a two-body orbit), then its modeling by a sequence of conics is not necessarily optimal in terms of its mathematical description and its resulting physical interpretation. Since, in principle a curve of any type can be represented as a sequence of points from a family of curves of any other type (Efroimsky 2005), alternate non-conic curves can be utilized to better describe the perturbed non-Keplerian motion of the body both mathematically and with a physically relevant interpretation. Non-Keplerian motion exists in both celestial mechanics and astrodynamics as evident by the complex interactions within star clusters and also as the result of a spacecraft accelerating via ion propulsion, solar sails and electro-dynamic tethers. For these cases, the sequence of simple orbits to describe the motion is not based on conics, but instead a family of spirals. The selection of spirals as the underlying simple motion is supported by the fact that it is unnecessary to describe the motion in terms of instantaneous orbits tangent to the actual trajectory (Efroimsky 2002, Newman & Efroimsky 2003) and at times there is an advantage to deviate from osculation, in order to greatly simplify the resulting mathematics via gauge freedom (Efroimsky & Goldreich 2003, Slabinski 2003, Gurfil 2004). From these two principles, (1) spirals as instantaneous orbits, and (2) controlled deviation from osculation, new planetary equations are derived for new non-osculating elements in the Frenet-Serret frame with the gauge function as a measure of non-osculation.

Evolutionary game theory using agent-based methods.

PubMed

Adami, Christoph; Schossau, Jory; Hintze, Arend

2016-12-01

Evolutionary game theory is a successful mathematical framework geared towards understanding the selective pressures that affect the evolution of the strategies of agents engaged in interactions with potential conflicts. While a mathematical treatment of the costs and benefits of decisions can predict the optimal strategy in simple settings, more realistic settings such as finite populations, non-vanishing mutations rates, stochastic decisions, communication between agents, and spatial interactions, require agent-based methods where each agent is modeled as an individual, carries its own genes that determine its decisions, and where the evolutionary outcome can only be ascertained by evolving the population of agents forward in time. While highlighting standard mathematical results, we compare those to agent-based methods that can go beyond the limitations of equations and simulate the complexity of heterogeneous populations and an ever-changing set of interactors. We conclude that agent-based methods can predict evolutionary outcomes where purely mathematical treatments cannot tread (for example in the weak selection-strong mutation limit), but that mathematics is crucial to validate the computational simulations. Copyright © 2016 Elsevier B.V. All rights reserved.

Selection by consequences, behavioral evolution, and the price equation.

PubMed

Baum, William M

2017-05-01

Price's equation describes evolution across time in simple mathematical terms. Although it is not a theory, but a derived identity, it is useful as an analytical tool. It affords lucid descriptions of genetic evolution, cultural evolution, and behavioral evolution (often called "selection by consequences") at different levels (e.g., individual vs. group) and at different time scales (local and extended). The importance of the Price equation for behavior analysis lies in its ability to precisely restate selection by consequences, thereby restating, or even replacing, the law of effect. Beyond this, the equation may be useful whenever one regards ontogenetic behavioral change as evolutionary change, because it describes evolutionary change in abstract, general terms. As an analytical tool, the behavioral Price equation is an excellent aid in understanding how behavior changes within organisms' lifetimes. For example, it illuminates evolution of response rate, analyses of choice in concurrent schedules, negative contingencies, and dilemmas of self-control. © 2017 Society for the Experimental Analysis of Behavior.

Undular bore theory for the Gardner equation

NASA Astrophysics Data System (ADS)

Kamchatnov, A. M.; Kuo, Y.-H.; Lin, T.-C.; Horng, T.-L.; Gou, S.-C.; Clift, R.; El, G. A.; Grimshaw, R. H. J.

2012-09-01

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.

Pattern of mathematic representation ability in magnetic electricity problem

NASA Astrophysics Data System (ADS)

Hau, R. R. H.; Marwoto, P.; Putra, N. M. D.

2018-03-01

The mathematic representation ability in solving magnetic electricity problem gives information about the way students understand magnetic electricity. Students have varied mathematic representation pattern ability in solving magnetic electricity problem. This study aims to determine the pattern of students' mathematic representation ability in solving magnet electrical problems.The research method used is qualitative. The subject of this study is the fourth semester students of UNNES Physics Education Study Program. The data collection is done by giving a description test that refers to the test of mathematical representation ability and interview about field line topic and Gauss law. The result of data analysis of student's mathematical representation ability in solving magnet electric problem is categorized into high, medium and low category. The ability of mathematical representations in the high category tends to use a pattern of making known and asked symbols, writing equations, using quantities of physics, substituting quantities into equations, performing calculations and final answers. The ability of mathematical representation in the medium category tends to use several patterns of writing the known symbols, writing equations, using quantities of physics, substituting quantities into equations, performing calculations and final answers. The ability of mathematical representations in the low category tends to use several patterns of making known symbols, writing equations, substituting quantities into equations, performing calculations and final answer.

An asymptotic membrane model for wrinkling of very thin films

NASA Astrophysics Data System (ADS)

Battista, Antonio; Hamdouni, Aziz; Millet, Olivier

2018-05-01

In this work, a formal deduction of a two-dimensional membrane theory, similar to Landau-Lifshitz model, is performed via an asymptotic development of the weak formulation of the three-dimensional equations of elasticity. Some interesting aspects of the deduced model are investigated, in particular the property of obtaining a hyperbolic equation for the out-of-plane displacement under a certain class of boundary conditions and loads. Some simple cases are analyzed to show the relevant aspects of the model and the phenomenology that can be addressed. In particular, it is shown how this mathematical formulation is capable to describe instabilities well known as wrinkling, often observed for the buckling of very thin membranes.

A semi-analytical method for the computation of the Lyapunov exponents of fractional-order systems

NASA Astrophysics Data System (ADS)

Caponetto, Riccardo; Fazzino, Stefano

2013-01-01

Fractional-order differential equations are interesting for their applications in the construction of mathematical models in finance, materials science or diffusion. In this paper, an application of a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equation is employed for calculating Lyapunov exponents of fractional order systems. It is known that the Lyapunov exponents, first introduced by Oseledec, play a crucial role in characterizing the behaviour of dynamical systems. They can be used to analyze the sensitive dependence on initial conditions and the presence of chaotic attractors. The results reveal that the proposed method is very effective and simple and leads to accurate, approximately convergent solutions.

Perspectives on the mathematics of biological patterning and morphogenesis

NASA Astrophysics Data System (ADS)

Garikipati, Krishna

2017-02-01

A central question in developmental biology is how size and position are determined. The genetic code carries instructions on how to control these properties in order to regulate the pattern and morphology of structures in the developing organism. Transcription and protein translation mechanisms implement these instructions. However, this cannot happen without some manner of sampling of epigenetic information on the current patterns and morphological forms of structures in the organism. Any rigorous description of space- and time-varying patterns and morphological forms reduces to one among various classes of spatio-temporal partial differential equations. Reaction-transport equations represent one such class. Starting from simple Fickian diffusion, the incorporation of reaction, phase segregation and advection terms can represent many of the patterns seen in the animal and plant kingdoms. Morphological form, requiring the development of three-dimensional structure, also can be represented by these equations of mass transport, albeit to a limited degree. The recognition that physical forces play controlling roles in shaping tissues leads to the conclusion that (nonlinear) elasticity governs the development of morphological form. In this setting, inhomogeneous growth drives the elasticity problem. The combination of reaction-transport equations with those of elasto-growth makes accessible a potentially unlimited spectrum of patterning and morphogenetic phenomena in developmental biology. This perspective communication is a survey of the partial differential equations of mathematical physics that have been proposed to govern patterning and morphogenesis in developmental biology. Several numerical examples are included to illustrate these equations and the corresponding physics, with the intention of providing physical insight wherever possible.

An innovative method for coordinate measuring machine one-dimensional self-calibration with simplified experimental process.

PubMed

Fang, Cheng; Butler, David Lee

2013-05-01

In this paper, an innovative method for CMM (Coordinate Measuring Machine) self-calibration is proposed. In contrast to conventional CMM calibration that relies heavily on a high precision reference standard such as a laser interferometer, the proposed calibration method is based on a low-cost artefact which is fabricated with commercially available precision ball bearings. By optimizing the mathematical model and rearranging the data sampling positions, the experimental process and data analysis can be simplified. In mathematical expression, the samples can be minimized by eliminating the redundant equations among those configured by the experimental data array. The section lengths of the artefact are measured at arranged positions, with which an equation set can be configured to determine the measurement errors at the corresponding positions. With the proposed method, the equation set is short of one equation, which can be supplemented by either measuring the total length of the artefact with a higher-precision CMM or calibrating the single point error at the extreme position with a laser interferometer. In this paper, the latter is selected. With spline interpolation, the error compensation curve can be determined. To verify the proposed method, a simple calibration system was set up on a commercial CMM. Experimental results showed that with the error compensation curve uncertainty of the measurement can be reduced to 50%.

A Walsh Function Module Users' Manual

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2014-01-01

The solution of partial differential equations (PDEs) with Walsh functions offers new opportunities to simulate many challenging problems in mathematical physics. The approach was developed to better simulate hypersonic flows with shocks on unstructured grids. It is unique in that integrals and derivatives are computed using simple matrix multiplication of series representations of functions without the need for divided differences. The product of any two Walsh functions is another Walsh function - a feature that radically changes an algorithm for solving PDEs. A FORTRAN module for supporting Walsh function simulations is documented. A FORTRAN code is also documented with options for solving time-dependent problems: an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the usage of the Walsh function module including such features as operator overloading, Fast Walsh Transforms in multi-dimensions, and a Fast Walsh reciprocal.

Analytical solution of the time-dependent Bloch NMR flow equations: a translational mechanical analysis

NASA Astrophysics Data System (ADS)

Awojoyogbe, O. B.

2004-08-01

Various biological and physiological properties of living tissue can be studied by means of nuclear magnetic resonance techniques. Unfortunately, the basic physics of extracting the relevant information from the solution of Bloch nuclear magnetic resource (NMR) equations to accurately monitor the clinical state of biological systems is still not yet fully understood. Presently, there are no simple closed solutions known to the Bloch equations for a general RF excitation. Therefore the translational mechanical analysis of the Bloch NMR equations presented in this study, which can be taken as definitions of new functions to be studied in detail may reveal very important information from which various NMR flow parameters can be derived. Fortunately, many of the most important but hidden applications of blood flow parameters can be revealed without too much difficulty if appropriate mathematical techniques are used to solve the equations. In this study we are concerned with a mathematical study of the laws of NMR physics from the point of view of translational mechanical theory. The important contribution of this study is that solutions to the Bloch NMR flow equations do always exist and can be found as accurately as desired. We shall restrict our attention to cases where the radio frequency field can be treated by simple analytical methods. First we shall derive a time dependant second-order non-homogeneous linear differential equation from the Bloch NMR equation in term of the equilibrium magnetization M0, RF B1( t) field, T1 and T2 relaxation times. Then, we would develop a general method of solving the differential equation for the cases when RF B1( t)=0, and when RF B1( t)≠0. This allows us to obtain the intrinsic or natural behavior of the NMR system as well as the response of the system under investigation to a specific influence of external force to the system. Specifically, we consider the case where the RF B1 varies harmonically with time. Here the complete motion of the system consists of two parts. The first part describes the motion of the transverse magnetization My in the absence of RF B( t) field. The second part of the motion described by the particular integral of the derived differential equation does not decay with time but continues its periodic behavior indefinitely. The complete motion of the NMR flow system is then quantitatively and qualitatively described.

Mathematical Modeling of the Dynamics of Salmonella Cerro Infection in a US Dairy Herd

NASA Astrophysics Data System (ADS)

Chapagain, Prem; van Kessel, Jo Ann; Karns, Jeffrey; Wolfgang, David; Schukken, Ynte; Grohn, Yrjo

2006-03-01

Salmonellosis has been one of the major causes of human foodborne illness in the US. The high prevalence of infections makes transmission dynamics of Salmonella in a farm environment of interest both from animal and human health perspectives. Mathematical modeling approaches are increasingly being applied to understand the dynamics of various infectious diseases in dairy herds. Here, we describe the transmission dynamics of Salmonella infection in a dairy herd with a set of non-linear differential equations. Although the infection dynamics of different serotypes of Salmonella in cattle are likely to be different, we find that a relatively simple SIR-type model can describe the observed dynamics of the Salmonella enterica serotype Cerro infection in the herd.

On determining important aspects of mathematical models: Application to problems in physics and chemistry

NASA Technical Reports Server (NTRS)

Rabitz, Herschel

1987-01-01

The use of parametric and functional gradient sensitivity analysis techniques is considered for models described by partial differential equations. By interchanging appropriate dependent and independent variables, questions of inverse sensitivity may be addressed to gain insight into the inversion of observational data for parameter and function identification in mathematical models. It may be argued that the presence of a subset of dominantly strong coupled dependent variables will result in the overall system sensitivity behavior collapsing into a simple set of scaling and self similarity relations amongst elements of the entire matrix of sensitivity coefficients. These general tools are generic in nature, but herein their application to problems arising in selected areas of physics and chemistry is presented.

How Students Process Equations in Solving Quantitative Synthesis Problems? Role of Mathematical Complexity in Students' Mathematical Performance

ERIC Educational Resources Information Center

Ibrahim, Bashirah; Ding, Lin; Heckler, Andrew F.; White, Daniel R.; Badeau, Ryan

2017-01-01

We examine students' mathematical performance on quantitative "synthesis problems" with varying mathematical complexity. Synthesis problems are tasks comprising multiple concepts typically taught in different chapters. Mathematical performance refers to the formulation, combination, and simplification of equations. Generally speaking,…

Order-of-magnitude physics of neutron stars. Estimating their properties from first principles

NASA Astrophysics Data System (ADS)

Reisenegger, Andreas; Zepeda, Felipe S.

2016-03-01

We use basic physics and simple mathematics accessible to advanced undergraduate students to estimate the main properties of neutron stars. We set the stage and introduce relevant concepts by discussing the properties of "everyday" matter on Earth, degenerate Fermi gases, white dwarfs, and scaling relations of stellar properties with polytropic equations of state. Then, we discuss various physical ingredients relevant for neutron stars and how they can be combined in order to obtain a couple of different simple estimates of their maximum mass, beyond which they would collapse, turning into black holes. Finally, we use the basic structural parameters of neutron stars to briefly discuss their rotational and electromagnetic properties.

Preservice Mathematics Teachers' Experiences about Function and Equation Concepts

ERIC Educational Resources Information Center

Dede, Yuksel; Soybas, Danyal

2011-01-01

The purpose of this study is to determine the experience of mathematics preservice teachers related to function and equation concepts and the relations between them. Determining preservice mathematics teachers' understanding of function and equation concepts has great importance since it directly affects their future teaching careers. Data were…

Mathematics anxiety in children with developmental dyscalculia.

PubMed

Rubinsten, Orly; Tannock, Rosemary

2010-07-15

Math anxiety, defined as a negative affective response to mathematics, is known to have deleterious effects on math performance in the general population. However, the assumption that math anxiety is directly related to math performance, has not yet been validated. Thus, our primary objective was to investigate the effects of math anxiety on numerical processing in children with specific deficits in the acquisition of math skills (Developmental Dyscalculia; DD) by using a novel affective priming task as an indirect measure. Participants (12 children with DD and 11 typically-developing peers) completed a novel priming task in which an arithmetic equation was preceded by one of four types of priming words (positive, neutral, negative or related to mathematics). Children were required to indicate whether the equation (simple math facts based on addition, subtraction, multiplication or division) was true or false. Typically, people respond to target stimuli more quickly after presentation of an affectively-related prime than after one that is unrelated affectively. Participants with DD responded faster to targets that were preceded by both negative primes and math-related primes. A reversed pattern was present in the control group. These results reveal a direct link between emotions, arithmetic and low achievement in math. It is also suggested that arithmetic-affective priming might be used as an indirect measure of math anxiety.

Determination of point of incidence for the case of reflection or refraction at spherical surface knowing two points lying on the ray.

PubMed

Mikš, Antonín; Novák, Pavel

2017-09-01

The paper is focused on the problem of determination of the point of incidence of a light ray for the case of reflection or refraction at the spherical optical surface, assuming that two fixed points in space that the sought light ray should go through are given. The requirement is that one of these points lies on the incident ray and the other point on the reflected/refracted ray. Although at first glance it seems to be a simple problem, it will be shown that it has no simple analytical solution. The basic idea of the solution is given, and it is shown that the problem leads to a nonlinear equation in one variable. The roots of the resulting nonlinear equation can be found by numerical methods of mathematical optimization. The proposed methods were implemented in MATLAB, and the proper function of these algorithms was verified on several examples.

A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler-Bernoulli beams

NASA Astrophysics Data System (ADS)

Rahimi, Zaher; Sumelka, Wojciech; Yang, Xiao-Jun

2017-11-01

The application of fractional calculus in fractional models (FMs) makes them more flexible than integer models inasmuch they can conclude all of integer and non-integer operators. In other words FMs let us use more potential of mathematics to modeling physical phenomena due to the use of both integer and fractional operators to present a better modeling of problems, which makes them more flexible and powerful. In the present work, a new fractional nonlocal model has been proposed, which has a simple form and can be used in different problems due to the simple form of numerical solutions. Then the model has been used to govern equations of the motion of the Timoshenko beam theory (TBT) and Euler-Bernoulli beam theory (EBT). Next, free vibration of the Timoshenko and Euler-Bernoulli simply-supported (S-S) beam has been investigated. The Galerkin weighted residual method has been used to solve the non-linear governing equations.

Numerical Modeling in Geodynamics: Success, Failure and Perspective

NASA Astrophysics Data System (ADS)

Ismail-Zadeh, A.

2005-12-01

A real success in numerical modeling of dynamics of the Earth can be achieved only by multidisciplinary research teams of experts in geodynamics, applied and pure mathematics, and computer science. The success in numerical modeling is based on the following basic, but simple, rules. (i) People need simplicity most, but they understand intricacies best (B. Pasternak, writer). Start from a simple numerical model, which describes basic physical laws by a set of mathematical equations, and move then to a complex model. Never start from a complex model, because you cannot understand the contribution of each term of the equations to the modeled geophysical phenomenon. (ii) Study the numerical methods behind your computer code. Otherwise it becomes difficult to distinguish true and erroneous solutions to the geodynamic problem, especially when your problem is complex enough. (iii) Test your model versus analytical and asymptotic solutions, simple 2D and 3D model examples. Develop benchmark analysis of different numerical codes and compare numerical results with laboratory experiments. Remember that the numerical tool you employ is not perfect, and there are small bugs in every computer code. Therefore the testing is the most important part of your numerical modeling. (iv) Prove (if possible) or learn relevant statements concerning the existence, uniqueness and stability of the solution to the mathematical and discrete problems. Otherwise you can solve an improperly-posed problem, and the results of the modeling will be far from the true solution of your model problem. (v) Try to analyze numerical models of a geological phenomenon using as less as possible tuning model variables. Already two tuning variables give enough possibilities to constrain your model well enough with respect to observations. The data fitting sometimes is quite attractive and can take you far from a principal aim of your numerical modeling: to understand geophysical phenomena. (vi) If the number of tuning model variables are greater than two, test carefully the effect of each of the variables on the modeled phenomenon. Remember: With four exponents I can fit an elephant (E. Fermi, physicist). (vii) Make your numerical model as accurate as possible, but never put the aim to reach a great accuracy: Undue precision of computations is the first symptom of mathematical illiteracy (N. Krylov, mathematician). How complex should be a numerical model? A model which images any detail of the reality is as useful as a map of scale 1:1 (J. Robinson, economist). This message is quite important for geoscientists, who study numerical models of complex geodynamical processes. I believe that geoscientists will never create a model of the real Earth dynamics, but we should try to model the dynamics such a way to simulate basic geophysical processes and phenomena. Does a particular model have a predictive power? Each numerical model has a predictive power, otherwise the model is useless. The predictability of the model varies with its complexity. Remember that a solution to the numerical model is an approximate solution to the equations, which have been chosen in believe that they describe dynamic processes of the Earth. Hence a numerical model predicts dynamics of the Earth as well as the mathematical equations describe this dynamics. What methodological advances are still needed for testable geodynamic modeling? Inverse (time-reverse) numerical modeling and data assimilation are new methodologies in geodynamics. The inverse modeling can allow to test geodynamic models forward in time using restored (from present-day observations) initial conditions instead of unknown conditions.

Process of .sup.196 Hg enrichment

DOEpatents

Grossman, Mark W.; Mellor, Charles E.

1993-01-01

A simple rate equation model shows that by increasing the length of the photochemical reactor and/or by increasing the photon intensity in said reactor, the feedstock utilization of .sup.196 Hg will be increased. Two preferred embodiments of the present invention are described, namely (1) long reactors using long photochemical lamps and vapor filters; and (2) quartz reactors with external UV reflecting films. These embodiments have each been constructed and operated, demonstrating the enhanced utilization process dictated by the mathematical model (also provided).

Summing up the noise in gene networks

NASA Astrophysics Data System (ADS)

Paulsson, Johan

2004-01-01

Random fluctuations in genetic networks are inevitable as chemical reactions are probabilistic and many genes, RNAs and proteins are present in low numbers per cell. Such `noise' affects all life processes and has recently been measured using green fluorescent protein (GFP). Two studies show that negative feedback suppresses noise, and three others identify the sources of noise in gene expression. Here I critically analyse these studies and present a simple equation that unifies and extends both the mathematical and biological perspectives.

Process of [sup 196]Hg enrichment

DOEpatents

Grossman, M.W.; Mellor, C.E.

1993-04-27

A simple rate equation model shows that by increasing the length of the photochemical reactor and/or by increasing the photon intensity in said reactor, the feedstock utilization of [sup 196]Hg will be increased. Two preferred embodiments of the present invention are described, namely (1) long reactors using long photochemical lamps and vapor filters; and (2) quartz reactors with external UV reflecting films. These embodiments have each been constructed and operated, demonstrating the enhanced utilization process dictated by the mathematical model (also provided).

New computer system simplifies programming of mathematical equations

NASA Technical Reports Server (NTRS)

Reinfelds, J.; Seitz, R. N.; Wood, L. H.

1966-01-01

Automatic Mathematical Translator /AMSTRAN/ permits scientists or engineers to enter mathematical equations in their natural mathematical format and to obtain an immediate graphical display of the solution. This automatic-programming, on-line, multiterminal computer system allows experienced programmers to solve nonroutine problems.

Measuring and modeling the oxygen profile in a nitrifying Moving Bed Biofilm Reactor.

PubMed

Masić, Alma; Bengtsson, Jessica; Christensson, Magnus

2010-09-01

In this paper we determine the oxygen profile in a biofilm on suspended carriers in two ways: firstly by microelectrode measurements and secondly by a simple mathematical model. The Moving Bed Biofilm Reactor is well-established for wastewater treatment where bacteria grow as a biofilm on the protective surfaces of suspended carriers. The flat shaped BiofilmChip P was developed to allow good conditions for transport of substrates into the biofilm. The oxygen profile was measured in situ the nitrifying biofilm with a microelectrode and it was simulated with a one-dimensional mathematical model. We extended the model by adding a CSTR equation, to connect the reactor to the biofilm through the boundary conditions. We showed the dependence of the thickness of the mass transfer boundary layer on the bulk flow rate. Finally, we estimated the erosion parameter lambda to increase the concordance between the measured and simulated profiles. This lead to a simple empirical relationship between lambda and the flow rate. The data gathered by in situ microelectrode measurements can, together with the mathematical model, be used in predictive modeling and give more insight in the design of new carriers, with the ambition of making process operation more energy efficient. Copyright 2010 Elsevier Inc. All rights reserved.

Design and analysis of unequal split Bagley power dividers

NASA Astrophysics Data System (ADS)

Abu-Alnadi, Omar; Dib, Nihad; Al-Shamaileh, Khair; Sheta, Abdelfattah

2015-03-01

In this article, we propose a general design procedure to develop unequal split Bagley power dividers (BPDs). Based on the mathematical approach carried out in the insight of simple circuit and transmission line theories, exact design equations for 3-way and 5-way BPDs are derived. Utilising the developed equations leads to power dividers with the ability of offering different output power ratios through a suitable choice of the characteristic impedances of the interconnecting transmission lines. For verification purposes, a 1:2:1 3-way, 1:2:1:2:1 5-way and 1:3:1:3:1 5-way BPDs are designed and fabricated. The experimental and full-wave simulation results prove the validity of the designed unequal split BPDs.

An ill-posed problem for the Black-Scholes equation for a profitable forecast of prices of stock options on real market data

NASA Astrophysics Data System (ADS)

Klibanov, Michael V.; Kuzhuget, Andrey V.; Golubnichiy, Kirill V.

2016-01-01

A new empirical mathematical model for the Black-Scholes equation is proposed to forecast option prices. This model includes new interval for the price of the underlying stock, new initial and new boundary conditions. Conventional notions of maturity time and strike prices are not used. The Black-Scholes equation is solved as a parabolic equation with the reversed time, which is an ill-posed problem. Thus, a regularization method is used to solve it. To verify the validity of our model, real market data for 368 randomly selected liquid options are used. A new trading strategy is proposed. Our results indicates that our method is profitable on those options. Furthermore, it is shown that the performance of two simple extrapolation-based techniques is much worse. We conjecture that our method might lead to significant profits of those financial insitutions which trade large amounts of options. We caution, however, that further studies are necessary to verify this conjecture.

Evolution in time of an N-atom system. I. A physical basis set for the projection of the master equation

NASA Astrophysics Data System (ADS)

Freedhoff, Helen

2004-01-01

We study an aggregate of N identical two-level atoms (TLA’s) coupled by the retarded interatomic interaction, using the Lehmberg-Agarwal master equation. First, we calculate the entangled eigenstates of the system; then, we use these eigenstates as a basis set for the projection of the master equation. We demonstrate that in this basis the equations of motion for the level populations, as well as the expressions for the emission and absorption spectra, assume a simple mathematical structure and allow for a transparent physical interpretation. To illustrate the use of the general theory in emission processes, we study an isosceles triangle of atoms, and present in the long wavelength limit the (cascade) emission spectrum for a hexagon of atoms fully excited at t=0. To illustrate its use for absorption processes, we tabulate (in the same limit) the biexciton absorption frequencies, linewidths, and relative intensities for polygons consisting of N=2,…,9 TLA’s.

High-order finite-volume solutions of the steady-state advection-diffusion equation with nonlinear Robin boundary conditions

NASA Astrophysics Data System (ADS)

Lin, Zhi; Zhang, Qinghai

2017-09-01

We propose high-order finite-volume schemes for numerically solving the steady-state advection-diffusion equation with nonlinear Robin boundary conditions. Although the original motivation comes from a mathematical model of blood clotting, the nonlinear boundary conditions may also apply to other scientific problems. The main contribution of this work is a generic algorithm for generating third-order, fourth-order, and even higher-order explicit ghost-filling formulas to enforce nonlinear Robin boundary conditions in multiple dimensions. Under the framework of finite volume methods, this appears to be the first algorithm of its kind. Numerical experiments on boundary value problems show that the proposed fourth-order formula can be much more accurate and efficient than a simple second-order formula. Furthermore, the proposed ghost-filling formulas may also be useful for solving other partial differential equations.

The Parker-Sochacki Method of Solving Differential Equations: Applications and Limitations

NASA Astrophysics Data System (ADS)

Rudmin, Joseph W.

2006-11-01

The Parker-Sochacki method is a powerful but simple technique of solving systems of differential equations, giving either analytical or numerical results. It has been in use for about 10 years now since its discovery by G. Edgar Parker and James Sochacki of the James Madison University Dept. of Mathematics and Statistics. It is being presented here because it is still not widely known and can benefit the listeners. It is a method of rapidly generating the Maclauren series to high order, non-iteratively. It has been successfully applied to more than a hundred systems of equations, including the classical many-body problem. Its advantages include its speed of calculation, its simplicity, and the fact that it uses only addition, subtraction and multiplication. It is not just a polynomial approximation, because it yields the Maclaurin series, and therefore exhibits the advantages and disadvantages of that series. A few applications will be presented.

Linear regression analysis and its application to multivariate chromatographic calibration for the quantitative analysis of two-component mixtures.

PubMed

Dinç, Erdal; Ozdemir, Abdil

2005-01-01

Multivariate chromatographic calibration technique was developed for the quantitative analysis of binary mixtures enalapril maleate (EA) and hydrochlorothiazide (HCT) in tablets in the presence of losartan potassium (LST). The mathematical algorithm of multivariate chromatographic calibration technique is based on the use of the linear regression equations constructed using relationship between concentration and peak area at the five-wavelength set. The algorithm of this mathematical calibration model having a simple mathematical content was briefly described. This approach is a powerful mathematical tool for an optimum chromatographic multivariate calibration and elimination of fluctuations coming from instrumental and experimental conditions. This multivariate chromatographic calibration contains reduction of multivariate linear regression functions to univariate data set. The validation of model was carried out by analyzing various synthetic binary mixtures and using the standard addition technique. Developed calibration technique was applied to the analysis of the real pharmaceutical tablets containing EA and HCT. The obtained results were compared with those obtained by classical HPLC method. It was observed that the proposed multivariate chromatographic calibration gives better results than classical HPLC.

Mathematical model for studying cyclist kinematics in vehicle-bicycle frontal collisions

NASA Astrophysics Data System (ADS)

Condrea, OA; Chiru, A.; Chiriac, RL; Vlase, S.

2017-10-01

For the development of effective vehicle related safety solutions to improve cyclist protection, kinematic predictions are essential. The objective of the paper was the elaboration of a simple mathematical model for predicting cyclist kinematics, with the advantage of yielding simple results for relatively complicated impact situations. Thus, the use of elaborated math software is not required and the calculation time is shortened. The paper presents a modelling framework to determine cyclist kinematic behaviour for the situations in which a M1 category vehicle frontally hits the rear part of a bicycle. After the primary impact between the vehicle front bumper and the bicycle, the cyclist hits the vehicle’s bonnet, the windscreen or both the vehicle’s bonnet and the windscreen in short succession. The head-windshield impact is often the most severe impact, causing serious and potentially lethal injuries. The cyclist is represented by a rigid segment and the equations of motion for the cyclist after the primary impact are obtained by applying Newton’s second law of motion. The impact time for the contact between the vehicle and the cyclist is yielded afterwards by formulating and intersecting the trajectories for two points positioned on the cyclist’s head/body and the vehicle’s windscreen/bonnet while assuming that the cyclist’s equations of motion after the primary impact remain the same. Postimpact kinematics for the secondary impact are yielded by applying linear and angular momentum conservation laws.

The IDEA model: A single equation approach to the Ebola forecasting challenge.

PubMed

Tuite, Ashleigh R; Fisman, David N

2018-03-01

Mathematical modeling is increasingly accepted as a tool that can inform disease control policy in the face of emerging infectious diseases, such as the 2014-2015 West African Ebola epidemic, but little is known about the relative performance of alternate forecasting approaches. The RAPIDD Ebola Forecasting Challenge (REFC) tested the ability of eight mathematical models to generate useful forecasts in the face of simulated Ebola outbreaks. We used a simple, phenomenological single-equation model (the "IDEA" model), which relies only on case counts, in the REFC. Model fits were performed using a maximum likelihood approach. We found that the model performed reasonably well relative to other more complex approaches, with performance metrics ranked on average 4th or 5th among participating models. IDEA appeared better suited to long- than short-term forecasts, and could be fit using nothing but reported case counts. Several limitations were identified, including difficulty in identifying epidemic peak (even retrospectively), unrealistically precise confidence intervals, and difficulty interpolating daily case counts when using a model scaled to epidemic generation time. More realistic confidence intervals were generated when case counts were assumed to follow a negative binomial, rather than Poisson, distribution. Nonetheless, IDEA represents a simple phenomenological model, easily implemented in widely available software packages that could be used by frontline public health personnel to generate forecasts with accuracy that approximates that which is achieved using more complex methodologies. Copyright © 2016 The Author(s). Published by Elsevier B.V. All rights reserved.

New extended (G'/G)-expansion method to solve nonlinear evolution equation: the (3 + 1)-dimensional potential-YTSF equation.

PubMed

Roshid, Harun-Or-; Akbar, M Ali; Alam, Md Nur; Hoque, Md Fazlul; Rahman, Nizhum

2014-01-01

In this article, a new extended (G'/G) -expansion method has been proposed for constructing more general exact traveling wave solutions of nonlinear evolution equations with the aid of symbolic computation. In order to illustrate the validity and effectiveness of the method, we pick the (3 + 1)-dimensional potential-YTSF equation. As a result, abundant new and more general exact solutions have been achieved of this equation. It has been shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in applied mathematics, engineering and mathematical physics.

Ordinary differential equations with applications in molecular biology.

PubMed

Ilea, M; Turnea, M; Rotariu, M

2012-01-01

Differential equations are of basic importance in molecular biology mathematics because many biological laws and relations appear mathematically in the form of a differential equation. In this article we presented some applications of mathematical models represented by ordinary differential equations in molecular biology. The vast majority of quantitative models in cell and molecular biology are formulated in terms of ordinary differential equations for the time evolution of concentrations of molecular species. Assuming that the diffusion in the cell is high enough to make the spatial distribution of molecules homogenous, these equations describe systems with many participating molecules of each kind. We propose an original mathematical model with small parameter for biological phospholipid pathway. All the equations system includes small parameter epsilon. The smallness of epsilon is relative to the size of the solution domain. If we reduce the size of the solution region the same small epsilon will result in a different condition number. It is clear that the solution for a smaller region is less difficult. We introduce the mathematical technique known as boundary function method for singular perturbation system. In this system, the small parameter is an asymptotic variable, different from the independent variable. In general, the solutions of such equations exhibit multiscale phenomena. Singularly perturbed problems form a special class of problems containing a small parameter which may tend to zero. Many molecular biology processes can be quantitatively characterized by ordinary differential equations. Mathematical cell biology is a very active and fast growing interdisciplinary area in which mathematical concepts, techniques, and models are applied to a variety of problems in developmental medicine and bioengineering. Among the different modeling approaches, ordinary differential equations (ODE) are particularly important and have led to significant advances. Ordinary differential equations are used to model biological processes on various levels ranging from DNA molecules or biosynthesis phospholipids on the cellular level.

Modeling the radiation pattern of LEDs.

PubMed

Moreno, Ivan; Sun, Ching-Cherng

2008-02-04

Light-emitting diodes (LEDs) come in many varieties and with a wide range of radiation patterns. We propose a general, simple but accurate analytic representation for the radiation pattern of the light emitted from an LED. To accurately render both the angular intensity distribution and the irradiance spatial pattern, a simple phenomenological model takes into account the emitting surfaces (chip, chip array, or phosphor surface), and the light redirected by both the reflecting cup and the encapsulating lens. Mathematically, the pattern is described as the sum of a maximum of two or three Gaussian or cosine-power functions. The resulting equation is widely applicable for any kind of LED of practical interest. We accurately model a wide variety of radiation patterns from several world-class manufacturers.

Dynamics and forecast in a simple model of sustainable development for rural populations.

PubMed

Angulo, David; Angulo, Fabiola; Olivar, Gerard

2015-02-01

Society is becoming more conscious on the need to preserve the environment. Sustainable development schemes have grown rapidly as a tool for managing, predicting and improving the growth path in different regions and economy sectors. We introduce a novel and simple mathematical model of ordinary differential equations (ODEs) in order to obtain a dynamical description for each one of the sustainability components (economy, social development and environment conservation), together with their dependence with demographic dynamics. The main part in the modeling task is inspired by the works by Cobb, Douglas, Brander and Taylor. This is completed through some new insights by the authors. A model application is presented for three specific geographical rural regions in Caldas (Colombia).

Technological pedagogical content knowledge of junior high school mathematics teachers in teaching linear equation

NASA Astrophysics Data System (ADS)

Wati, S.; Fitriana, L.; Mardiyana

2018-04-01

Linear equation is one of the topics in mathematics that are considered difficult. Student difficulties of understanding linear equation can be caused by lack of understanding this concept and the way of teachers teach. TPACK is a way to understand the complex relationships between teaching and content taught through the use of specific teaching approaches and supported by the right technology tools. This study aims to identify TPACK of junior high school mathematics teachers in teaching linear equation. The method used in the study was descriptive. In the first phase, a survey using a questionnaire was carried out on 45 junior high school mathematics teachers in teaching linear equation. While in the second phase, the interview involved three teachers. The analysis of data used were quantitative and qualitative technique. The result PCK revealed teachers emphasized developing procedural and conceptual knowledge through reliance on traditional in teaching linear equation. The result of TPK revealed teachers’ lower capacity to deal with the general information and communications technologies goals across the curriculum in teaching linear equation. The result indicated that PowerPoint constitutes TCK modal technological capability in teaching linear equation. The result of TPACK seems to suggest a low standard in teachers’ technological skills across a variety of mathematics education goals in teaching linear equation. This means that the ability of teachers’ TPACK in teaching linear equation still needs to be improved.

Mathematical Modeling for Scrub Typhus and Its Implications for Disease Control.

PubMed

Min, Kyung Duk; Cho, Sung Il

2018-03-19

The incidence rate of scrub typhus has been increasing in the Republic of Korea. Previous studies have suggested that this trend may have resulted from the effects of climate change on the transmission dynamics among vectors and hosts, but a clear explanation of the process is still lacking. In this study, we applied mathematical models to explore the potential factors that influence the epidemiology of tsutsugamushi disease. We developed mathematical models of ordinary differential equations including human, rodent and mite groups. Two models, including simple and complex models, were developed, and all parameters employed in the models were adopted from previous articles that represent epidemiological situations in the Republic of Korea. The simulation results showed that the force of infection at the equilibrium state under the simple model was 0.236 (per 100,000 person-months), and that in the complex model was 26.796 (per 100,000 person-months). Sensitivity analyses indicated that the most influential parameters were rodent and mite populations and contact rate between them for the simple model, and trans-ovarian transmission for the complex model. In both models, contact rate between humans and mites is more influential than morality rate of rodent and mite group. The results indicate that the effect of controlling either rodents or mites could be limited, and reducing the contact rate between humans and mites is more practical and effective strategy. However, the current level of control would be insufficient relative to the growing mite population. © 2018 The Korean Academy of Medical Sciences.

Saturation behavior: a general relationship described by a simple second-order differential equation.

PubMed

Kepner, Gordon R

2010-04-13

The numerous natural phenomena that exhibit saturation behavior, e.g., ligand binding and enzyme kinetics, have been approached, to date, via empirical and particular analyses. This paper presents a mechanism-free, and assumption-free, second-order differential equation, designed only to describe a typical relationship between the variables governing these phenomena. It develops a mathematical model for this relation, based solely on the analysis of the typical experimental data plot and its saturation characteristics. Its utility complements the traditional empirical approaches. For the general saturation curve, described in terms of its independent (x) and dependent (y) variables, a second-order differential equation is obtained that applies to any saturation phenomena. It shows that the driving factor for the basic saturation behavior is the probability of the interactive site being free, which is described quantitatively. Solving the equation relates the variables in terms of the two empirical constants common to all these phenomena, the initial slope of the data plot and the limiting value at saturation. A first-order differential equation for the slope emerged that led to the concept of the effective binding rate at the active site and its dependence on the calculable probability the interactive site is free. These results are illustrated using specific cases, including ligand binding and enzyme kinetics. This leads to a revised understanding of how to interpret the empirical constants, in terms of the variables pertinent to the phenomenon under study. The second-order differential equation revealed the basic underlying relations that describe these saturation phenomena, and the basic mathematical properties of the standard experimental data plot. It was shown how to integrate this differential equation, and define the common basic properties of these phenomena. The results regarding the importance of the slope and the new perspectives on the empirical constants governing the behavior of these phenomena led to an alternative perspective on saturation behavior kinetics. Their essential commonality was revealed by this analysis, based on the second-order differential equation.

Students’ difficulties in solving linear equation problems

NASA Astrophysics Data System (ADS)

Wati, S.; Fitriana, L.; Mardiyana

2018-03-01

A linear equation is an algebra material that exists in junior high school to university. It is a very important material for students in order to learn more advanced mathematics topics. Therefore, linear equation material is essential to be mastered. However, the result of 2016 national examination in Indonesia showed that students’ achievement in solving linear equation problem was low. This fact became a background to investigate students’ difficulties in solving linear equation problems. This study used qualitative descriptive method. An individual written test on linear equation tasks was administered, followed by interviews. Twenty-one sample students of grade VIII of SMPIT Insan Kamil Karanganyar did the written test, and 6 of them were interviewed afterward. The result showed that students with high mathematics achievement donot have difficulties, students with medium mathematics achievement have factual difficulties, and students with low mathematics achievement have factual, conceptual, operational, and principle difficulties. Based on the result there is a need of meaningfulness teaching strategy to help students to overcome difficulties in solving linear equation problems.

Mathematics anxiety in children with developmental dyscalculia

PubMed Central

2010-01-01

Background Math anxiety, defined as a negative affective response to mathematics, is known to have deleterious effects on math performance in the general population. However, the assumption that math anxiety is directly related to math performance, has not yet been validated. Thus, our primary objective was to investigate the effects of math anxiety on numerical processing in children with specific deficits in the acquisition of math skills (Developmental Dyscalculia; DD) by using a novel affective priming task as an indirect measure. Methods Participants (12 children with DD and 11 typically-developing peers) completed a novel priming task in which an arithmetic equation was preceded by one of four types of priming words (positive, neutral, negative or related to mathematics). Children were required to indicate whether the equation (simple math facts based on addition, subtraction, multiplication or division) was true or false. Typically, people respond to target stimuli more quickly after presentation of an affectively-related prime than after one that is unrelated affectively. Result Participants with DD responded faster to targets that were preceded by both negative primes and math-related primes. A reversed pattern was present in the control group. Conclusion These results reveal a direct link between emotions, arithmetic and low achievement in math. It is also suggested that arithmetic-affective priming might be used as an indirect measure of math anxiety. PMID:20633269

Assessing Quantitative Learning With The Math You Need When You Need It

NASA Astrophysics Data System (ADS)

Wenner, J. M.; Baer, E. M.; Burn, H.

2008-12-01

We present new data from a pilot project using the The Math You Need, When You Need It (TMYN) web resources in conjunction with several introductory geoscience courses. TMYN is a series of NSF-supported, NAGT-sponsored, web-based modular resources designed to help students learn (or relearn) mathematical skills essential for success in introductory geoscience courses. TMYN presents mathematical topics that are relevant to introductory geoscience based on a survey of more than 75 geoscience faculty members. To date, modules include unit conversions, many aspects of graphing, density calculations, rearranging equations and other simple mathematical concepts commonly used in the geosciences. The modular nature of the resources make it simple to select the units that are appropriate for a given course. In the fall of 2008, nine TMYN modules were tested in three courses taught at Highline Community College (Geology 101) and University of Wisconsin Oshkosh (Physical and Environmental Geology). Over 300 students participated in the study by taking pre- and post-tests and completing modules relevant to their course. Feedback about the use of these modules has been mixed. Initial results confirm anecdotal evidence that students initially have difficulty applying mathematical concepts to geologic problems. Furthermore, pre- test results indicate that, although instructors assume that students can perform simple mathematical manipulations, many students arrive in courses without the skills to apply mathematical concepts in problem solving situations. TMYN resources effectively provide support for learning quantitative problem solving and a mechanism for students to engage in self-teaching. Although we have seen mixed results due to a range of instructor engagement with the material, TMYN can have significant effect on students who are math phobic or "can't do math" because they can work at their own pace to overcome affective obstacles such as fear and dislike of mathematics. TMYN is most effective when instructors make explicit connections between material in the modules and course content. Instructors who participated in the study in Fall 2008 reacted positively to the use of TMYN in introductory geoscience courses because the resources require minimal class and prep time. Furthermore, when instructors can hold students responsible for the quantitative concepts covered with TMYN, they feel more comfortable including quantitative information without significant loss of geologic content.

Secondary School Advanced Mathematics, Chapter 8, Systems of Equations. Student's Text.

ERIC Educational Resources Information Center

Stanford Univ., CA. School Mathematics Study Group.

This text is the last of five in the Secondary School Advanced Mathematics (SSAM) series which was designed to meet the needs of students who have completed the Secondary School Mathematics (SSM) program, and wish to continue their study of mathematics. In this volume the solution of systems of linear and quadratic equations and inequalities in…

Explicating mathematical thinking in differential equations using a computer algebra system

NASA Astrophysics Data System (ADS)

Zeynivandnezhad, Fereshteh; Bates, Rachel

2018-07-01

The importance of developing students' mathematical thinking is frequently highlighted in literature regarding the teaching and learning of mathematics. Despite this importance, most curricula and instructional activities for undergraduate mathematics fail to bring the learner beyond the mathematics. The purpose of this study was to enhance students' mathematical thinking by implementing a computer algebra system and active learning pedagogical approaches. students' mathematical thinking processes were analyzed while completing specific differential equations tasks based on posed prompts and questions and Instrumental Genesis. Data were collected from 37 engineering students in a public Malaysian university. This study used the descriptive and interpretive qualitative research design to investigate the students' perspectives of emerging mathematical understanding and approaches to learning mathematics in an undergraduate differential equations course. Results of this study concluded that students used a variety of mathematical thinking processes in a non-sequential manner. Additionally, the outcomes provide justification for continued use of technologies such as computer algebra systems in undergraduate mathematics courses and the need for further studies to uncover the various processes students utilize to complete specific mathematical tasks.

RF tumour ablation: computer simulation and mathematical modelling of the effects of electrical and thermal conductivity.

PubMed

Lobo, S M; Liu, Z-J; Yu, N C; Humphries, S; Ahmed, M; Cosman, E R; Lenkinski, R E; Goldberg, W; Goldberg, S N

2005-05-01

This study determined the effects of thermal conductivity on RF ablation tissue heating using mathematical modelling and computer simulations of RF heating coupled to thermal transport. Computer simulation of the Bio-Heat equation coupled with temperature-dependent solutions for RF electric fields (ETherm) was used to generate temperature profiles 2 cm away from a 3 cm internally-cooled electrode. Multiple conditions of clinically relevant electrical conductivities (0.07-12 S m-1) and 'tumour' radius (5-30 mm) at a given background electrical conductivity (0.12 S m-1) were studied. Temperature response surfaces were plotted for six thermal conductivities, ranging from 0.3-2 W m-1 degrees C (the range of anticipated clinical and experimental systems). A temperature response surface was obtained for each thermal conductivity at 25 electrical conductivities and 17 radii (n=425 temperature data points). The simulated temperature response was fit to a mathematical model derived from prior phantom data. This mathematical model is of the form (T=a+bRc exp(dR) s(f) exp(g)(s)) for RF generator-energy dependent situations and (T=h+k exp(mR)+n?exp(p)(s)) for RF generator-current limited situations, where T is the temperature (degrees C) 2 cm from the electrode and a, b, c, d, f, g, h, k, m, n and p are fitting parameters. For each of the thermal conductivity temperature profiles generated, the mathematical model fit the response surface to an r2 of 0.97-0.99. Parameters a, b, c, d, f, k and m were highly correlated to thermal conductivity (r2=0.96-0.99). The monotonic progression of fitting parameters permitted their mathematical expression using simple functions. Additionally, the effect of thermal conductivity simplified the above equation to the extent that g, h, n and p were found to be invariant. Thus, representation of the temperature response surface could be accurately expressed as a function of electrical conductivity, radius and thermal conductivity. As a result, the non-linear temperature response of RF induced heating can be adequately expressed mathematically as a function of electrical conductivity, radius and thermal conductivity. Hence, thermal conductivity accounts for some of the previously unexplained variance. Furthermore, the addition of this variable into the mathematical model substantially simplifies the equations and, as such, it is expected that this will permit improved prediction of RF ablation induced temperatures in clinical practice.

Compartmental analysis and its manifold applications to pharmacokinetics.

PubMed

Rescigno, Aldo

2010-03-01

In this paper, I show how the concept of compartment evolved from the simple dilution of a substance in a physiological volume to its distribution in a network of interconnected spaces. The differential equations describing the fate of a substance in a living being can be solved, qualitatively and quantitatively, with the help of a number of mathematical techniques. A number of parameters of pharmacokinetic interest can be computed from the experimental data; often, the data available are not sufficient to determine some parameters, but it is possible to determine their range.

On the dynamical vs. thermodynamical performance of a β-type Stirling engine

NASA Astrophysics Data System (ADS)

Reséndiz-Antonio, Margarita; Santillán, Moisés

2014-09-01

In this work we present a simple mathematical model for a β-type Stirling engine. Despite its simplicity, the model considers all the engine’s relevant thermodynamic and mechanical aspects. The dynamic behavior of the model equation of motion is analyzed in order to obtain the sufficient conditions for engine cycling and to study the stability of the stationary regime. The performance of the engine’s thermodynamic part is also investigated. As a matter of fact, we found that it corresponds to a Carnot engine.

A review on principles, theory and practices of 2D-QSAR.

PubMed

Roy, Kunal; Das, Rudra Narayan

2014-01-01

The central axiom of science purports the explanation of every natural phenomenon using all possible logics coming from pure as well as mixed scientific background. The quantitative structure-activity relationship (QSAR) analysis is a study correlating the behavioral manifestation of compounds with their structures employing the interdisciplinary knowledge of chemistry, mathematics, biology as well as physics. Several studies have attempted to mathematically correlate the chemistry and property (physicochemical/ biological/toxicological) of molecules using various computationally or experimentally derived quantitative parameters termed as descriptors. The dimensionality of the descriptors depends on the type of algorithm employed and defines the nature of QSAR analysis. The most interesting feature of predictive QSAR models is that the behavior of any new or even hypothesized molecule can be predicted by the use of the mathematical equations. The phrase "2D-QSAR" signifies development of QSAR models using 2D-descriptors. Such predictor variables are the most widely practised ones because of their simple and direct mathematical algorithmic nature involving no time consuming energy computations and having reproducible operability. 2D-descriptors have a deluge of contributions in extracting chemical attributes and they are also capable of representing the 3D molecular features to some extent; although in no case they should be considered as the ultimate one, since they often suffer from the problems of intercorrelation, insufficient chemical information as well as lack of interpretation. However, by following rational approaches, novel 2D-descriptors may be developed to obviate various existing problems giving potential 2D-QSAR equations, thereby solving the innumerable chemical mysteries still unexplored.

Partial differential equation techniques for analysing animal movement: A comparison of different methods.

PubMed

Wang, Yi-Shan; Potts, Jonathan R

2017-03-07

Recent advances in animal tracking have allowed us to uncover the drivers of movement in unprecedented detail. This has enabled modellers to construct ever more realistic models of animal movement, which aid in uncovering detailed patterns of space use in animal populations. Partial differential equations (PDEs) provide a popular tool for mathematically analysing such models. However, their construction often relies on simplifying assumptions which may greatly affect the model outcomes. Here, we analyse the effect of various PDE approximations on the analysis of some simple movement models, including a biased random walk, central-place foraging processes and movement in heterogeneous landscapes. Perhaps the most commonly-used PDE method dates back to a seminal paper of Patlak from 1953. However, our results show that this can be a very poor approximation in even quite simple models. On the other hand, more recent methods, based on transport equation formalisms, can provide more accurate results, as long as the kernel describing the animal's movement is sufficiently smooth. When the movement kernel is not smooth, we show that both the older and newer methods can lead to quantitatively misleading results. Our detailed analysis will aid future researchers in the appropriate choice of PDE approximation for analysing models of animal movement. Copyright © 2017 Elsevier Ltd. All rights reserved.

Variational method enabling simplified solutions to the linearized Boltzmann equation for oscillatory gas flows

NASA Astrophysics Data System (ADS)

Ladiges, Daniel R.; Sader, John E.

2018-05-01

Nanomechanical resonators and sensors, operated in ambient conditions, often generate low-Mach-number oscillating rarefied gas flows. Cercignani [C. Cercignani, J. Stat. Phys. 1, 297 (1969), 10.1007/BF01007482] proposed a variational principle for the linearized Boltzmann equation, which can be used to derive approximate analytical solutions of steady (time-independent) flows. Here we extend and generalize this principle to unsteady oscillatory rarefied flows and thus accommodate resonating nanomechanical devices. This includes a mathematical approach that facilitates its general use and allows for systematic improvements in accuracy. This formulation is demonstrated for two canonical flow problems: oscillatory Couette flow and Stokes' second problem. Approximate analytical formulas giving the bulk velocity and shear stress, valid for arbitrary oscillation frequency, are obtained for Couette flow. For Stokes' second problem, a simple system of ordinary differential equations is derived which may be solved to obtain the desired flow fields. Using this framework, a simple and accurate formula is provided for the shear stress at the oscillating boundary, again for arbitrary frequency, which may prove useful in application. These solutions are easily implemented on any symbolic or numerical package, such as Mathematica or matlab, facilitating the characterization of flows produced by nanomechanical devices and providing insight into the underlying flow physics.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ketterer, S.P.

This manual is designed as a comprehensive hands-on instructional manual for learning the T{sub E}X* computer typesetting program in a classroom environment. Each section presents a new concept in careful detail, concluding with an exercise (T{sub E}Xercise) to reinforce the learning of the concept. The manual introduces the novice T{sub E}X user to the program's basic command structure, along with the concepts of grouping, producing accents, making font changes, and generating mathematical symbols. The T{sub E}Xercises guide the new user in generating text containing footnotes, multilevel lists, and hanging indentations, as well as in magnifying text for viewgraphs. Once themore » basic text generation is defined, the more advanced topics of formatting math equations and tables are explained. A full range of math capabilities is presented --- beginning with simple one-line equations, progressing through complex numbered and aligned equations, and concluding with matrices. The sections on table generation present the basic concepts in T{sub E}X's table-formatting program and then build on them. The new user first learns to construct simple tables, and with careful explanations and guidance, learns to add one new table enhancement at a time. By the conclusion of these sections, the user can construct tables with horizontal and vertical rules and with column entries that are paragraphs. 1 ref.« less

Predicting earthquakes by analyzing accelerating precursory seismic activity

USGS Publications Warehouse

Varnes, D.J.

1989-01-01

During 11 sequences of earthquakes that in retrospect can be classed as foreshocks, the accelerating rate at which seismic moment is released follows, at least in part, a simple equation. This equation (1) is {Mathematical expression},where {Mathematical expression} is the cumulative sum until time, t, of the square roots of seismic moments of individual foreshocks computed from reported magnitudes;C and n are constants; and tfis a limiting time at which the rate of seismic moment accumulation becomes infinite. The possible time of a major foreshock or main shock, tf,is found by the best fit of equation (1), or its integral, to step-like plots of {Mathematical expression} versus time using successive estimates of tfin linearized regressions until the maximum coefficient of determination, r2,is obtained. Analyzed examples include sequences preceding earthquakes at Cremasta, Greece, 2/5/66; Haicheng, China 2/4/75; Oaxaca, Mexico, 11/29/78; Petatlan, Mexico, 3/14/79; and Central Chile, 3/3/85. In 29 estimates of main-shock time, made as the sequences developed, the errors in 20 were less than one-half and in 9 less than one tenth the time remaining between the time of the last data used and the main shock. Some precursory sequences, or parts of them, yield no solution. Two sequences appear to include in their first parts the aftershocks of a previous event; plots using the integral of equation (1) show that the sequences are easily separable into aftershock and foreshock segments. Synthetic seismic sequences of shocks at equal time intervals were constructed to follow equation (1), using four values of n. In each series the resulting distributions of magnitudes closely follow the linear Gutenberg-Richter relation log N=a-bM, and the product n times b for each series is the same constant. In various forms and for decades, equation (1) has been used successfully to predict failure times of stressed metals and ceramics, landslides in soil and rock slopes, and volcanic eruptions. Results of more recent experiments and theoretical studies on crack propagation, fault mechanics, and acoustic emission can be closely reproduced by equation (1). Rate-process theory and continuum damage mechanics offer leads toward understanding the physical processes. ?? 1989 Birkha??user Verlag.

On the origin of Hawking mini black-holes and the cold early universe

NASA Technical Reports Server (NTRS)

Canuto, V.

1978-01-01

A simple argument is outlined leading to the result that the mass of mini black holes exploding today is 10 to the 15th power g. A mathematical model is discussed which indicates that the equation of state is greatly softened in the high-density regime and a phase transition may exist, such that any length (particularly very small sizes) will grow with time irrespective of its relation to the size of the particle horizon. It is shown that the effect of spin-2 mesons with respect to the equation of state is to soften the pressure and make it negative. An analytical expression is given for the probability that any particular region in a hot early universe will evolve into a black hole.

The Equation, the Whole Equation, and Nothing but the Equation! One Approach to the Teaching of Linear Equations.

ERIC Educational Resources Information Center

Pirie, Susan E. B.; Martin, Lyndon

1997-01-01

Presents the results of a case study which looked at the mathematics classroom of one teacher trying to teach mathematics with meaning to pupils or lower ability at the secondary level. Contrasts methods of teaching linear equations to a variety of ability levels and uses the Pirie-Kieren model to account for the successful growth in understanding…

Transport through a network of capillaries from ultrametric diffusion equation with quadratic nonlinearity

NASA Astrophysics Data System (ADS)

Oleschko, K.; Khrennikov, A.

2017-10-01

This paper is about a novel mathematical framework to model transport (of, e.g., fluid or gas) through networks of capillaries. This framework takes into account the tree structure of the networks of capillaries. (Roughly speaking, we use the tree-like system of coordinates.) As is well known, tree-geometry can be topologically described as the geometry of an ultrametric space, i.e., a metric space in which the metric satisfies the strong triangle inequality: in each triangle, the third side is less than or equal to the maximum of two other sides. Thus transport (e.g., of oil or emulsion of oil and water in porous media, or blood and air in biological organisms) through networks of capillaries can be mathematically modelled as ultrametric diffusion. Such modelling was performed in a series of recently published papers of the authors. However, the process of transport through capillaries can be only approximately described by the linear diffusion, because the concentration of, e.g., oil droplets, in a capillary can essentially modify the dynamics. Therefore nonlinear dynamical equations provide a more adequate model of transport in a network of capillaries. We consider a nonlinear ultrametric diffusion equation with quadratic nonlinearity - to model transport in such a network. Here, as in the linear case, we apply the theory of ultrametric wavelets. The paper also contains a simple introduction to theory of ultrametric spaces and analysis on them.

Mastering algebra retrains the visual system to perceive hierarchical structure in equations.

PubMed

Marghetis, Tyler; Landy, David; Goldstone, Robert L

2016-01-01

Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual system-in particular, object-based attention-is retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressions-but only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception.

Davidenko’s Method for the Solution of Nonlinear Operator Equations.

DTIC Science & Technology

NONLINEAR DIFFERENTIAL EQUATIONS, NUMERICAL INTEGRATION), OPERATORS(MATHEMATICS), BANACH SPACE , MAPPING (TRANSFORMATIONS), NUMERICAL METHODS AND PROCEDURES, INTEGRALS, SET THEORY, CONVERGENCE, MATRICES(MATHEMATICS)

Induced Eddy Currents in Simple Conductive Geometries: Mathematical Formalism Describes the Excitation of Electrical Eddy Currents in a Time-Varying Magnetic Field

DOE PAGES

Nagel, James R.

2017-12-22

In this paper, a complete mathematical formalism is introduced to describe the excitation of electrical eddy currents due to a time-varying magnetic field. The process works by applying a quasistatic approximation to Ampere's law and then segregating the magnetic field into impressed and induced terms. The result is a nonhomogeneous vector Helmholtz equation that can be analytically solved for many practical geometries. Four demonstration cases are then solved under a constant excitation field over all space—an infinite slab in one dimension, a longitudinal cylinder in two dimensions, a transverse cylinder in two dimensions, and a sphere in three dimensions. Numericalmore » simulations are also performed in parallel with analytic computations, all of which verify the accuracy of the derived expressions.« less

Induced Eddy Currents in Simple Conductive Geometries: Mathematical Formalism Describes the Excitation of Electrical Eddy Currents in a Time-Varying Magnetic Field

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nagel, James R.

In this paper, a complete mathematical formalism is introduced to describe the excitation of electrical eddy currents due to a time-varying magnetic field. The process works by applying a quasistatic approximation to Ampere's law and then segregating the magnetic field into impressed and induced terms. The result is a nonhomogeneous vector Helmholtz equation that can be analytically solved for many practical geometries. Four demonstration cases are then solved under a constant excitation field over all space—an infinite slab in one dimension, a longitudinal cylinder in two dimensions, a transverse cylinder in two dimensions, and a sphere in three dimensions. Numericalmore » simulations are also performed in parallel with analytic computations, all of which verify the accuracy of the derived expressions.« less

Nonlinear Boltzmann equation for the homogeneous isotropic case: Some improvements to deterministic methods and applications to relaxation towards local equilibrium

NASA Astrophysics Data System (ADS)

Asinari, P.

2011-03-01

Boltzmann equation is one the most powerful paradigms for explaining transport phenomena in fluids. Since early fifties, it received a lot of attention due to aerodynamic requirements for high altitude vehicles, vacuum technology requirements and nowadays, micro-electro-mechanical systems (MEMs). Because of the intrinsic mathematical complexity of the problem, Boltzmann himself started his work by considering first the case when the distribution function does not depend on space (homogeneous case), but only on time and the magnitude of the molecular velocity (isotropic collisional integral). The interest with regards to the homogeneous isotropic Boltzmann equation goes beyond simple dilute gases. In the so-called econophysics, a Boltzmann type model is sometimes introduced for studying the distribution of wealth in a simple market. Another recent application of the homogeneous isotropic Boltzmann equation is given by opinion formation modeling in quantitative sociology, also called socio-dynamics or sociophysics. The present work [1] aims to improve the deterministic method for solving homogenous isotropic Boltzmann equation proposed by Aristov [2] by two ideas: (a) the homogeneous isotropic problem is reformulated first in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium).

The stability issues in problems of mathematical modeling

NASA Astrophysics Data System (ADS)

Mokin, A. Yu.; Savenkova, N. P.; Udovichenko, N. S.

2018-03-01

In the paper it is briefly considered various aspects of stability concepts, which are used in physics, mathematics and numerical methods of solution. The interrelation between these concepts is described, the questions of preliminary stability research before the numerical solution of the problem and the correctness of the mathematical statement of the physical problem are discussed. Examples of concrete mathematical statements of individual physical problems are given: a nonlocal problem for the heat equation, the Korteweg-de Fries equation with boundary conditions at infinity, the sine-Gordon equation, the problem of propagation of femtosecond light pulses in an area with a cubic nonlinearity.

Comments on "Drill-string horizontal dynamics with uncertainty on the frictional force" by T.G. Ritto, M.R. Escalante, Rubens Sampaio, M.B. Rosales [J. Sound Vib. 332 (2013) 145-153

NASA Astrophysics Data System (ADS)

Li, Zifeng

2016-12-01

This paper analyzes the mechanical and mathematical models in "Ritto et al. (2013) [1]". The results are that: (1) the mechanical model is obviously incorrect; (2) the mathematical model is not complete; (3) the differential equation is obviously incorrect; (4) the finite element equation is obviously not discretized from the corresponding mathematical model above, and is obviously incorrect. A mathematical model of dynamics should include the differential equations, the boundary conditions and the initial conditions.

What is behind small deviations of quantum mechanics theory from experiments? Observer's mathematics point of view

NASA Astrophysics Data System (ADS)

Khots, Boris; Khots, Dmitriy

2014-12-01

Certain results that have been predicted by Quantum Mechanics (QM) theory are not always supported by experiments. This defines a deep crisis in contemporary physics and, in particular, quantum mechanics. We believe that, in fact, the mathematical apparatus employed within today's physics is a possible reason. In particular, we consider the concept of infinity that exists in today's mathematics as the root cause of this problem. We have created Observer's Mathematics that offers an alternative to contemporary mathematics. This paper is an attempt to relay how Observer's Mathematics may explain some of the contradictions in QM theory results. We consider the Hamiltonian Mechanics, Newton equation, Schrodinger equation, two slit interference, wave-particle duality for single photons, uncertainty principle, Dirac equations for free electron in a setting of arithmetic, algebra, and topology provided by Observer's Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to solution of these problems are provided.

Coupled Particle Transport and Pattern Formation in a Nonlinear Leaky-Box Model

NASA Technical Reports Server (NTRS)

Barghouty, A. F.; El-Nemr, K. W.; Baird, J. K.

2009-01-01

Effects of particle-particle coupling on particle characteristics in nonlinear leaky-box type descriptions of the acceleration and transport of energetic particles in space plasmas are examined in the framework of a simple two-particle model based on the Fokker-Planck equation in momentum space. In this model, the two particles are assumed coupled via a common nonlinear source term. In analogy with a prototypical mathematical system of diffusion-driven instability, this work demonstrates that steady-state patterns with strong dependence on the magnetic turbulence but a rather weak one on the coupled particles attributes can emerge in solutions of a nonlinearly coupled leaky-box model. The insight gained from this simple model may be of wider use and significance to nonlinearly coupled leaky-box type descriptions in general.

Biomat development in soil treatment units for on-site wastewater treatment.

PubMed

Winstanley, H F; Fowler, A C

2013-10-01

We provide a simple mathematical model of the bioremediation of contaminated wastewater leaching into the subsoil below a septic tank percolation system. The model comprises a description of the percolation system's flows, together with equations describing the growth of biomass and the uptake of an organic contaminant concentration. By first rendering the model dimensionless, it can be partially solved, to provide simple insights into the processes which control the efficacy of the system. In particular, we provide quantitative insight into the effect of a near surface biomat on subsoil permeability; this can lead to trench ponding, and thus propagation of effluent further down the trench. Using the computed vadose zone flow field, the model can be simply extended to include reactive transport of other contaminants of interest.

Secondary School Advanced Mathematics, Chapter 4, Equations, Inequalities, and Radicals, Chapter 5, Circles and Spheres. Student's Text.

ERIC Educational Resources Information Center

Stanford Univ., CA. School Mathematics Study Group.

This text is the third of five in the Secondary School Advanced Mathematics (SSAM) series which was designed to meet the needs of students who have completed the Secondary School Mathematics (SSM) program, and wish to continue their study of mathematics. The first of the two chapters in this text deals with equations, inequalities and radicals.…

Student Perceptions of Writing Projects in a University Differential-Equations Course

ERIC Educational Resources Information Center

Latulippe, Christine; Latulippe, Joe

2014-01-01

This qualitative study surveyed 102 differential-equations students in order to investigate how students participating in writing projects in university-level mathematics courses perceive the benefits of writing in the mathematics classroom. Based on previous literature on writing in mathematics, students were asked specifically about the benefits…

Mathematical Modeling of Chemical Stoichiometry

ERIC Educational Resources Information Center

Croteau, Joshua; Fox, William P.; Varazo, Kristofoland

2007-01-01

In beginning chemistry classes, students are taught a variety of techniques for balancing chemical equations. The most common method is inspection. This paper addresses using a system of linear mathematical equations to solve for the stoichiometric coefficients. Many linear algebra books carry the standard balancing of chemical equations as an…

Differential Equations Models to Study Quorum Sensing.

PubMed

Pérez-Velázquez, Judith; Hense, Burkhard A

2018-01-01

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

Prediction and Production of Simple Mathematical Equations: Evidence from Visual World Eye-Tracking.

PubMed

Hintz, Florian; Meyer, Antje S

2015-01-01

The relationship between the production and the comprehension systems has recently become a topic of interest for many psycholinguists. It has been argued that these systems are tightly linked and in particular that listeners use the production system to predict upcoming content. In this study, we tested how similar production and prediction processes are in a novel version of the visual world paradigm. Dutch speaking participants (native speakers in Experiment 1; German-Dutch bilinguals in Experiment 2) listened to mathematical equations while looking at a clock face featuring the numbers 1 to 12. On alternating trials, they either heard a complete equation ("three plus eight is eleven") or they heard the first part ("three plus eight is") and had to produce the result ("eleven") themselves. Participants were encouraged to look at the relevant numbers throughout the trial. Their eye movements were recorded and analyzed. We found that the participants' eye movements in the two tasks were overall very similar. They fixated the first and second number of the equations shortly after they were mentioned, and fixated the result number well before they named it on production trials and well before the recorded speaker named it on comprehension trials. However, all fixation latencies were shorter on production than on comprehension trials. These findings suggest that the processes involved in planning to say a word and anticipating hearing a word are quite similar, but that people are more aroused or engaged when they intend to respond than when they merely listen to another person.

Biology As a Source for Algebra Equations: Insects

ERIC Educational Resources Information Center

Horak, Virginia M.

2005-01-01

The activity developed in an integrated high school course that was team-taught by both mathematics and science teachers examines linear equations developed from relationships in biology. These equations provide students with opportunities to see the way mathematics could be used to describe biological relationships, and then apply to solve…

Secondary School Advanced Mathematics, Chapter 6, The Complex Number System, Chapter 7, Equations of the First and Second Degree in Two Variables. Student's Text.

ERIC Educational Resources Information Center

Stanford Univ., CA. School Mathematics Study Group.

This text is the fourth of five in the Secondary School Advanced Mathematics (SSAM) series which was designed to meet the needs of students who have completed the Secondary School Mathematics (SSM) program, and wish to continue their study of mathematics. This text begins with a brief discussion of quadratic equations which motivates the…

Control of Solar Power Plants Connected Grid with Simple Calculation Method on Residential Homes

NASA Astrophysics Data System (ADS)

Kananda, Kiki; Nazir, Refdinal

2017-12-01

One of the most compatible renewable energy in all regions to apply is solar energy. Solar power plants can be built connected to existing or stand-alone power grids. In assisting the residential electricity in which there is a power grid, then a small scale solar energy power plants is very appropriate. However, the general constraint of solar energy power plants is still low in terms of efficiency. Therefore, this study will explain how to control the power of solar power plants more optimally, which is expected to reactive power to zero to raise efficiency. This is a continuation of previous research using Newton Rapshon control method. In this study we introduce a simple method by using ordinary mathematical calculations of solar-related equations. In this model, 10 PV modules type of ND T060M1 with a 60 Wp capacity are used. The calculations performed using MATLAB Simulink provide excellent value. For PCC voltage values obtained a stable quantity of approximately 220 V. At a maximum irradiation condition of 1000 W / m2, the reactive power value of Q solar generating system maximum 20.48 Var and maximum active power of 417.5 W. In the condition of lower irradiation, value of reactive power Q almost close to zero 0.77Var. This simple mathematical method can provide excellent quality control power values.

Localized light waves: Paraxial and exact solutions of the wave equation (a review)

NASA Astrophysics Data System (ADS)

Kiselev, A. P.

2007-04-01

Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “ X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.

Haldane, Waddington and recombinant inbred lines: extension of their work to any number of genes.

PubMed

Samal, Areejit; Martin, Olivier C

2017-11-01

In the early 1930s, J. B. S. Haldane and C. H. Waddington collaborated on the consequences of genetic linkage and inbreeding. One elegant mathematical genetics problem solved by them concerns recombinant inbred lines (RILs) produced via repeated self or brother-sister mating. In this classic contribution, Haldane and Waddington derived an analytical formula for the probabilities of 2-locus and 3-locus RIL genotypes. Specifically, the Haldane-Waddington formula gives the recombination rate R in such lines as a simple function of the per generation recombination rate r. Interestingly, for more than 80 years, an extension of this result to four or more loci remained elusive. In 2015, we generalized the Haldane-Waddington self-mating result to any number of loci. Our solution used self-consistent equations of the multi-locus probabilities 'for an infinite number of generations' and solved these by simple algebraic operations. In practice, our approach provides a quantum leap in the systems that can be handled: the cases of up to six loci can be solved by hand while a computer program implementing our mathematical formalism tackles up to 20 loci on standard desktop computers.

Stimulating Mathematical Reasoning with Simple Open-Ended Tasks

ERIC Educational Resources Information Center

West, John

2018-01-01

The importance of mathematical reasoning is unquestioned and providing opportunities for students to become involved in mathematical reasoning is paramount. The open-ended tasks presented incorporate mathematical content explored through the contexts of problem solving and reasoning. This article presents a number of simple tasks that may be…

Rare events in finite and infinite dimensions

NASA Astrophysics Data System (ADS)

Reznikoff, Maria G.

Thermal noise introduces stochasticity into deterministic equations and makes possible events which are never seen in the zero temperature setting. The driving force behind the thesis work is a desire to bring analysis and probability to bear on a class of relevant and intriguing physical problems, and in so doing, to allow applications to drive the development of new mathematical theory. The unifying theme is the study of rare events under the influence of small, random perturbations, and the manifold mathematical problems which ensue. In the first part, we apply large deviation theory and prefactor estimates to a coherent rotation micromagnetic model in order to analyze thermally activated magnetic switching. We consider recent physical experiments and the mathematical questions "asked" by them. A stochastic resonance type phenomenon is discovered, leading to the definition of finite temperature astroids. Non-Arrhenius behavior is discussed. The analysis is extended to ramped astroids. In addition, we discover that for low damping and ultrashort pulses, deterministic effects can override thermal effects, in accord with very recent ultrashort pulse experiments. Even more interesting, perhaps, is the study of large deviations in the infinite dimensional context, i.e. in spatially extended systems. Inspired by recent numerical investigations, we study the stochastically perturbed Allen Cahn and Cahn Hilliard equations. For the Allen Cahn equation, we study the action minimization problem (a deterministic variational problem) and prove the action scaling in four parameter regimes, via upper and lower bounds. The sharp interface limit is studied. We formally derive a reduced action functional which lends insight into the connection between action minimization and curvature flow. For the Cahn Hilliard equation, we prove upper and lower bounds for the scaling of the energy barrier in the nucleation and growth regime. Finally, we consider rare events in large or infinite domains, in one spatial dimension. We introduce a natural reference measure through which to analyze the invariant measure of stochastically perturbed, nonlinear partial differential equations. Also, for noisy reaction diffusion equations with an asymmetric potential, we discover how to rescale space and time in order to map the dynamics in the zero temperature limit to the Poisson Model, a simple version of the Johnson-Mehl-Avrami-Kolmogorov model for nucleation and growth.

Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gamba, Irene M.; Tharkabhushanam, Sri Harsha

We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation (BTE) in d-dimensions for variable hard sphere (VHS) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computation is reduced to a separate integral over the unit sphere S{sup d-1}. The conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibilitymore » (inelastic interactions) or elastic models of slowing down process. Our simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation, both for elastic and inelastic VHS interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in Bobylev et al. [A.V. Bobylev, C. Cercignani, G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, Journal of Statistical Physics 111 (2003) 403-417] and generalized to a wide range of related models in Bobylev et al. [A.V. Bobylev, C. Cercignani, I.M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Communication in Mathematical Physics, in press. URL: ()]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods [T.V. Noije, M. Ernst, Velocity distributions in homogeneously cooling and heated granular fluids, Granular Matter 1(57) (1998); M.H. Ernst, R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, Journal of Statistical Physics 109 (2002) 407-432; S.J. Moon, M.D. Shattuck, J. Swift, Velocity distributions and correlations in homogeneously heated granular media, Physical Review E 64 (2001) 031303; I.M. Gamba, S. Rjasanow, W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Mathematical and Computer Modelling 42 (2005) 683-700] and rigorously proven in Gamba et al. [I.M. Gamba, V. Panferov, C. Villani, On the Boltzmann equation for diffusively excited granular media, Communications in Mathematical Physics 246 (2004) 503-541(39)] and [A.V. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, Journal of Statistical Physics 116 (2004) 1651-1682].« less

A systematic literature review of Burgers' equation with recent advances

NASA Astrophysics Data System (ADS)

Bonkile, Mayur P.; Awasthi, Ashish; Lakshmi, C.; Mukundan, Vijitha; Aswin, V. S.

2018-06-01

Even if numerical simulation of the Burgers' equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers' equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective-diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers' equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers' equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers' equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers' equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.

Optimal shielding design for minimum materials cost or mass

DOE PAGES

Woolley, Robert D.

2015-12-02

The mathematical underpinnings of cost optimal radiation shielding designs based on an extension of optimal control theory are presented, a heuristic algorithm to iteratively solve the resulting optimal design equations is suggested, and computational results for a simple test case are discussed. A typical radiation shielding design problem can have infinitely many solutions, all satisfying the problem's specified set of radiation attenuation requirements. Each such design has its own total materials cost. For a design to be optimal, no admissible change in its deployment of shielding materials can result in a lower cost. This applies in particular to very smallmore » changes, which can be restated using the calculus of variations as the Euler-Lagrange equations. Furthermore, the associated Hamiltonian function and application of Pontryagin's theorem lead to conditions for a shield to be optimal.« less

AN ANALYTIC MODEL OF DUSTY, STRATIFIED, SPHERICAL H ii REGIONS

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rodríguez-Ramírez, J. C.; Raga, A. C.; Lora, V.

2016-12-20

We study analytically the effect of radiation pressure (associated with photoionization processes and with dust absorption) on spherical, hydrostatic H ii regions. We consider two basic equations, one for the hydrostatic balance between the radiation-pressure components and the gas pressure, and another for the balance among the recombination rate, the dust absorption, and the ionizing photon rate. Based on appropriate mathematical approximations, we find a simple analytic solution for the density stratification of the nebula, which is defined by specifying the radius of the external boundary, the cross section of dust absorption, and the luminosity of the central star. Wemore » compare the analytic solution with numerical integrations of the model equations of Draine, and find a wide range of the physical parameters for which the analytic solution is accurate.« less

A thermal analysis of a spirally wound battery using a simple mathematical model

NASA Technical Reports Server (NTRS)

Evans, T. I.; White, R. E.

1989-01-01

A two-dimensional thermal model for spirally wound batteries has been developed. The governing equation of the model is the energy balance. Convective and insulated boundary conditions are used, and the equations are solved using a finite element code called TOPAZ2D. The finite element mesh is generated using a preprocessor to TOPAZ2D called MAZE. The model is used to estimate temperature profiles within a spirally wound D-size cell. The model is applied to the lithium/thionyl chloride cell because of the thermal management problems that this cell exhibits. Simplified one-dimensional models are presented that can be used to predict best and worst temperature profiles. The two-dimensional model is used to predict the regions of maximum temperature within the spirally wound cell. Normal discharge as well as thermal runaway conditions are investigated.

Decomposition of the compound Atwood machine

NASA Astrophysics Data System (ADS)

Lopes Coelho, R.

2017-11-01

Non-standard solving strategies for the compound Atwood machine problem have been proposed. The present strategy is based on a very simple idea. Taking an Atwood machine and replacing one of its bodies by another Atwood machine, we have a compound machine. As this operation can be repeated, we can construct any compound Atwood machine. This rule of construction is transferred to a mathematical model, whereby the equations of motion are obtained. The only difference between the machine and its model is that instead of pulleys and bodies, we have reference frames that move solidarily with these objects. This model provides us with the accelerations in the non-inertial frames of the bodies, which we will use to obtain the equations of motion. This approach to the problem will be justified by the Lagrange method and exemplified by machines with six and eight bodies.

A simple mathematical method to estimate ammonia emission from in-house windrowing of poultry litter.

PubMed

Ro, Kyoung S; Szogi, Ariel A; Moore, Philip A

2018-05-12

In-house windrowing between flocks is an emerging sanitary management practice to partially disinfect the built-up litter in broiler houses. However, this practice may also increase ammonia (NH 3 ) emission from the litter due to the increase in litter temperature. The objectives of this study were to develop mathematical models to estimate NH 3 emission rates from broiler houses practicing in-house windrowing between flocks. Equations to estimate mass-transfer areas form different shapes windrowed litter (triangular, rectangular, and semi-cylindrical prisms) were developed. Using these equations, the heights of windrows yielding the smallest mass-transfer area were estimated. Smaller mass-transfer area is preferred as it reduces both emission rates and heat loss. The heights yielding the minimum mass-transfer area were 0.8 and 0.5 m for triangular and rectangular windrows, respectively. Only one height (0.6 m) was theoretically possible for semi-cylindrical windrows because the base and the height were not independent. Mass-transfer areas were integrated with published process-based mathematical models to estimate the total house NH 3 emission rates during in-house windrowing of poultry litter. The NH 3 emission rate change calculated from the integrated model compared well with the observed values except for the very high NH 3 initial emission rate from mechanically disturbing the litter to form the windrows. This approach can be used to conveniently estimate broiler house NH 3 emission rates during in-house windrowing between flocks by simply measuring litter temperatures.

Bush Physics for the 21st Century, A Distance Delivery Physics Course to Bridge the Gap in Rural Alaska and Across the North

NASA Astrophysics Data System (ADS)

Solie, D. J.; Spencer, V.

2009-12-01

Bush Physics for the 21st Century brings physics that is culturally connected, engaging to modern youth, and mathematically rigorous, to high school and college students in the remote and often road-less villages of Alaska. The primary goal of the course is to prepare rural (predominantly Alaska Native) students for success in university science and engineering degree programs and ultimately STEM careers. The course is currently delivered via video conference and web based electronic blackboard tailored to the needs of remote students. Practical, culturally relevant kinetic examples from traditional and modern northern life are used to engage students, and a rigorous and mathematical focus is stressed to strengthen problem solving skills. Simple hands-on-lab experiments are delivered to the students with the exercises completed on-line. In addition, students are teamed and required to perform a much more involved experimental study with the results presented by teams at the conclusion of the course. Connecting abstract mathematical symbols and equations to real physical objects and problems is one of the most difficult things to master in physics. Greek symbols are traditionally used in equations, however, to strengthen the visual/conceptual connection with symbol and encourage an indigenous connection to the concepts we have introduced Inuktitut symbols to complement the traditional Greek symbols. Results and observations from the first two pilot semesters (spring 2008 and 2009) will be presented.

Bush Physics for the 21st Century, A Distance Delivery Physics Course Targeting Students in Rural Alaska and Across the North

NASA Astrophysics Data System (ADS)

Solie, D. J.; Spencer, V. K.

2010-12-01

Bush Physics for the 21st Century brings physics that is engaging to modern youth, and mathematically rigorous, to high school and college students in the remote and often road-less villages of Alaska where the opportunity to take a physics course has been nearly nonexistent. The primary goal of the course is to prepare rural (predominantly Alaska Native) students for success in university science and engineering degree programs and ultimately STEM careers. The course is delivered via video conference and web based electronic blackboard tailored to the needs of remote students. Kinetic, practical and culturally relevant place-based examples from traditional and modern northern life are used to engage students, and a rigorous and mathematical focus is stressed to strengthen problem solving skills. Simple hands-on-lab experiment kits are shipped to the students. In addition students conduct a Collaborative Research Experiment where they coordinate times of sun angle measurements with teams in other villages to determine their latitude and longitude as well as an estimate of the circumference of the earth. Connecting abstract mathematical symbols and equations to real physical objects and problems is one of the most difficult things to master in physics. We introduce Inuktitut symbols to complement the traditional Greek symbols in equations to strengthen the visual/conceptual connection with symbol and encourage an indigenous connection to the physical concepts. Results and observations from the first three pilot semesters (spring 2008, 2009 and 2010) will be presented.

Differential equations with applications in cancer diseases.

PubMed

Ilea, M; Turnea, M; Rotariu, M

2013-01-01

Mathematical modeling is a process by which a real world problem is described by a mathematical formulation. The cancer modeling is a highly challenging problem at the frontier of applied mathematics. A variety of modeling strategies have been developed, each focusing on one or more aspects of cancer. The vast majority of mathematical models in cancer diseases biology are formulated in terms of differential equations. We propose an original mathematical model with small parameter for the interactions between these two cancer cell sub-populations and the mathematical model of a vascular tumor. We work on the assumption that, the quiescent cells' nutrient consumption is long. One the equations system includes small parameter epsilon. The smallness of epsilon is relative to the size of the solution domain. MATLAB simulations obtained for transition rate from the quiescent cells' nutrient consumption is long, we show a similar asymptotic behavior for two solutions of the perturbed problem. In this system, the small parameter is an asymptotic variable, different from the independent variable. The graphical output for a mathematical model of a vascular tumor shows the differences in the evolution of the tumor populations of proliferating, quiescent and necrotic cells. The nutrient concentration decreases sharply through the viable rim and tends to a constant level in the core due to the nearly complete necrosis in this region. Many mathematical models can be quantitatively characterized by ordinary differential equations or partial differential equations. The use of MATLAB in this article illustrates the important role of informatics in research in mathematical modeling. The study of avascular tumor growth cells is an exciting and important topic in cancer research and will profit considerably from theoretical input. Interpret these results to be a permanent collaboration between math's and medical oncologists.

Human sleep and circadian rhythms: a simple model based on two coupled oscillators.

PubMed

Strogatz, S H

1987-01-01

We propose a model of the human circadian system. The sleep-wake and body temperature rhythms are assumed to be driven by a pair of coupled nonlinear oscillators described by phase variables alone. The novel aspect of the model is that its equations may be solved analytically. Computer simulations are used to test the model against sleep-wake data pooled from 15 studies of subjects living for weeks in unscheduled, time-free environments. On these tests the model performs about as well as the existing models, although its mathematical structure is far simpler.

The modified alternative (G'/G)-expansion method to nonlinear evolution equation: application to the (1+1)-dimensional Drinfel'd-Sokolov-Wilson equation.

PubMed

Akbar, M Ali; Mohd Ali, Norhashidah Hj; Mohyud-Din, Syed Tauseef

2013-01-01

Over the years, (G'/G)-expansion method is employed to generate traveling wave solutions to various wave equations in mathematical physics. In the present paper, the alternative (G'/G)-expansion method has been further modified by introducing the generalized Riccati equation to construct new exact solutions. In order to illustrate the novelty and advantages of this approach, the (1+1)-dimensional Drinfel'd-Sokolov-Wilson (DSW) equation is considered and abundant new exact traveling wave solutions are obtained in a uniform way. These solutions may be imperative and significant for the explanation of some practical physical phenomena. It is shown that the modified alternative (G'/G)-expansion method an efficient and advance mathematical tool for solving nonlinear partial differential equations in mathematical physics.

Validation the use of refractometer and mathematic equations to measure dietary formula contents for clinical application.

PubMed

Chang, W-K; Chao, Y-C; Mcclave, S-A; Yeh, M-K

2005-10-01

Gastric residual volumes are widely used to evaluate gastric emptying for patients receiving enteral feeding, but controversy exists about what constitutes gastric residual volume. We have developed a method by using refractometer and derived mathematical equations to calculate the formula concentration, total residual volume (TRV), and formula volume. In this study, we like to validate these mathematical equations before they can be implemented for clinical patient care. Four dietary formulas were evaluated in two consecutive validation experiments. Firstly, dietary formula volume of 50, 100, 200, and 400 ml were diluted with 50 ml water, and then the Brix value (BV) was measured by the refractometer. Secondly, 50 ml of water, then 100 ml of dietary formula were infused into a beaker, and followed by the BV measurement. After this, 50 ml of water was infused and followed by the second BV measurement. The entire procedure of infusing of dietary formula (100 ml) and waster (50 ml) was repeated twice and followed by the BV measurement. The formula contents (formula concentration, TRV, and formula volume) were calculated by mathematical equations. The calculated formula concentrations, TRVs, and formula volumes measured from mathematic equations were strongly close to the true values in the first and second validation experiments (R2>0.98, P<0.001). Refractometer and the derived mathematical equations may be used to accurately measure the formula concentration, TRV, and formula volume and served as a tool to monitor gastric emptying for patients receiving enteral feeding.

What is behind small deviations of quantum mechanics theory from experiments? Observer's mathematics point of view

DOE Office of Scientific and Technical Information (OSTI.GOV)

Khots, Boris, E-mail: bkhots@cccglobal.com; Khots, Dmitriy, E-mail: dkhots@imathconsulting.com

2014-12-10

Certain results that have been predicted by Quantum Mechanics (QM) theory are not always supported by experiments. This defines a deep crisis in contemporary physics and, in particular, quantum mechanics. We believe that, in fact, the mathematical apparatus employed within today's physics is a possible reason. In particular, we consider the concept of infinity that exists in today's mathematics as the root cause of this problem. We have created Observer's Mathematics that offers an alternative to contemporary mathematics. This paper is an attempt to relay how Observer's Mathematics may explain some of the contradictions in QM theory results. We considermore » the Hamiltonian Mechanics, Newton equation, Schrodinger equation, two slit interference, wave-particle duality for single photons, uncertainty principle, Dirac equations for free electron in a setting of arithmetic, algebra, and topology provided by Observer's Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to solution of these problems are provided.« less

A boundary integral approach in primitive variables for free surface flows

NASA Astrophysics Data System (ADS)

Casciola, C.; Piva, R.

The boundary integral formulation, very efficient for free surface potential flows, was considered for its possible extension to rotational flows either inviscid or viscous. We first analyze a general formulation for unsteady Navier-Stokes equations in primitive variables, which reduces to a representation for the Euler equations in the limiting case of Reynolds infinity. A first simplified model for rotational flows, obtained by decoupling kinematics and dynamics, reduces the integral equations to a known kinematical form whose mathematical and numerical properties have been studied. The dynamics equations to complete the model are obtained for the free surface and the wake. A simple and efficient scheme for the study of the non linear evolution of the wave system and its interaction with the body wake is presented. A steady state version for the calculation of the wave resistance is also reported. A second model was proposed for the simulation of rotational separated regions, by coupling the integral equations in velocity with an integral equation for the vorticity at the body boundary. The same procedure may be extended to include the diffusion of the vorticity in the flowfield. The vortex shedding from a cylindrical body in unsteady motion is discussed, as a first application of the model.

Stable time filtering of strongly unstable spatially extended systems

PubMed Central

Grote, Marcus J.; Majda, Andrew J.

2006-01-01

Many contemporary problems in science involve making predictions based on partial observation of extremely complicated spatially extended systems with many degrees of freedom and with physical instabilities on both large and small scale. Various new ensemble filtering strategies have been developed recently for these applications, and new mathematical issues arise. Because ensembles are extremely expensive to generate, one such issue is whether it is possible under appropriate circumstances to take long time steps in an explicit difference scheme and violate the classical Courant–Friedrichs–Lewy (CFL)-stability condition yet obtain stable accurate filtering by using the observations. These issues are explored here both through elementary mathematical theory, which provides simple guidelines, and the detailed study of a prototype model. The prototype model involves an unstable finite difference scheme for a convection–diffusion equation, and it is demonstrated below that appropriate observations can result in stable accurate filtering of this strongly unstable spatially extended system. PMID:16682626

Stable time filtering of strongly unstable spatially extended systems.

PubMed

Grote, Marcus J; Majda, Andrew J

2006-05-16

Many contemporary problems in science involve making predictions based on partial observation of extremely complicated spatially extended systems with many degrees of freedom and with physical instabilities on both large and small scale. Various new ensemble filtering strategies have been developed recently for these applications, and new mathematical issues arise. Because ensembles are extremely expensive to generate, one such issue is whether it is possible under appropriate circumstances to take long time steps in an explicit difference scheme and violate the classical Courant-Friedrichs-Lewy (CFL)-stability condition yet obtain stable accurate filtering by using the observations. These issues are explored here both through elementary mathematical theory, which provides simple guidelines, and the detailed study of a prototype model. The prototype model involves an unstable finite difference scheme for a convection-diffusion equation, and it is demonstrated below that appropriate observations can result in stable accurate filtering of this strongly unstable spatially extended system.

Analysis of Mathematical Modelling on Potentiometric Biosensors

PubMed Central

Mehala, N.; Rajendran, L.

2014-01-01

A mathematical model of potentiometric enzyme electrodes for a nonsteady condition has been developed. The model is based on the system of two coupled nonlinear time-dependent reaction diffusion equations for Michaelis-Menten formalism that describes the concentrations of substrate and product within the enzymatic layer. Analytical expressions for the concentration of substrate and product and the corresponding flux response have been derived for all values of parameters using the new homotopy perturbation method. Furthermore, the complex inversion formula is employed in this work to solve the boundary value problem. The analytical solutions obtained allow a full description of the response curves for only two kinetic parameters (unsaturation/saturation parameter and reaction/diffusion parameter). Theoretical descriptions are given for the two limiting cases (zero and first order kinetics) and relatively simple approaches for general cases are presented. All the analytical results are compared with simulation results using Scilab/Matlab program. The numerical results agree with the appropriate theories. PMID:25969765

Analysis of mathematical modelling on potentiometric biosensors.

PubMed

Mehala, N; Rajendran, L

2014-01-01

A mathematical model of potentiometric enzyme electrodes for a nonsteady condition has been developed. The model is based on the system of two coupled nonlinear time-dependent reaction diffusion equations for Michaelis-Menten formalism that describes the concentrations of substrate and product within the enzymatic layer. Analytical expressions for the concentration of substrate and product and the corresponding flux response have been derived for all values of parameters using the new homotopy perturbation method. Furthermore, the complex inversion formula is employed in this work to solve the boundary value problem. The analytical solutions obtained allow a full description of the response curves for only two kinetic parameters (unsaturation/saturation parameter and reaction/diffusion parameter). Theoretical descriptions are given for the two limiting cases (zero and first order kinetics) and relatively simple approaches for general cases are presented. All the analytical results are compared with simulation results using Scilab/Matlab program. The numerical results agree with the appropriate theories.

On Two-Scale Modelling of Heat and Mass Transfer

NASA Astrophysics Data System (ADS)

Vala, J.; Št'astník, S.

2008-09-01

Modelling of macroscopic behaviour of materials, consisting of several layers or components, whose microscopic (at least stochastic) analysis is available, as well as (more general) simulation of non-local phenomena, complicated coupled processes, etc., requires both deeper understanding of physical principles and development of mathematical theories and software algorithms. Starting from the (relatively simple) example of phase transformation in substitutional alloys, this paper sketches the general formulation of a nonlinear system of partial differential equations of evolution for the heat and mass transfer (useful in mechanical and civil engineering, etc.), corresponding to conservation principles of thermodynamics, both at the micro- and at the macroscopic level, and suggests an algorithm for scale-bridging, based on the robust finite element techniques. Some existence and convergence questions, namely those based on the construction of sequences of Rothe and on the mathematical theory of two-scale convergence, are discussed together with references to useful generalizations, required by new technologies.

A caveat concerning center of resistance

PubMed Central

Nägerl, Hans; Kubein-Meesenburg, Dietmar

2013-01-01

The center of resistance is a concept in theoretical orthodontics used to describe tooth movement under loads. It is commonly used to qualitatively predict tooth movement without recourse to complex equations or simulations. We start with a survey of the historical origin of the technical term. After this, the periodontal ligament is idealized as a linear elastic suspension. The mathematical formalism of vector and tensor calculus will clarify our reasoning. We show that a point such as the center of resistance basically only exists in two dimensions or in very special symmetric spatial configurations. In three dimensions, a simple counterexample of a suspension without a center of resistance is given. A second more tooth-like example illustrates the magnitude of the effects in question in dentistry. In conclusion, the center of resistance should be replaced by a newer and wider mathematical concept, the “center of elasticity,” together with a limiting parameter, the “radius of resistance.” PMID:24019849

Equating Two Forms of a Criterion-Referenced Test by Using Norm Referenced Data: An Illustration of Two Methods.

ERIC Educational Resources Information Center

Garcia-Quintana, Roan A.; Johnson, Lynne M.

Three different computational procedures for equating two forms of a test were applied to a pair of mathematics tests to compare the results of the three procedures. The tests that were being equated were two forms of the SRA Mastery Mathematics Tests. The common, linking test used for equating was the Comprehensive Tests of Basic Skills, Form S,…

Sine-Gordon equation and its application to tectonic stress transfer

NASA Astrophysics Data System (ADS)

Bykov, Victor G.

2014-07-01

An overview is given on remarkable progress that has been made in theoretical studies of solitons and other nonlinear wave patterns, excited during the deformation of fault block (fragmented) geological media. The models that are compliant with the classical and perturbed sine-Gordon equations have only been chosen. In these mathematical models, the rotation angle of blocks (fragments) and their translatory displacement of the medium are used as dynamic variables. A brief description of the known models and their geophysical and geodynamic applications is given. These models reproduce the kinematic and dynamic features of the traveling deformation front (kink, soliton) generated in the fragmented media. It is demonstrated that the sine-Gordon equation is applicable to the description of series of the observed seismic data, modeling of strain waves, as well as the features related to fault dynamics and the subduction slab, including slow earthquakes, periodicity of episodic tremor and slow slip (ETS) events, and migration pattern of tremors. The study shows that simple heuristic models and analytical and numerical computations can explain triggering of seismicity by transient processes, such as stress changes associated with solitary strain waves in crustal faults. The need to develop the above-mentioned new (nonlinear) mathematical models of the deformed fault and fragmented media was caused by the reason that it is impossible to explain a lot of the observed effects, particularly, slow redistribution and migration of stresses in the lithosphere, within the framework of the linear elasticity theory.

Spheroidal Integral Equations for Geodetic Inversion of Geopotential Gradients

NASA Astrophysics Data System (ADS)

Novák, Pavel; Šprlák, Michal

2018-03-01

The static Earth's gravitational field has traditionally been described in geodesy and geophysics by the gravitational potential (geopotential for short), a scalar function of 3-D position. Although not directly observable, geopotential functionals such as its first- and second-order gradients are routinely measured by ground, airborne and/or satellite sensors. In geodesy, these observables are often used for recovery of the static geopotential at some simple reference surface approximating the actual Earth's surface. A generalized mathematical model is represented by a surface integral equation which originates in solving Dirichlet's boundary-value problem of the potential theory defined for the harmonic geopotential, spheroidal boundary and globally distributed gradient data. The mathematical model can be used for combining various geopotential gradients without necessity of their re-sampling or prior continuation in space. The model extends the apparatus of integral equations which results from solving boundary-value problems of the potential theory to all geopotential gradients observed by current ground, airborne and satellite sensors. Differences between spherical and spheroidal formulations of integral kernel functions of Green's kind are investigated. Estimated differences reach relative values at the level of 3% which demonstrates the significance of spheroidal approximation for flattened bodies such as the Earth. The observation model can be used for combined inversion of currently available geopotential gradients while exploring their spectral and stochastic characteristics. The model would be even more relevant to gravitational field modelling of other bodies in space with more pronounced spheroidal geometry than that of the Earth.

Cross Coursing in Mathematics: Physical Modelling in Differential Equations Crossing to Discrete Dynamical Systems

ERIC Educational Resources Information Center

Winkel, Brian

2012-01-01

We give an example of cross coursing in which a subject or approach in one course in undergraduate mathematics is used in a completely different course. This situation crosses falling body modelling in an upper level differential equations course into a modest discrete dynamical systems unit of a first-year mathematics course. (Contains 1 figure.)

Reflections on the Notion of Culture in the History of Mathematics: The Example of "Geometrical Equations".

PubMed

Lê, François

2016-09-01

Argument This paper challenges the use of the notion of "culture" to describe a particular organization of mathematical knowledge, shared by a few mathematicians over a short period of time in the second half of the nineteenth century. This knowledge relates to "geometrical equations," objects that proved crucial for the mechanisms of encounters between equation theory, substitution theory, and geometry at that time, although they were not well-defined mathematical objects. The description of the mathematical collective activities linked to "geometrical equations," and especially the technical aspects of these activities, is made on the basis of a sociological definition of "culture." More precisely, after an examination of the social organization of the group of mathematicians, I argue that these activities form an intricate system of patterns, symbols, and values, for which I suggest a characterization as a "cultural system."

Nonlinear mathematical modeling and sensitivity analysis of hydraulic drive unit

NASA Astrophysics Data System (ADS)

Kong, Xiangdong; Yu, Bin; Quan, Lingxiao; Ba, Kaixian; Wu, Liujie

2015-09-01

The previous sensitivity analysis researches are not accurate enough and also have the limited reference value, because those mathematical models are relatively simple and the change of the load and the initial displacement changes of the piston are ignored, even experiment verification is not conducted. Therefore, in view of deficiencies above, a nonlinear mathematical model is established in this paper, including dynamic characteristics of servo valve, nonlinear characteristics of pressure-flow, initial displacement of servo cylinder piston and friction nonlinearity. The transfer function block diagram is built for the hydraulic drive unit closed loop position control, as well as the state equations. Through deriving the time-varying coefficient items matrix and time-varying free items matrix of sensitivity equations respectively, the expression of sensitivity equations based on the nonlinear mathematical model are obtained. According to structure parameters of hydraulic drive unit, working parameters, fluid transmission characteristics and measured friction-velocity curves, the simulation analysis of hydraulic drive unit is completed on the MATLAB/Simulink simulation platform with the displacement step 2 mm, 5 mm and 10 mm, respectively. The simulation results indicate that the developed nonlinear mathematical model is sufficient by comparing the characteristic curves of experimental step response and simulation step response under different constant load. Then, the sensitivity function time-history curves of seventeen parameters are obtained, basing on each state vector time-history curve of step response characteristic. The maximum value of displacement variation percentage and the sum of displacement variation absolute values in the sampling time are both taken as sensitivity indexes. The sensitivity indexes values above are calculated and shown visually in histograms under different working conditions, and change rules are analyzed. Then the sensitivity indexes values of four measurable parameters, such as supply pressure, proportional gain, initial position of servo cylinder piston and load force, are verified experimentally on test platform of hydraulic drive unit, and the experimental research shows that the sensitivity analysis results obtained through simulation are approximate to the test results. This research indicates each parameter sensitivity characteristics of hydraulic drive unit, the performance-affected main parameters and secondary parameters are got under different working conditions, which will provide the theoretical foundation for the control compensation and structure optimization of hydraulic drive unit.

Noticing relevant problem features: activating prior knowledge affects problem solving by guiding encoding

PubMed Central

Crooks, Noelle M.; Alibali, Martha W.

2013-01-01

This study investigated whether activating elements of prior knowledge can influence how problem solvers encode and solve simple mathematical equivalence problems (e.g., 3 + 4 + 5 = 3 + __). Past work has shown that such problems are difficult for elementary school students (McNeil and Alibali, 2000). One possible reason is that children's experiences in math classes may encourage them to think about equations in ways that are ultimately detrimental. Specifically, children learn a set of patterns that are potentially problematic (McNeil and Alibali, 2005a): the perceptual pattern that all equations follow an “operations = answer” format, the conceptual pattern that the equal sign means “calculate the total”, and the procedural pattern that the correct way to solve an equation is to perform all of the given operations on all of the given numbers. Upon viewing an equivalence problem, knowledge of these patterns may be reactivated, leading to incorrect problem solving. We hypothesized that these patterns may negatively affect problem solving by influencing what people encode about a problem. To test this hypothesis in children would require strengthening their misconceptions, and this could be detrimental to their mathematical development. Therefore, we tested this hypothesis in undergraduate participants. Participants completed either control tasks or tasks that activated their knowledge of the three patterns, and were then asked to reconstruct and solve a set of equivalence problems. Participants in the knowledge activation condition encoded the problems less well than control participants. They also made more errors in solving the problems, and their errors resembled the errors children make when solving equivalence problems. Moreover, encoding performance mediated the effect of knowledge activation on equivalence problem solving. Thus, one way in which experience may affect equivalence problem solving is by influencing what students encode about the equations. PMID:24324454

Improving Teaching Quality and Problem Solving Ability through Contextual Teaching and Learning in Differential Equations: A Lesson Study Approach

ERIC Educational Resources Information Center

Khotimah, Rita Pramujiyanti; Masduki

2016-01-01

Differential equations is a branch of mathematics which is closely related to mathematical modeling that arises in real-world problems. Problem solving ability is an essential component to solve contextual problem of differential equations properly. The purposes of this study are to describe contextual teaching and learning (CTL) model in…

Evolutionary game theory for physical and biological scientists. II. Population dynamics equations can be associated with interpretations

PubMed Central

Liao, David; Tlsty, Thea D.

2014-01-01

The use of mathematical equations to analyse population dynamics measurements is being increasingly applied to elucidate complex dynamic processes in biological systems, including cancer. Purely ‘empirical’ equations may provide sufficient accuracy to support predictions and therapy design. Nevertheless, interpretation of fitting equations in terms of physical and biological propositions can provide additional insights that can be used both to refine models that prove inconsistent with data and to understand the scope of applicability of models that validate. The purpose of this tutorial is to assist readers in mathematically associating interpretations with equations and to provide guidance in choosing interpretations and experimental systems to investigate based on currently available biological knowledge, techniques in mathematical and computational analysis and methods for in vitro and in vivo experiments. PMID:25097752

The comparison of Missouri mathematics project and teams games tournament viewed from emotional quotient eight grade student of junior school

NASA Astrophysics Data System (ADS)

Setyawan, Indra; Budiyono, Slamet, Isnandar

2017-08-01

This research was a quasi-experimental research with 2 × 3 factorial design. It aimed to determine the learning model between Missouri Mathematics Project (MMP) and Teams Games Tournament (TGT) that gave the best achievement on mathematics subject viewed from emotional quotient. The population of this research were all of Junior High School students at the 8th grade in Surakarta City, Central Java, Indonesia in academic year 2016/2017 which applied KTSP curriculum. The sample was taken by using stratified cluster random sampling. The data were collected by using methods of documentation, emotional quotient questionnaires, and mathematics achievement test. Data analysis technique used two ways analysis of variance (ANOVA) with unequal cell. According to the research findings, it could be concluded that: (1) students' mathematics achievement which were taught by using MMP is as good as emotional quotient achievement which were taught by using TGT in straight-line equation material, (2) mathematics achievement of students with high emotional quotient is better than students with medium and low emotional quotient, and mathematics achievement of students with medium emotional quotient is as good as students with low emotional quotient in straight-line equation material, (3) in each learning model, mathematics achievement of students with high emotional quotient is better than students with medium and low emotional quotient, and mathematics achievement of students with medium emotional quotient is as good as students with low emotional quotient in straight-line equation material (4) in each category of high and medium emotional quotient, student's mathematics achievement which were taught by using MMP is better than student's mathematics achievement which were taught by using TGT and in low emotional quotient student's mathematics achievement which were taught by using MMP is as good as student's mathematics achievement which were taught by using TGT in straight-line equation material.

Local equilibrium solutions in simple anisotropic cosmological models, as described by relativistic fluid dynamics

NASA Astrophysics Data System (ADS)

Shogin, Dmitry; Amund Amundsen, Per

2016-10-01

We test the physical relevance of the full and the truncated versions of the Israel-Stewart (IS) theory of irreversible thermodynamics in a cosmological setting. Using a dynamical systems method, we determine the asymptotic future of plane symmetric Bianchi type I spacetimes with a viscous mathematical fluid, keeping track of the magnitude of the relative dissipative fluxes, which determines the applicability of the IS theory. We consider the situations where the dissipative mechanisms of shear and bulk viscosity are involved separately and simultaneously. It is demonstrated that the only case in the given model when the fluid asymptotically approaches local thermal equilibrium, and the underlying assumptions of the IS theory are therefore not violated, is that of a dissipative fluid with vanishing bulk viscosity. The truncated IS equations for shear viscosity are found to produce solutions which manifest pathological dynamical features and, in addition, to be strongly sensitive to the choice of initial conditions. Since these features are observed already in the case of an oversimplified mathematical fluid model, we have no reason to assume that the truncation of the IS transport equations will produce relevant results for physically more realistic fluids. The possible role of bulk and shear viscosity in cosmological evolution is also discussed.

Remote sensing techniques for prediction of watershed runoff

NASA Technical Reports Server (NTRS)

Blanchard, B. J.

1975-01-01

Hydrologic parameters of watersheds for use in mathematical models and as design criteria for flood detention structures are sometimes difficult to quantify using conventional measuring systems. The advent of remote sensing devices developed in the past decade offers the possibility that watershed characteristics such as vegetative cover, soils, soil moisture, etc., may be quantified rapidly and economically. Experiments with visible and near infrared data from the LANDSAT-1 multispectral scanner indicate a simple technique for calibration of runoff equation coefficients is feasible. The technique was tested on 10 watersheds in the Chickasha area and test results show more accurate runoff coefficients were obtained than with conventional methods. The technique worked equally as well using a dry fall scene. The runoff equation coefficients were then predicted for 22 subwatersheds with flood detention structures. Predicted values were again more accurate than coefficients produced by conventional methods.

A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance

NASA Astrophysics Data System (ADS)

Witte, J. H.; Reisinger, C.

2010-09-01

We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desireable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to an order of O(1/ρ), where ρ>0 is the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.

Optimal Assignment Problem Applications of Finite Mathematics to Business and Economics. [and] Difference Equations with Applications. Applications of Difference Equations to Economics and Social Sciences. [and] Selected Applications of Mathematics to Finance and Investment. Applications of Elementary Algebra to Finance. [and] Force of Interest. Applications of Calculus to Finance. UMAP Units 317, 322, 381, 382.

ERIC Educational Resources Information Center

Gale, David; And Others

Four units make up the contents of this document. The first examines applications of finite mathematics to business and economies. The user is expected to learn the method of optimization in optimal assignment problems. The second module presents applications of difference equations to economics and social sciences, and shows how to: 1) interpret…

The mathematical relationship between Zipf’s law and the hierarchical scaling law

NASA Astrophysics Data System (ADS)

Chen, Yanguang

2012-06-01

The empirical studies of city-size distribution show that Zipf's law and the hierarchical scaling law are linked in many ways. The rank-size scaling and hierarchical scaling seem to be two different sides of the same coin, but their relationship has never been revealed by strict mathematical proof. In this paper, the Zipf's distribution of cities is abstracted as a q-sequence. Based on this sequence, a self-similar hierarchy consisting of many levels is defined and the numbers of cities in different levels form a geometric sequence. An exponential distribution of the average size of cities is derived from the hierarchy. Thus we have two exponential functions, from which follows a hierarchical scaling equation. The results can be statistically verified by simple mathematical experiments and observational data of cities. A theoretical foundation is then laid for the conversion from Zipf's law to the hierarchical scaling law, and the latter can show more information about city development than the former. Moreover, the self-similar hierarchy provides a new perspective for studying networks of cities as complex systems. A series of mathematical rules applied to cities such as the allometric growth law, the 2n principle and Pareto's law can be associated with one another by the hierarchical organization.

Predicting Relationships between Mathematics Anxiety, Mathematics Teaching Anxiety, Self-Efficacy Beliefs towards Mathematics and Mathematics Teaching

ERIC Educational Resources Information Center

Unlu, Melihan; Ertekin, Erhan; Dilmac, Bulent

2017-01-01

The purpose of the research is to investigate the relationships between self-efficacy beliefs toward mathematics, mathematics anxiety and self-efficacy beliefs toward mathematics teaching, mathematics teaching anxiety variables and testing the relationships between these variables with structural equation model. The sample of the research, which…

Simple taper: Taper equations for the field forester

Treesearch

David R. Larsen

2017-01-01

"Simple taper" is set of linear equations that are based on stem taper rates; the intent is to provide taper equation functionality to field foresters. The equation parameters are two taper rates based on differences in diameter outside bark at two points on a tree. The simple taper equations are statistically equivalent to more complex equations. The linear...

Transport of reacting solutes in porous media: Relation between mathematical nature of problem formulation and chemical nature of reactions

USGS Publications Warehouse

Rubin, Jacob

1983-01-01

Examples involving six broad reaction classes show that the nature of transport-affecting chemistry may have a profound effect on the mathematical character of solute transport problem formulation. Substantive mathematical diversity among such formulations is brought about principally by reaction properties that determine whether (1) the reaction can be regarded as being controlled by local chemical equilibria or whether it must be considered as being controlled by kinetics, (2) the reaction is homogeneous or heterogeneous, (3) the reaction is a surface reaction (adsorption, ion exchange) or one of the reactions of classical chemistry (e.g., precipitation, dissolution, oxidation, reduction, complex formation). These properties, as well as the choice of means to describe them, stipulate, for instance, (1) the type of chemical entities for which a formulation's basic, mass-balance equations should be written; (2) the nature of mathematical transformations needed to change the problem's basic equations into operational ones. These and other influences determine such mathematical features of problem formulations as the nature of the operational transport-equation system (e.g., whether it involves algebraic, partial-differential, or integro-partial-differential simultaneous equations), the type of nonlinearities of such a system, and the character of the boundaries (e.g., whether they are stationary or moving). Exploration of the reasons for the dependence of transport mathematics on transport chemistry suggests that many results of this dependence stem from the basic properties of the reactions' chemical-relation (i.e., equilibrium or rate) equations.

Exact traveling wave solutions for system of nonlinear evolution equations.

PubMed

Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H

2016-01-01

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

Hammett equation and generalized Pauling's electronegativity equation.

PubMed

Liu, Lei; Fu, Yao; Liu, Rui; Li, Rui-Qiong; Guo, Qing-Xiang

2004-01-01

Substituent interaction energy (SIE) was defined as the energy change of the isodesmic reaction X-spacer-Y + H-spacer-H --> X-spacer-H + H-spacer-Y. It was found that this SIE followed a simple equation, SIE(X,Y) = -ksigma(X)sigma(Y), where k was a constant dependent on the system and sigma was a certain scale of electronic substituent constant. It was demonstrated that the equation was applicable to disubstituted bicyclo[2.2.2]octanes, benzenes, ethylenes, butadienes, and hexatrienes. It was also demonstrated that Hammett's equation was a derivative form of the above equation. Furthermore, it was found that when spacer = nil the above equation was mathematically the same as Pauling's electronegativity equation. Thus it was shown that Hammett's equation was a derivative form of the generalized Pauling's electronegativity equation and that a generalized Pauling's electronegativity equation could be utilized for diverse X-spacer-Y systems. In addition, the total electronic substituent effects were successfully separated into field/inductive and resonance effects in the equation SIE(X,Y) = -k(1)F(X)F(Y) - k(2)R(X)R(Y) - k(3)(F(X)R(Y) + R(X)F(Y)). The existence of the cross term (i.e., F(X)R(Y) and R(X)F(Y)) suggested that the field/inductive effect was not orthogonal to the resonance effect because the field/inductive effect from one substituent interacted with the resonance effect from the other. Further studies on multi-substituted systems suggested that the electronic substituent effects should be pairwise and additive. Hence, the SIE in a multi-substituted system could be described using the equation SIE(X1, X2, ..., Xn) = Sigma(n-1)(i=1)Sigma(n)(j=i+1)k(ij)sigma(X)isigma(X)j.

Adiabatic dynamics of one-dimensional classical Hamiltonian dissipative systems

NASA Astrophysics Data System (ADS)

Pritula, G. M.; Petrenko, E. V.; Usatenko, O. V.

2018-02-01

A linearized plane pendulum with the slowly varying mass and length of string and the suspension point moving at a slowly varying speed is presented as an example of a simple 1D mechanical system described by the generalized harmonic oscillator equation, which is a basic model in discussion of the adiabatic dynamics and geometric phase. The expression for the pendulum geometric phase is obtained by three different methods. The pendulum is shown to be canonically equivalent to the damped harmonic oscillator. This supports the mathematical conclusion, not widely accepted in physical community, of no difference between the dissipative and Hamiltonian 1D systems.

A MODEL FOR FISSION-GAS RELEASE FROM POROUS FUELS IN LOW-PERMEABILITY CONTAINERS

DOE Office of Scientific and Technical Information (OSTI.GOV)

Prados, J.W.

1961-08-25

A simple mathematical model was developed to describe the steady-state release rate of gaseous fission products from porous ceramic fuels in low- permeability containers. The resulting equations are used to analyze experimental release rate results obtained from a UC/sub 2/-fueled graphite fuel body enclosed in a low-permeability impregnated graphite container. The relative release rates of the fission-product species Kr/sup 85m/, Kr/sup 88/, and Xe/sup 133/ were predicted with reasonable success. Absolute-rate predictions were not possible due to lack of information on true permeability and porosity profiles in the graphite container. (auth)

On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics

NASA Astrophysics Data System (ADS)

Motsepa, Tanki; Masood Khalique, Chaudry

2018-05-01

In this paper, we study a (2+1) dimensional KdV-mKdV equation, which models many physical phenomena of mathematical physics. This equation has two integral terms in it. By an appropriate substitution, we convert this equation into two partial differential equations, which do not have integral terms and are equivalent to the original equation. We then work with the system of two equations and obtain its exact travelling wave solutions in form of Jacobi elliptic functions. Furthermore, we employ the multiplier method to construct conservation laws for the system. Finally, we revert the results obtained into the original variables of the (2+1) dimensional KdV-mKdV equation.

A Complex-Valued Firing-Rate Model That Approximates the Dynamics of Spiking Networks

PubMed Central

Schaffer, Evan S.; Ostojic, Srdjan; Abbott, L. F.

2013-01-01

Firing-rate models provide an attractive approach for studying large neural networks because they can be simulated rapidly and are amenable to mathematical analysis. Traditional firing-rate models assume a simple form in which the dynamics are governed by a single time constant. These models fail to replicate certain dynamic features of populations of spiking neurons, especially those involving synchronization. We present a complex-valued firing-rate model derived from an eigenfunction expansion of the Fokker-Planck equation and apply it to the linear, quadratic and exponential integrate-and-fire models. Despite being almost as simple as a traditional firing-rate description, this model can reproduce firing-rate dynamics due to partial synchronization of the action potentials in a spiking model, and it successfully predicts the transition to spike synchronization in networks of coupled excitatory and inhibitory neurons. PMID:24204236

Reactive extraction of lactic acid with trioctylamine/methylene chloride/n-hexane

DOE Office of Scientific and Technical Information (OSTI.GOV)

Han, D.H.; Hong, W.H.

The trioctylamine (TOA)/methylene chloride (MC)/n-hexane system was used as the extraction agent for the extraction of lactic acid. Curves of equilibrium and hydration were obtained at various temperatures and concentrations of TOA. A modified mass action model was proposed to interpret the equilibrium and the hydration curves. The reaction mechanism and the corresponding parameters which best represent the equilibrium data were estimated, and the concentration of water in the organic phase was predicted by inserting the parameters into the simple mathematical equation of the modified model. The concentration of MC and the change of temperature were important factors for themore » extraction and the stripping process. The stripping was performed by a simple distillation which was a combination of temperature-swing regeneration and diluent-swing regeneration. The type of inactive diluent has no influence on the stripping. The stripping efficiencies were about 70%.« less

Analytical drafting curves provide exact equations for plotted data

NASA Technical Reports Server (NTRS)

Stewart, R. B.

1967-01-01

Analytical drafting curves provide explicit mathematical expressions for any numerical data that appears in the form of graphical plots. The curves each have a reference coordinate axis system indicated on the curve as well as the mathematical equation from which the curve was generated.

Entropy and convexity for nonlinear partial differential equations

PubMed Central

Ball, John M.; Chen, Gui-Qiang G.

2013-01-01

Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue. PMID:24249768

Geometric model of pseudo-distance measurement in satellite location systems

NASA Astrophysics Data System (ADS)

Panchuk, K. L.; Lyashkov, A. A.; Lyubchinov, E. V.

2018-04-01

The existing mathematical model of pseudo-distance measurement in satellite location systems does not provide a precise solution of the problem, but rather an approximate one. The existence of such inaccuracy, as well as bias in measurement of distance from satellite to receiver, results in inaccuracy level of several meters. Thereupon, relevance of refinement of the current mathematical model becomes obvious. The solution of the system of quadratic equations used in the current mathematical model is based on linearization. The objective of the paper is refinement of current mathematical model and derivation of analytical solution of the system of equations on its basis. In order to attain the objective, geometric analysis is performed; geometric interpretation of the equations is given. As a result, an equivalent system of equations, which allows analytical solution, is derived. An example of analytical solution implementation is presented. Application of analytical solution algorithm to the problem of pseudo-distance measurement in satellite location systems allows to improve the accuracy such measurements.

Entropy and convexity for nonlinear partial differential equations.

PubMed

Ball, John M; Chen, Gui-Qiang G

2013-12-28

Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.

Equivalence of equations describing trace element distribution during equilibrium partial melting

NASA Technical Reports Server (NTRS)

Consolmagno, G. J.; Drake, M. J.

1976-01-01

It is shown that four equations used for calculating the evolution of trace-element abundances during equilibrium partial melting are mathematically equivalent. The equations include those of Hertogen and Gijbels (1976), Shaw (1970), Schilling (1971), and O'Nions and Clarke (1972). The general form to which all these equations reduce is presented, and an analysis is performed to demonstrate their mathematical equivalence. It is noted that the utility of the general equation flows from the nature of equilibrium (i.e., the final state is independent of the path by which that state is attained).

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rasouli, C.; Abbasi Davani, F.; Rokrok, B.

Plasma confinement using external magnetic field is one of the successful ways leading to the controlled nuclear fusion. Development and validation of the solution process for plasma equilibrium in the experimental toroidal fusion devices is the main subject of this work. Solution of the nonlinear 2D stationary problem as posed by the Grad-Shafranov equation gives quantitative information about plasma equilibrium inside the vacuum chamber of hot fusion devices. This study suggests solving plasma equilibrium equation which is essential in toroidal nuclear fusion devices, using a mesh-free method in a condition that the plasma boundary is unknown. The Grad-Shafranov equation hasmore » been solved numerically by the point interpolation collocation mesh-free method. Important features of this approach include truly mesh free, simple mathematical relationships between points and acceptable precision in comparison with the parametric results. The calculation process has been done by using the regular and irregular nodal distribution and support domains with different points. The relative error between numerical and analytical solution is discussed for several test examples such as small size Damavand tokamak, ITER-like equilibrium, NSTX-like equilibrium, and typical Spheromak.« less

Thermodynamics of Highly Concentrated Aqueous Electrolytes: Based on Boltzmann's eponymous equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ally, Moonis Raza

This sharply focused book invites the reader to explore the chemical thermodynamics of highly concentrated aqueous electrolytes from a different vantage point than traditional methods. The book's foundation is deeply rooted in Ludwig Boltzmann's eponymous equation. The pathway from micro to macro thermodynamics is explained heuristically, in a step-by-step approach. Concepts and mathematical formalism are explained in detail to captivate and maintain interest as the algebra twists and turns. Every significant result is derived in a lucid and piecemeal fashion. Application of the theory is exemplified with examples. It is amazing to realize that Boltamann's simple equation contains sufficient informationmore » from which such an elaborate theory can emerge. This book is suitable for undergraduate and graduate level classes in chemical engineering, chemistry, geochemistry, environmental sciences, and those studying aerosol particles in the troposphere. Students interested in understanding how thermodynamic theories may be developed would be inspired by the methodology. The author wishes that readers get as much excitement reading this book as he did writing it.« less

Fun with maths: exploring implications of mathematical models for malaria eradication.

PubMed

Eckhoff, Philip A; Bever, Caitlin A; Gerardin, Jaline; Wenger, Edward A

2014-12-11

Mathematical analyses and modelling have an important role informing malaria eradication strategies. Simple mathematical approaches can answer many questions, but it is important to investigate their assumptions and to test whether simple assumptions affect the results. In this note, four examples demonstrate both the effects of model structures and assumptions and also the benefits of using a diversity of model approaches. These examples include the time to eradication, the impact of vaccine efficacy and coverage, drug programs and the effects of duration of infections and delays to treatment, and the influence of seasonality and migration coupling on disease fadeout. An excessively simple structure can miss key results, but simple mathematical approaches can still achieve key results for eradication strategy and define areas for investigation by more complex models.

Estimation of true serum thyroglobulin concentration using simultaneous measurement of serum antithyroglobulin antibody.

PubMed

Ahn, Byeong-Cheol; Lee, Won Kee; Jeong, Shin Young; Lee, Sang-Woo; Lee, Jaetae

2013-01-01

We investigated the analytical interference of antithyroglobulin antibody (TgAb) to thyroglobulin (Tg) measurement and tried to convert measured Tg concentration to true Tg concentration using a mathematical equation which includes a concentration of TgAb. Methods. Tg was measured by immunoradiometric assay and TgAb by radioimmunoassy. Experimental samples were produced by mixing Tg and TgAb standard solutions or mixing patients' serum with high Tg or high TgAb. Mathematical equations for prediction of expected Tg concentration with measured Tg and TgAb concentrations were deduced. The Tg concentration calculated using the equations was compared with the expected Tg concentration. Results. Measured Tg concentrations of samples having high TgAb were significantly lower than their expected Tg concentration. Magnitude of TgAb interference with the Tg assay showed a positive correlation with concentration of TgAb. Mathematical equations for estimation of expected Tg concentration using measured Tg and TgAb concentrations were successfully deduced and the calculated Tg concentration showed excellent correlation with expected Tg concentration. Conclusions. A mathematic equation for estimation of true Tg concentration using measured Tg and TgAb concentration was deduced. Tg concentration calculated by use of the equation might be more valuable than measured Tg concentration in patients with differentiated thyroid cancer.

Iontophoretic transdermal drug delivery: a multi-layered approach.

PubMed

Pontrelli, Giuseppe; Lauricella, Marco; Ferreira, José A; Pena, Gonçalo

2017-12-11

We present a multi-layer mathematical model to describe the transdermal drug release from an iontophoretic system. The Nernst-Planck equation describes the basic convection-diffusion process, with the electric potential obtained by solving the Laplace's equation. These equations are complemented with suitable interface and boundary conditions in a multi-domain. The stability of the mathematical problem is discussed in different scenarios and a finite-difference method is used to solve the coupled system. Numerical experiments are included to illustrate the drug dynamics under different conditions. © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Problems Relating Mathematics and Science in the High School.

ERIC Educational Resources Information Center

Morrow, Richard; Beard, Earl

This document contains various science problems which require a mathematical solution. The problems are arranged under two general areas. The first (algebra I) contains biology, chemistry, and physics problems which require solutions related to linear equations, exponentials, and nonlinear equations. The second (algebra II) contains physics…

Biological Applications in the Mathematics Curriculum

ERIC Educational Resources Information Center

Marland, Eric; Palmer, Katrina M.; Salinas, Rene A.

2008-01-01

In this article we provide two detailed examples of how we incorporate biological examples into two mathematics courses: Linear Algebra and Ordinary Differential Equations. We use Leslie matrix models to demonstrate the biological properties of eigenvalues and eigenvectors. For Ordinary Differential Equations, we show how using a logistic growth…

Undergraduate Students' Perceptions of Collaborative Learning in a Differential Equations Mathematics Course

ERIC Educational Resources Information Center

Hajra, Sayonita Ghosh; Das, Ujjaini

2015-01-01

This paper uses collaborative learning strategies to examine students' perceptions in a differential equations mathematics course. Students' perceptions were analyzed using three collaborative learning strategies including collaborative activity, group-quiz and online discussion. The study results show that students identified both strengths and…

Student perceptions of writing projects in a university differential-equations course

NASA Astrophysics Data System (ADS)

Latulippe, Christine; Latulippe, Joe

2014-01-01

This qualitative study surveyed 102 differential-equations students in order to investigate how students participating in writing projects in university-level mathematics courses perceive the benefits of writing in the mathematics classroom. Based on previous literature on writing in mathematics, students were asked specifically about the benefits of writing projects as a means to explore practical uses of mathematics, deepen content knowledge, and strengthen communication. Student responses indicated an awareness of these benefits, supporting justifications commonly cited by instructors assigning writing projects. Open-ended survey responses highlighted additional themes which students associated with writing in mathematics, including using software programs and technology, working in groups, and stimulating interest in mathematics. This study provides student feedback to support the use of writing projects in mathematics, as well as student input, which can be utilized to strengthen the impact of writing projects in mathematics.

Interpreting experimental data on egg production--applications of dynamic differential equations.

PubMed

France, J; Lopez, S; Kebreab, E; Dijkstra, J

2013-09-01

This contribution focuses on applying mathematical models based on systems of ordinary first-order differential equations to synthesize and interpret data from egg production experiments. Models based on linear systems of differential equations are contrasted with those based on nonlinear systems. Regression equations arising from analytical solutions to linear compartmental schemes are considered as candidate functions for describing egg production curves, together with aspects of parameter estimation. Extant candidate functions are reviewed, a role for growth functions such as the Gompertz equation suggested, and a function based on a simple new model outlined. Structurally, the new model comprises a single pool with an inflow and an outflow. Compartmental simulation models based on nonlinear systems of differential equations, and thus requiring numerical solution, are next discussed, and aspects of parameter estimation considered. This type of model is illustrated in relation to development and evaluation of a dynamic model of calcium and phosphorus flows in layers. The model consists of 8 state variables representing calcium and phosphorus pools in the crop, stomachs, plasma, and bone. The flow equations are described by Michaelis-Menten or mass action forms. Experiments that measure Ca and P uptake in layers fed different calcium concentrations during shell-forming days are used to evaluate the model. In addition to providing a useful management tool, such a simulation model also provides a means to evaluate feeding strategies aimed at reducing excretion of potential pollutants in poultry manure to the environment.

Mathematics, thermodynamics, and modeling to address ten common misconceptions about protein structure, folding, and stability.

PubMed

Robic, Srebrenka

2010-01-01

To fully understand the roles proteins play in cellular processes, students need to grasp complex ideas about protein structure, folding, and stability. Our current understanding of these topics is based on mathematical models and experimental data. However, protein structure, folding, and stability are often introduced as descriptive, qualitative phenomena in undergraduate classes. In the process of learning about these topics, students often form incorrect ideas. For example, by learning about protein folding in the context of protein synthesis, students may come to an incorrect conclusion that once synthesized on the ribosome, a protein spends its entire cellular life time in its fully folded native confirmation. This is clearly not true; proteins are dynamic structures that undergo both local fluctuations and global unfolding events. To prevent and address such misconceptions, basic concepts of protein science can be introduced in the context of simple mathematical models and hands-on explorations of publicly available data sets. Ten common misconceptions about proteins are presented, along with suggestions for using equations, models, sequence, structure, and thermodynamic data to help students gain a deeper understanding of basic concepts relating to protein structure, folding, and stability.

A mathematical function for the description of nutrient-response curve

PubMed Central

Ahmadi, Hamed

2017-01-01

Several mathematical equations have been proposed to modeling nutrient-response curve for animal and human justified on the goodness of fit and/or on the biological mechanism. In this paper, a functional form of a generalized quantitative model based on Rayleigh distribution principle for description of nutrient-response phenomena is derived. The three parameters governing the curve a) has biological interpretation, b) may be used to calculate reliable estimates of nutrient response relationships, and c) provide the basis for deriving relationships between nutrient and physiological responses. The new function was successfully applied to fit the nutritional data obtained from 6 experiments including a wide range of nutrients and responses. An evaluation and comparison were also done based simulated data sets to check the suitability of new model and four-parameter logistic model for describing nutrient responses. This study indicates the usefulness and wide applicability of the new introduced, simple and flexible model when applied as a quantitative approach to characterizing nutrient-response curve. This new mathematical way to describe nutritional-response data, with some useful biological interpretations, has potential to be used as an alternative approach in modeling nutritional responses curve to estimate nutrient efficiency and requirements. PMID:29161271

Scale-dependent behavior of scale equations.

PubMed

Kim, Pilwon

2009-09-01

We propose a new mathematical framework to formulate scale structures of general systems. Stack equations characterize a system in terms of accumulative scales. Their behavior at each scale level is determined independently without referring to other levels. Most standard geometries in mathematics can be reformulated in such stack equations. By involving interaction between scales, we generalize stack equations into scale equations. Scale equations are capable to accommodate various behaviors at different scale levels into one integrated solution. On contrary to standard geometries, such solutions often reveal eccentric scale-dependent figures, providing a clue to understand multiscale nature of the real world. Especially, it is suggested that the Gaussian noise stems from nonlinear scale interactions.

A model for closing the inviscid form of the average-passage equation system

NASA Technical Reports Server (NTRS)

Adamczyk, J. J.; Mulac, R. A.; Celestina, M. L.

1985-01-01

A mathematical model is proposed for closing or mathematically completing the system of equations which describes the time average flow field through the blade passages of multistage turbomachinery. These equations referred to as the average passage equation system govern a conceptual model which has proven useful in turbomachinery aerodynamic design and analysis. The closure model is developed so as to insure a consistency between these equations and the axisymmetric through flow equations. The closure model was incorporated into a computer code for use in simulating the flow field about a high speed counter rotating propeller and a high speed fan stage. Results from these simulations are presented.

Structural Equation Model to Validate: Mathematics-Computer Interaction, Computer Confidence, Mathematics Commitment, Mathematics Motivation and Mathematics Confidence

ERIC Educational Resources Information Center

Garcia-Santillán, Arturo; Moreno-Garcia, Elena; Escalera-Chávez, Milka E.; Rojas-Kramer, Carlos A.; Pozos-Texon, Felipe

2016-01-01

Most mathematics students show a definite tendency toward an attitudinal deficiency, which can be primarily understood as intolerance to the matter, affecting their scholar performance adversely. In addition, information and communication technologies have been gradually included within the process of teaching mathematics. Such adoption of…

Partial regularity of viscosity solutions for a class of Kolmogorov equations arising from mathematical finance

NASA Astrophysics Data System (ADS)

Rosestolato, M.; Święch, A.

2017-02-01

We study value functions which are viscosity solutions of certain Kolmogorov equations. Using PDE techniques we prove that they are C 1 + α regular on special finite dimensional subspaces. The problem has origins in hedging derivatives of risky assets in mathematical finance.

The Root of the Problem

ERIC Educational Resources Information Center

Grosser-Clarkson, Dana L.

2015-01-01

The Common Core State Standards for Mathematics expect students to build on their knowledge of the number system, expressions and equations, and functions throughout school mathematics. For example, students learn that they can add something to both sides of an equation and that doing so will not affect the equivalency; however, squaring both…

A Conceptual Approach to Absolute Value Equations and Inequalities

ERIC Educational Resources Information Center

Ellis, Mark W.; Bryson, Janet L.

2011-01-01

The absolute value learning objective in high school mathematics requires students to solve far more complex absolute value equations and inequalities. When absolute value problems become more complex, students often do not have sufficient conceptual understanding to make any sense of what is happening mathematically. The authors suggest that the…

Some Fundamental Issues of Mathematical Simulation in Biology

NASA Astrophysics Data System (ADS)

Razzhevaikin, V. N.

2018-02-01

Some directions of simulation in biology leading to original formulations of mathematical problems are overviewed. Two of them are discussed in detail: the correct solvability of first-order linear equations with unbounded coefficients and the construction of a reaction-diffusion equation with nonlinear diffusion for a model of genetic wave propagation.

Investigating Students' Mathematical Difficulties with Quadratic Equations

ERIC Educational Resources Information Center

O'Connor, Bronwyn Reid; Norton, Stephen

2016-01-01

This paper examines the factors that hinder students' success in working with and understanding the mathematics of quadratic equations using a case study analysis of student error patterns. Twenty-five Year 11 students were administered a written test to examine their understanding of concepts and procedures associated with this topic. The…

The Influence of Symbols and Equations on Understanding Mathematical Equivalence

ERIC Educational Resources Information Center

Powell, Sarah R.

2015-01-01

Students with mathematics difficulty demonstrate lower mathematics performance than typical-performing peers. One contributing factor to lower mathematics performance may be misunderstanding of mathematics symbols. In several studies related to the equal sign (=), students who received explicit instruction on the relational definition (i.e.,…

How Long is my Toilet Roll?--A Simple Exercise in Mathematical Modelling

ERIC Educational Resources Information Center

Johnston, Peter R.

2013-01-01

The simple question of how much paper is left on my toilet roll is studied from a mathematical modelling perspective. As is typical with applied mathematics, models of increasing complexity are introduced and solved. Solutions produced at each step are compared with the solution from the previous step. This process exposes students to the typical…

A Comparative Analysis of a Generalized Lanchester Equation Model and a Stochastic Computer Simulation Model.

DTIC Science & Technology

1987-03-01

model is one in which words or numerical descriptions are used to represent an entity or process. An example of a symbolic model is a mathematical ...are the third type of model used in modeling combat attrition. Analytical models are symbolic models which use mathematical symbols and equations to...simplicity and the ease of tracing through the mathematical computations. In this section I will discuss some of the shortcoming which have been

A model of a fishery with fish stock involving delay equations.

PubMed

Auger, P; Ducrot, Arnaud

2009-12-13

The aim of this paper is to provide a new mathematical model for a fishery by including a stock variable for the resource. This model takes the form of an infinite delay differential equation. It is mathematically studied and a bifurcation analysis of the steady states is fulfilled. Depending on the different parameters of the problem, we show that Hopf bifurcation may occur leading to oscillating behaviours of the system. The mathematical results are finally discussed.

Mathematical modelling of tissue formation in chondrocyte filter cultures.

PubMed

Catt, C J; Schuurman, W; Sengers, B G; van Weeren, P R; Dhert, W J A; Please, C P; Malda, J

2011-12-17

In the field of cartilage tissue engineering, filter cultures are a frequently used three-dimensional differentiation model. However, understanding of the governing processes of in vitro growth and development of tissue in these models is limited. Therefore, this study aimed to further characterise these processes by means of an approach combining both experimental and applied mathematical methods. A mathematical model was constructed, consisting of partial differential equations predicting the distribution of cells and glycosaminoglycans (GAGs), as well as the overall thickness of the tissue. Experimental data was collected to allow comparison with the predictions of the simulation and refinement of the initial models. Healthy mature equine chondrocytes were expanded and subsequently seeded on collagen-coated filters and cultured for up to 7 weeks. Resulting samples were characterised biochemically, as well as histologically. The simulations showed a good representation of the experimentally obtained cell and matrix distribution within the cultures. The mathematical results indicate that the experimental GAG and cell distribution is critically dependent on the rate at which the cell differentiation process takes place, which has important implications for interpreting experimental results. This study demonstrates that large regions of the tissue are inactive in terms of proliferation and growth of the layer. In particular, this would imply that higher seeding densities will not significantly affect the growth rate. A simple mathematical model was developed to predict the observed experimental data and enable interpretation of the principal underlying mechanisms controlling growth-related changes in tissue composition.

Mathematical models of ABE fermentation: review and analysis.

PubMed

Mayank, Rahul; Ranjan, Amrita; Moholkar, Vijayanand S

2013-12-01

Among different liquid biofuels that have emerged in the recent past, biobutanol produced via fermentation processes is of special interest due to very similar properties to that of gasoline. For an effective design, scale-up, and optimization of the acetone-butanol-ethanol (ABE) fermentation process, it is necessary to have insight into the micro- and macro-mechanisms of the process. The mathematical models for ABE fermentation are efficient tools for this purpose, which have evolved from simple stoichiometric fermentation equations in the 1980s to the recent sophisticated and elaborate kinetic models based on metabolic pathways. In this article, we have reviewed the literature published in the area of mathematical modeling of the ABE fermentation. We have tried to present an analysis of these models in terms of their potency in describing the overall physiology of the process, design features, mode of operation along with comparison and validation with experimental results. In addition, we have also highlighted important facets of these models such as metabolic pathways, basic kinetics of different metabolites, biomass growth, inhibition modeling and other additional features such as cell retention and immobilized cultures. Our review also covers the mathematical modeling of the downstream processing of ABE fermentation, i.e. recovery and purification of solvents through flash distillation, liquid-liquid extraction, and pervaporation. We believe that this review will be a useful source of information and analysis on mathematical models for ABE fermentation for both the appropriate scientific and engineering communities.

Mathematical modeling on obligate mutualism: Interactions between leaf-cutter ants and their fungus garden.

PubMed

Kang, Yun; Clark, Rebecca; Makiyama, Michael; Fewell, Jennifer

2011-11-21

We propose a simple mathematical model by applying Michaelis-Menton equations of enzyme kinetics to study the mutualistic interaction between the leaf cutter ant and its fungus garden at the early stage of colony expansion. We derive sufficient conditions on the extinction and coexistence of these two species. In addition, we give a region of initial condition that leads to the extinction of two species when the model has an interior attractor. Our global analysis indicates that the division of labor by worker ants and initial conditions are two important factors that determine whether leaf cutter ants' colonies and their fungus garden can survive and grow or not. We validate the model by comparing model simulations and data on fungal and ant colony growth rates under laboratory conditions. We perform sensitive analysis of the model based on the experimental data to gain more biological insights on ecological interactions between leaf-cutter ants and their fungus garden. Finally, we give conclusions and discuss potential future work. Published by Elsevier Ltd.

It's all just mathematics

NASA Astrophysics Data System (ADS)

Tegmark, Max

2014-02-01

The world can be described using mathematical equations and numbers, but why does maths do it so well? In his new book Our Mathematical Universe, a section of which is abridged and edited here, Max Tegmark makes the radical proposal that our reality isn't just described by mathematics - it is mathematics.

Explicating Mathematical Thinking in Differential Equations Using a Computer Algebra System

ERIC Educational Resources Information Center

Zeynivandnezhad, Fereshteh; Bates, Rachel

2018-01-01

The importance of developing students' mathematical thinking is frequently highlighted in literature regarding the teaching and learning of mathematics. Despite this importance, most curricula and instructional activities for undergraduate mathematics fail to bring the learner beyond the mathematics. The purpose of this study was to enhance…

How Mathematics Describes Life

NASA Astrophysics Data System (ADS)

Teklu, Abraham

2017-01-01

The circle of life is something we have all heard of from somewhere, but we don't usually try to calculate it. For some time we have been working on analyzing a predator-prey model to better understand how mathematics can describe life, in particular the interaction between two different species. The model we are analyzing is called the Holling-Tanner model, and it cannot be solved analytically. The Holling-Tanner model is a very common model in population dynamics because it is a simple descriptor of how predators and prey interact. The model is a system of two differential equations. The model is not specific to any particular set of species and so it can describe predator-prey species ranging from lions and zebras to white blood cells and infections. One thing all these systems have in common are critical points. A critical point is a value for both populations that keeps both populations constant. It is important because at this point the differential equations are equal to zero. For this model there are two critical points, a predator free critical point and a coexistence critical point. Most of the analysis we did is on the coexistence critical point because the predator free critical point is always unstable and frankly less interesting than the coexistence critical point. What we did is consider two regimes for the differential equations, large B and small B. B, A, and C are parameters in the differential equations that control the system where B measures how responsive the predators are to change in the population, A represents predation of the prey, and C represents the satiation point of the prey population. For the large B case we were able to approximate the system of differential equations by a single scalar equation. For the small B case we were able to predict the limit cycle. The limit cycle is a process of the predator and prey populations growing and shrinking periodically. This model has a limit cycle in the regime of small B, that we solved for numerically. With some assumptions to reduce the differential equations we were able to create a system of equations and unknowns to predict the behavior of the limit cycle for small B.

A new mathematical adjoint for the modified SAAF -SN equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schunert, Sebastian; Wang, Yaqi; Martineau, Richard

2015-01-01

We present a new adjoint FEM weak form, which can be directly used for evaluating the mathematical adjoint, suitable for perturbation calculations, of the self-adjoint angular flux SN equations (SAAF -SN) without construction and transposition of the underlying coefficient matrix. Stabilization schemes incorporated in the described SAAF -SN method make the mathematical adjoint distinct from the physical adjoint, i.e. the solution of the continuous adjoint equation with SAAF -SN . This weak form is implemented into RattleSnake, the MOOSE (Multiphysics Object-Oriented Simulation Environment) based transport solver. Numerical results verify the correctness of the implementation and show its utility both formore » fixed source and eigenvalue problems.« less

Microcomputer-Assisted Mathematics: From Simple Interest to e.

ERIC Educational Resources Information Center

Kimberling, Clark

1985-01-01

The progression from simple interest to compound interest leads naturally and quickly to the number e, involving mathematical discovery learning through writing programs. Several programs are given, with suggestions for a teaching sequence. (MNS)

Teaching the Mathematics of Radioactive Dating.

ERIC Educational Resources Information Center

Shea, James H.

2001-01-01

Describes a method used to teach the concept of radiometric dating using mathematical equations. Explores the lack of information in textbooks on how to solve radiometric dating problems using mathematical concepts. (SAH)

Mathematical simulation of efficiency of various shapes of solar panels for NASA geostationary satellites

NASA Astrophysics Data System (ADS)

Pandya, Raaghav; Raja, Hammad; Enriquez-Torres, Delfino; Serey-Roman, Maria Ignacia; Hassebo, Yasser; Marciniak, Małgorzata

2018-02-01

The purpose of this research is to analyze mathematically cylindrical shapes of flexible solar panels and compare their efficiency to the flat panels. The efficiency is defined to be the flux density, which is the ratio of the mathematical flux and the surface area. In addition we describe the trajectory of the Sun at specific locations: the North Pole, The Equator and a geostationary satellite above the Equator. The calculations were performed with software: Maple, Mathematica, and MATLAB.

Some Aspects of Mathematical Model of Collaborative Learning

ERIC Educational Resources Information Center

Nakamura, Yasuyuki; Yasutake, Koichi; Yamakawa, Osamu

2012-01-01

There are some mathematical learning models of collaborative learning, with which we can learn how students obtain knowledge and we expect to design effective education. We put together those models and classify into three categories; model by differential equations, so-called Ising spin and a stochastic process equation. Some of the models do not…

Fibonacci Numbers Revisited: Technology-Motivated Inquiry into a Two-Parametric Difference Equation

ERIC Educational Resources Information Center

Abramovich, Sergei; Leonov, Gennady A.

2008-01-01

This article demonstrates how within an educational context, supported by the notion of hidden mathematics curriculum and enhanced by the use of technology, new mathematical knowledge can be discovered. More specifically, proceeding from the well-known representation of Fibonacci numbers through a second-order difference equation, this article…

Equations, Functions, Critical Aspects and Mathematical Communication

ERIC Educational Resources Information Center

Olteanu, Constanta; Olteanu, Lucian

2012-01-01

The purpose of this paper is to present the mechanism for effective communication when the mathematical objects of learning are equations and functions. The presentation is based on data collected while the same object of learning is presented in two classes, and it includes two teachers and 45 students. Among other things, the data consists of…

A GENERALIZED MATHEMATICAL SCHEME TO ANALYTICALLY SOLVE THE ATMOSPHERIC DIFFUSION EQUATION WITH DRY DEPOSITION. (R825689C072)

EPA Science Inventory

Abstract

A generalized mathematical scheme is developed to simulate the turbulent dispersion of pollutants which are adsorbed or deposit to the ground. The scheme is an analytical (exact) solution of the atmospheric diffusion equation with height-dependent wind speed a...

The What and Where of Adding Channel Noise to the Hodgkin-Huxley Equations

PubMed Central

Goldwyn, Joshua H.; Shea-Brown, Eric

2011-01-01

Conductance-based equations for electrically active cells form one of the most widely studied mathematical frameworks in computational biology. This framework, as expressed through a set of differential equations by Hodgkin and Huxley, synthesizes the impact of ionic currents on a cell's voltage—and the highly nonlinear impact of that voltage back on the currents themselves—into the rapid push and pull of the action potential. Later studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations or their counterparts. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the equations of Hodgkin-Huxley type. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic equations of Hodgkin-Huxley type as well as to more modern models of ion channel dynamics. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly MATLAB simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html. PMID:22125479

Descriptions of Free and Freeware Software in the Mathematics Teaching

NASA Astrophysics Data System (ADS)

Antunes de Macedo, Josue; Neves de Almeida, Samara; Voelzke, Marcos Rincon

2016-05-01

This paper presents the analysis and the cataloging of free and freeware mathematical software available on the internet, a brief explanation of them, and types of licenses for use in teaching and learning. The methodology is based on the qualitative research. Among the different types of software found, it stands out in algebra, the Winmat, that works with linear algebra, matrices and linear systems. In geometry, the GeoGebra, which can be used in the study of functions, plan and spatial geometry, algebra and calculus. For graphing, can quote the Graph and Graphequation. With Graphmatica software, it is possible to build various graphs of mathematical equations on the same screen, representing cartesian equations, inequalities, parametric among other functions. The Winplot allows the user to build graphics in two and three dimensions functions and mathematical equations. Thus, this work aims to present the teachers some free math software able to be used in the classroom.

The Effect of Tutoring With Nonstandard Equations for Students With Mathematics Difficulty.

PubMed

Powell, Sarah R; Driver, Melissa K; Julian, Tyler E

2015-01-01

Students often misinterpret the equal sign (=) as operational instead of relational. Research indicates misinterpretation of the equal sign occurs because students receive relatively little exposure to equations that promote relational understanding of the equal sign. No study, however, has examined effects of nonstandard equations on the equation solving and equal-sign understanding of students with mathematics difficulty (MD). In the present study, second-grade students with MD (n = 51) were randomly assigned to standard equations tutoring, combined tutoring (standard and nonstandard equations), and no-tutoring control. Combined tutoring students demonstrated greater gains on equation-solving assessments and equal-sign tasks compared to the other two conditions. Standard tutoring students demonstrated improved skill on equation solving over control students, but combined tutoring students' performance gains were significantly larger. Results indicate that exposure to and practice with nonstandard equations positively influence student understanding of the equal sign. © Hammill Institute on Disabilities 2013.

A theoretical analysis of fluid flow and energy transport in hydrothermal systems

USGS Publications Warehouse

Faust, Charles R.; Mercer, James W.

1977-01-01

A mathematical derivation for fluid flow and energy transport in hydrothermal systems is presented. Specifically, the mathematical model describes the three-dimensional flow of both single- and two-phase, single-component water and the transport of heat in porous media. The derivation begins with the point balance equations for mass, momentum, and energy. These equations are then averaged over a finite volume to obtain the macroscopic balance equations for a porous medium. The macroscopic equations are combined by appropriate constitutive relationships to form two similified partial differential equations posed in terms of fluid pressure and enthalpy. A two-dimensional formulation of the simplified equations is also derived by partial integration in the vertical dimension. (Woodard-USGS)

A mathematical model of physiological processes and its application to the study of aging

NASA Technical Reports Server (NTRS)

Hibbs, A. R.; Walford, R. L.

1989-01-01

The behavior of a physiological system which, after displacement, returns by homeostatic mechanisms to its original condition can be described by a simple differential equation in which the "recovery time" is a parameter. Two such systems, which influence one another, can be linked mathematically by the use of "coupling" or "feedback" coefficients. These concepts are the basis for many mathematical models of physiological behavior, and we describe the general nature of such models. Next, we introduce the concept of a "fatal limit" for the displacement of a physiological system, and show how measures of such limits can be included in mathematical models. We show how the numerical values of such limits depend on the values of other system parameters, i.e., recovery times and coupling coefficients, and suggest ways of measuring all these parameters experimentally, for example by monitoring changes induced by X-irradiation. Next, we discuss age-related changes in these parameters, and show how the parameters of mortality statistics, such as the famous Gompertz parameters, can be derived from experimentally measurable changes. Concepts of onset-of-aging, critical or fatal limits, equilibrium value (homeostasis), recovery times and coupling constants are involved. Illustrations are given using published data from mouse and rat populations. We believe that this method of deriving survival patterns from model that is experimentally testable is unique.

Stability Analysis of Finite Difference Schemes for Hyperbolic Systems, and Problems in Applied and Computational Linear Algebra.

DTIC Science & Technology

FINITE DIFFERENCE THEORY, * LINEAR ALGEBRA , APPLIED MATHEMATICS, APPROXIMATION(MATHEMATICS), BOUNDARY VALUE PROBLEMS, COMPUTATIONS, HYPERBOLAS, MATHEMATICAL MODELS, NUMERICAL ANALYSIS, PARTIAL DIFFERENTIAL EQUATIONS, STABILITY.

Explaining formation of Astronomical Jets using Dynamic Universe Model

NASA Astrophysics Data System (ADS)

Naga Parameswara Gupta, Satyavarapu

2016-07-01

Astronomical jets are observed from the centres of many Galaxies including our own Milkyway. The formation of such jet is explained using SITA simulations of Dynamic Universe Model. For this purpose the path traced by a test neutron is calculated and depicted using a set up of one densemass of the mass equivalent to mass of Galaxy center, 90 stars with similar masses of stars near Galaxy center, mass equivalents of 23 Globular Cluster groups, 16 Milkyway parts, Andromeda and Triangulum Galaxies at appropriate distances. Five different kinds of theoretical simulations gave positive results The path travelled by this test neutron was found to be an astronomical jet emerging from Galaxy center. This is another result from Dynamic Universe Model. It solves new problems like a. Variable Mass Rocket Trajectory Problem b. Explaining Very long baseline interferometry (VLBI) observations c. Astronomical jets observed from Milkyway Center d. Prediction of Blue shifted Galaxies e. Explaining Pioneer Anomaly f. Prediction of New Horizons satellite trajectory etc. Dynamic Universe Model never reduces to General relativity on any condition. It uses a different type of mathematics based on Newtonian physics. This mathematics used here is simple and straightforward. As there are no differential equations present in Dynamic Universe Model, the set of equations give single solution in x y z Cartesian coordinates for every point mass for every time step

Qualitative-Modeling-Based Silicon Neurons and Their Networks

PubMed Central

Kohno, Takashi; Sekikawa, Munehisa; Li, Jing; Nanami, Takuya; Aihara, Kazuyuki

2016-01-01

The ionic conductance models of neuronal cells can finely reproduce a wide variety of complex neuronal activities. However, the complexity of these models has prompted the development of qualitative neuron models. They are described by differential equations with a reduced number of variables and their low-dimensional polynomials, which retain the core mathematical structures. Such simple models form the foundation of a bottom-up approach in computational and theoretical neuroscience. We proposed a qualitative-modeling-based approach for designing silicon neuron circuits, in which the mathematical structures in the polynomial-based qualitative models are reproduced by differential equations with silicon-native expressions. This approach can realize low-power-consuming circuits that can be configured to realize various classes of neuronal cells. In this article, our qualitative-modeling-based silicon neuron circuits for analog and digital implementations are quickly reviewed. One of our CMOS analog silicon neuron circuits can realize a variety of neuronal activities with a power consumption less than 72 nW. The square-wave bursting mode of this circuit is explained. Another circuit can realize Class I and II neuronal activities with about 3 nW. Our digital silicon neuron circuit can also realize these classes. An auto-associative memory realized on an all-to-all connected network of these silicon neurons is also reviewed, in which the neuron class plays important roles in its performance. PMID:27378842

Mathematical modeling of a process the rolling delivery

NASA Astrophysics Data System (ADS)

Stepanov, Mikhail A.; Korolev, Andrey A.

2018-03-01

An adduced analysis of the scientific researches in a domain of the rolling equipments, also research of properties the working material. A one of perspective direction of scientific research this is mathematical modeling. That is broadly used in many scientific disciplines and especially at the technical, applied sciences. With the aid of mathematical modeling it can be study of physical properties of the researching objects and systems. A research of the rolling delivery and transporting devices realized with the aid of a construction of mathematical model of appropriate process. To be described the basic principles and conditions of a construction of mathematical models of the real objects. For example to be consider a construction of mathematical model the rolling delivery device. For a construction that is model used system of the equations, which consist of: Lagrange’s equation of a motion, describing of the law conservation of energy of a mechanical system, and the Navier - Stokes equations, which characterize of the flow of a continuous non-compressed fluid. A construction of mathematical model the rolling deliver to let determined of a total energy of device, and therefore to got the dependence upon the power of drive to a gap between of rolls. A corroborate the hypothesis about laminar the flow of a material into the rolling gap of deliver.

Derivation and computation of discrete-delay and continuous-delay SDEs in mathematical biology.

PubMed

Allen, Edward J

2014-06-01

Stochastic versions of several discrete-delay and continuous-delay differential equations, useful in mathematical biology, are derived from basic principles carefully taking into account the demographic, environmental, or physiological randomness in the dynamic processes. In particular, stochastic delay differential equation (SDDE) models are derived and studied for Nicholson's blowflies equation, Hutchinson's equation, an SIS epidemic model with delay, bacteria/phage dynamics, and glucose/insulin levels. Computational methods for approximating the SDDE models are described. Comparisons between computational solutions of the SDDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations and of the computational methods.

Mathematical model of one-man air revitalization system

NASA Technical Reports Server (NTRS)

1976-01-01

A mathematical model was developed for simulating the steady state performance in electrochemical CO2 concentrators which utilize (NMe4)2 CO3 (aq.) electrolyte. This electrolyte, which accommodates a wide range of air relative humidity, is most suitable for one-man air revitalization systems. The model is based on the solution of coupled nonlinear ordinary differential equations derived from mass transport and rate equations for the processes which take place in the cell. The boundary conditions are obtained by solving the mass and energy transport equations. A shooting method is used to solve the differential equations.

A structural equation modeling analysis of students' understanding in basic mathematics

NASA Astrophysics Data System (ADS)

Oktavia, Rini; Arif, Salmawaty; Ferdhiana, Ridha; Yuni, Syarifah Meurah; Ihsan, Mahyus

2017-11-01

This research, in general, aims to identify incoming students' understanding and misconceptions of several basic concepts in mathematics. The participants of this study are the 2015 incoming students of Faculty of Mathematics and Natural Science of Syiah Kuala University, Indonesia. Using an instrument that were developed based on some anecdotal and empirical evidences on students' misconceptions, a survey involving 325 participants was administered and several quantitative and qualitative analysis of the survey data were conducted. In this article, we discuss the confirmatory factor analysis using Structural Equation Modeling (SEM) on factors that determine the new students' overall understanding of basic mathematics. The results showed that students' understanding on algebra, arithmetic, and geometry were significant predictors for their overall understanding of basic mathematics. This result supported that arithmetic and algebra are not the only predictors of students' understanding of basic mathematics.

Electro-osmotic flow of semidilute polyelectrolyte solutions.

PubMed

Uematsu, Yuki; Araki, Takeaki

2013-09-07

We investigate electro-osmosis in aqueous solutions of polyelectrolytes using mean-field equations. A solution of positively charged polyelectrolytes is confined between two negatively charged planar surfaces, and an electric field is applied parallel to the surfaces. When electrostatic attraction between the polymer and the surface is strong, the polymers adhere to the surface, forming a highly viscous adsorption layer that greatly suppresses the electro-osmosis. Conversely, electro-osmosis is enhanced by depleting the polymers from the surfaces. We also found that the electro-osmotic flow is invertible when the electrostatic potential decays to its bulk value with the opposite sign. These behaviors are well explained by a simple mathematical form of the electro-osmotic coefficient.

Dynamics of Social Group Competition: Modeling the Decline of Religious Affiliation

NASA Astrophysics Data System (ADS)

Abrams, Daniel M.; Yaple, Haley A.; Wiener, Richard J.

2011-08-01

When social groups compete for members, the resulting dynamics may be understandable with mathematical models. We demonstrate that a simple ordinary differential equation (ODE) model is a good fit for religious shift by comparing it to a new international data set tracking religious nonaffiliation. We then generalize the model to include the possibility of nontrivial social interaction networks and examine the limiting case of a continuous system. Analytical and numerical predictions of this generalized system, which is robust to polarizing perturbations, match those of the original ODE model and justify its agreement with real-world data. The resulting predictions highlight possible causes of social shift and suggest future lines of research in both physics and sociology.

Modelling Evolutionary Algorithms with Stochastic Differential Equations.

PubMed

Heredia, Jorge Pérez

2017-11-20

There has been renewed interest in modelling the behaviour of evolutionary algorithms (EAs) by more traditional mathematical objects, such as ordinary differential equations or Markov chains. The advantage is that the analysis becomes greatly facilitated due to the existence of well established methods. However, this typically comes at the cost of disregarding information about the process. Here, we introduce the use of stochastic differential equations (SDEs) for the study of EAs. SDEs can produce simple analytical results for the dynamics of stochastic processes, unlike Markov chains which can produce rigorous but unwieldy expressions about the dynamics. On the other hand, unlike ordinary differential equations (ODEs), they do not discard information about the stochasticity of the process. We show that these are especially suitable for the analysis of fixed budget scenarios and present analogues of the additive and multiplicative drift theorems from runtime analysis. In addition, we derive a new more general multiplicative drift theorem that also covers non-elitist EAs. This theorem simultaneously allows for positive and negative results, providing information on the algorithm's progress even when the problem cannot be optimised efficiently. Finally, we provide results for some well-known heuristics namely Random Walk (RW), Random Local Search (RLS), the (1+1) EA, the Metropolis Algorithm (MA), and the Strong Selection Weak Mutation (SSWM) algorithm.

Unified Theory for Decoding the Signals from X-Ray Florescence and X-Ray Diffraction of Mixtures.

PubMed

Chung, Frank H

2017-05-01

For research and development or for solving technical problems, we often need to know the chemical composition of an unknown mixture, which is coded and stored in the signals of its X-ray fluorescence (XRF) and X-ray diffraction (XRD). X-ray fluorescence gives chemical elements, whereas XRD gives chemical compounds. The major problem in XRF and XRD analyses is the complex matrix effect. The conventional technique to deal with the matrix effect is to construct empirical calibration lines with standards for each element or compound sought, which is tedious and time-consuming. A unified theory of quantitative XRF analysis is presented here. The idea is to cancel the matrix effect mathematically. It turns out that the decoding equation for quantitative XRF analysis is identical to that for quantitative XRD analysis although the physics of XRD and XRF are fundamentally different. The XRD work has been published and practiced worldwide. The unified theory derives a new intensity-concentration equation of XRF, which is free from the matrix effect and valid for a wide range of concentrations. The linear decoding equation establishes a constant slope for each element sought, hence eliminating the work on calibration lines. The simple linear decoding equation has been verified by 18 experiments.

Representing the Electromagnetic Field: How Maxwell's Mathematics Empowered Faraday's Field Theory

NASA Astrophysics Data System (ADS)

Tweney, Ryan D.

2011-07-01

James Clerk Maxwell `translated' Michael Faraday's experimentally-based field theory into the mathematical representation now known as `Maxwell's Equations.' Working with a variety of mathematical representations and physical models Maxwell extended the reach of Faraday's theory and brought it into consistency with other results in the physics of electricity and magnetism. Examination of Maxwell's procedures opens many issues about the role of mathematical representation in physics and the learning background required for its success. Specifically, Maxwell's training in `Cambridge University' mathematical physics emphasized the use of analogous equations across fields of physics and the repeated solving of extremely difficult problems in physics. Such training develops an array of overlearned mathematical representations supported by highly sophisticated cognitive mechanisms for the retrieval of relevant information from long term memory. For Maxwell, mathematics constituted a new form of representation in physics, enhancing the formal derivational and calculational role of mathematics and opening a cognitive means for the conduct of `experiments in the mind' and for sophisticated representations of theory.

Problem-solving rubrics revisited: Attending to the blending of informal conceptual and formal mathematical reasoning

NASA Astrophysics Data System (ADS)

Hull, Michael M.; Kuo, Eric; Gupta, Ayush; Elby, Andrew

2013-06-01

Much research in engineering and physics education has focused on improving students’ problem-solving skills. This research has led to the development of step-by-step problem-solving strategies and grading rubrics to assess a student’s expertise in solving problems using these strategies. These rubrics value “communication” between the student’s qualitative description of the physical situation and the student’s formal mathematical descriptions (usually equations) at two points: when initially setting up the equations, and when evaluating the final mathematical answer for meaning and plausibility. We argue that (i) neither the rubrics nor the associated problem-solving strategies explicitly value this kind of communication during mathematical manipulations of the chosen equations, and (ii) such communication is an aspect of problem-solving expertise. To make this argument, we present a case study of two students, Alex and Pat, solving the same kinematics problem in clinical interviews. We argue that Pat’s solution, which connects manipulation of equations to their physical interpretation, is more expertlike than Alex’s solution, which uses equations more algorithmically. We then show that the types of problem-solving rubrics currently available do not discriminate between these two types of solutions. We conclude that problem-solving rubrics should be revised or repurposed to more accurately assess problem-solving expertise.

Two-dimensional coupled mathematical modeling of fluvial processes with intense sediment transport and rapid bed evolution

NASA Astrophysics Data System (ADS)

Yue, Zhiyuan; Cao, Zhixian; Li, Xin; Che, Tao

2008-09-01

Alluvial rivers may experience intense sediment transport and rapid bed evolution under a high flow regime, for which traditional decoupled mathematical river models based on simplified conservation equations are not applicable. A two-dimensional coupled mathematical model is presented, which is generally applicable to the fluvial processes with either intense or weak sediment transport. The governing equations of the model comprise the complete shallow water hydrodynamic equations closed with Manning roughness for boundary resistance and empirical relationships for sediment exchange with the erodible bed. The second-order Total-Variation-Diminishing version of the Weighted-Average-Flux method, along with the HLLC approximate Riemann Solver, is adapted to solve the governing equations, which can properly resolve shock waves and contact discontinuities. The model is applied to the pilot study of the flooding due to a sudden outburst of a real glacial-lake.

Mathematics for Physics

NASA Astrophysics Data System (ADS)

Stone, Michael; Goldbart, Paul

2009-07-01

Preface; 1. Calculus of variations; 2. Function spaces; 3. Linear ordinary differential equations; 4. Linear differential operators; 5. Green functions; 6. Partial differential equations; 7. The mathematics of real waves; 8. Special functions; 9. Integral equations; 10. Vectors and tensors; 11. Differential calculus on manifolds; 12. Integration on manifolds; 13. An introduction to differential topology; 14. Group and group representations; 15. Lie groups; 16. The geometry of fibre bundles; 17. Complex analysis I; 18. Applications of complex variables; 19. Special functions and complex variables; Appendixes; Reference; Index.

A mathematical approach for evaluating nickel-hydrogen cells

NASA Technical Reports Server (NTRS)

Leibecki, H. F.

1986-01-01

A mathematical equation is presented which gives a quantitative relationship between time-voltage discharge curves, when a cell's ampere-hour capacity is determined at a constant discharge current. In particular the equation quantifies the initial exponential voltage decay; the rate of voltage decay; the overall voltage shift of the curve and the total capacity of the cell at the given discharge current. The results of 12 nickel-hydrogen boiler plate cells cycled to 80 percent depth-of-discharge (DOD) are discussed in association with these equations.

Introduction to Numerical Methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schoonover, Joseph A.

2016-06-14

These are slides for a lecture for the Parallel Computing Summer Research Internship at the National Security Education Center. This gives an introduction to numerical methods. Repetitive algorithms are used to obtain approximate solutions to mathematical problems, using sorting, searching, root finding, optimization, interpolation, extrapolation, least squares regresion, Eigenvalue problems, ordinary differential equations, and partial differential equations. Many equations are shown. Discretizations allow us to approximate solutions to mathematical models of physical systems using a repetitive algorithm and introduce errors that can lead to numerical instabilities if we are not careful.

Chemical Equation Balancing.

ERIC Educational Resources Information Center

Blakley, G. R.

1982-01-01

Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)

Algebraic Manipulation as Motion within a Landscape

ERIC Educational Resources Information Center

Wittmann, Michael C.; Flood, Virginia J.; Black, Katrina E.

2013-01-01

We show that students rearranging the terms of a mathematical equation in order to separate variables prior to integration use gestures and speech to manipulate the mathematical terms on the page. They treat the terms of the equation as physical objects in a landscape, capable of being moved around. We analyze our results within the tradition of…

Load influence on gear noise. [mathematical model for determining acoustic pressure level as function of load

NASA Technical Reports Server (NTRS)

Merticaru, V.

1974-01-01

An original mathematical model is proposed to derive equations for calculation of gear noise. These equations permit the acoustic pressure level to be determined as a function of load. Application of this method to three parallel gears is reported. The logical calculation scheme is given, as well as the results obtained.

General existence principles for Stieltjes differential equations with applications to mathematical biology

NASA Astrophysics Data System (ADS)

López Pouso, Rodrigo; Márquez Albés, Ignacio

2018-04-01

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper.

A three-dimensional model of corotating streams in the solar wind. 1: Theoretical foundations

NASA Technical Reports Server (NTRS)

Pizzo, V. J.

1978-01-01

The theoretical and mathematical background pertinent to the study of steady, corotating solar wind structure in all three spatial dimensions (3-D) is discussed. The dynamical evolution of the plasma in interplanetary space (defined as the region beyond roughly 35 solar radii where the flow is supersonic) is approximately described by the nonlinear, single fluid, polytropic (magneto-) hydrodynamic equations. Efficient numerical techniques for solving this complex system of coupled, hyperbolic partial differential equations are outlined. The formulation is inviscid and nonmagnetic, but methods allow for the potential inclusion of both features with only modest modifications. One simple, highly idealized, hydrodynamic model stream is examined to illustrate the fundamental processes involved in the 3-D dynamics of stream evolution. Spatial variations in the rotational stream interaction mechanism were found to produce small nonradial flows on a global scale that lead to the transport of mass, energy, and momentum away from regions of relative compression and into regions of relative rarefaction.

A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives

NASA Astrophysics Data System (ADS)

Abro, Kashif Ali; Memon, Anwar Ahmed; Uqaili, Muhammad Aslam

2018-03-01

This research article is analyzed for the comparative study of RL and RC electrical circuits by employing newly presented Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. The governing ordinary differential equations of RL and RC electrical circuits have been fractionalized in terms of fractional operators in the range of 0 ≤ ξ ≤ 1 and 0 ≤ η ≤ 1. The analytic solutions of fractional differential equations for RL and RC electrical circuits have been solved by using the Laplace transform with its inversions. General solutions have been investigated for periodic and exponential sources by implementing the Atangana-Baleanu and Caputo-Fabrizio fractional operators separately. The investigated solutions have been expressed in terms of simple elementary functions with convolution product. On the basis of newly fractional derivatives with and without singular kernel, the voltage and current have interesting behavior with several similarities and differences for the periodic and exponential sources.

Large-aspect-ratio limit of neoclassical transport theory.

PubMed

Wong, S K; Chan, V S

2003-06-01

This paper presents a comprehensive description of neoclassical transport theory in the banana regime for large-aspect-ratio flux surfaces of arbitrary shapes. The method of matched-asymptotic expansions is used to obtain analytical solutions for plasma distribution functions and to compute transport coefficients. The method provides justification for retaining only the part of the Fokker-Planck operator that involves the second derivative with respect to the cosine of the pitch angle for the trapped and barely circulating particles. It leads to a simple equation for the freely circulating particles with boundary conditions that embody a discontinuity separating particles moving in opposite directions. Corrections to the transport coefficients are obtained by generalizing an existing boundary layer analysis. The system of moment and field equations is consistently taken in the cylinder limit, which facilitates the discussion of the treatment of dynamical constraints. It is shown that the nonlocal nature of Ohm's law in neoclassical theory renders the mathematical problem of plasma transport with changing flux surfaces nonstandard.

Influence of friction forces on the motion of VTOL aircraft during landing operations on ships at sea

NASA Technical Reports Server (NTRS)

Howard, J. C.; Chin, D. O.

1981-01-01

Equations describing the friction forces generated during landing operations on ships at sea were formulated. These forces depend on the platform reaction and the coefficient of friction. The platform reaction depends on the relative sink rate and the shock absorbing capability of the landing gear. The friction coefficient varies with the surface condition of the landing platform and the angle of yaw of the aircraft relative to the landing platform. Landings by VTOL aircraft, equipped with conventional oleopneumatic landing gears are discussed. Simplifications are introduced to reduce the complexity of the mathematical description of the tire and shock strut characteristics. Approximating the actual complicated force deflection characteristic of the tire by linear relationship is adequate. The internal friction forces in the shock strut are included in the landing gear model. A set of relatively simple equations was obtained by including only those tire and shock strut characteristics that contribute significantly to the generation of landing gear forces.

Langevin modelling of high-frequency Hang-Seng index data

NASA Astrophysics Data System (ADS)

Tang, Lei-Han

2003-06-01

Accurate statistical characterization of financial time series, such as compound stock indices, foreign currency exchange rates, etc., is fundamental to investment risk management, pricing of derivative products and financial decision making. Traditionally, such data were analyzed and modeled from a purely statistics point of view, with little concern on the specifics of financial markets. Increasingly, however, attention has been paid to the underlying economic forces and the collective behavior of investors. Here we summarize a novel approach to the statistical modeling of a major stock index (the Hang Seng index). Based on mathematical results previously derived in the fluid turbulence literature, we show that a Langevin equation with a variable noise amplitude correctly reproduces the ubiquitous fat tails in the probability distribution of intra-day price moves. The form of the Langevin equation suggests that, despite the extremely complex nature of financial concerns and investment strategies at the individual's level, there exist simple universal rules governing the high-frequency price move in a stock market.

Phase transition solutions in geometrically constrained magnetic domain wall models

NASA Astrophysics Data System (ADS)

Chen, Shouxin; Yang, Yisong

2010-02-01

Recent work on magnetic phase transition in nanoscale systems indicates that new physical phenomena, in particular, the Bloch wall width narrowing, arise as a consequence of geometrical confinement of magnetization and leads to the introduction of geometrically constrained domain wall models. In this paper, we present a systematic mathematical analysis on the existence of the solutions of the basic governing equations in such domain wall models. We show that, when the cross section of the geometric constriction is a simple step function, the solutions may be obtained by minimizing the domain wall energy over the constriction and solving the Bogomol'nyi equation outside the constriction. When the cross section and potential density are both even, we establish the existence of an odd domain wall solution realizing the phase transition process between two adjacent domain phases. When the cross section satisfies a certain integrability condition, we prove that a domain wall solution always exists which links two arbitrarily designated domain phases.

Full-envelope aerodynamic modeling of the Harrier aircraft

NASA Technical Reports Server (NTRS)

Mcnally, B. David

1986-01-01

A project to identify a full-envelope model of the YAV-8B Harrier using flight-test and parameter identification techniques is described. As part of the research in advanced control and display concepts for V/STOL aircraft, a full-envelope aerodynamic model of the Harrier is identified, using mathematical model structures and parameter identification methods. A global-polynomial model structure is also used as a basis for the identification of the YAV-8B aerodynamic model. State estimation methods are used to ensure flight data consistency prior to parameter identification.Equation-error methods are used to identify model parameters. A fixed-base simulator is used extensively to develop flight test procedures and to validate parameter identification software. Using simple flight maneuvers, a simulated data set was created covering the YAV-8B flight envelope from about 0.3 to 0.7 Mach and about -5 to 15 deg angle of attack. A singular value decomposition implementation of the equation-error approach produced good parameter estimates based on this simulated data set.

Onset of Turbulence in a Pipe

NASA Astrophysics Data System (ADS)

Böberg, L.; Brösa, U.

1988-09-01

Turbulence in a pipe is derived directly from the Navier-Stokes equation. Analysis of numerical simulations revealed that small disturbances called 'mothers' induce other much stronger disturbances called 'daughters'. Daughters determine the look of turbulence, while mothers control the transfer of energy from the basic flow to the turbulent motion. From a practical point of view, ruling mothers means ruling turbulence. For theory, the mother-daughter process represents a mechanism permitting chaotic motion in a linearly stable system. The mechanism relies on a property of the linearized problem according to which the eigenfunctions become more and more collinear as the Reynolds number increases. The mathematical methods are described, comparisons with experiments are made, mothers and daughters are analyzed, also graphically, with full particulars, and the systematic construction of small systems of differential equations to mimic the non-linear process by means as simple as possible is explained. We suggest that more then 20 but less than 180 essential degrees of freedom take part in the onset of turbulence.

Homogenization of Large-Scale Movement Models in Ecology

USGS Publications Warehouse

Garlick, M.J.; Powell, J.A.; Hooten, M.B.; McFarlane, L.R.

2011-01-01

A difficulty in using diffusion models to predict large scale animal population dispersal is that individuals move differently based on local information (as opposed to gradients) in differing habitat types. This can be accommodated by using ecological diffusion. However, real environments are often spatially complex, limiting application of a direct approach. Homogenization for partial differential equations has long been applied to Fickian diffusion (in which average individual movement is organized along gradients of habitat and population density). We derive a homogenization procedure for ecological diffusion and apply it to a simple model for chronic wasting disease in mule deer. Homogenization allows us to determine the impact of small scale (10-100 m) habitat variability on large scale (10-100 km) movement. The procedure generates asymptotic equations for solutions on the large scale with parameters defined by small-scale variation. The simplicity of this homogenization procedure is striking when compared to the multi-dimensional homogenization procedure for Fickian diffusion,and the method will be equally straightforward for more complex models. ?? 2010 Society for Mathematical Biology.

A new parametric method to smooth time-series data of metabolites in metabolic networks.

PubMed

Miyawaki, Atsuko; Sriyudthsak, Kansuporn; Hirai, Masami Yokota; Shiraishi, Fumihide

2016-12-01

Mathematical modeling of large-scale metabolic networks usually requires smoothing of metabolite time-series data to account for measurement or biological errors. Accordingly, the accuracy of smoothing curves strongly affects the subsequent estimation of model parameters. Here, an efficient parametric method is proposed for smoothing metabolite time-series data, and its performance is evaluated. To simplify parameter estimation, the method uses S-system-type equations with simple power law-type efflux terms. Iterative calculation using this method was found to readily converge, because parameters are estimated stepwise. Importantly, smoothing curves are determined so that metabolite concentrations satisfy mass balances. Furthermore, the slopes of smoothing curves are useful in estimating parameters, because they are probably close to their true behaviors regardless of errors that may be present in the actual data. Finally, calculations for each differential equation were found to converge in much less than one second if initial parameters are set at appropriate (guessed) values. Copyright © 2016 Elsevier Inc. All rights reserved.

Using Mathematical Algorithms to Modify Glomerular Filtration Rate Estimation Equations

PubMed Central

Zhu, Bei; Wu, Jianqing; Zhu, Jin; Zhao, Weihong

2013-01-01

Background The equations provide a rapid and low-cost method of evaluating glomerular filtration rate (GFR). Previous studies indicated that the Modification of Diet in Renal Disease (MDRD), Chronic Kidney Disease-Epidemiology (CKD-EPI) and MacIsaac equations need further modification for application in Chinese population. Thus, this study was designed to modify the three equations, and compare the diagnostic accuracy of the equations modified before and after. Methodology With the use of 99 mTc-DTPA renal dynamic imaging as the reference GFR (rGFR), the MDRD, CKD-EPI and MacIsaac equations were modified by two mathematical algorithms: the hill-climbing and the simulated-annealing algorithms. Results A total of 703 Chinese subjects were recruited, with the average rGFR 77.14±25.93 ml/min. The entire modification process was based on a random sample of 80% of subjects in each GFR level as a training sample set, the rest of 20% of subjects as a validation sample set. After modification, the three equations performed significant improvement in slop, intercept, correlated coefficient, root mean square error (RMSE), total deviation index (TDI), and the proportion of estimated GFR (eGFR) within 10% and 30% deviation of rGFR (P10 and P30). Of the three modified equations, the modified CKD-EPI equation showed the best accuracy. Conclusions Mathematical algorithms could be a considerable tool to modify the GFR equations. Accuracy of all the three modified equations was significantly improved in which the modified CKD-EPI equation could be the optimal one. PMID:23472113

Three dimensional thermal pollution models. Volume 1: Review of mathematical formulations. [waste heat discharge from power plants and effects on ecosystems

NASA Technical Reports Server (NTRS)

Lee, S. S.; Sengupta, S.

1978-01-01

A mathematical model package for thermal pollution analyses and prediction is presented. These models, intended as user's manuals, are three dimensional and time dependent using the primitive equation approach. Although they have sufficient generality for application at sites with diverse topographical features; they also present specific instructions regarding data preparation for program execution and sample problems. The mathematical formulation of these models is presented including assumptions, approximations, governing equations, boundary and initial conditions, numerical method of solution, and same results.

Diffuse sorption modeling.

PubMed

Pivovarov, Sergey

2009-04-01

This work presents a simple solution for the diffuse double layer model, applicable to calculation of surface speciation as well as to simulation of ionic adsorption within the diffuse layer of solution in arbitrary salt media. Based on Poisson-Boltzmann equation, the Gaines-Thomas selectivity coefficient for uni-bivalent exchange on clay, K(GT)(Me(2+)/M(+))=(Q(Me)(0.5)/Q(M)){M(+)}/{Me(2+)}(0.5), (Q is the equivalent fraction of cation in the exchange capacity, and {M(+)} and {Me(2+)} are the ionic activities in solution) may be calculated as [surface charge, mueq/m(2)]/0.61. The obtained solution of the Poisson-Boltzmann equation was applied to calculation of ionic exchange on clays and to simulation of the surface charge of ferrihydrite in 0.01-6 M NaCl solutions. In addition, a new model of acid-base properties was developed. This model is based on assumption that the net proton charge is not located on the mathematical surface plane but diffusely distributed within the subsurface layer of the lattice. It is shown that the obtained solution of the Poisson-Boltzmann equation makes such calculations possible, and that this approach is more efficient than the original diffuse double layer model.

Exact traveling wave solutions of the KP-BBM equation by using the new approach of generalized (G'/G)-expansion method.

PubMed

Alam, Md Nur; Akbar, M Ali

2013-01-01

The new approach of the generalized (G'/G)-expansion method is an effective and powerful mathematical tool in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in science, engineering and mathematical physics. In this article, the new approach of the generalized (G'/G)-expansion method is applied to construct traveling wave solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. By means of this scheme, we found some new traveling wave solutions of the above mentioned equation.

Solution of some types of differential equations: operational calculus and inverse differential operators.

PubMed

Zhukovsky, K

2014-01-01

We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated.

Differences between Expected Answers and the Answers Given by Computer Algebra Systems to School Equations

ERIC Educational Resources Information Center

Tonisson, Eno

2015-01-01

Sometimes Computer Algebra Systems (CAS) offer an answer that is somewhat different from the answer that is probably expected by the student or teacher. These (somewhat unexpected) answers could serve as a catalyst for rich mathematical discussion. In this study, over 120 equations from school mathematics were solved using 8 different CAS. Many…

A simplified rotor system mathematical model for piloted flight dynamics simulation

NASA Technical Reports Server (NTRS)

Chen, R. T. N.

1979-01-01

The model was developed for real-time pilot-in-the-loop investigation of helicopter flying qualities. The mathematical model included the tip-path plane dynamics and several primary rotor design parameters, such as flapping hinge restraint, flapping hinge offset, blade Lock number, and pitch-flap coupling. The model was used in several exploratory studies of the flying qualities of helicopters with a variety of rotor systems. The basic assumptions used and the major steps involved in the development of the set of equations listed are described. The equations consisted of the tip-path plane dynamic equation, the equations for the main rotor forces and moments, and the equation for control phasing required to achieve decoupling in pitch and roll due to cyclic inputs.

Laplace and the era of differential equations

NASA Astrophysics Data System (ADS)

Weinberger, Peter

2012-11-01

Between about 1790 and 1850 French mathematicians dominated not only mathematics, but also all other sciences. The belief that a particular physical phenomenon has to correspond to a single differential equation originates from the enormous influence Laplace and his contemporary compatriots had in all European learned circles. It will be shown that at the beginning of the nineteenth century Newton's "fluxionary calculus" finally gave way to a French-type notation of handling differential equations. A heated dispute in the Philosophical Magazine between Challis, Airy and Stokes, all three of them famous Cambridge professors of mathematics, then serves to illustrate the era of differential equations. A remark about Schrödinger and his equation for the hydrogen atom finally will lead back to present times.

A mathematical model for simulating noise suppression of lined ejectors

NASA Technical Reports Server (NTRS)

Watson, Willie R.

1994-01-01

A mathematical model containing the essential features embodied in the noise suppression of lined ejectors is presented. Although some simplification of the physics is necessary to render the model mathematically tractable, the current model is the most versatile and technologically advanced at the current time. A system of linearized equations and the boundary conditions governing the sound field are derived starting from the equations of fluid dynamics. A nonreflecting boundary condition is developed. In view of the complex nature of the equations, a parametric study requires the use of numerical techniques and modern computers. A finite element algorithm that solves the differential equations coupled with the boundary condition is then introduced. The numerical method results in a matrix equation with several hundred thousand degrees of freedom that is solved efficiently on a supercomputer. The model is validated by comparing results either with exact solutions or with approximate solutions from other works. In each case, excellent correlations are obtained. The usefulness of the model as an optimization tool and the importance of variable impedance liners as a mechanism for achieving broadband suppression within a lined ejector are demonstrated.

A Reduced Ejector Equation,

DTIC Science & Technology

Ejectors , * Thrust augmentation , * Thrust augmentor nozzles, *Mathematical models, Equations, Supersonic characteristics, Inlets, Exits, Aerodynamics, Vertical takeoff aircraft, Short takeoff aircraft, Workshops

The art of spacecraft design: A multidisciplinary challenge

NASA Technical Reports Server (NTRS)

Abdi, F.; Ide, H.; Levine, M.; Austel, L.

1989-01-01

Actual design turn-around time has become shorter due to the use of optimization techniques which have been introduced into the design process. It seems that what, how and when to use these optimization techniques may be the key factor for future aircraft engineering operations. Another important aspect of this technique is that complex physical phenomena can be modeled by a simple mathematical equation. The new powerful multilevel methodology reduces time-consuming analysis significantly while maintaining the coupling effects. This simultaneous analysis method stems from the implicit function theorem and system sensitivity derivatives of input variables. Use of the Taylor's series expansion and finite differencing technique for sensitivity derivatives in each discipline makes this approach unique for screening dominant variables from nondominant variables. In this study, the current Computational Fluid Dynamics (CFD) aerodynamic and sensitivity derivative/optimization techniques are applied for a simple cone-type forebody of a high-speed vehicle configuration to understand basic aerodynamic/structure interaction in a hypersonic flight condition.

Effective Algorithm for Detection and Correction of the Wave Reconstruction Errors Caused by the Tilt of Reference Wave in Phase-shifting Interferometry

NASA Astrophysics Data System (ADS)

Xu, Xianfeng; Cai, Luzhong; Li, Dailin; Mao, Jieying

2010-04-01

In phase-shifting interferometry (PSI) the reference wave is usually supposed to be an on-axis plane wave. But in practice a slight tilt of reference wave often occurs, and this tilt will introduce unexpected errors of the reconstructed object wave-front. Usually the least-square method with iterations, which is time consuming, is employed to analyze the phase errors caused by the tilt of reference wave. Here a simple effective algorithm is suggested to detect and then correct this kind of errors. In this method, only some simple mathematic operation is used, avoiding using least-square equations as needed in most methods reported before. It can be used for generalized phase-shifting interferometry with two or more frames for both smooth and diffusing objects, and the excellent performance has been verified by computer simulations. The numerical simulations show that the wave reconstruction errors can be reduced by 2 orders of magnitude.

Light-Stimulated Synaptic Devices Utilizing Interfacial Effect of Organic Field-Effect Transistors.

PubMed

Dai, Shilei; Wu, Xiaohan; Liu, Dapeng; Chu, Yingli; Wang, Kai; Yang, Ben; Huang, Jia

2018-06-14

Synaptic transistors stimulated by light waves or photons may offer advantages to the devices, such as wide bandwidth, ultrafast signal transmission, and robustness. However, previously reported light-stimulated synaptic devices generally require special photoelectric properties from the semiconductors and sophisticated device's architectures. In this work, a simple and effective strategy for fabricating light-stimulated synaptic transistors is provided by utilizing interface charge trapping effect of organic field-effect transistors (OFETs). Significantly, our devices exhibited highly synapselike behaviors, such as excitatory postsynaptic current (EPSC) and pair-pulse facilitation (PPF), and presented memory and learning ability. The EPSC decay, PPF curves, and forgetting behavior can be well expressed by mathematical equations for synaptic devices, indicating that interfacial charge trapping effect of OFETs can be utilized as a reliable strategy to realize organic light-stimulated synapses. Therefore, this work provides a simple and effective strategy for fabricating light-stimulated synaptic transistors with both memory and learning ability, which enlightens a new direction for developing neuromorphic devices.

Ever since Gompertz.

PubMed

Olshansky, S J; Carnes, B A

1997-02-01

In 1825 British actuary Benjamin Gompertz made a simple but important observation that a law of geometrical progression pervades large portions of different tables of mortality for humans. The simple formula he derived describing the exponential rise in death rates between sexual maturity and old age is commonly, referred to as the Gompertz equation-a formula that remains a valuable tool in demography and in other scientific disciplines. Gompertz's observation of a mathematical regularity in the life table led him to believe in the presence of a low of mortality that explained why common age patterns of death exist. This law of mortality has captured the attention of scientists for the past 170 years because it was the first among what are now several reliable empirical tools for describing the dying-out process of many living organisms during a significant portion of their life spans. In this paper we review the literature on Gompertz's law of mortality and discuss the importance of his observations and insights in light of research on aging that has taken place since then.

Unified approach for incompressible flows

NASA Astrophysics Data System (ADS)

Chang, Tyne-Hsien

1993-12-01

An unified approach for solving both compressible and incompressible flows was investigated in this study. The difference in CFD code development between incompressible and compressible flows is due to the mathematical characteristics. However, if one can modify the continuity equation for incompressible flows by introducing pseudocompressibility, the governing equations for incompressible flows would have the same mathematical characters as compressible flows. The application of a compressible flow code to solve incompressible flows becomes feasible. Among numerical algorithms developed for compressible flows, the Centered Total Variation Diminishing (CTVD) schemes possess better mathematical properties to damp out the spurious oscillations while providing high-order accuracy for high speed flows. It leads us to believe that CTVD schemes can equally well solve incompressible flows. In this study, the governing equations for incompressible flows include the continuity equation and momentum equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. Thus, the CTVD schemes can be implemented. In addition, the boundary conditions including physical and numerical boundary conditions must be properly specified to obtain accurate solution. The CFD code for this research is currently in progress. Flow past a circular cylinder will be used for numerical experiments to determine the accuracy and efficiency of the code before applying this code to more specific applications.

Comparison of Mathematical Equation and Neural Network Modeling for Drying Kinetic of Mendong in Microwave Oven

NASA Astrophysics Data System (ADS)

Maulidah, Rifa'atul; Purqon, Acep

2016-08-01

Mendong (Fimbristylis globulosa) has a potentially industrial application. We investigate a predictive model for heat and mass transfer in drying kinetics during drying a Mendong. We experimentally dry the Mendong by using a microwave oven. In this study, we analyze three mathematical equations and feed forward neural network (FNN) with back propagation to describe the drying behavior of Mendong. Our results show that the experimental data and the artificial neural network model has a good agreement and better than a mathematical equation approach. The best FNN for the prediction is 3-20-1-1 structure with Levenberg- Marquardt training function. This drying kinetics modeling is potentially applied to determine the optimal parameters during mendong drying and to estimate and control of drying process.

Differences between quadratic equations and functions: Indonesian pre-service secondary mathematics teachers’ views

NASA Astrophysics Data System (ADS)

Aziz, T. A.; Pramudiani, P.; Purnomo, Y. W.

2018-01-01

Difference between quadratic equation and quadratic function as perceived by Indonesian pre-service secondary mathematics teachers (N = 55) who enrolled at one private university in Jakarta City was investigated. Analysis of participants’ written responses and interviews were conducted consecutively. Participants’ written responses highlighted differences between quadratic equation and function by referring to their general terms, main characteristics, processes, and geometrical aspects. However, they showed several obstacles in describing the differences such as inappropriate constraints and improper interpretations. Implications of the study are discussed.

A mathematical simulation model of a 1985-era tilt-rotor passenger aircraft

NASA Technical Reports Server (NTRS)

Mcveigh, M. A.; Widdison, C. A.

1976-01-01

A mathematical model for use in real-time piloted simulation of a 1985-era tilt rotor passenger aircraft is presented. The model comprises the basic six degrees-of-freedom equations of motion, and a large angle of attack representation of the airframe and rotor aerodynamics, together with equations and functions used to model turbine engine performance, aircraft control system and stability augmentation system. A complete derivation of the primary equations is given together with a description of the modeling techniques used. Data for the model is included in an appendix.

Some operational tools for solving fractional and higher integer order differential equations: A survey on their mutual relations

NASA Astrophysics Data System (ADS)

Kiryakova, Virginia S.

2012-11-01

The Laplace Transform (LT) serves as a basis of the Operational Calculus (OC), widely explored by engineers and applied scientists in solving mathematical models for their practical needs. This transform is closely related to the exponential and trigonometric functions (exp, cos, sin) and to the classical differentiation and integration operators, reducing them to simple algebraic operations. Thus, the classical LT and the OC give useful tool to handle differential equations and systems with constant coefficients. Several generalizations of the LT have been introduced to allow solving, in a similar way, of differential equations with variable coefficients and of higher integer orders, as well as of fractional (arbitrary non-integer) orders. Note that fractional order mathematical models are recently widely used to describe better various systems and phenomena of the real world. This paper surveys briefly some of our results on classes of such integral transforms, that can be obtained from the LT by means of "transmutations" which are operators of the generalized fractional calculus (GFC). On the list of these Laplace-type integral transforms, we consider the Borel-Dzrbashjan, Meijer, Krätzel, Obrechkoff, generalized Obrechkoff (multi-index Borel-Dzrbashjan) transforms, etc. All of them are G- and H-integral transforms of convolutional type, having as kernels Meijer's G- or Fox's H-functions. Besides, some special functions (also being G- and H-functions), among them - the generalized Bessel-type and Mittag-Leffler (M-L) type functions, are generating Gel'fond-Leontiev (G-L) operators of generalized differentiation and integration, which happen to be also operators of GFC. Our integral transforms have operational properties analogous to those of the LT - they do algebrize the G-L generalized integrations and differentiations, and thus can serve for solving wide classes of differential equations with variable coefficients of arbitrary, including non-integer order. Throughout the survey, we illustrate the parallels in the relationships: Laplace type integral transforms - special functions as kernels - operators of generalized integration and differentiation generated by special functions - special functions as solutions of related differential equations. The role of the so-called Special Functions of Fractional Calculus is emphasized.

Time and frequency domain analysis of sampled data controllers via mixed operation equations

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1981-01-01

Specification of the mathematical equations required to define the dynamic response of a linear continuous plant, subject to sampled data control, is complicated by the fact that the digital components of the control system cannot be modeled via linear ordinary differential equations. This complication can be overcome by introducing two new mathematical operations; namely, the operation of zero order hold and digial delay. It is shown that by direct utilization of these operations, a set of linear mixed operation equations can be written and used to define the dynamic response characteristics of the controlled system. It also is shown how these linear mixed operation equations lead, in an automatable manner, directly to a set of finite difference equations which are in a format compatible with follow on time and frequency domain analysis methods.

A Structural Equation Model Explaining 8th Grade Students' Mathematics Achievements

ERIC Educational Resources Information Center

Yurt, Eyüp; Sünbül, Ali Murat

2014-01-01

The purpose of this study is to investigate, via a model, the explanatory and predictive relationships among the following variables: Mathematical Problem Solving and Reasoning Skills, Sources of Mathematics Self-Efficacy, Spatial Ability, and Mathematics Achievements of Secondary School 8th Grade Students. The sample group of the study, itself…

More than Motivation: The Combined Effects of Critical Motivational Variables on Middle School Mathematics Achievement

ERIC Educational Resources Information Center

Middleton, James A.

2013-01-01

The role of mathematical interest, identity, utility, self-efficacy, and effort was examined as a set of interdependent factors leading to students' mathematics achievement. A structural equations model, testing a hypothesized structure of motivation variables and their impact on middle school mathematics achievement was developed utilizing the…

The Art of Teaching the Art of Applying Mathematics

ERIC Educational Resources Information Center

Lighthill, M. J.

1971-01-01

This article (the 1971 Presidential Address to the Mathematical Association of U. K.) makes a plea that college teachers of mathematics should be better aware of the applications of mathematics in other subjects. As an example, the author describes the use of differential equations to solve certain vibrational problems in mechanical engineering.…

Students’ mathematical representations on secondary school in solving trigonometric problems

NASA Astrophysics Data System (ADS)

Istadi; Kusmayadi, T. A.; Sujadi, I.

2017-06-01

This research aimed to analyse students’ mathematical representations on secondary school in solving trigonometric problems. This research used qualitative method. The participants were 4 students who had high competence of knowledge taken from 20 students of 12th natural-science grade SMAN-1 Kota Besi, Central Kalimantan. Data validation was carried out using time triangulation. Data analysis used Huberman and Miles stages. The results showed that their answers were not only based on the given figure, but also used the definition of trigonometric ratio on verbal representations. On the other hand, they were able to determine the object positions to be observed. However, they failed to determine the position of the angle of depression at the sketches made on visual representations. Failure in determining the position of the angle of depression to cause an error in using the mathematical equation. Finally, they were unsuccessful to use the mathematical equation properly on symbolic representations. From this research, we could recommend the importance of translations between mathematical problems and mathematical representations as well as translations among mathematical representaions (verbal, visual, and symbolic) in learning mathematics in the classroom.

Methodology of Thermodynamics

ERIC Educational Resources Information Center

Mohan, Gyan

1969-01-01

Presents a systematization of the mathematical formulae in thermodynamics. From the set of thermodynamic variables, four equations are derived which contain the total mathematical jargon of thermodynamics. (LC)

Earth Sciences Push Radiative Transfer Theory

NASA Astrophysics Data System (ADS)

Davis, Anthony; Mishchenko, Michael

2009-12-01

2009 International Conference on Advances in Mathematics, Computational Methods, and Reactor Physics; Saratoga Springs, New York, 4-7 May 2009; The theories of radiative transfer and particle—particularly neutron—transport are grounded in distinctive microscale physics that deals with either optics or particle dynamics. However, it is not practical to track every wave or particle in macroscopic systems, nor do all of these details matter. That is why Newton's laws, which describe individual particles, are replaced by those of Euler, Navier-Stokes, Maxwell, Boltzmann, Gibbs, and others, which describe the collective behavior of vast numbers of particles. And that is why the radiative transfer (RT) equation is used to describe the flow of radiation through geophysical-scale systems, leaving to Maxwell's wave equations only the task of providing the optical properties of the medium, be it air, water, snow, ice, or biomass. Interestingly, particle transport is determined by the linear transport equation, which is mathematically identical to the RT equation, so geophysicists and nuclear scientists are interested in the same mathematics and computational techniques.

Theory of linear sweep voltammetry with diffuse charge: Unsupported electrolytes, thin films, and leaky membranes

NASA Astrophysics Data System (ADS)

Yan, David; Bazant, Martin Z.; Biesheuvel, P. M.; Pugh, Mary C.; Dawson, Francis P.

2017-03-01

Linear sweep and cyclic voltammetry techniques are important tools for electrochemists and have a variety of applications in engineering. Voltammetry has classically been treated with the Randles-Sevcik equation, which assumes an electroneutral supported electrolyte. In this paper, we provide a comprehensive mathematical theory of voltammetry in electrochemical cells with unsupported electrolytes and for other situations where diffuse charge effects play a role, and present analytical and simulated solutions of the time-dependent Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions for a 1:1 electrolyte and a simple reaction. Using these solutions, we construct theoretical and simulated current-voltage curves for liquid and solid thin films, membranes with fixed background charge, and cells with blocking electrodes. The full range of dimensionless parameters is considered, including the dimensionless Debye screening length (scaled to the electrode separation), Damkohler number (ratio of characteristic diffusion and reaction times), and dimensionless sweep rate (scaled to the thermal voltage per diffusion time). The analysis focuses on the coupling of Faradaic reactions and diffuse charge dynamics, although capacitive charging of the electrical double layers is also studied, for early time transients at reactive electrodes and for nonreactive blocking electrodes. Our work highlights cases where diffuse charge effects are important in the context of voltammetry, and illustrates which regimes can be approximated using simple analytical expressions and which require more careful consideration.

Mathematics Literacy of Secondary Students in Solving Simultanenous Linear Equations

NASA Astrophysics Data System (ADS)

Sitompul, R. S. I.; Budayasa, I. K.; Masriyah

2018-01-01

This study examines the profile of secondary students’ mathematical literacy in solving simultanenous linear equations problems in terms of cognitive style of visualizer and verbalizer. This research is a descriptive research with qualitative approach. The subjects in this research consist of one student with cognitive style of visualizer and one student with cognitive style of verbalizer. The main instrument in this research is the researcher herself and supporting instruments are cognitive style tests, mathematics skills tests, problem-solving tests and interview guidelines. Research was begun by determining the cognitive style test and mathematics skill test. The subjects chosen were given problem-solving test about simultaneous linear equations and continued with interview. To ensure the validity of the data, the researcher conducted data triangulation; the steps of data reduction, data presentation, data interpretation, and conclusion drawing. The results show that there is a similarity of visualizer and verbalizer-cognitive style in identifying and understanding the mathematical structure in the process of formulating. There are differences in how to represent problems in the process of implementing, there are differences in designing strategies and in the process of interpreting, and there are differences in explaining the logical reasons.

Unified approach for incompressible flows

NASA Astrophysics Data System (ADS)

Chang, Tyne-Hsien

1995-07-01

A unified approach for solving incompressible flows has been investigated in this study. The numerical CTVD (Centered Total Variation Diminishing) scheme used in this study was successfully developed by Sanders and Li for compressible flows, especially for the high speed. The CTVD scheme possesses better mathematical properties to damp out the spurious oscillations while providing high-order accuracy for high speed flows. It leads us to believe that the CTVD scheme can equally well apply to solve incompressible flows. Because of the mathematical difference between the governing equations for incompressible and compressible flows, the scheme can not directly apply to the incompressible flows. However, if one can modify the continuity equation for incompressible flows by introducing pseudo-compressibility, the governing equations for incompressible flows would have the same mathematical characters as compressible flows. The application of the algorithm to incompressible flows thus becomes feasible. In this study, the governing equations for incompressible flows comprise continuity equation and momentum equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. Thus, the CTVD schemes can be implemented. In addition, the physical and numerical boundary conditions are properly implemented by the characteristic boundary conditions. Accordingly, a CFD code has been developed for this research and is currently under testing. Flow past a circular cylinder was chosen for numerical experiments to determine the accuracy and efficiency of the code. The code has shown some promising results.

Unified approach for incompressible flows

NASA Technical Reports Server (NTRS)

Chang, Tyne-Hsien

1995-01-01

A unified approach for solving incompressible flows has been investigated in this study. The numerical CTVD (Centered Total Variation Diminishing) scheme used in this study was successfully developed by Sanders and Li for compressible flows, especially for the high speed. The CTVD scheme possesses better mathematical properties to damp out the spurious oscillations while providing high-order accuracy for high speed flows. It leads us to believe that the CTVD scheme can equally well apply to solve incompressible flows. Because of the mathematical difference between the governing equations for incompressible and compressible flows, the scheme can not directly apply to the incompressible flows. However, if one can modify the continuity equation for incompressible flows by introducing pseudo-compressibility, the governing equations for incompressible flows would have the same mathematical characters as compressible flows. The application of the algorithm to incompressible flows thus becomes feasible. In this study, the governing equations for incompressible flows comprise continuity equation and momentum equations. The continuity equation is modified by adding a time-derivative of the pressure term containing the artificial compressibility. The modified continuity equation together with the unsteady momentum equations forms a hyperbolic-parabolic type of time-dependent system of equations. Thus, the CTVD schemes can be implemented. In addition, the physical and numerical boundary conditions are properly implemented by the characteristic boundary conditions. Accordingly, a CFD code has been developed for this research and is currently under testing. Flow past a circular cylinder was chosen for numerical experiments to determine the accuracy and efficiency of the code. The code has shown some promising results.

Automation of reliability evaluation procedures through CARE - The computer-aided reliability estimation program.

NASA Technical Reports Server (NTRS)

Mathur, F. P.

1972-01-01

Description of an on-line interactive computer program called CARE (Computer-Aided Reliability Estimation) which can model self-repair and fault-tolerant organizations and perform certain other functions. Essentially CARE consists of a repository of mathematical equations defining the various basic redundancy schemes. These equations, under program control, are then interrelated to generate the desired mathematical model to fit the architecture of the system under evaluation. The mathematical model is then supplied with ground instances of its variables and is then evaluated to generate values for the reliability-theoretic functions applied to the model.

Progress in recognizing typeset mathematics

NASA Astrophysics Data System (ADS)

Fateman, Richard J.; Tokuyasu, Taku A.

1996-03-01

Printed mathematics has a number of features which distinguish it from conventional text. These include structure in two dimensions (fractions, exponents, limits), frequent font changes, symbols with variable shape (quotient bars), and substantially differing notational conventions from source to source. When compounded with more generic problems such as noise and merged or broken characters, printed mathematics offers a challenging arena for recognition. Our project was initially driven by the goal of scanning and parsing some 5,000 pages of elaborate mathematics (tables of definite integrals). While our prototype system demonstrates success on translating noise-free typeset equations into Lisp expressions appropriate for further processing, a more semantic top-down approach appears necessary for higher levels of performance. Such an approach may benefit the incorporation of these programs into a more general document processing viewpoint. We intend to release to the public our somewhat refined prototypes as utility programs in the hope that they will be of general use in the construction of custom OCR packages. These utilities are quite fast even as originally prototyped in Lisp, where they may be of particular interest to those working on 'intelligent' optical processing. Some routines have been re-written in C++ as well. Additional programs providing formula recognition and parsing also form a part of this system. It is important however to realize that distinct conflicting grammars are needed to cover variations in contemporary and historical typesetting, and thus a single simple solution is not possible.

A Mathematical Model of a Simple Amplifier Using a Ferroelectric Transistor

NASA Technical Reports Server (NTRS)

Sayyah, Rana; Hunt, Mitchell; MacLeod, Todd C.; Ho, Fat D.

2009-01-01

This paper presents a mathematical model characterizing the behavior of a simple amplifier using a FeFET. The model is based on empirical data and incorporates several variables that affect the output, including frequency, load resistance, and gate-to-source voltage. Since the amplifier is the basis of many circuit configurations, a mathematical model that describes the behavior of a FeFET-based amplifier will help in the integration of FeFETs into many other circuits.

A stochastic model for the normal tissue complication probability (NTCP) and applicationss.

PubMed

Stocks, Theresa; Hillen, Thomas; Gong, Jiafen; Burger, Martin

2017-12-11

The normal tissue complication probability (NTCP) is a measure for the estimated side effects of a given radiation treatment schedule. Here we use a stochastic logistic birth-death process to define an organ-specific and patient-specific NTCP. We emphasize an asymptotic simplification which relates the NTCP to the solution of a logistic differential equation. This framework is based on simple modelling assumptions and it prepares a framework for the use of the NTCP model in clinical practice. As example, we consider side effects of prostate cancer brachytherapy such as increase in urinal frequency, urinal retention and acute rectal dysfunction. © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Seismic migration in generalized coordinates

NASA Astrophysics Data System (ADS)

Arias, C.; Duque, L. F.

2017-06-01

Reverse time migration (RTM) is a technique widely used nowadays to obtain images of the earth’s sub-surface, using artificially produced seismic waves. This technique has been developed for zones with flat surface and when applied to zones with rugged topography some corrections must be introduced in order to adapt it. This can produce defects in the final image called artifacts. We introduce a simple mathematical map that transforms a scenario with rugged topography into a flat one. The three steps of the RTM can be applied in a way similar to the conventional ones just by changing the Laplacian in the acoustic wave equation for a generalized one. We present a test of this technique using the Canadian foothills SEG velocity model.

Adaptive nonlinear control for autonomous ground vehicles

NASA Astrophysics Data System (ADS)

Black, William S.

We present the background and motivation for ground vehicle autonomy, and focus on uses for space-exploration. Using a simple design example of an autonomous ground vehicle we derive the equations of motion. After providing the mathematical background for nonlinear systems and control we present two common methods for exactly linearizing nonlinear systems, feedback linearization and backstepping. We use these in combination with three adaptive control methods: model reference adaptive control, adaptive sliding mode control, and extremum-seeking model reference adaptive control. We show the performances of each combination through several simulation results. We then consider disturbances in the system, and design nonlinear disturbance observers for both single-input-single-output and multi-input-multi-output systems. Finally, we show the performance of these observers with simulation results.

Modeling of chemical inhibition from amyloid protein aggregation kinetics.

PubMed

Vázquez, José Antonio

2014-02-27

The process of amyloid proteins aggregation causes several human neuropathologies. In some cases, e.g. fibrillar deposits of insulin, the problems are generated in the processes of production and purification of protein and in the pump devices or injectable preparations for diabetics. Experimental kinetics and adequate modelling of chemical inhibition from amyloid aggregation are of practical importance in order to study the viable processing, formulation and storage as well as to predict and optimize the best conditions to reduce the effect of protein nucleation. In this manuscript, experimental data of insulin, Aβ42 amyloid protein and apomyoglobin fibrillation from recent bibliography were selected to evaluate the capability of a bivariate sigmoid equation to model them. The mathematical functions (logistic combined with Weibull equation) were used in reparameterized form and the effect of inhibitor concentrations on kinetic parameters from logistic equation were perfectly defined and explained. The surfaces of data were accurately described by proposed model and the presented analysis characterized the inhibitory influence on the protein aggregation by several chemicals. Discrimination between true and apparent inhibitors was also confirmed by the bivariate equation. EGCG for insulin (working at pH = 7.4/T = 37°C) and taiwaniaflavone for Aβ42 were the compounds studied that shown the greatest inhibition capacity. An accurate, simple and effective model to investigate the inhibition of chemicals on amyloid protein aggregation has been developed. The equation could be useful for the clear quantification of inhibitor potential of chemicals and rigorous comparison among them.

Full dimensional computer simulations to study pulsatile blood flow in vessels, aortic arch and bifurcated veins: Investigation of blood viscosity and turbulent effects.

PubMed

Sultanov, Renat A; Guster, Dennis

2009-01-01

We report computational results of blood flow through a model of the human aortic arch and a vessel of actual diameter and length. A realistic pulsatile flow is used in all simulations. Calculations for bifurcation type vessels are also carried out and presented. Different mathematical methods for numerical solution of the fluid dynamics equations have been considered. The non-Newtonian behaviour of the human blood is investigated together with turbulence effects. A detailed time-dependent mathematical convergence test has been carried out. The results of computer simulations of the blood flow in vessels of three different geometries are presented: for pressure, strain rate and velocity component distributions we found significant disagreements between our results obtained with realistic non-Newtonian treatment of human blood and the widely used method in the literature: a simple Newtonian approximation. A significant increase of the strain rate and, as a result, the wall shear stress distribution, is found in the region of the aortic arch. Turbulent effects are found to be important, particularly in the case of bifurcation vessels.

What Physicists Mean By the Equals Sign in Undergraduate Education

NASA Astrophysics Data System (ADS)

Kornick, Kellianne; Alaee, Dina; Sayre, Eleanor; Franklin, Scott

2017-01-01

Mathematical syntax allows for the description of meaningful concepts in the physical sciences, and having nuanced proficiency in mathematical formalism is closely tied to communication and understanding of physical principles. The concept of equality is especially important, as it constrains and dictates the relationships between two equated expressions, and a student with detailed understanding of these relationships can derive physical meaning from syntactical expressions mediated by equals signs by knowing the ``meaning'' of equals signs. We delineate types of equals signs as used in undergraduate textbooks and develop a categorization scheme in order to investigate how equals signs are used paradigmatically and culturally in textbooks to convey physical meaning. We classify equals signs into general clusters (causal, definitional, assignment, balancing, and ``just math''), each cluster containing more detailed types. We investigate differences across various topics and between introductory and upper-division textbooks. We found that upper division textbooks are more likely to use balancing, definitional, and more complex kinds of assignment forms, while introductory texts have much higher frequencies of simple assignment and ``just math'' types.

Animal models for studying transport across the blood-brain barrier.

PubMed

Bonate, P L

1995-01-01

There are many reasons for wishing to determine the rate of uptake of a drug from blood into brain parenchyma. However, when faced with doing so for the first time, choosing a method can be a formidable task. There are at least 7 methods from which to choose: indicator dilution, brain uptake index, microdialysis, external registration, PET scanning, in situ perfusion, and compartmental modeling. Each method has advantages and disadvantages. Some methods require very little equipment while others require equipment that can cost millions of dollars. Some methods require very little technical experience whereas others require complex surgical manipulation. The mathematics alone for the various methods range from simple algebra to complex integral calculus and differential equations. Like most things in science, as the complexity of the technique increases, so does the quantity of information it provides. This review is meant to serve as a starting point for the researcher who wishes to study transport and uptake across the blood-brain barrier in animal models. An overview of the mathematical theory, as well as an introduction to the techniques, is presented.

Secondary School Advanced Mathematics, Chapter 8, Systems of Equations. Teacher's Commentary.

ERIC Educational Resources Information Center

Stanford Univ., CA. School Mathematics Study Group.

This manual was designed for use with the last of five texts in the Secondary School Advanced Mathematics (SSAM) series. Developed for students who have completed the Secondary School Mathematics (SSM) program and wish to continue their studies in mathematics, this series is designed to review, strengthen, and fill gaps in the material covered in…

Notes for Applied Mathematics in Trigonometry and Earth Geometry/Navigation

ERIC Educational Resources Information Center

Faulkner, Peter

2004-01-01

As time has progressed, the role of applied mathematics has become increasingly important. Indeed there are now more students enrolled in applied mathematics courses in senior high schools and colleges than in pure mathematics. Such courses become more relevant both to the student and to future employers, if the same constants and equations that…

Order-of-magnitude estimates of latency (time to appearance) and refill time of a cancer from a single cancer 'stem' cell compared by an exponential and a logistic equation.

PubMed

Anderson, Ken M; Rubenstein, Marvin; Guinan, Patrick; Patel, Minu

2012-01-01

The time required before a mass of cancer cells considered to have originated from a single malignantly transformed cancer 'stem' cell reaches a certain number has not been studied. Applications might include determination of the time the cell mass reaches a size that can be detected by X-rays or physical examination or modeling growth rates in vitro in order to compare with other models or established data. We employed a simple logarithmic equation and a common logistic equation incorporating 'feedback' for unknown variables of cell birth, growth, division, and death that can be used to model cell proliferation. It can be used in association with free or commercial statistical software. Results with these two equations, varying the proliferation rate, nominally reduced by generational cell loss, are presented in two tables. The resulting equation, instructions, examples, and necessary mathematical software are available in the online appendix, where several parameters of interest can be modified by the reader www.uic.edu/nursing/publicationsupplements/tobillion_Anderson_Rubenstein_Guinan_Patel1.pdf. Reducing the proliferation rate by whatever alterations employed, markedly increases the time to reach 10(9) cells originating from an initial progenitor. In thinking about multistep oncogenesis, it is useful to consider the profound effect that variations in the effective proliferation rate may have during cancer development. This can be approached with the proposed equation, which is easy to use and available to further peer fine-tuning to be used in future modeling of cell growth.

Why does shear banding behave like first-order phase transitions? Derivation of a potential from a mechanical constitutive model.

PubMed

Sato, K; Yuan, X-F; Kawakatsu, T

2010-02-01

Numerous numerical and experimental evidence suggest that shear banding behavior looks like first-order phase transitions. In this paper, we demonstrate that this correspondence is actually established in the so-called non-local diffusive Johnson-Segalman model (the DJS model), a typical mechanical constitutive model that has been widely used for describing shear banding phenomena. In the neighborhood of the critical point, we apply the reduction procedure based on the center manifold theory to the governing equations of the DJS model. As a result, we obtain a time evolution equation of the flow field that is equivalent to the time-dependent Ginzburg-Landau (TDGL) equations for modeling thermodynamic first-order phase transitions. This result, for the first time, provides a mathematical proof that there is an analogy between the mechanical instability and thermodynamic phase transition at least in the vicinity of the critical point of the shear banding of DJS model. Within this framework, we can clearly distinguish the metastable branch in the stress-strain rate curve around the shear banding region from the globally stable branch. A simple extension of this analysis to a class of more general constitutive models is also discussed. Numerical simulations for the original DJS model and the reduced TDGL equation is performed to confirm the range of validity of our reduction theory.

Breakdown of the reaction-diffusion master equation with nonelementary rates

NASA Astrophysics Data System (ADS)

Smith, Stephen; Grima, Ramon

2016-05-01

The chemical master equation (CME) is the exact mathematical formulation of chemical reactions occurring in a dilute and well-mixed volume. The reaction-diffusion master equation (RDME) is a stochastic description of reaction-diffusion processes on a spatial lattice, assuming well mixing only on the length scale of the lattice. It is clear that, for the sake of consistency, the solution of the RDME of a chemical system should converge to the solution of the CME of the same system in the limit of fast diffusion: Indeed, this has been tacitly assumed in most literature concerning the RDME. We show that, in the limit of fast diffusion, the RDME indeed converges to a master equation but not necessarily the CME. We introduce a class of propensity functions, such that if the RDME has propensities exclusively of this class, then the RDME converges to the CME of the same system, whereas if the RDME has propensities not in this class, then convergence is not guaranteed. These are revealed to be elementary and nonelementary propensities, respectively. We also show that independent of the type of propensity, the RDME converges to the CME in the simultaneous limit of fast diffusion and large volumes. We illustrate our results with some simple example systems and argue that the RDME cannot generally be an accurate description of systems with nonelementary rates.

Shotgun Canceling.

ERIC Educational Resources Information Center

Szymanski, Theodore

1999-01-01

Discusses a common misunderstanding demonstrated by many students in basic mathematics courses: not knowing how to properly "cancel" factors in simplifying mathematical equations. Asserts that "crossing-out" or "canceling" is not a valid mathematical operation, and that instructors should be wary about using these terms because of the ease with…

A Cryptological Way of Teaching Mathematics

ERIC Educational Resources Information Center

Caballero-Gil, Pino; Bruno-Castaneda, Carlos

2007-01-01

This work addresses the subject of mathematics education at secondary schools from a current and stimulating point of view intimately related to computational science. Cryptology is a captivating way of introducing into the classroom different mathematical subjects such as functions, matrices, modular arithmetic, combinatorics, equations,…

New Challenges in the Teaching of Mathematics.

ERIC Educational Resources Information Center

Bourguignon, Jean Pierre

The manifold but discrete presence of mathematics in many objects or services imposes new constraints to the teaching of mathematics. If citizens need to be comfortable in various situations with a variety of mathematical tools, the learning of mathematics requires that one starts with simple concepts. This paper proposes some solutions to solve…

Teaching Mathematical Modelling for Earth Sciences via Case Studies

NASA Astrophysics Data System (ADS)

Yang, Xin-She

2010-05-01

Mathematical modelling is becoming crucially important for earth sciences because the modelling of complex systems such as geological, geophysical and environmental processes requires mathematical analysis, numerical methods and computer programming. However, a substantial fraction of earth science undergraduates and graduates may not have sufficient skills in mathematical modelling, which is due to either limited mathematical training or lack of appropriate mathematical textbooks for self-study. In this paper, we described a detailed case-study-based approach for teaching mathematical modelling. We illustrate how essential mathematical skills can be developed for students with limited training in secondary mathematics so that they are confident in dealing with real-world mathematical modelling at university level. We have chosen various topics such as Airy isostasy, greenhouse effect, sedimentation and Stokes' flow,free-air and Bouguer gravity, Brownian motion, rain-drop dynamics, impact cratering, heat conduction and cooling of the lithosphere as case studies; and we use these step-by-step case studies to teach exponentials, logarithms, spherical geometry, basic calculus, complex numbers, Fourier transforms, ordinary differential equations, vectors and matrix algebra, partial differential equations, geostatistics and basic numeric methods. Implications for teaching university mathematics for earth scientists for tomorrow's classroom will also be discussed. Refereces 1) D. L. Turcotte and G. Schubert, Geodynamics, 2nd Edition, Cambridge University Press, (2002). 2) X. S. Yang, Introductory Mathematics for Earth Scientists, Dunedin Academic Press, (2009).

The living Drake equation of the Tau Zero Foundation

NASA Astrophysics Data System (ADS)

Maccone, Claudio

2011-03-01

The living Drake equation is our statistical generalization of the Drake equation such that it can take into account any number of factors. This new result opens up the possibility to enrich the equation by inserting more new factors as long as the scientific learning increases. The adjective "Living" refers just to this continuous enrichment of the Drake equation and is the goal of a new research project that the Tau Zero Foundation has entrusted to this author as the discoverer of the statistical Drake equation described hereafter. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be arbitrarily distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov form of the CLT, or the Lindeberg form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the lognormal distribution. Then, the mean value, standard deviation, mode, median and all the moments of this lognormal N can be derived from the means and standard deviations of the seven input random variables. In fact, the seven factors in the ordinary Drake equation now become seven independent positive random variables. The probability distribution of each random variable may be arbitrary. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. An application of our statistical Drake equation then follows. The (average) distance between any two neighbouring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, this distance now becomes a new random variable. We derive the relevant probability density function, apparently previously unknown (dubbed "Maccone distribution" by Paul Davies). Data Enrichment Principle. It should be noticed that any positive number of random variables in the statistical Drake equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the statistical Drake equation we call the "Data Enrichment Principle", and regard as the key to more profound, future results in Astrobiology and SETI.

Conference on Non-linear Phenomena in Mathematical Physics: Dedicated to Cathleen Synge Morawetz on her 85th Birthday. The Fields Institute, Toronto, Canada September 18-20, 2008. Sponsors: Association for Women in Mathematics, Inc. and The Fields Institute

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lewis, Jennifer

2012-10-15

This scientific meeting focused on the legacy of Cathleen S. Morawetz and the impact that her scientific work on transonic flow and the non-linear wave equation has had in recent progress on different aspects of analysis for non-linear wave, kinetic and quantum transport problems associated to mathematical physics. These are areas where the elements of continuum, statistical and stochastic mechanics, and their interplay, have counterparts in the theory of existence, uniqueness and stability of the associated systems of equations and geometric constraints. It was a central event for the applied and computational analysis community focusing on Partial Differential Equations. Themore » goal of the proposal was to honor Cathleen Morawetz, a highly successful woman in mathematics, while encouraging beginning researchers. The conference was successful in show casing the work of successful women, enhancing the visibility of women in the profession and providing role models for those just beginning their careers. The two-day conference included seven 45-minute lectures and one day of six 45-minute lectures, and a poster session for junior participants. The conference program included 19 distinguished speakers, 10 poster presentations, about 70 junior and senior participants and, of course, the participation of Cathleen Synge Morawetz. The conference celebrated Morawetz's paramount contributions to the theory of non-linear equations in gas dynamics and their impact in the current trends of nonlinear phenomena in mathematical physics, but also served as an awareness session of current women's contribution to mathematics.« less

Computer modeling of heat pipe performance

NASA Technical Reports Server (NTRS)

Peterson, G. P.

1983-01-01

A parametric study of the defining equations which govern the steady state operational characteristics of the Grumman monogroove dual passage heat pipe is presented. These defining equations are combined to develop a mathematical model which describes and predicts the operational and performance capabilities of a specific heat pipe given the necessary physical characteristics and working fluid. Included is a brief review of the current literature, a discussion of the governing equations, and a description of both the mathematical and computer model. Final results of preliminary test runs of the model are presented and compared with experimental tests on actual prototypes.

Soliton-type solutions for two models in mathematical physics

NASA Astrophysics Data System (ADS)

Al-Ghafri, K. S.

2018-04-01

In this paper, the generalised Klein-Gordon and Kadomtsov-Petviashvili Benjamin-Bona-Mahony equations with power law nonlinearity are investigated. Our study is based on reducing the form of both equations to a first-order ordinary differential equation having the travelling wave solutions. Subsequently, soliton-type solutions such as compacton and solitary pattern solutions are obtained analytically. Additionally, the peaked soliton has been derived where it exists under a specific restrictions. In addition to the soliton solutions, the mathematical method which is exploited in this work also creates a few amount of travelling wave solutions.

Stochastic differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sobczyk, K.

1990-01-01

This book provides a unified treatment of both regular (or random) and Ito stochastic differential equations. It focuses on solution methods, including some developed only recently. Applications are discussed, in particular an insight is given into both the mathematical structure, and the most efficient solution methods (analytical as well as numerical). Starting from basic notions and results of the theory of stochastic processes and stochastic calculus (including Ito's stochastic integral), many principal mathematical problems and results related to stochastic differential equations are expounded here for the first time. Applications treated include those relating to road vehicles, earthquake excitations and offshoremore » structures.« less

Heat Source/Sink in a Magneto-Hydrodynamic Non-Newtonian Fluid Flow in a Porous Medium: Dual Solutions.

PubMed

Hayat, Tasawar; Awais, Muhammad; Imtiaz, Amna

2016-01-01

This communication deals with the properties of heat source/sink in a magneto-hydrodynamic flow of a non-Newtonian fluid immersed in a porous medium. Shrinking phenomenon along with the permeability of the wall is considered. Mathematical modelling is performed to convert the considered physical process into set of coupled nonlinear mathematical equations. Suitable transformations are invoked to convert the set of partial differential equations into nonlinear ordinary differential equations which are tackled numerically for the solution computations. It is noted that dual solutions for various physical parameters exist which are analyzed in detail.

Middle school students' reading comprehension of mathematical texts and algebraic equations

NASA Astrophysics Data System (ADS)

Duru, Adem; Koklu, Onder

2011-06-01

In this study, middle school students' abilities to translate mathematical texts into algebraic representations and vice versa were investigated. In addition, students' difficulties in making such translations and the potential sources for these difficulties were also explored. Both qualitative and quantitative methods were used to collect data for this study: questionnaire and clinical interviews. The questionnaire consisted of two general types of items: (1) selected-response (multiple-choice) items for which the respondent selects from multiple options and (2) open-ended items for which the respondent constructs a response. In order to further investigate the students' strategies while they were translating the given mathematical texts to algebraic equations and vice versa, five randomly chosen (n = 5) students were interviewed. Data were collected in the 2007-2008 school year from 185 middle-school students in five teachers' classrooms in three different schools in the city of Adıyaman, Turkey. After the analysis of data, it was found that students who participated in this study had difficulties in translating the mathematical texts into algebraic equations by using symbols. It was also observed that these students had difficulties in translating the symbolic representations into mathematical texts because of their weak reading comprehension. In addition, finding of this research revealed that students' difficulties in translating the given mathematical texts into symbolic representations or vice versa come from different sources.

Semiempirical equations for modeling solid-state kinetics based on a Maxwell-Boltzmann distribution of activation energies: applications to a polymorphic transformation under crystallization slurry conditions and to the thermal decomposition of AgMnO4 crystals.

PubMed

Skrdla, Peter J; Robertson, Rebecca T

2005-06-02

Many solid-state reactions and phase transformations performed under isothermal conditions give rise to asymmetric, sigmoidally shaped conversion-time (x-t) profiles. The mathematical treatment of such curves, as well as their physical interpretation, is often challenging. In this work, the functional form of a Maxwell-Boltzmann (M-B) distribution is used to describe the distribution of activation energies for the reagent solids, which, when coupled with an integrated first-order rate expression, yields a novel semiempirical equation that may offer better success in the modeling of solid-state kinetics. In this approach, the Arrhenius equation is used to relate the distribution of activation energies to a corresponding distribution of rate constants for the individual molecules in the reagent solids. This distribution of molecular rate constants is then correlated to the (observable) reaction time in the derivation of the model equation. In addition to providing a versatile treatment for asymmetric, sigmoidal reaction curves, another key advantage of our equation over other models is that the start time of conversion is uniquely defined at t = 0. We demonstrate the ability of our simple, two-parameter equation to successfully model the experimental x-t data for the polymorphic transformation of a pharmaceutical compound under crystallization slurry (i.e., heterogeneous) conditions. Additionally, we use a modification of this equation to model the kinetics of a historically significant, homogeneous solid-state reaction: the thermal decomposition of AgMnO4 crystals. The potential broad applicability of our statistical (i.e., dispersive) kinetic approach makes it a potentially attractive alternative to existing models/approaches.

Feynman-Kac equation for anomalous processes with space- and time-dependent forces

NASA Astrophysics Data System (ADS)

Cairoli, Andrea; Baule, Adrian

2017-04-01

Functionals of a stochastic process Y(t) model many physical time-extensive observables, for instance particle positions, local and occupation times or accumulated mechanical work. When Y(t) is a normal diffusive process, their statistics are obtained as the solution of the celebrated Feynman-Kac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic second-order partial differential equations. When Y(t) is non-Brownian, e.g. an anomalous diffusive process, generalizations of the Feynman-Kac equation that incorporate power-law or more general waiting time distributions of the underlying random walk have recently been derived. A general representation of such waiting times is provided in terms of a Lévy process whose Laplace exponent is directly related to the memory kernel appearing in the generalized Feynman-Kac equation. The corresponding anomalous processes have been shown to capture nonlinear mean square displacements exhibiting crossovers between different scaling regimes, which have been observed in numerous experiments on biological systems like migrating cells or diffusing macromolecules in intracellular environments. However, the case where both space- and time-dependent forces drive the dynamics of the generalized anomalous process has not been solved yet. Here, we present the missing derivation of the Feynman-Kac equation in such general case by using the subordination technique. Furthermore, we discuss its extension to functionals explicitly depending on time, which are of particular relevance for the stochastic thermodynamics of anomalous diffusive systems. Exact results on the work fluctuations of a simple non-equilibrium model are obtained. An additional aim of this paper is to provide a pedagogical introduction to Lévy processes, semimartingales and their associated stochastic calculus, which underlie the mathematical formulation of anomalous diffusion as a subordinated process.

Hydrodynamic water impact. [Apollo spacecraft waterlanding

NASA Technical Reports Server (NTRS)

Kettleborough, C. F.

1972-01-01

The hydrodynamic impact of a falling body upon a viscous incompressible fluid was investigated by numerically solving the equations of motion. Initially the mathematical model simulated the axisymmetric impact of a rigid right circular cylinder upon the initially quiescent free surface of a fluid. A compressible air layer exists between the falling cylinder and the liquid free surface. The mathematical model was developed by applying the Navier-Stokes equations to the incompressible air layer and the incompressible fluid. Assuming the flow to be one dimensional within the air layer, the average velocity, pressure and density distributions were calculated. The liquid free surface was allowed to deform as the air pressure acting on it increases. For the liquid the normalized equations were expressed in two-dimensional cylindrical coordinates. The governing equations for the air layer and the liquid were expressed in finite difference form and solved numerically. For the liquid a modified version of the Marker-and-Cell method was used. The mathematical model has been reexamined and a new approach has recently been initiated. Essentially this consists of examining the impact of an inclined plate onto a quiesent water surface with the equations now formulated in cartesian coordinates.

Equivalent model of a dually-fed machine for electric drive control systems

NASA Astrophysics Data System (ADS)

Ostrovlyanchik, I. Yu; Popolzin, I. Yu

2018-05-01

The article shows that the mathematical model of a dually-fed machine is complicated because of the presence of a controlled voltage source in the rotor circuit. As a method of obtaining a mathematical model, the method of a generalized two-phase electric machine is applied and a rotating orthogonal coordinate system is chosen that is associated with the representing vector of a stator current. In the chosen coordinate system in the operator form the differential equations of electric equilibrium for the windings of the generalized machine (the Kirchhoff equation) are written together with the expression for the moment, which determines the electromechanical energy transformation in the machine. Equations are transformed so that they connect the currents of the windings, that determine the moment of the machine, and the voltages on these windings. The structural diagram of the machine is assigned to the written equations. Based on the written equations and accepted assumptions, expressions were obtained for the balancing the EMF of windings, and on the basis of these expressions an equivalent mathematical model of a dually-fed machine is proposed, convenient for use in electric drive control systems.

Building Mathematical Models of Simple Harmonic and Damped Motion.

ERIC Educational Resources Information Center

Edwards, Thomas

1995-01-01

By developing a sequence of mathematical models of harmonic motion, shows that mathematical models are not right or wrong, but instead are better or poorer representations of the problem situation. (MKR)

Fractional Diffusion Equations and Anomalous Diffusion

NASA Astrophysics Data System (ADS)

Evangelista, Luiz Roberto; Kaminski Lenzi, Ervin

2018-01-01

Preface; 1. Mathematical preliminaries; 2. A survey of the fractional calculus; 3. From normal to anomalous diffusion; 4. Fractional diffusion equations: elementary applications; 5. Fractional diffusion equations: surface effects; 6. Fractional nonlinear diffusion equation; 7. Anomalous diffusion: anisotropic case; 8. Fractional Schrödinger equations; 9. Anomalous diffusion and impedance spectroscopy; 10. The Poisson–Nernst–Planck anomalous (PNPA) models; References; Index.

Is it Code Imperfection or 'garbage in Garbage Out'? Outline of Experiences from a Comprehensive Adr Code Verification

NASA Astrophysics Data System (ADS)

Zamani, K.; Bombardelli, F. A.

2013-12-01

ADR equation describes many physical phenomena of interest in the field of water quality in natural streams and groundwater. In many cases such as: density driven flow, multiphase reactive transport, and sediment transport, either one or a number of terms in the ADR equation may become nonlinear. For that reason, numerical tools are the only practical choice to solve these PDEs. All numerical solvers developed for transport equation need to undergo code verification procedure before they are put in to practice. Code verification is a mathematical activity to uncover failures and check for rigorous discretization of PDEs and implementation of initial/boundary conditions. In the context computational PDE verification is not a well-defined procedure on a clear path. Thus, verification tests should be designed and implemented with in-depth knowledge of numerical algorithms and physics of the phenomena as well as mathematical behavior of the solution. Even test results need to be mathematically analyzed to distinguish between an inherent limitation of algorithm and a coding error. Therefore, it is well known that code verification is a state of the art, in which innovative methods and case-based tricks are very common. This study presents full verification of a general transport code. To that end, a complete test suite is designed to probe the ADR solver comprehensively and discover all possible imperfections. In this study we convey our experiences in finding several errors which were not detectable with routine verification techniques. We developed a test suit including hundreds of unit tests and system tests. The test package has gradual increment in complexity such that tests start from simple and increase to the most sophisticated level. Appropriate verification metrics are defined for the required capabilities of the solver as follows: mass conservation, convergence order, capabilities in handling stiff problems, nonnegative concentration, shape preservation, and spurious wiggles. Thereby, we provide objective, quantitative values as opposed to subjective qualitative descriptions as 'weak' or 'satisfactory' agreement with those metrics. We start testing from a simple case of unidirectional advection, then bidirectional advection and tidal flow and build up to nonlinear cases. We design tests to check nonlinearity in velocity, dispersivity and reactions. For all of the mentioned cases we conduct mesh convergence tests. These tests compare the results' order of accuracy versus the formal order of accuracy of discretization. The concealing effect of scales (Peclet and Damkohler numbers) on the mesh convergence study and appropriate remedies are also discussed. For the cases in which the appropriate benchmarks for mesh convergence study are not available we utilize Symmetry, Complete Richardson Extrapolation and Method of False Injection to uncover bugs. Detailed discussions of capabilities of the mentioned code verification techniques are given. Auxiliary subroutines for automation of the test suit and report generation are designed. All in all, the test package is not only a robust tool for code verification but also it provides comprehensive insight on the ADR solvers capabilities. Such information is essential for any rigorous computational modeling of ADR equation for surface/subsurface pollution transport.

Breaking the Equation

ERIC Educational Resources Information Center

Stylianides, Andreas J.

2009-01-01

A proof's potential to promote understanding and conviction is one of the main reasons for which proof is so important for students' learning of mathematics. Unless students realise the limitations of empirical arguments as methods for validating mathematical generalisations, they are unlikely to appreciate the importance of proof in mathematics.…

Representing the Electromagnetic Field: How Maxwell's Mathematics Empowered Faraday's Field Theory

ERIC Educational Resources Information Center

Tweney, Ryan D.

2011-01-01

James Clerk Maxwell "translated" Michael Faraday's experimentally-based field theory into the mathematical representation now known as "Maxwell's Equations." Working with a variety of mathematical representations and physical models Maxwell extended the reach of Faraday's theory and brought it into consistency with other…

Framing the Structural Role of Mathematics in Physics Lectures: A Case Study on Electromagnetism

ERIC Educational Resources Information Center

Karam, Ricardo

2014-01-01

Physics education research has shown that students tend to struggle when trying to use mathematics in a meaningful way in physics (e.g., mathematizing a physical situation or making sense of equations). Concerning the possible reasons for these difficulties, little attention has been paid to the way mathematics is treated in physics instruction.…

A brief survey of symmetry in mathematics

PubMed Central

Mostow, G. D.

1996-01-01

This paper presents a brief survey of the idea of symmetry in mathematics, as exemplified by some particular developments in algebra, differential equations, topology, and number theory. PMID:11607716

A mathematical solution for the parameters of three interfering resonances

NASA Astrophysics Data System (ADS)

Han, X.; Shen, C. P.

2018-04-01

The multiple-solution problem in determining the parameters of three interfering resonances from a fit to an experimentally measured distribution is considered from a mathematical viewpoint. It is shown that there are four numerical solutions for a fit with three coherent Breit-Wigner functions. Although explicit analytical formulae cannot be derived in this case, we provide some constraint equations between the four solutions. For the cases of nonrelativistic and relativistic Breit-Wigner forms of amplitude functions, a numerical method is provided to derive the other solutions from that already obtained, based on the obtained constraint equations. In real experimental measurements with more complicated amplitude forms similar to Breit-Wigner functions, the same method can be deduced and performed to get numerical solutions. The good agreement between the solutions found using this mathematical method and those directly from the fit verifies the correctness of the constraint equations and mathematical methodology used. Supported by National Natural Science Foundation of China (NSFC) (11575017, 11761141009), the Ministry of Science and Technology of China (2015CB856701) and the CAS Center for Excellence in Particle Physics (CCEPP)

Traveling wave solutions of the Boussinesq equation via the new approach of generalized (G'/G)-expansion method.

PubMed

Alam, Md Nur; Akbar, M Ali; Roshid, Harun-Or-

2014-01-01

Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this work, the exact traveling wave solutions of the Boussinesq equation is studied by using the new generalized (G'/G)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G'/G)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations in mathematical physics and engineering. 05.45.Yv, 02.30.Jr, 02.30.Ik.

Applications of the Soave-Redlich-Kwong Equations of State Using Mathematic

NASA Astrophysics Data System (ADS)

Sun, Lanyi; Zhai, Cheng; Zhang, Hui

The application of the Peng-Robinson equations of state (PR EOS) using Matlab and Mathematic has already been demonstrated. In this paper, using Mathematic to solve Soave-Redlich-Kwong (SRK) EOS, as well as the estimation of pure component properties, plotting of vapor-liquid equilibrium (VLE) diagram and calculation of chemical equilibrium, is presented. First the SRK EOS is used to predict several pure-component properties, such as liquid and gas molar volumes for isobutane. The vapor-liquid isobaric diagram is then plotted for a binary mixture composed of n-pentane and n-hexane under the pressures of 2*10^5 and 8*10^5 Pa respectively.

Applying a Conceptual Mini Game for Supporting Simple Mathematical Calculation Skills: Students' Perceptions and Considerations

ERIC Educational Resources Information Center

Panagiotakopoulos, Chris T.

2011-01-01

Mathematics is an area of study that particularly lacks student enthusiasm. Nevertheless, with the help of educational games, any phobias concerning mathematics can be considerably decreased and mathematics can become more appealing. In this study, an educational game addressing mathematics was designed, developed and evaluated by a sample of 33…

Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations

NASA Astrophysics Data System (ADS)

Arqub, Omar Abu; El-Ajou, Ahmad; Momani, Shaher

2015-07-01

Building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics, physics, and engineering. In this work, a new analytical technique for constructing and predicting solitary pattern solutions of time-fractional dispersive partial differential equations is proposed based on the generalized Taylor series formula and residual error function. The new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software. For method evaluation and validation, the proposed technique was applied to three different models and compared with some of the well-known methods. The resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct, as well as the prediction of solitary pattern solutions for time-fractional dispersive partial differential equations.

Stability analysis for a delay differential equations model of a hydraulic turbine speed governor

NASA Astrophysics Data System (ADS)

Halanay, Andrei; Safta, Carmen A.; Dragoi, Constantin; Piraianu, Vlad F.

2017-01-01

The paper aims to study the dynamic behavior of a speed governor for a hydraulic turbine using a mathematical model. The nonlinear mathematical model proposed consists in a system of delay differential equations (DDE) to be compared with already established mathematical models of ordinary differential equations (ODE). A new kind of nonlinearity is introduced as a time delay. The delays can characterize different running conditions of the speed governor. For example, it is considered that spool displacement of hydraulic amplifier might be blocked due to oil impurities in the oil supply system and so the hydraulic amplifier has a time delay in comparison to the time control. Numerical simulations are presented in a comparative manner. A stability analysis of the hydraulic control system is performed, too. Conclusions of the dynamic behavior using the DDE model of a hydraulic turbine speed governor are useful in modeling and controlling hydropower plants.

A Persistent Feature of Multiple Scattering of Waves in the Time-Domain: A Tutorial

NASA Technical Reports Server (NTRS)

Lock, James A.; Mishchenko, Michael I.

2015-01-01

The equations for frequency-domain multiple scattering are derived for a scalar or electromagnetic plane wave incident on a collection of particles at known positions, and in the time-domain for a plane wave pulse incident on the same collection of particles. The calculation is carried out for five different combinations of wave types and particle types of increasing geometrical complexity. The results are used to illustrate and discuss a number of physical and mathematical characteristics of multiple scattering in the frequency- and time-domains. We argue that frequency-domain multiple scattering is a purely mathematical construct since there is no temporal sequencing information in the frequency-domain equations and since the multi-particle path information can be dispelled by writing the equations in another mathematical form. However, multiple scattering becomes a definite physical phenomenon in the time-domain when the collection of particles is illuminated by an appropriately short localized pulse.

Multiphysics Simulation of Welding-Arc and Nozzle-Arc System: Mathematical-Model, Solution-Methodology and Validation

NASA Astrophysics Data System (ADS)

Pawar, Sumedh; Sharma, Atul

2018-01-01

This work presents mathematical model and solution methodology for a multiphysics engineering problem on arc formation during welding and inside a nozzle. A general-purpose commercial CFD solver ANSYS FLUENT 13.0.0 is used in this work. Arc formation involves strongly coupled gas dynamics and electro-dynamics, simulated by solution of coupled Navier-Stoke equations, Maxwell's equations and radiation heat-transfer equation. Validation of the present numerical methodology is demonstrated with an excellent agreement with the published results. The developed mathematical model and the user defined functions (UDFs) are independent of the geometry and are applicable to any system that involves arc-formation, in 2D axisymmetric coordinates system. The high-pressure flow of SF6 gas in the nozzle-arc system resembles arc chamber of SF6 gas circuit breaker; thus, this methodology can be extended to simulate arcing phenomenon during current interruption.

Numerical modelling in biosciences using delay differential equations

NASA Astrophysics Data System (ADS)

Bocharov, Gennadii A.; Rihan, Fathalla A.

2000-12-01

Our principal purposes here are (i) to consider, from the perspective of applied mathematics, models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics, (ii) to review the application of numerical techniques to investigate these models. We show that there are prima facie reasons for using such models: (i) they have a richer mathematical framework (compared with ordinary differential equations) for the analysis of biosystem dynamics, (ii) they display better consistency with the nature of certain biological processes and predictive results. We analyze both the qualitative and quantitative role that delays play in basic time-lag models proposed in population dynamics, epidemiology, physiology, immunology, neural networks and cell kinetics. We then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences, comparing them with those implemented by the bio-modellers.

Chaos and insect ecology

Treesearch

Jesse A. Logan; Fred P. Hain

1990-01-01

Recent advances in applied mathematical analysis have uncovered a fascinating and unexpected dynamical richness that underlies behavior of even the simplest non-linear mathematical models. Due to the complexity of solutions to these non-linear equations, a new mathematical term, chaos, has been coined to describe the resulting dynamics. This term captures the notion...

Effects of Background and School Factors on the Mathematics Achievement.

ERIC Educational Resources Information Center

Papanastasiou, Constantinos

2002-01-01

Using a structural equation model, this study investigated the mathematics achievement of eighth graders in Cyprus enrolled in the year 1994-1995. The model considered two exogenous constructs related to student background and five endogenous constructs. Although attitudes, teaching, and beliefs had direct effect on mathematics outcomes, these…

Leading Undergraduate Research Projects in Mathematical Modeling

ERIC Educational Resources Information Center

Seshaiyer, Padmanabhan

2017-01-01

In this article, we provide some useful perspectives and experiences in mentoring students in undergraduate research (UR) in mathematical modeling using differential equations. To engage students in this topic, we present a systematic approach to the creation of rich problems from real-world phenomena; present mathematical models that are derived…

Effects of General and Broad Cognitive Abilities on Mathematics Achievement

ERIC Educational Resources Information Center

Taub, Gordon E.; Keith, Timothy Z.; Floyd, Randy G.; Mcgrew, Kevin S.

2008-01-01

This study investigated the direct and indirect effects of general intelligence and 7 broad cognitive abilities on mathematics achievement. Structural equation modeling was used to investigate the simultaneous effects of both general and broad cognitive abilities on students' mathematics achievement. A hierarchical model of intelligence derived…

Modelling with Difference Equations Supported by GeoGebra: Exploring the Kepler Problem

ERIC Educational Resources Information Center

Kovacs, Zoltan

2010-01-01

The use of difference and differential equations in the modelling is a topic usually studied by advanced students in mathematics. However difference and differential equations appear in the school curriculum in many direct or hidden ways. Difference equations first enter in the curriculum when studying arithmetic sequences. Moreover Newtonian…

Successfully Transitioning to Linear Equations

ERIC Educational Resources Information Center

Colton, Connie; Smith, Wendy M.

2014-01-01

The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…

Complexity of life via collective mind

NASA Technical Reports Server (NTRS)

Zak, Michail

2004-01-01

e mind is introduced as a set of simple intelligent units (say, neurons, or interacting agents), which can communicate by exchange of information without explicit global control. Incomplete information is compensated by a sequence of random guesses symmetrically distributed around expectations with prescribed variances. Both the expectations and variances are the invariants characterizing the whole class of agents. These invariants are stored as parameters of the collective mind, while they contribute into dynamical formalism of the agents' evolution, and in particular, into the reflective chains of their nested abstract images of the selves and non-selves. The proposed model consists of the system of stochastic differential equations in the Langevin form representing the motor dynamics, and the corresponding Fokker-Planck equation representing the mental dynamics (Motor dynamics describes the motion in physical space, while mental dynamics simulates the evolution of initial errors in terms of the probability density). The main departure of this model from Newtonian and statistical physics is due to a feedback from the mental to the motor dynamics which makes the Fokker-Planck equation nonlinear. Interpretation of this model from mathematical and physical viewpoints, as well as possible interpretation from biological, psychological, and social viewpoints are discussed. The model is illustrated by the dynamics of a dialog.

Temperature-viscosity models reassessed.

PubMed

Peleg, Micha

2017-05-04

The temperature effect on viscosity of liquid and semi-liquid foods has been traditionally described by the Arrhenius equation, a few other mathematical models, and more recently by the WLF and VTF (or VFT) equations. The essence of the Arrhenius equation is that the viscosity is proportional to the absolute temperature's reciprocal and governed by a single parameter, namely, the energy of activation. However, if the absolute temperature in K in the Arrhenius equation is replaced by T + b where both T and the adjustable b are in °C, the result is a two-parameter model, which has superior fit to experimental viscosity-temperature data. This modified version of the Arrhenius equation is also mathematically equal to the WLF and VTF equations, which are known to be equal to each other. Thus, despite their dissimilar appearances all three equations are essentially the same model, and when used to fit experimental temperature-viscosity data render exactly the same very high regression coefficient. It is shown that three new hybrid two-parameter mathematical models, whose formulation bears little resemblance to any of the conventional models, can also have excellent fit with r 2 ∼ 1. This is demonstrated by comparing the various models' regression coefficients to published viscosity-temperature relationships of 40% sucrose solution, soybean oil, and 70°Bx pear juice concentrate at different temperature ranges. Also compared are reconstructed temperature-viscosity curves using parameters calculated directly from 2 or 3 data points and fitted curves obtained by nonlinear regression using a larger number of experimental viscosity measurements.

Secondary School Advanced Mathematics, Chapter 4, Equations, Inequalities, and Radicals, Chapter 5, Circles and Spheres. Teacher's Commentary.

ERIC Educational Resources Information Center

Stanford Univ., CA. School Mathematics Study Group.

This manual was designed for use with the third of five texts in the Secondary School Advanced Mathematics (SSAM) series. Developed for students who have completed the Secondary School Mathematics (SSM) program and wish to continue their studies in mathematics, this series is designed to review, strengthen, and fill gaps in the material covered in…

An analytic performance model of disk arrays and its application

NASA Technical Reports Server (NTRS)

Lee, Edward K.; Katz, Randy H.

1991-01-01

As disk arrays become widely used, tools for understanding and analyzing their performance become increasingly important. In particular, performance models can be invaluable in both configuring and designing disk arrays. Accurate analytic performance models are desirable over other types of models because they can be quickly evaluated, are applicable under a wide range of system and workload parameters, and can be manipulated by a range of mathematical techniques. Unfortunately, analytical performance models of disk arrays are difficult to formulate due to the presence of queuing and fork-join synchronization; a disk array request is broken up into independent disk requests which must all complete to satisfy the original request. We develop, validate, and apply an analytic performance model for disk arrays. We derive simple equations for approximating their utilization, response time, and throughput. We then validate the analytic model via simulation and investigate the accuracy of each approximation used in deriving the analytical model. Finally, we apply the analytical model to derive an equation for the optimal unit of data striping in disk arrays.

Statistical mechanics in the context of special relativity. II.

PubMed

Kaniadakis, G

2005-09-01

The special relativity laws emerge as one-parameter (light speed) generalizations of the corresponding laws of classical physics. These generalizations, imposed by the Lorentz transformations, affect both the definition of the various physical observables (e.g., momentum, energy, etc.), as well as the mathematical apparatus of the theory. Here, following the general lines of [Phys. Rev. E 66, 056125 (2002)], we show that the Lorentz transformations impose also a proper one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits us to construct a coherent and self-consistent relativistic statistical theory, preserving the main features of the ordinary statistical theory, which is recovered in the classical limit. The predicted distribution function is a one-parameter continuous deformation of the classical Maxwell-Boltzmann distribution and has a simple analytic form, showing power law tails in accordance with the experimental evidence. Furthermore, this statistical mechanics can be obtained as the stationary case of a generalized kinetic theory governed by an evolution equation obeying the H theorem and reproducing the Boltzmann equation of the ordinary kinetics in the classical limit.

A porous media theory for characterization of membrane blood oxygenation devices

NASA Astrophysics Data System (ADS)

Sano, Yoshihiko; Adachi, Jun; Nakayama, Akira

2013-07-01

A porous media theory has been proposed to characterize oxygen transport processes associated with membrane blood oxygenation devices. For the first time, a rigorous mathematical procedure based a volume averaging procedure has been presented to derive a complete set of the governing equations for the blood flow field and oxygen concentration field. As a first step towards a complete three-dimensional numerical analysis, one-dimensional steady case is considered to model typical membrane blood oxygenator scenarios, and to validate the derived equations. The relative magnitudes of oxygen transport terms are made clear, introducing a dimensionless parameter which measures the distance the oxygen gas travels to dissolve in the blood as compared with the blood dispersion length. This dimensionless number is found so large that the oxygen diffusion term can be neglected in most cases. A simple linear relationship between the blood flow rate and total oxygen transfer rate is found for oxygenators with sufficiently large membrane surface areas. Comparison of the one-dimensional analytic results and available experimental data reveals the soundness of the present analysis.

High-order asynchrony-tolerant finite difference schemes for partial differential equations

NASA Astrophysics Data System (ADS)

Aditya, Konduri; Donzis, Diego A.

2017-12-01

Synchronizations of processing elements (PEs) in massively parallel simulations, which arise due to communication or load imbalances between PEs, significantly affect the scalability of scientific applications. We have recently proposed a method based on finite-difference schemes to solve partial differential equations in an asynchronous fashion - synchronization between PEs is relaxed at a mathematical level. While standard schemes can maintain their stability in the presence of asynchrony, their accuracy is drastically affected. In this work, we present a general methodology to derive asynchrony-tolerant (AT) finite difference schemes of arbitrary order of accuracy, which can maintain their accuracy when synchronizations are relaxed. We show that there are several choices available in selecting a stencil to derive these schemes and discuss their effect on numerical and computational performance. We provide a simple classification of schemes based on the stencil and derive schemes that are representative of different classes. Their numerical error is rigorously analyzed within a statistical framework to obtain the overall accuracy of the solution. Results from numerical experiments are used to validate the performance of the schemes.

New Approaches to Coding Information using Inverse Scattering Transform

NASA Astrophysics Data System (ADS)

Frumin, L. L.; Gelash, A. A.; Turitsyn, S. K.

2017-06-01

Remarkable mathematical properties of the integrable nonlinear Schrödinger equation (NLSE) can offer advanced solutions for the mitigation of nonlinear signal distortions in optical fiber links. Fundamental optical soliton, continuous, and discrete eigenvalues of the nonlinear spectrum have already been considered for the transmission of information in fiber-optic channels. Here, we propose to apply signal modulation to the kernel of the Gelfand-Levitan-Marchenko equations that offers the advantage of a relatively simple decoder design. First, we describe an approach based on exploiting the general N -soliton solution of the NLSE for simultaneous coding of N symbols involving 4 ×N coding parameters. As a specific elegant subclass of the general schemes, we introduce a soliton orthogonal frequency division multiplexing (SOFDM) method. This method is based on the choice of identical imaginary parts of the N -soliton solution eigenvalues, corresponding to equidistant soliton frequencies, making it similar to the conventional OFDM scheme, thus, allowing for the use of the efficient fast Fourier transform algorithm to recover the data. Then, we demonstrate how to use this new approach to control signal parameters in the case of the continuous spectrum.

A cumulative energy demand indicator (CED), life cycle based, for industrial waste management decision making.

PubMed

Puig, Rita; Fullana-I-Palmer, Pere; Baquero, Grau; Riba, Jordi-Roger; Bala, Alba

2013-12-01

Life cycle thinking is a good approach to be used for environmental decision-support, although the complexity of the Life Cycle Assessment (LCA) studies sometimes prevents their wide use. The purpose of this paper is to show how LCA methodology can be simplified to be more useful for certain applications. In order to improve waste management in Catalonia (Spain), a Cumulative Energy Demand indicator (LCA-based) has been used to obtain four mathematical models to help the government in the decision of preventing or allowing a specific waste from going out of the borders. The conceptual equations and all the subsequent developments and assumptions made to obtain the simplified models are presented. One of the four models is discussed in detail, presenting the final simplified equation to be subsequently used by the government in decision making. The resulting model has been found to be scientifically robust, simple to implement and, above all, fulfilling its purpose: the limitation of waste transport out of Catalonia unless the waste recovery operations are significantly better and justify this transport. Copyright © 2013. Published by Elsevier Ltd.

Rescriptive and Descriptive Gauge Symmetry in Finite-Dimensional Dynamical Systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gurfil, Pini

2007-02-07

Gauge theories in physics constitute a fundamental tool for modeling interactions among electromagnetic, weak and strong forces. They have been used in a myriad of fields, ranging from sub-atomic physics to cosmology. The basic mathematical tool generating the gauge theories is that of symmetry, i.e. a redundancy in the description of the system. Although symmetries have long been recognized as a fundamental tool for solving ordinary differential equations, they have not been formally categorized as gauge theories. In this paper, we show how simple systems described by ordinary differential equations are prone to exhibit gauge symmetry, and discuss a fewmore » practical applications of this approach. In particular, we utilize the notion of gauge symmetry to question some common engineering misconceptions of chaotic and stochastic phenomena, and show that seemingly 'disordered' (deterministic) or 'random' (stochastic) behaviors can be 'ordered'. This brings into play the notion of observation; we show that temporal observations may be misleading when used for chaos detection. From a practical standpoint, we use gauge symmetry to considerably mitigate the numerical truncation error of numerical integrations.« less

The Physics of Decision Making:. Stochastic Differential Equations as Models for Neural Dynamics and Evidence Accumulation in Cortical Circuits

NASA Astrophysics Data System (ADS)

Holmes, Philip; Eckhoff, Philip; Wong-Lin, K. F.; Bogacz, Rafal; Zacksenhouse, Miriam; Cohen, Jonathan D.

2010-03-01

We describe how drift-diffusion (DD) processes - systems familiar in physics - can be used to model evidence accumulation and decision-making in two-alternative, forced choice tasks. We sketch the derivation of these stochastic differential equations from biophysically-detailed models of spiking neurons. DD processes are also continuum limits of the sequential probability ratio test and are therefore optimal in the sense that they deliver decisions of specified accuracy in the shortest possible time. This leaves open the critical balance of accuracy and speed. Using the DD model, we derive a speed-accuracy tradeoff that optimizes reward rate for a simple perceptual decision task, compare human performance with this benchmark, and discuss possible reasons for prevalent sub-optimality, focussing on the question of uncertain estimates of key parameters. We present an alternative theory of robust decisions that allows for uncertainty, and show that its predictions provide better fits to experimental data than a more prevalent account that emphasises a commitment to accuracy. The article illustrates how mathematical models can illuminate the neural basis of cognitive processes.

Burton-Miller-type singular boundary method for acoustic radiation and scattering

NASA Astrophysics Data System (ADS)

Fu, Zhuo-Jia; Chen, Wen; Gu, Yan

2014-08-01

This paper proposes the singular boundary method (SBM) in conjunction with Burton and Miller's formulation for acoustic radiation and scattering. The SBM is a strong-form collocation boundary discretization technique using the singular fundamental solutions, which is mathematically simple, easy-to-program, meshless and introduces the concept of source intensity factors (SIFs) to eliminate the singularities of the fundamental solutions. Therefore, it avoids singular numerical integrals in the boundary element method (BEM) and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions (MFS). In the present method, we derive the SIFs of exterior Helmholtz equation by means of the SIFs of exterior Laplace equation owing to the same order of singularities between the Laplace and Helmholtz fundamental solutions. In conjunction with the Burton-Miller formulation, the SBM enhances the quality of the solution, particularly in the vicinity of the corresponding interior eigenfrequencies. Numerical illustrations demonstrate efficiency and accuracy of the present scheme on some benchmark examples under 2D and 3D unbounded domains in comparison with the analytical solutions, the boundary element solutions and Dirichlet-to-Neumann finite element solutions.

Adsorption of trichloroethylene and benzene vapors onto hypercrosslinked polymeric resin.

PubMed

Liu, Peng; Long, Chao; Li, Qifen; Qian, Hongming; Li, Aimin; Zhang, Quanxing

2009-07-15

In this research, the adsorption equilibria of trichloroethylene (TCE) and benzene vapors onto hypercrosslinked polymeric resin (NDA201) were investigated by the column adsorption method in the temperature range from 303 to 333 K and pressures up to 8 kPa for TCE, 12 kPa for benzene. The Toth and Dubinin-Astakov (D-A) equations were tested to correlate experimental isotherms, and the experimental data were found to fit well by them. The good fits and characteristic curves of D-A equation provided evidence that a pore-filling phenomenon was involved during the adsorption of TCE and benzene onto NDA-201. Moreover, thermodynamic properties such as the Henry's constant and the isosteric enthalpy of adsorption were calculated. The isosteric enthalpy curves varied with the surface loading for each adsorbate, indicating that the hypercrosslinked polymeric resin has an energetically heterogeneous surface. In addition, a simple mathematic model developed by Yoon and Nelson was applied to investigate the breakthrough behavior on a hypercrosslinked polymeric resin column at 303 K and the calculated breakthrough curves were in high agreement with corresponding experimental data.

ABSIM. Simulation of Absorption Systems in Flexible and Modular Form

DOE Office of Scientific and Technical Information (OSTI.GOV)

Grossman, G.

1994-06-01

The computer code has been developed for simulation of absorption systems at steady-state in a flexible and modular form, making it possible to investigate various cycle configurations with different working fluids. The code is based on unit subroutines containing the governing equations for the system`s components. When all the equations have been established, a mathematical solver routine is employed to solve them simultaneously. Property subroutines contained in a separate data base serve to provide thermodynamic properties of the working fluids. The code is user-oriented and requires a relatively simple input containing the given operating conditions and the working fluid atmore » each state point. the user conveys to the computer an image of the cycle by specifying the different components and their interconnections. Based on this information, the program calculates the temperature, flowrate, concentration, pressure and vapor fraction at each state point in the system and the heat duty at each unit, from which the coefficient of performance may be determined. A graphical user-interface is provided to facilitate interactive input and study of the output.« less

ABSIM. Simulation of Absorption Systems in Flexible and Modular Form

DOE Office of Scientific and Technical Information (OSTI.GOV)

Grossman, G.

1994-06-01

The computer code has been developed for simulation of absorption systems at steady-state in a flexible and modular form, making it possible to investigate various cycle configurations with different working fluids. The code is based on unit subroutines containing the governing equations for the system's components. When all the equations have been established, a mathematical solver routine is employed to solve them simultaneously. Property subroutines contained in a separate data base serve to provide thermodynamic properties of the working fluids. The code is user-oriented and requires a relatively simple input containing the given operating conditions and the working fluid atmore » each state point. the user conveys to the computer an imagev of the cycle by specifying the different components and their interconnections. Based on this information, the program calculates the temperature, flowrate, concentration, pressure and vapor fraction at each state point in the system and the heat duty at each unit, from which the coefficient of performance may be determined. A graphical user-interface is provided to fcilitate interactive input and study of the output.« less

Techniques for estimating selected streamflow characteristics of rural unregulated streams in Ohio

USGS Publications Warehouse

Koltun, G.F.; Whitehead, Matthew T.

2002-01-01

This report provides equations for estimating mean annual streamflow, mean monthly streamflows, harmonic mean streamflow, and streamflow quartiles (the 25th-, 50th-, and 75th-percentile streamflows) as a function of selected basin characteristics for rural, unregulated streams in Ohio. The equations were developed from streamflow statistics and basin-characteristics data for as many as 219 active or discontinued streamflow-gaging stations on rural, unregulated streams in Ohio with 10 or more years of homogenous daily streamflow record. Streamflow statistics and basin-characteristics data for the 219 stations are presented in this report. Simple equations (based on drainage area only) and best-fit equations (based on drainage area and at least two other basin characteristics) were developed by means of ordinary least-squares regression techniques. Application of the best-fit equations generally involves quantification of basin characteristics that require or are facilitated by use of a geographic information system. In contrast, the simple equations can be used with information that can be obtained without use of a geographic information system; however, the simple equations have larger prediction errors than the best-fit equations and exhibit geographic biases for most streamflow statistics. The best-fit equations should be used instead of the simple equations whenever possible.

Anticancer Agents Based on a New Class of Transition- State Analog Inhibitors for Serine and Cysteine Proteases

DTIC Science & Technology

1999-08-01

electrostatic repulsion between the het- eroatom and the ketone. Swain and Lupton31 have constructed a modified Hammett equation (eq 2) in which they...determined by nonlinear fit to the Michaelis-Menten equation for competitive inhibition using simple weighing. Competitive inhibition was confirmed... equation for competitive inhibition using simple weighing. Competitive inhibition was confirmed by Lineweaver - Burk analysis using simple

Heat Source/Sink in a Magneto-Hydrodynamic Non-Newtonian Fluid Flow in a Porous Medium: Dual Solutions

PubMed Central

Hayat, Tasawar; Awais, Muhammad; Imtiaz, Amna

2016-01-01

This communication deals with the properties of heat source/sink in a magneto-hydrodynamic flow of a non-Newtonian fluid immersed in a porous medium. Shrinking phenomenon along with the permeability of the wall is considered. Mathematical modelling is performed to convert the considered physical process into set of coupled nonlinear mathematical equations. Suitable transformations are invoked to convert the set of partial differential equations into nonlinear ordinary differential equations which are tackled numerically for the solution computations. It is noted that dual solutions for various physical parameters exist which are analyzed in detail. PMID:27598314

New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified KdV-Zakharov-Kuznetsov equations and their applications

NASA Astrophysics Data System (ADS)

Lu, Dianchen; Seadawy, A. R.; Arshad, M.; Wang, Jun

In this paper, new exact solitary wave, soliton and elliptic function solutions are constructed in various forms of three dimensional nonlinear partial differential equations (PDEs) in mathematical physics by utilizing modified extended direct algebraic method. Soliton solutions in different forms such as bell and anti-bell periodic, dark soliton, bright soliton, bright and dark solitary wave in periodic form etc are obtained, which have large applications in different branches of physics and other areas of applied sciences. The obtained solutions are also presented graphically. Furthermore, many other nonlinear evolution equations arising in mathematical physics and engineering can also be solved by this powerful, reliable and capable method. The nonlinear three dimensional extended Zakharov-Kuznetsov dynamica equation and (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov equation are selected to show the reliability and effectiveness of the current method.

Mathematical nonlinear optics

NASA Astrophysics Data System (ADS)

McLaughlin, David W.

1995-08-01

The principal investigator, together with a post-doctoral fellows Tetsuji Ueda and Xiao Wang, several graduate students, and colleagues, has applied the modern mathematical theory of nonlinear waves to problems in nonlinear optics and to equations directly relevant to nonlinear optics. Projects included the interaction of laser light with nematic liquid crystals and chaotic, homoclinic, small dispersive, and random behavior of solutions of the nonlinear Schroedinger equation. In project 1, the extremely strong nonlinear response of a continuous wave laser beam in a nematic liquid crystal medium has produced striking undulation and filamentation of the laser beam which has been observed experimentally and explained theoretically. In project 2, qualitative properties of the nonlinear Schroedinger equation (which is the fundamental equation for nonlinear optics) have been identified and studied. These properties include optical shocking behavior in the limit of very small dispersion, chaotic and homoclinic behavior in discretizations of the partial differential equation, and random behavior.

Toward a Nonlinear Acoustic Analogy: Turbulence as a Source of Sound and Nonlinear Propagation

NASA Technical Reports Server (NTRS)

Miller, Steven A. E.

2015-01-01

An acoustic analogy is proposed that directly includes nonlinear propagation effects. We examine the Lighthill acoustic analogy and replace the Green's function of the wave equation with numerical solutions of the generalized Burgers' equation. This is justified mathematically by using similar arguments that are the basis of the solution of the Lighthill acoustic analogy. This approach is superior to alternatives because propagation is accounted for directly from the source to the far-field observer instead of from an arbitrary intermediate point. Validation of a numerical solver for the generalized Burgers' equation is performed by comparing solutions with the Blackstock bridging function and measurement data. Most importantly, the mathematical relationship between the Navier- Stokes equations, the acoustic analogy that describes the source, and canonical nonlinear propagation equations is shown. Example predictions are presented for nonlinear propagation of jet mixing noise at the sideline angle

Toward a Nonlinear Acoustic Analogy: Turbulence as a Source of Sound and Nonlinear Propagation

NASA Technical Reports Server (NTRS)

Miller, Steven A. E.

2015-01-01

An acoustic analogy is proposed that directly includes nonlinear propagation effects. We examine the Lighthill acoustic analogy and replace the Green's function of the wave equation with numerical solutions of the generalized Burgers' equation. This is justified mathematically by using similar arguments that are the basis of the solution of the Lighthill acoustic analogy. This approach is superior to alternatives because propagation is accounted for directly from the source to the far-field observer instead of from an arbitrary intermediate point. Validation of a numerical solver for the generalized Burgers' equation is performed by comparing solutions with the Blackstock bridging function and measurement data. Most importantly, the mathematical relationship between the Navier-Stokes equations, the acoustic analogy that describes the source, and canonical nonlinear propagation equations is shown. Example predictions are presented for nonlinear propagation of jet mixing noise at the sideline angle.

Mathematical Model of the Jet Engine Fuel System

NASA Astrophysics Data System (ADS)

Klimko, Marek

2015-05-01

The paper discusses the design of a simplified mathematical model of the jet (turbo-compressor) engine fuel system. The solution will be based on the regulation law, where the control parameter is a fuel mass flow rate and the regulated parameter is the rotational speed. A differential equation of the jet engine and also differential equations of other fuel system components (fuel pump, throttle valve, pressure regulator) will be described, with respect to advanced predetermined simplifications.

Multirate Integration Properties of Waveform Relaxation with Applications to Circuit Simulation and Parallel Computation

DTIC Science & Technology

1985-11-18

Greenberg and K. Sakallah at Digital Equipment Corporation, and C-F. Chen, L Nagel, and P. ,. Subrahmanyam at AT&T Bell Laboratories, both for providing...Circuit Theory McGraw-Hill, 1969. [37] R. Courant and D. Hilbert , Partial Differential Equations, Vol. 2 of Methods of Mathematical Physics...McGraw-Hill, N.Y., 1965. Page 161 [44) R. Courant and D. Hilbert , Partial Differential Equations, Vol. 2 of Methods of Mathematical Physics

Some remarks on quantum physics, stochastic processes, and nonlinear filtering theory

NASA Astrophysics Data System (ADS)

Balaji, Bhashyam

2016-05-01

The mathematical similarities between quantum mechanics and stochastic processes has been studied in the literature. Some of the major results are reviewed, such as the relationship between the Fokker-Planck equation and the Schrödinger equation. Also reviewed are more recent results that show the mathematical similarities between quantum many particle systems and concepts in other areas of applied science, such as stochastic Petri nets. Some connections to filtering theory are discussed.

Direct integration of the inverse Radon equation for X-ray computed tomography.

PubMed

Libin, E E; Chakhlov, S V; Trinca, D

2016-11-22

A new mathematical appoach using the inverse Radon equation for restoration of images in problems of linear two-dimensional x-ray tomography is formulated. In this approach, Fourier transformation is not used, and it gives the chance to create the practical computing algorithms having more reliable mathematical substantiation. Results of software implementation show that for especially for low number of projections, the described approach performs better than standard X-ray tomographic reconstruction algorithms.

Optimization of Thermal Object Nonlinear Control Systems by Energy Efficiency Criterion.

NASA Astrophysics Data System (ADS)

Velichkin, Vladimir A.; Zavyalov, Vladimir A.

2018-03-01

This article presents the results of thermal object functioning control analysis (heat exchanger, dryer, heat treatment chamber, etc.). The results were used to determine a mathematical model of the generalized thermal control object. The appropriate optimality criterion was chosen to make the control more energy-efficient. The mathematical programming task was formulated based on the chosen optimality criterion, control object mathematical model and technological constraints. The “maximum energy efficiency” criterion helped avoid solving a system of nonlinear differential equations and solve the formulated problem of mathematical programming in an analytical way. It should be noted that in the case under review the search for optimal control and optimal trajectory reduces to solving an algebraic system of equations. In addition, it is shown that the optimal trajectory does not depend on the dynamic characteristics of the control object.

On space-based SETI

NASA Technical Reports Server (NTRS)

Stuiver, Willem

1990-01-01

Space-based antenna systems for the search of signals from extra-terrestrial intelligence are discussed. Independent studies of the ecliptic solar-sailing transfer problem from the geosynchronous departure orbit to Sun-Earth collinear transterrestrial liberation point were conducted. They were based on a relatively simple mathematical model describing attitude-controlled spacecraft motion in the ecliptic plane as governed by solar and terrestrial gravitational attraction together with the solar radiation pressure. The resulting equations of motion were integrated numerically for a relevant range of values of spacecraft area-to-mass ratio and for an appropriate spacecraft attitude-control law known to lead to Earth escape. Experimentation with varying initial conditions in the departure orbit, and with attitude-control law modification after having achieved Earth escape, established the feasibility of component deployment by means of solar sailing. Details are presented.

Search times and probability of detection in time-limited search

NASA Astrophysics Data System (ADS)

Wilson, David; Devitt, Nicole; Maurer, Tana

2005-05-01

When modeling the search and target acquisition process, probability of detection as a function of time is important to war games and physical entity simulations. Recent US Army RDECOM CERDEC Night Vision and Electronics Sensor Directorate modeling of search and detection has focused on time-limited search. Developing the relationship between detection probability and time of search as a differential equation is explored. One of the parameters in the current formula for probability of detection in time-limited search corresponds to the mean time to detect in time-unlimited search. However, the mean time to detect in time-limited search is shorter than the mean time to detect in time-unlimited search and the relationship between them is a mathematical relationship between these two mean times. This simple relationship is derived.

Investigation of a New Flux-Modulated Permanent Magnet Brushless Motor for EVs

PubMed Central

Gu, Lingling; Luo, Yong; Han, Xuedong

2014-01-01

This paper presents a flux-modulated direct drive (FMDD) motor. The key is to integrate the magnetic gear with the PM motor while removing the gear inner-rotor. Hence, the proposed FMDD motor can achieve the low-speed high-torque output and high-speed compact design requirements as well as high-torque density with a simple structure. The output power equation is analytically derived. By using finite element analysis (FEA), the static characteristics of the proposed motor are obtained. Based on these characteristics, the system mathematical model can be established. Hence, the evaluation of system performances is conducted by computer simulation using the Matlab/Simulink. A prototype is designed and built for experimentation. Experimental results are given to verify the theoretical analysis and simulation. PMID:24883405

Autonomous propulsion of nanorods trapped in an acoustic field

NASA Astrophysics Data System (ADS)

Sader, John; Collis, Jesse; Chakraborty, Debadi

2017-11-01

Recent measurements demonstrate that nanorods trapped in acoustic fields generate autonomous propulsion, with their direction and speed controlled by both the particle's shape and density distribution. In this talk, we investigate the physical mechanisms underlying this combined density/shape induced phenomenon by developing a simple yet rigorous mathematical framework for arbitrary axisymmetric particles. This only requires solution of the (linear) unsteady Stokes equations. Geometric and density asymmetries in the particle generate axial jets that can produce motion in either direction. Strikingly, the propulsion direction is found to reverse with increasing frequency, an effect that is yet to be reported experimentally. The general theory and mechanism described here enable the a priori design and fabrication of nano-motors in fluid for transport of small-scale payloads and robotic applications.

Optimal design of reverse osmosis module networks

DOE Office of Scientific and Technical Information (OSTI.GOV)

Maskan, F.; Wiley, D.E.; Johnston, L.P.M.

2000-05-01

The structure of individual reverse osmosis modules, the configuration of the module network, and the operating conditions were optimized for seawater and brackish water desalination. The system model included simple mathematical equations to predict the performance of the reverse osmosis modules. The optimization problem was formulated as a constrained multivariable nonlinear optimization. The objective function was the annual profit for the system, consisting of the profit obtained from the permeate, capital cost for the process units, and operating costs associated with energy consumption and maintenance. Optimization of several dual-stage reverse osmosis systems were investigated and compared. It was found thatmore » optimal network designs are the ones that produce the most permeate. It may be possible to achieve economic improvements by refining current membrane module designs and their operating pressures.« less

Insertion device calculations with mathematica

DOE Office of Scientific and Technical Information (OSTI.GOV)

Carr, R.; Lidia, S.

1995-02-01

The design of accelerator insertion devices such as wigglers and undulators has usually been aided by numerical modeling on digital computers, using code in high level languages like Fortran. In the present era, there are higher level programming environments like IDL{reg_sign}, MatLab{reg_sign}, and Mathematica{reg_sign} in which these calculations may be performed by writing much less code, and in which standard mathematical techniques are very easily used. The authors present a suite of standard insertion device modeling routines in Mathematica to illustrate the new techniques. These routines include a simple way to generate magnetic fields using blocks of CSEM materials, trajectorymore » solutions from the Lorentz force equations for given magnetic fields, Bessel function calculations of radiation for wigglers and undulators and general radiation calculations for undulators.« less

Investigation of a new flux-modulated permanent magnet brushless motor for EVs.

PubMed

Fan, Ying; Gu, Lingling; Luo, Yong; Han, Xuedong; Cheng, Ming

2014-01-01

This paper presents a flux-modulated direct drive (FMDD) motor. The key is to integrate the magnetic gear with the PM motor while removing the gear inner-rotor. Hence, the proposed FMDD motor can achieve the low-speed high-torque output and high-speed compact design requirements as well as high-torque density with a simple structure. The output power equation is analytically derived. By using finite element analysis (FEA), the static characteristics of the proposed motor are obtained. Based on these characteristics, the system mathematical model can be established. Hence, the evaluation of system performances is conducted by computer simulation using the Matlab/Simulink. A prototype is designed and built for experimentation. Experimental results are given to verify the theoretical analysis and simulation.

Structure preserving noise and dissipation in the Toda lattice

NASA Astrophysics Data System (ADS)

Arnaudon, Alexis

2018-05-01

In this paper, we use Flaschka’s change of variables of the open Toda lattice and its interpretation in terms of the group structure of the LU factorisation as a coadjoint motion on a certain dual of the Lie algebra to implement a structure preserving noise and dissipation. Both preserve the structure of the coadjoint orbit, that is the space of symmetric tri-diagonal matrices and arise as a new type of multiplicative noise and nonlinear dissipation of the Toda lattice. We investigate some of the properties of these deformations and, in particular, the continuum limit as a stochastic Burger equation with a nonlinear viscosity. This work is meant to be exploratory, and open more questions that we can answer with simple mathematical tools and without numerical simulations.

Modeling the Transmission of Piscirickettsia salmonis in Farmed Salmon

NASA Astrophysics Data System (ADS)

Cisternas, Jaime; Moreno, Adolfo

2007-05-01

Farming Atlantic salmon is an economic activity of growing relevance in the southern regions of Chile. The need to increase efficiency and reach production goals, as well as restrictions on the use of water resources, had led in recent years to certain practices that proved prone to bacterial infections among the fish. Our study focuses on the impact of rickettsial bacteria in farmed salmon, and the possibility of controlling its incidence once it is established along the salmon life cicle. We used compartmental models to separate fish in their maturation stages and health status. The mathematical analysis will involve differential equations with and without delays, and linear stability principles. Our goal was to build a simple model that explains the basic mechanisms at work and provides predictions on the outcome of different control strategies.

Building Conceptual Understanding in a Remedial College Mathematics Classroom: A Study of Effectiveness

ERIC Educational Resources Information Center

Bachman, Rachel Marie

2013-01-01

This study investigated the effectiveness of two remedial mathematics courses that aimed to (a) present topics conceptually, (b) construct adequate schemata, and (c) introduce students to the culture of mathematics. The topics covered during the two courses were word problems, equivalence, variables and expressions, equations and inequalities, and…

Equations and Inequalities: Making Mathematics Accessible to All. PISA

ERIC Educational Resources Information Center

Piacentini, Mario; Monticone, Chiara

2016-01-01

More than ever, students need to engage with mathematics concepts, think quantitatively and analytically, and communicate using mathematics. All these skills are central to a young person's preparedness to tackle problems that arise at work and in life beyond the classroom. But the reality is that many students are not familiar with basic…

Explorations in Elementary Mathematical Modeling

ERIC Educational Resources Information Center

Shahin, Mazen

2010-01-01

In this paper we will present the methodology and pedagogy of Elementary Mathematical Modeling as a one-semester course in the liberal arts core. We will focus on the elementary models in finance and business. The main mathematical tools in this course are the difference equations and matrix algebra. We also integrate computer technology and…

A Conversion Tool for Mathematical Expressions in Web XML Files.

ERIC Educational Resources Information Center

Ohtake, Nobuyuki; Kanahori, Toshihiro

2003-01-01

This article discusses the conversion of mathematical equations into Extensible Markup Language (XML) on the World Wide Web for individuals with visual impairments. A program is described that converts the presentation markup style to the content markup style in MathML to allow browsers to render mathematical expressions without other programs.…

Creating Printed Materials for Mathematics with a Macintosh Computer.

ERIC Educational Resources Information Center

Mahler, Philip

This document gives instructions on how to use a Macintosh computer to create printed materials for mathematics. A Macintosh computer, Microsoft Word, and objected-oriented (Draw-type) art program, and a function-graphing program are capable of producing high quality printed instructional materials for mathematics. Word 5.1 has an equation editor…

Mathematical form models of tree trunks

Treesearch

Rudolfs Ozolins

2000-01-01

Assortment structure analysis of tree trunks is a characteristic and proper problem that can be solved by using mathematical modeling and standard computer programs. Mathematical form model of tree trunks consists of tapering curve equations and their parameters. Parameters for nine species were obtained by processing measurements of 2,794 model trees and studying the...

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

NASA Astrophysics Data System (ADS)

Peshkov, Ilya; Pavelka, Michal; Romenski, Evgeniy; Grmela, Miroslav

2018-01-01

Continuum mechanics with dislocations, with the Cattaneo-type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov-type system of the first-order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).

Measurement Uncertainty Budget of the PMV Thermal Comfort Equation

NASA Astrophysics Data System (ADS)

Ekici, Can

2016-05-01

Fanger's predicted mean vote (PMV) equation is the result of the combined quantitative effects of the air temperature, mean radiant temperature, air velocity, humidity activity level and clothing thermal resistance. PMV is a mathematical model of thermal comfort which was developed by Fanger. The uncertainty budget of the PMV equation was developed according to GUM in this study. An example is given for the uncertainty model of PMV in the exemplification section of the study. Sensitivity coefficients were derived from the PMV equation. Uncertainty budgets can be seen in the tables. A mathematical model of the sensitivity coefficients of Ta, hc, T_{mrt}, T_{cl}, and Pa is given in this study. And the uncertainty budgets for hc, T_{cl}, and Pa are given in this study.

Mathematical Gossip: Relevance and Context in the Mathematics Classroom

ERIC Educational Resources Information Center

Callingham, Rosemary

2004-01-01

Using mathematical gossip in the classroom allows teachers to expand their students' horizons, and provide pathways to improvement of understanding. The expansion of a simple idea into another mathematical context can enrich a student's learning. In particular it may help to bridge the gap between purely procedural approaches and a conceptual…

Numerical Modeling of Saturated Boiling in a Heated Tube

NASA Technical Reports Server (NTRS)

Majumdar, Alok; LeClair, Andre; Hartwig, Jason

2017-01-01

This paper describes a mathematical formulation and numerical solution of boiling in a heated tube. The mathematical formulation involves a discretization of the tube into a flow network consisting of fluid nodes and branches and a thermal network consisting of solid nodes and conductors. In the fluid network, the mass, momentum and energy conservation equations are solved and in the thermal network, the energy conservation equation of solids is solved. A pressure-based, finite-volume formulation has been used to solve the equations in the fluid network. The system of equations is solved by a hybrid numerical scheme which solves the mass and momentum conservation equations by a simultaneous Newton-Raphson method and the energy conservation equation by a successive substitution method. The fluid network and thermal network are coupled through heat transfer between the solid and fluid nodes which is computed by Chen's correlation of saturated boiling heat transfer. The computer model is developed using the Generalized Fluid System Simulation Program and the numerical predictions are compared with test data.

A consistent modelling methodology for secondary settling tanks in wastewater treatment.

PubMed

Bürger, Raimund; Diehl, Stefan; Nopens, Ingmar

2011-03-01

The aim of this contribution is partly to build consensus on a consistent modelling methodology (CMM) of complex real processes in wastewater treatment by combining classical concepts with results from applied mathematics, and partly to apply it to the clarification-thickening process in the secondary settling tank. In the CMM, the real process should be approximated by a mathematical model (process model; ordinary or partial differential equation (ODE or PDE)), which in turn is approximated by a simulation model (numerical method) implemented on a computer. These steps have often not been carried out in a correct way. The secondary settling tank was chosen as a case since this is one of the most complex processes in a wastewater treatment plant and simulation models developed decades ago have no guarantee of satisfying fundamental mathematical and physical properties. Nevertheless, such methods are still used in commercial tools to date. This particularly becomes of interest as the state-of-the-art practice is moving towards plant-wide modelling. Then all submodels interact and errors propagate through the model and severely hamper any calibration effort and, hence, the predictive purpose of the model. The CMM is described by applying it first to a simple conversion process in the biological reactor yielding an ODE solver, and then to the solid-liquid separation in the secondary settling tank, yielding a PDE solver. Time has come to incorporate established mathematical techniques into environmental engineering, and wastewater treatment modelling in particular, and to use proven reliable and consistent simulation models. Copyright © 2011 Elsevier Ltd. All rights reserved.

Separation of Variables and Superintegrability; The symmetry of solvable systems

NASA Astrophysics Data System (ADS)

Kalnins, Ernest G.; Kress, Jonathan M.; Miller, Willard, Jr.

2018-06-01

Separation of variables methods for solving partial differential equations are of immense theoretical and practical importance in mathematical physics. They are the most powerful tool known for obtaining explicit solutions of the partial differential equations of mathematical physics. The purpose of this book is to give an up-to-date presentation of the theory of separation of variables and its relation to superintegrability. Collating and presenting it in a unified, updated and a more accessible manner, the results scattered in the literature that the authors have prepared is an invaluable resource for mathematicians and mathematical physicists in particular, as well as science, engineering, geological and biological researchers interested in explicit solutions.

Nonlinear Dynamics of Biofilm Growth on Sediment Surfaces

NASA Astrophysics Data System (ADS)

Molz, F. J.; Murdoch, L. C.; Faybishenko, B.

2013-12-01

Bioclogging often begins with the establishment of small colonies (microcolonies), which then form biofilms on the surfaces of a porous medium. These biofilm-porous media surfaces are not simple coatings of single microbes, but complex assemblages of cooperative and competing microbes, interacting with their chemical environment. This leads one to ask: what are the underlying dynamics involved with biofilm growth? To begin answering this question, we have extended the work of Kot et al. (1992, Bull. Mathematical Bio.) from a fully mixed chemostat to an idealized, one-dimensional, biofilm environment, taking into account a simple predator-prey microbial competition, with the prey feeding on a specified food source. With a variable (periodic) food source, Kot et al. (1992) were able to demonstrate chaotic dynamics in the coupled substrate-prey-predator system. Initially, deterministic chaos was thought by many to be mainly a mathematical phenomenon. However, several recent publications (e.g., Becks et al, 2005, Nature Letters; Graham et al. 2007, Int. Soc Microb. Eco. J.; Beninca et al., 2008, Nature Letters; Saleh, 2011, IJBAS) have brought together, using experimental studies and relevant mathematics, a breakthrough discovery that deterministic chaos is present in relatively simple biochemical systems. Two of us (Faybishenko and Molz, 2013, Procedia Environ. Sci)) have numerically analyzed a mathematical model of rhizosphere dynamics (Kravchenko et al., 2004, Microbiology) and detected patterns of nonlinear dynamical interactions supporting evidence of synchronized synergetic oscillations of microbial populations, carbon and oxygen concentrations driven by root exudation into a fully mixed system. In this study, we have extended the application of the Kot et al. model to investigate a spatially-dependent biofilm system. We will present the results of numerical simulations obtained using COMSOL Multi-Physics software, which we used to determine the nature of the complex dynamics. We found that complex dynamics occur even with a constant food supply maintained at the upstream boundary of the biofilm. Results will be presented along with a description of the model, including 3 coupled partial differential equations and examples of the localized and propagating nonlinear dynamics inherent in the system. Contrary to a common opinion that chaos in many mechanical systems is a type of instability, appearing when energy is added, we hypothesize, based on the results of our modeling, that chaos in biofilm dynamics and other microbial ecosystems is driven by a competitive decrease in the food supply (i.e., chemical energy). We also hypothesize that, somewhat paradoxically, this, in turn, may support a long-term system stability that could cause bioclogging in porous media.

Historical Notes. A Madness in the Methods. Cubic and Quartic Equations: Are the General Solving Techniques Still Important?

ERIC Educational Resources Information Center

Francis, Richard L.

1991-01-01

Described is an outline for a school mathematics project dealing with the theory of equations, specifically solutions to polynomials of the third and of the fourth degree. Cardano's method for solution of cubic equations and Ferrari's method for solution of quartic equations are included with examples. (JJK)

Secondary School Advanced Mathematics, Chapter 6, The Complex Number System, Chapter 7, Equations of the First and Second Degree in Two Variables. Teacher's Commentary.

ERIC Educational Resources Information Center

Stanford Univ., CA. School Mathematics Study Group.

This manual was designed for use with the fourth of five texts in the Secondary School Advanced Mathematics (SSAM) series. Developed for students who have completed the Secondary School Mathematics (SSM) program and wish to continue their studies in mathematics, this series is designed to review, strengthen, and fill gaps in the material covered…

Parameter Estimation for Simultaneous Saccharification and Fermentation of Food Waste Into Ethanol Using Matlab Simulink

NASA Astrophysics Data System (ADS)

Davis, Rebecca Anne

The increase in waste disposal and energy costs has provided an incentive to convert carbohydrate-rich food waste streams into fuel. For example, dining halls and restaurants discard foods that require tipping fees for removal. An effective use of food waste may be the enzymatic hydrolysis of the waste to simple sugars and fermentation of the sugars to ethanol. As these wastes have complex compositions which may change day-to-day, experiments were carried out to test fermentability of two different types of food waste at 27° C using Saccharomyces cerevisiae yeast (ATCC4124) and Genencor's STARGEN™ enzyme in batch simultaneous saccharification and fermentation (SSF) experiments. A mathematical model of SSF based on experimentally matched rate equations for enzyme hydrolysis and yeast fermentation was developed in Matlab Simulink®. Using Simulink® parameter estimation 1.1.3, parameters for hydrolysis and fermentation were estimated through modified Michaelis-Menten and Monod-type equations with the aim of predicting changes in the levels of ethanol and glycerol from different initial concentrations of glucose, fructose, maltose, and starch. The model predictions and experimental observations agree reasonably well for the two food waste streams and a third validation dataset. The approach of using Simulink® as a dynamic visual model for SSF represents a simple method which can be applied to a variety of biological pathways and may be very useful for systems approaches in metabolic engineering in the future.

Highly accurate symplectic element based on two variational principles

NASA Astrophysics Data System (ADS)

Qing, Guanghui; Tian, Jia

2018-02-01

For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element (NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses.

Measurement-based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry

NASA Astrophysics Data System (ADS)

Xu, Peiliang

2018-06-01

The numerical integration method has been routinely used by major institutions worldwide, for example, NASA Goddard Space Flight Center and German Research Center for Geosciences (GFZ), to produce global gravitational models from satellite tracking measurements of CHAMP and/or GRACE types. Such Earth's gravitational products have found widest possible multidisciplinary applications in Earth Sciences. The method is essentially implemented by solving the differential equations of the partial derivatives of the orbit of a satellite with respect to the unknown harmonic coefficients under the conditions of zero initial values. From the mathematical and statistical point of view, satellite gravimetry from satellite tracking is essentially the problem of estimating unknown parameters in the Newton's nonlinear differential equations from satellite tracking measurements. We prove that zero initial values for the partial derivatives are incorrect mathematically and not permitted physically. The numerical integration method, as currently implemented and used in mathematics and statistics, chemistry and physics, and satellite gravimetry, is groundless, mathematically and physically. Given the Newton's nonlinear governing differential equations of satellite motion with unknown equation parameters and unknown initial conditions, we develop three methods to derive new local solutions around a nominal reference orbit, which are linked to measurements to estimate the unknown corrections to approximate values of the unknown parameters and the unknown initial conditions. Bearing in mind that satellite orbits can now be tracked almost continuously at unprecedented accuracy, we propose the measurement-based perturbation theory and derive global uniformly convergent solutions to the Newton's nonlinear governing differential equations of satellite motion for the next generation of global gravitational models. Since the solutions are global uniformly convergent, theoretically speaking, they are able to extract smallest possible gravitational signals from modern and future satellite tracking measurements, leading to the production of global high-precision, high-resolution gravitational models. By directly turning the nonlinear differential equations of satellite motion into the nonlinear integral equations, and recognizing the fact that satellite orbits are measured with random errors, we further reformulate the links between satellite tracking measurements and the global uniformly convergent solutions to the Newton's governing differential equations as a condition adjustment model with unknown parameters, or equivalently, the weighted least squares estimation of unknown differential equation parameters with equality constraints, for the reconstruction of global high-precision, high-resolution gravitational models from modern (and future) satellite tracking measurements.

Teaching Mathematics to Civil Engineers

ERIC Educational Resources Information Center

Sharp, J. J.; Moore, E.

1977-01-01

This paper outlines a technique for teaching a rigorous course in calculus and differential equations which stresses applicability of the mathematics to problems in civil engineering. The method involves integration of subject matter and team teaching. (SD)

Evaluation of Uncertainty in Runoff Analysis Incorporating Theory of Stochastic Process

NASA Astrophysics Data System (ADS)

Yoshimi, Kazuhiro; Wang, Chao-Wen; Yamada, Tadashi

2015-04-01

The aim of this paper is to provide a theoretical framework of uncertainty estimate on rainfall-runoff analysis based on theory of stochastic process. SDE (stochastic differential equation) based on this theory has been widely used in the field of mathematical finance due to predict stock price movement. Meanwhile, some researchers in the field of civil engineering have investigated by using this knowledge about SDE (stochastic differential equation) (e.g. Kurino et.al, 1999; Higashino and Kanda, 2001). However, there have been no studies about evaluation of uncertainty in runoff phenomenon based on comparisons between SDE (stochastic differential equation) and Fokker-Planck equation. The Fokker-Planck equation is a partial differential equation that describes the temporal variation of PDF (probability density function), and there is evidence to suggest that SDEs and Fokker-Planck equations are equivalent mathematically. In this paper, therefore, the uncertainty of discharge on the uncertainty of rainfall is explained theoretically and mathematically by introduction of theory of stochastic process. The lumped rainfall-runoff model is represented by SDE (stochastic differential equation) due to describe it as difference formula, because the temporal variation of rainfall is expressed by its average plus deviation, which is approximated by Gaussian distribution. This is attributed to the observed rainfall by rain-gauge station and radar rain-gauge system. As a result, this paper has shown that it is possible to evaluate the uncertainty of discharge by using the relationship between SDE (stochastic differential equation) and Fokker-Planck equation. Moreover, the results of this study show that the uncertainty of discharge increases as rainfall intensity rises and non-linearity about resistance grows strong. These results are clarified by PDFs (probability density function) that satisfy Fokker-Planck equation about discharge. It means the reasonable discharge can be estimated based on the theory of stochastic processes, and it can be applied to the probabilistic risk of flood management.

Outcomes of a service teaching module on ODEs for physics students

NASA Astrophysics Data System (ADS)

Hyland, Diarmaid; van Kampen, Paul; Nolan, Brien C.

2018-07-01

This paper reports on the first part of a multiphase research project that seeks to identify and address the difficulties encountered by physics students when studying differential equations. Differential equations are used extensively by undergraduate physics students, particularly in the advanced modules of their degree. It is, therefore, necessary that students develop conceptual understanding of differential equations in addition to procedural skills. We have investigated the difficulties encountered by third-year students at Dublin City University in an introductory differential equations module. We developed a survey to identify these difficulties and administered it to students who had recently completed the module. We found that students' mathematical ability in relation to procedural competence is an issue in their study of differential equations, but not as severe an issue as their conceptual understanding. Mathematical competence alone is insufficient if we expect our students to be able to recognize the need for differential equations in a physical context and to be able to set up, solve and interpret the solutions of such equations. We discuss the implications of these results for the next stages of the research project.

Heavy use of equations impedes communication among biologists.

PubMed

Fawcett, Tim W; Higginson, Andrew D

2012-07-17

Most research in biology is empirical, yet empirical studies rely fundamentally on theoretical work for generating testable predictions and interpreting observations. Despite this interdependence, many empirical studies build largely on other empirical studies with little direct reference to relevant theory, suggesting a failure of communication that may hinder scientific progress. To investigate the extent of this problem, we analyzed how the use of mathematical equations affects the scientific impact of studies in ecology and evolution. The density of equations in an article has a significant negative impact on citation rates, with papers receiving 28% fewer citations overall for each additional equation per page in the main text. Long, equation-dense papers tend to be more frequently cited by other theoretical papers, but this increase is outweighed by a sharp drop in citations from nontheoretical papers (35% fewer citations for each additional equation per page in the main text). In contrast, equations presented in an accompanying appendix do not lessen a paper's impact. Our analysis suggests possible strategies for enhancing the presentation of mathematical models to facilitate progress in disciplines that rely on the tight integration of theoretical and empirical work.

A Unified Mathematical Framework for Coding Time, Space, and Sequences in the Hippocampal Region

PubMed Central

MacDonald, Christopher J.; Tiganj, Zoran; Shankar, Karthik H.; Du, Qian; Hasselmo, Michael E.; Eichenbaum, Howard

2014-01-01

The medial temporal lobe (MTL) is believed to support episodic memory, vivid recollection of a specific event situated in a particular place at a particular time. There is ample neurophysiological evidence that the MTL computes location in allocentric space and more recent evidence that the MTL also codes for time. Space and time represent a similar computational challenge; both are variables that cannot be simply calculated from the immediately available sensory information. We introduce a simple mathematical framework that computes functions of both spatial location and time as special cases of a more general computation. In this framework, experience unfolding in time is encoded via a set of leaky integrators. These leaky integrators encode the Laplace transform of their input. The information contained in the transform can be recovered using an approximation to the inverse Laplace transform. In the temporal domain, the resulting representation reconstructs the temporal history. By integrating movements, the equations give rise to a representation of the path taken to arrive at the present location. By modulating the transform with information about allocentric velocity, the equations code for position of a landmark. Simulated cells show a close correspondence to neurons observed in various regions for all three cases. In the temporal domain, novel secondary analyses of hippocampal time cells verified several qualitative predictions of the model. An integrated representation of spatiotemporal context can be computed by taking conjunctions of these elemental inputs, leading to a correspondence with conjunctive neural representations observed in dorsal CA1. PMID:24672015

Human evaluation in association to the mathematical analysis of arch forms: Two-dimensional study.

PubMed

Zabidin, Nurwahidah; Mohamed, Alizae Marny; Zaharim, Azami; Marizan Nor, Murshida; Rosli, Tanti Irawati

2018-03-01

To evaluate the relationship between human evaluation of the dental-arch form, to complete a mathematical analysis via two different methods in quantifying the arch form, and to establish agreement with the fourth-order polynomial equation. This study included 64 sets of digitised maxilla and mandible dental casts obtained from a sample of dental arch with normal occlusion. For human evaluation, a convenient sample of orthodontic practitioners ranked the photo images of dental cast from the most tapered to the less tapered (square). In the mathematical analysis, dental arches were interpolated using the fourth-order polynomial equation with millimetric acetate paper and AutoCAD software. Finally, the relations between human evaluation and mathematical objective analyses were evaluated. Human evaluations were found to be generally in agreement, but only at the extremes of tapered and square arch forms; this indicated general human error and observer bias. The two methods used to plot the arch form were comparable. The use of fourth-order polynomial equation may be facilitative in obtaining a smooth curve, which can produce a template for individual arch that represents all potential tooth positions for the dental arch. Copyright © 2018 CEO. Published by Elsevier Masson SAS. All rights reserved.

Digital computer simulation of inductor-energy-storage dc-to-dc converters with closed-loop regulators

NASA Technical Reports Server (NTRS)

Ohri, A. K.; Owen, H. A.; Wilson, T. G.; Rodriguez, G. E.

1974-01-01

The simulation of converter-controller combinations by means of a flexible digital computer program which produces output to a graphic display is discussed. The procedure is an alternative to mathematical analysis of converter systems. The types of computer programming involved in the simulation are described. Schematic diagrams, state equations, and output equations are displayed for four basic forms of inductor-energy-storage dc to dc converters. Mathematical models are developed to show the relationship of the parameters.

Numerical solution of linear and nonlinear Fredholm integral equations by using weighted mean-value theorem.

PubMed

Altürk, Ahmet

2016-01-01

Mean value theorems for both derivatives and integrals are very useful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. In this article, a semi-analytical method that is based on weighted mean-value theorem for obtaining solutions for a wide class of Fredholm integral equations of the second kind is introduced. Illustrative examples are provided to show the significant advantage of the proposed method over some existing techniques.

Using "Tracker" to Prove the Simple Harmonic Motion Equation

ERIC Educational Resources Information Center

Kinchin, John

2016-01-01

Simple harmonic motion (SHM) is a common topic for many students to study. Using the free, though versatile, motion tracking software; "Tracker", we can extend the students experience and show that the general equation for SHM does lead to the correct period of a simple pendulum.

Solving Simple Kinetics without Integrals

ERIC Educational Resources Information Center

de la Pen~a, Lisandro Herna´ndez

2016-01-01

The solution of simple kinetic equations is analyzed without referencing any topic from differential equations or integral calculus. Guided by the physical meaning of the rate equation, a systematic procedure is used to generate an approximate solution that converges uniformly to the exact solution in the case of zero, first, and second order…

Traveling wave solutions of the nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Akbari-Moghanjoughi, M.

2017-10-01

In this paper, we investigate the traveling soliton and the periodic wave solutions of the nonlinear Schrödinger equation (NLSE) with generalized nonlinear functionality. We also explore the underlying close connection between the well-known KdV equation and the NLSE. It is remarked that both one-dimensional KdV and NLSE models share the same pseudoenergy spectrum. We also derive the traveling wave solutions for two cases of weakly nonlinear mathematical models, namely, the Helmholtz and the Duffing oscillators' potentials. It is found that these models only allow gray-type NLSE solitary propagations. It is also found that the pseudofrequency ratio for the Helmholtz potential between the nonlinear periodic carrier and the modulated sinusoidal waves is always in the range 0.5 ≤ Ω/ω ≤ 0.537285 regardless of the potential parameter values. The values of Ω/ω = {0.5, 0.537285} correspond to the cnoidal waves modulus of m = {0, 1} for soliton and sinusoidal limits and m = 0.5, respectively. Moreover, the current NLSE model is extended to fully NLSE (FNLSE) situation for Sagdeev oscillator pseudopotential which can be derived using a closed set of hydrodynamic fluid equations with a fully integrable Hamiltonian system. The generalized quasi-three-dimensional traveling wave solution is also derived. The current simple hydrodynamic plasma model may also be generalized to two dimensions and other complex situations including different charged species and cases with magnetic or gravitational field effects.

Mathematical thinking and origami

NASA Astrophysics Data System (ADS)

Wares, Arsalan

2016-01-01

The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and calculus.

Students' conceptual performance on synthesis physics problems with varying mathematical complexity

NASA Astrophysics Data System (ADS)

Ibrahim, Bashirah; Ding, Lin; Heckler, Andrew F.; White, Daniel R.; Badeau, Ryan

2017-06-01

A body of research on physics problem solving has focused on single-concept problems. In this study we use "synthesis problems" that involve multiple concepts typically taught in different chapters. We use two types of synthesis problems, sequential and simultaneous synthesis tasks. Sequential problems require a consecutive application of fundamental principles, and simultaneous problems require a concurrent application of pertinent concepts. We explore students' conceptual performance when they solve quantitative synthesis problems with varying mathematical complexity. Conceptual performance refers to the identification, follow-up, and correct application of the pertinent concepts. Mathematical complexity is determined by the type and the number of equations to be manipulated concurrently due to the number of unknowns in each equation. Data were collected from written tasks and individual interviews administered to physics major students (N =179 ) enrolled in a second year mechanics course. The results indicate that mathematical complexity does not impact students' conceptual performance on the sequential tasks. In contrast, for the simultaneous problems, mathematical complexity negatively influences the students' conceptual performance. This difference may be explained by the students' familiarity with and confidence in particular concepts coupled with cognitive load associated with manipulating complex quantitative equations. Another explanation pertains to the type of synthesis problems, either sequential or simultaneous task. The students split the situation presented in the sequential synthesis tasks into segments but treated the situation in the simultaneous synthesis tasks as a single event.

Modeling Scientific Processes with Mathematics Equations Enhances Student Qualitative Conceptual Understanding and Quantitative Problem Solving

ERIC Educational Resources Information Center

Schuchardt, Anita M.; Schunn, Christian D.

2016-01-01

Amid calls for integrating science, technology, engineering, and mathematics (iSTEM) in K-12 education, there is a pressing need to uncover productive methods of integration. Prior research has shown that increasing contextual linkages between science and mathematics is associated with student problem solving and conceptual understanding. However,…

Teachers' Concerns and Efficacy Beliefs about Implementing a Mathematics Curriculum Reform: Integrating Two Lines of Inquiry

ERIC Educational Resources Information Center

Charalambous, Charalambos Y.; Philippou, George N.

2010-01-01

This study brings together two lines of research on teachers' affective responses toward mathematics curriculum reforms: their concerns and their efficacy beliefs. Using structural equation modeling to analyze data on 151 elementary mathematics teachers' concerns and efficacy beliefs 5 years into a mandated curriculum reform on problem solving,…

Using a Functional Model to Develop a Mathematical Formula

ERIC Educational Resources Information Center

Otto, Charlotte A.; Everett, Susan A.; Luera, Gail R.

2008-01-01

The unifying theme of models was incorporated into a required Science Capstone course for pre-service elementary teachers based on national standards in science and mathematics. A model of a teeter-totter was selected for use as an example of a functional model for gathering data as well as a visual model of a mathematical equation for developing…

Improving Mathematics: An Examination of the Effects of Specific Cognitive Abilities on College-Age Students' Mathematics Achievement

ERIC Educational Resources Information Center

Taub, Gordon E.; Benson, Nicholas; Szente, Judit

2014-01-01

This study investigated the effects of general intelligence and seven specific cognitive abilities on college-age students' mathematics achievement. The present investigation went beyond previous research by employing structural equation modeling. It also represents the first study to examine the direct and indirect effects of general and specific…

Transient Mathematical Modeling for Liquid Rocket Engine Systems: Methods, Capabilities, and Experience

NASA Technical Reports Server (NTRS)

Seymour, David C.; Martin, Michael A.; Nguyen, Huy H.; Greene, William D.

2005-01-01

The subject of mathematical modeling of the transient operation of liquid rocket engines is presented in overview form from the perspective of engineers working at the NASA Marshall Space Flight Center. The necessity of creating and utilizing accurate mathematical models as part of liquid rocket engine development process has become well established and is likely to increase in importance in the future. The issues of design considerations for transient operation, development testing, and failure scenario simulation are discussed. An overview of the derivation of the basic governing equations is presented along with a discussion of computational and numerical issues associated with the implementation of these equations in computer codes. Also, work in the field of generating usable fluid property tables is presented along with an overview of efforts to be undertaken in the future to improve the tools use for the mathematical modeling process.

Transient Mathematical Modeling for Liquid Rocket Engine Systems: Methods, Capabilities, and Experience

NASA Technical Reports Server (NTRS)

Martin, Michael A.; Nguyen, Huy H.; Greene, William D.; Seymout, David C.

2003-01-01

The subject of mathematical modeling of the transient operation of liquid rocket engines is presented in overview form from the perspective of engineers working at the NASA Marshall Space Flight Center. The necessity of creating and utilizing accurate mathematical models as part of liquid rocket engine development process has become well established and is likely to increase in importance in the future. The issues of design considerations for transient operation, development testing, and failure scenario simulation are discussed. An overview of the derivation of the basic governing equations is presented along with a discussion of computational and numerical issues associated with the implementation of these equations in computer codes. Also, work in the field of generating usable fluid property tables is presented along with an overview of efforts to be undertaken in the future to improve the tools use for the mathematical modeling process.

The Effect of Tutoring with Nonstandard Equations for Students with Mathematics Difficulty

ERIC Educational Resources Information Center

Powell, Sarah R.; Driver, Melissa K.; Julian, Tyler E.

2015-01-01

Students often misinterpret the equal sign (=) as operational instead of relational. Research indicates misinterpretation of the equal sign occurs because students receive relatively little exposure to equations that promote relational understanding of the equal sign. No study, however, has examined effects of nonstandard equations on the equation…

Equational Sentence Structure in Eskimo.

ERIC Educational Resources Information Center

Hofmann, Th. R.

A comparison of the syntactic characteristics of mathematical equations and Eskimo syntax is made, and a proposal that Eskimo has a level of structure similar to that of equations is described. P:t performative contrast is reanalyzed. Questions and speculations on the formal treatment of this type of structure in transformational grammar, and its…

New explicit global asymptotic stability criteria for higher order difference equations

NASA Astrophysics Data System (ADS)

El-Morshedy, Hassan A.

2007-12-01

New explicit sufficient conditions for the asymptotic stability of the zero solution of higher order difference equations are obtained. These criteria can be applied to autonomous and nonautonomous equations. The celebrated Clark asymptotic stability criterion is improved. Also, applications to models from mathematical biology and macroeconomics are given.

Preprocessing Inconsistent Linear System for a Meaningful Least Squares Solution

NASA Technical Reports Server (NTRS)

Sen, Syamal K.; Shaykhian, Gholam Ali

2011-01-01

Mathematical models of many physical/statistical problems are systems of linear equations. Due to measurement and possible human errors/mistakes in modeling/data, as well as due to certain assumptions to reduce complexity, inconsistency (contradiction) is injected into the model, viz. the linear system. While any inconsistent system irrespective of the degree of inconsistency has always a least-squares solution, one needs to check whether an equation is too much inconsistent or, equivalently too much contradictory. Such an equation will affect/distort the least-squares solution to such an extent that renders it unacceptable/unfit to be used in a real-world application. We propose an algorithm which (i) prunes numerically redundant linear equations from the system as these do not add any new information to the model, (ii) detects contradictory linear equations along with their degree of contradiction (inconsistency index), (iii) removes those equations presumed to be too contradictory, and then (iv) obtain the minimum norm least-squares solution of the acceptably inconsistent reduced linear system. The algorithm presented in Matlab reduces the computational and storage complexities and also improves the accuracy of the solution. It also provides the necessary warning about the existence of too much contradiction in the model. In addition, we suggest a thorough relook into the mathematical modeling to determine the reason why unacceptable contradiction has occurred thus prompting us to make necessary corrections/modifications to the models - both mathematical and, if necessary, physical.

Preprocessing in Matlab Inconsistent Linear System for a Meaningful Least Squares Solution

NASA Technical Reports Server (NTRS)

Sen, Symal K.; Shaykhian, Gholam Ali

2011-01-01

Mathematical models of many physical/statistical problems are systems of linear equations Due to measurement and possible human errors/mistakes in modeling/data, as well as due to certain assumptions to reduce complexity, inconsistency (contradiction) is injected into the model, viz. the linear system. While any inconsistent system irrespective of the degree of inconsistency has always a least-squares solution, one needs to check whether an equation is too much inconsistent or, equivalently too much contradictory. Such an equation will affect/distort the least-squares solution to such an extent that renders it unacceptable/unfit to be used in a real-world application. We propose an algorithm which (i) prunes numerically redundant linear equations from the system as these do not add any new information to the model, (ii) detects contradictory linear equations along with their degree of contradiction (inconsistency index), (iii) removes those equations presumed to be too contradictory, and then (iv) obtain the . minimum norm least-squares solution of the acceptably inconsistent reduced linear system. The algorithm presented in Matlab reduces the computational and storage complexities and also improves the accuracy of the solution. It also provides the necessary warning about the existence of too much contradiction in the model. In addition, we suggest a thorough relook into the mathematical modeling to determine the reason why unacceptable contradiction has occurred thus prompting us to make necessary corrections/modifications to the models - both mathematical and, if necessary, physical.

Determination of minimum enzymatic decolorization time of reactive dye solution by spectroscopic & mathematical approach.

PubMed

Celebi, Mithat; Ozdemir, Zafer Omer; Eroglu, Emre; Altikatoglu, Melda; Guney, Ibrahim

2015-02-01

Synthetic dyes are very important for textile dyeing, paper printing, color photography and petroleum products. Traditional methods of dye removal include biodegradation, precipitation, adsorption, chemical degradation, photo degradation, and chemical coagulation. Dye decolorization with enzymatic reaction is an important issue for several research field (chemistry, environment) In this study, minimum decolorization time of Remazol Brilliant Blue R dye with Horseradish peroxidase enzyme was calculated using with mathematical equation depending on experimental data. Dye decolorization was determined by monitoring the absorbance decrease at the specific maximum wavelength for dye. All experiments were carried out with different initial dye concentrations of Remazol Brilliant Blue R at 25 degrees C constant temperature for 30 minutes. The development of the least squares estimators for a nonlinear model brings about complications not encountered in the case of the linear model. Decolorization times for completely removal of dye were calculated according to equation. It was shown that mathematical equation was conformed exponential curve for dye degradation.

Variational symmetries, conserved quantities and identities for several equations of mathematical physics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Donchev, Veliko, E-mail: velikod@ie.bas.bg

2014-03-15

We find variational symmetries, conserved quantities and identities for several equations: envelope equation, Böcher equation, the propagation of sound waves with losses, flow of a gas with losses, and the nonlinear Schrödinger equation with losses or gains, and an electro-magnetic interaction. Most of these equations do not have a variational description with the classical variational principle and we find such a description with the generalized variational principle of Herglotz.

Modelling food and population dynamics in honey bee colonies.

PubMed

Khoury, David S; Barron, Andrew B; Myerscough, Mary R

2013-01-01

Honey bees (Apis mellifera) are increasingly in demand as pollinators for various key agricultural food crops, but globally honey bee populations are in decline, and honey bee colony failure rates have increased. This scenario highlights a need to understand the conditions in which colonies flourish and in which colonies fail. To aid this investigation we present a compartment model of bee population dynamics to explore how food availability and bee death rates interact to determine colony growth and development. Our model uses simple differential equations to represent the transitions of eggs laid by the queen to brood, then hive bees and finally forager bees, and the process of social inhibition that regulates the rate at which hive bees begin to forage. We assume that food availability can influence both the number of brood successfully reared to adulthood and the rate at which bees transition from hive duties to foraging. The model predicts complex interactions between food availability and forager death rates in shaping colony fate. Low death rates and high food availability results in stable bee populations at equilibrium (with population size strongly determined by forager death rate) but consistently increasing food reserves. At higher death rates food stores in a colony settle at a finite equilibrium reflecting the balance of food collection and food use. When forager death rates exceed a critical threshold the colony fails but residual food remains. Our model presents a simple mathematical framework for exploring the interactions of food and forager mortality on colony fate, and provides the mathematical basis for more involved simulation models of hive performance.

Verifying a Computer Algorithm Mathematically.

ERIC Educational Resources Information Center

Olson, Alton T.

1986-01-01

Presents an example of mathematics from an algorithmic point of view, with emphasis on the design and verification of this algorithm. The program involves finding roots for algebraic equations using the half-interval search algorithm. The program listing is included. (JN)

Kinematic equations for resolved-rate control of an industrial robot arm

NASA Technical Reports Server (NTRS)

Barker, L. K.

1983-01-01

An operator can use kinematic, resolved-rate equations to dynamically control a robot arm by watching its response to commanded inputs. Known resolved-rate equations for the control of a particular six-degree-of-freedom industrial robot arm and proceeds to simplify the equations for faster computations are derived. Methods for controlling the robot arm in regions which normally cause mathematical singularities in the resolved-rate equations are discussed.

The Employment Equation: Why Our Young People Need More Maths for Today's Jobs

ERIC Educational Resources Information Center

Hodgen, Jeremy; Marks, Rachel

2013-01-01

This report reviews over 50 research studies to consider the level and type of mathematical skills needed by employers in today's economy. It considers five key questions: (1) What mathematics (level and content) is required in the workplace today?; (2) How and why have the mathematical needs of the workplace changed over time?; (3) In what ways…

Messy but Meaningful: Exploring the Transition to Reform-Based Pedagogy with Teachers of Mathematics and Coordinators in Ontario, Canada

ERIC Educational Resources Information Center

Jarvis, Daniel

2016-01-01

The RE4MUL8 Project involved the creation of an online/mobile resource for Intermediate Division (Grade 7 and 8) teachers of mathematics. This resource showcases video documentaries of seven key mathematics topic lessons (fractions, integers, proportional reasoning, composite shapes and solids, solving equations, and, patterning and algebraic…

Solving the Common Core Equation: Teaching Mathematics CCSS to Students with Moderate and Severe Disabilities

ERIC Educational Resources Information Center

Saunders, Alicia F.; Bethune, Keri S.; Spooner, Fred; Browder, Diane

2013-01-01

The Common Core State Standards (CCSS) in mathematics were created to help all students become prepared for the demands of future careers and life in an age of technology. Similarly, students with moderate and severe disability will need these skills to meet these changing expectations. Although mathematics instruction could focus on a few of the…

Influence of Teacher Support and Personal Relevance on Academic Self-Efficacy and Enjoyment of Mathematics Lessons: A Structural Equation Modeling Approach

ERIC Educational Resources Information Center

Aldridge, Jill M.; Afari, Ernest; Fraser, Barry J.

2012-01-01

The purpose of our study was to examine the effects of two psychosocial features of the classroom environment (teacher support and personal relevance) on college students' academic self-efficacy and enjoyment of mathematics lessons. Data collected from 352 mathematics students attending three higher education institutions in the United Arab…

Between Politics and Equations: Teaching Critical Mathematics in a Remedial Secondary Classroom

ERIC Educational Resources Information Center

Brantlinger, Andrew

2013-01-01

Proponents of critical mathematics (CM) argue that it has the potential to be more equitable and socially empowering than other approaches to mathematics education. In this article, the author presents results from a practitioner research study of his own teaching of CM to low-income students of color in a U.S. context. The results pertain to the…

A simple mathematical model of gradual Darwinian evolution: emergence of a Gaussian trait distribution in adaptation along a fitness gradient.

PubMed

Biktashev, Vadim N

2014-04-01

We consider a simple mathematical model of gradual Darwinian evolution in continuous time and continuous trait space, due to intraspecific competition for common resource in an asexually reproducing population in constant environment, while far from evolutionary stable equilibrium. The model admits exact analytical solution. In particular, Gaussian distribution of the trait emerges from generic initial conditions.

An Assessment of the Level of Mathematics in Introductory Meteorology Textbooks.

NASA Astrophysics Data System (ADS)

Ulanski, Stan L.

1992-10-01

A review of introductory meteorology textbooks shows a wide difference in the level of mathematical treatment of atmospheric principles-from virtually none to fairly high. Particular deficiencies include lack of equations integrated into the text, problem-solving examples, and paucity of end-of-chapter questions requiring mathematical reasoning. These issues are raised in order to generate discussion among the meteorological community with regard to the degree of interaction between mathematics and meteorology in introductory courses.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Thompson, S.

This report describes the use of several subroutines from the CORLIB core mathematical subroutine library for the solution of a model fluid flow problem. The model consists of the Euler partial differential equations. The equations are spatially discretized using the method of pseudo-characteristics. The resulting system of ordinary differential equations is then integrated using the method of lines. The stiff ordinary differential equation solver LSODE (2) from CORLIB is used to perform the time integration. The non-stiff solver ODE (4) is used to perform a related integration. The linear equation solver subroutines DECOMP and SOLVE are used to solve linearmore » systems whose solutions are required in the calculation of the time derivatives. The monotone cubic spline interpolation subroutines PCHIM and PCHFE are used to approximate water properties. The report describes the use of each of these subroutines in detail. It illustrates the manner in which modules from a standard mathematical software library such as CORLIB can be used as building blocks in the solution of complex problems of practical interest. 9 refs., 2 figs., 4 tabs.« less

A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel

NASA Astrophysics Data System (ADS)

Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru

2018-02-01

The mathematical model of breaking of non-linear dispersive water waves with memory effect is very important in mathematical physics. In the present article, we examine a novel fractional extension of the non-linear Fornberg-Whitham equation occurring in wave breaking. We consider the most recent theory of differentiation involving the non-singular kernel based on the extended Mittag-Leffler-type function to modify the Fornberg-Whitham equation. We examine the existence of the solution of the non-linear Fornberg-Whitham equation of fractional order. Further, we show the uniqueness of the solution. We obtain the numerical solution of the new arbitrary order model of the non-linear Fornberg-Whitham equation with the aid of the Laplace decomposition technique. The numerical outcomes are displayed in the form of graphs and tables. The results indicate that the Laplace decomposition algorithm is a very user-friendly and reliable scheme for handling such type of non-linear problems of fractional order.

Solving Differential Equations Analytically. Elementary Differential Equations. Modules and Monographs in Undergraduate Mathematics and Its Applications Project. UMAP Unit 335.

ERIC Educational Resources Information Center

Goldston, J. W.

This unit introduces analytic solutions of ordinary differential equations. The objective is to enable the student to decide whether a given function solves a given differential equation. Examples of problems from biology and chemistry are covered. Problem sets, quizzes, and a model exam are included, and answers to all items are provided. The…

Approximating a nonlinear advanced-delayed equation from acoustics

NASA Astrophysics Data System (ADS)

Teodoro, M. Filomena

2016-10-01

We approximate the solution of a particular non-linear mixed type functional differential equation from physiology, the mucosal wave model of the vocal oscillation during phonation. The mathematical equation models a superficial wave propagating through the tissues. The numerical scheme is adapted from the work presented in [1, 2, 3], using homotopy analysis method (HAM) to solve the non linear mixed type equation under study.

Introduction to the Difference Calculus through the Fibonacci Numbers

ERIC Educational Resources Information Center

Shannon, A. G.; Atanassov, K. T.

2002-01-01

This note explores ways in which the Fibonacci numbers can be used to introduce difference equations as a prelude to differential equations. The rationale is that the formal aspects of discrete mathematics can provide a concrete introduction to the mechanisms of solving difference and differential equations without the distractions of the analytic…

Maneuverability Estimation of High-Speed Craft

DTIC Science & Technology

2015-06-01

derived based on equations by Lewandowski and Denny- Hubble in order to find the fundamental maneuvering characteristics. The model is developed in...characteristic of high- speed craft. A mathematical model is derived based on equations by Lewandowski and Denny- Hubble in order to find the fundamental...33 C. EQUATIONS BY DENNY AND HUBBLE ................................................43 D. NOMOTO

Forced Convection Heat Transfer in Circular Pipes

ERIC Educational Resources Information Center

Tosun, Ismail

2007-01-01

One of the pitfalls of engineering education is to lose the physical insight of the problem while tackling the mathematical part. Forced convection heat transfer (the Graetz-Nusselt problem) certainly falls into this category. The equation of energy together with the equation of motion leads to a partial differential equation subject to various…

Monograph - The Numerical Integration of Ordinary Differential Equations.

ERIC Educational Resources Information Center

Hull, T. E.

The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this…

Solving Cubic Equations by Polynomial Decomposition

ERIC Educational Resources Information Center

Kulkarni, Raghavendra G.

2011-01-01

Several mathematicians struggled to solve cubic equations, and in 1515 Scipione del Ferro reportedly solved the cubic while participating in a local mathematical contest, but did not bother to publish his method. Then it was Cardano (1539) who first published the solution to the general cubic equation in his book "The Great Art, or, The Rules of…

The Local Brewery: A Project for Use in Differential Equations Courses

ERIC Educational Resources Information Center

Starling, James K.; Povich, Timothy J.; Findlay, Michael

2016-01-01

We describe a modeling project designed for an ordinary differential equations (ODEs) course using first-order and systems of first-order differential equations to model the fermentation process in beer. The project aims to expose the students to the modeling process by creating and solving a mathematical model and effectively communicating their…

The Empirical Derivation of Equations for Predicting Subjective Textual Information. Final Report.

ERIC Educational Resources Information Center

Kauffman, Dan; And Others

A study was made to derive an equation for predicting the "subjective" textual information contained in a text of material written in the English language. Specifically, this investigation describes, by a mathematical equation, the relationship between the "subjective" information content of written textual material and the relative number of…

Solution of the equations for one-dimensional, two-phase, immiscible flow by geometric methods

NASA Astrophysics Data System (ADS)

Boronin, Ivan; Shevlyakov, Andrey

2018-03-01

Buckley-Leverett equations describe non viscous, immiscible, two-phase filtration, which is often of interest in modelling of oil production. For many parameters and initial conditions, the solutions of these equations exhibit non-smooth behaviour, namely discontinuities in form of shock waves. In this paper we obtain a novel method for the solution of Buckley-Leverett equations, which is based on geometry of differential equations. This method is fast, accurate, stable, and describes non-smooth phenomena. The main idea of the method is that classic discontinuous solutions correspond to the continuous surfaces in the space of jets - the so-called multi-valued solutions (Bocharov et al., Symmetries and conservation laws for differential equations of mathematical physics. American Mathematical Society, Providence, 1998). A mapping of multi-valued solutions from the jet space onto the plane of the independent variables is constructed. This mapping is not one-to-one, and its singular points form a curve on the plane of the independent variables, which is called the caustic. The real shock occurs at the points close to the caustic and is determined by the Rankine-Hugoniot conditions.

The problem of exact interior solutions for rotating rigid bodies in general relativity

NASA Technical Reports Server (NTRS)

Wahlquist, H. D.

1993-01-01

The (3 + 1) dyadic formalism for timelike congruences is applied to derive interior solutions for stationary, axisymmetric, rigidly rotating bodies. In this approach the mathematics is formulated in terms of three-space-covariant, first-order, vector-dyadic, differential equations for a and Omega, the acceleration and angular velocity three-vectors of the rigid body; for T, the stress dyadic of the matter; and for A and B, the 'electric' and 'magnetic' Weyl curvature dyadics which describe the gravitational field. It is shown how an appropriate ansatz for the forms of these dyadics can be used to discover exact rotating interior solutions such as the perfect fluid solution first published in 1968. By incorporating anisotropic stresses, a generalization is found of that previous solution and, in addition, a very simple new solution that can only exist in toroidal configurations.

Using explanatory crop models to develop simple tools for Advanced Life Support system studies

NASA Technical Reports Server (NTRS)

Cavazzoni, J.

2004-01-01

System-level analyses for Advanced Life Support require mathematical models for various processes, such as for biomass production and waste management, which would ideally be integrated into overall system models. Explanatory models (also referred to as mechanistic or process models) would provide the basis for a more robust system model, as these would be based on an understanding of specific processes. However, implementing such models at the system level may not always be practicable because of their complexity. For the area of biomass production, explanatory models were used to generate parameters and multivariable polynomial equations for basic models that are suitable for estimating the direction and magnitude of daily changes in canopy gas-exchange, harvest index, and production scheduling for both nominal and off-nominal growing conditions. c2004 COSPAR. Published by Elsevier Ltd. All rights reserved.

Random walk, diffusion and mixing in simulations of scalar transport in fluid flows

NASA Astrophysics Data System (ADS)

Klimenko, A. Y.

2008-12-01

Physical similarity and mathematical equivalence of continuous diffusion and particle random walk form one of the cornerstones of modern physics and the theory of stochastic processes. In many applied models used in simulation of turbulent transport and turbulent combustion, mixing between particles is used to reflect the influence of the continuous diffusion terms in the transport equations. We show that the continuous scalar transport and diffusion can be accurately specified by means of mixing between randomly walking Lagrangian particles with scalar properties and assess errors associated with this scheme. This gives an alternative formulation for the stochastic process which is selected to represent the continuous diffusion. This paper focuses on statistical errors and deals with relatively simple cases, where one-particle distributions are sufficient for a complete description of the problem.

Preliminary Development of an Object-Oriented Optimization Tool

NASA Technical Reports Server (NTRS)

Pak, Chan-gi

2011-01-01

The National Aeronautics and Space Administration Dryden Flight Research Center has developed a FORTRAN-based object-oriented optimization (O3) tool that leverages existing tools and practices and allows easy integration and adoption of new state-of-the-art software. The object-oriented framework can integrate the analysis codes for multiple disciplines, as opposed to relying on one code to perform analysis for all disciplines. Optimization can thus take place within each discipline module, or in a loop between the central executive module and the discipline modules, or both. Six sample optimization problems are presented. The first four sample problems are based on simple mathematical equations; the fifth and sixth problems consider a three-bar truss, which is a classical example in structural synthesis. Instructions for preparing input data for the O3 tool are presented.

Games for Developing Mathematical Strategy.

ERIC Educational Resources Information Center

Williford, Harold

1992-01-01

The three middle school games described in this article focus on problem solving and mathematical reasoning skills. Included are Solomon's game, nine mens' morris, and nim-with-cards which all offer not only high student interest and strong mathematical value but also simple rules, ease of construction, and attractiveness for repeated play.…

Student Attrition in Mathematics E-Learning

ERIC Educational Resources Information Center

Smith, Glenn Gordon; Ferguson, David

2005-01-01

Qualitative studies indicate that mathematics does not work well in e-learning. The current study used quantitative methods to investigate more objectively the extent of problems with mathematics in e-learning. The authors used student attrition as a simple measure of student satisfaction and course viability in two studies, one investigating…

Connecting Dynamic Representations of Simple Mathematical Objects with the Construction and Exploration of Conic Sections

ERIC Educational Resources Information Center

Santos-Trigo, Manuel; Espinosa-Perez, Hugo; Reyes-Rodriguez, Aaron

2008-01-01

Different technological artefacts may offer distinct opportunities for students to develop resources and strategies to formulate, comprehend and solve mathematical problems. In particular, the use of dynamic software becomes relevant to assemble geometric configurations that may help students reconstruct and examine mathematical relationships. In…

Research on the application of a decoupling algorithm for structure analysis

NASA Technical Reports Server (NTRS)

Denman, E. D.

1980-01-01

The mathematical theory for decoupling mth-order matrix differential equations is presented. It is shown that the decoupling precedure can be developed from the algebraic theory of matrix polynomials. The role of eigenprojectors and latent projectors in the decoupling process is discussed and the mathematical relationships between eigenvalues, eigenvectors, latent roots, and latent vectors are developed. It is shown that the eigenvectors of the companion form of a matrix contains the latent vectors as a subset. The spectral decomposition of a matrix and the application to differential equations is given.

Learning to Calculate and Learning Mathematics.

ERIC Educational Resources Information Center

Fearnley-Sander, Desmond

1980-01-01

A calculator solution of a simple computational problem is discussed with emphasis on its ramifications for the understanding of some fundamental theorems of pure mathematics and techniques of computing. (Author/MK)

Solving Differential Equations in R

EPA Science Inventory

Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...

Mathematical toy model inspired by the problem of the adaptive origins of the sexual orientation continuum

NASA Astrophysics Data System (ADS)

Skinner, Brian

2016-09-01

Same-sex sexual behaviour is ubiquitous in the animal kingdom, but its adaptive origins remain a prominent puzzle. Here, I suggest the possibility that same-sex sexual behaviour arises as a consequence of the competition between an evolutionary drive for a wide diversity in traits, which improves the adaptability of a population, and a drive for sexual dichotomization of traits, which promotes opposite-sex attraction and increases the rate of reproduction. This trade-off is explored via a simple mathematical `toy model'. The model exhibits a number of interesting features and suggests a simple mathematical form for describing the sexual orientation continuum.

Using mathematical software to design power electronic converters

NASA Astrophysics Data System (ADS)

Hinov, Nikolay; Hranov, Tsveti

2017-12-01

In the paper is presented mathematical software, which was used for design of power electronic devices. Examined to different example, which are applied to designing electronic converters. In this way, it is possible to play different combinations of the circuit elements by simple means, thus optimizing according to certain criteria and limitations. Free software with a simple and intuitive interface is selected. No special user training is required to work with it and no further training is required. The use of mathematical software greatly facilitates the design, assists and makes it attractive and accessible to a wider range of students and specialists in power electronics training.