A Unified Introduction to Ordinary Differential Equations

ERIC Educational Resources Information Center

Lutzer, Carl V.

2006-01-01

This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

PubMed Central

Yu, Zhang; Zhang, Yufeng

2009-01-01

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings. PMID:20084092

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.

PubMed

Yu, Zhang; Zhang, Yufeng

2009-01-15

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.

Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Morozov, Oleg I.

2018-06-01

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the PDE s under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer-Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones.

Symmetric linear systems - An application of algebraic systems theory

NASA Technical Reports Server (NTRS)

Hazewinkel, M.; Martin, C.

1983-01-01

Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.

Partial Fractions via Calculus

ERIC Educational Resources Information Center

Bauldry, William C.

2018-01-01

The standard technique taught in calculus courses for partial fraction expansions uses undetermined coefficients to generate a system of linear equations; we present a derivative-based technique that calculus and differential equations instructors can use to reinforce connections to calculus. Simple algebra shows that we can use the derivative to…

pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

DOE PAGES

Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul; ...

2017-12-20

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

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

Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

Parallel Element Agglomeration Algebraic Multigrid and Upscaling Library

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

Barker, Andrew T.; Benson, Thomas R.; Lee, Chak Shing

ParELAG is a parallel C++ library for numerical upscaling of finite element discretizations and element-based algebraic multigrid solvers. It provides optimal complexity algorithms to build multilevel hierarchies and solvers that can be used for solving a wide class of partial differential equations (elliptic, hyperbolic, saddle point problems) on general unstructured meshes. Additionally, a novel multilevel solver for saddle point problems with divergence constraint is implemented.

Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras

PubMed Central

Gazizov, R. K.

2017-01-01

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures. PMID:28265184

Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras.

PubMed

Gainetdinova, A A; Gazizov, R K

2017-01-01

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.

Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics

NASA Astrophysics Data System (ADS)

Motsepa, Tanki; Aziz, Taha; Fatima, Aeeman; Khalique, Chaudry Masood

2018-03-01

The optimal investment-consumption problem under the constant elasticity of variance (CEV) model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition. Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws.

Discovery and Optimization of Low-Storage Runge-Kutta Methods

DTIC Science & Technology

2015-06-01

NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS DISCOVERY AND OPTIMIZATION OF LOW-STORAGE RUNGE-KUTTA METHODS by Matthew T. Fletcher June 2015... methods are an important family of iterative methods for approximating the solutions of ordinary differential equations (ODEs) and differential...algebraic equations (DAEs). It is common to use an RK method to discretize in time when solving time dependent partial differential equations (PDEs) with a

Analysis and synthesis of distributed-lumped-active networks by digital computer

NASA Technical Reports Server (NTRS)

1973-01-01

The use of digital computational techniques in the analysis and synthesis of DLA (distributed lumped active) networks is considered. This class of networks consists of three distinct types of elements, namely, distributed elements (modeled by partial differential equations), lumped elements (modeled by algebraic relations and ordinary differential equations), and active elements (modeled by algebraic relations). Such a characterization is applicable to a broad class of circuits, especially including those usually referred to as linear integrated circuits, since the fabrication techniques for such circuits readily produce elements which may be modeled as distributed, as well as the more conventional lumped and active ones.

Iterative methods for elliptic finite element equations on general meshes

NASA Technical Reports Server (NTRS)

Nicolaides, R. A.; Choudhury, Shenaz

1986-01-01

Iterative methods for arbitrary mesh discretizations of elliptic partial differential equations are surveyed. The methods discussed are preconditioned conjugate gradients, algebraic multigrid, deflated conjugate gradients, an element-by-element techniques, and domain decomposition. Computational results are included.

Symmetry groups of integro-differential equations for linear thermoviscoelastic materials with memory

NASA Astrophysics Data System (ADS)

Zhou, L.-Q.; Meleshko, S. V.

2017-07-01

The group analysis method is applied to a system of integro-differential equations corresponding to a linear thermoviscoelastic model. A recently developed approach for calculating the symmetry groups of such equations is used. The general solution of the determining equations for the system is obtained. Using subalgebras of the admitted Lie algebra, two classes of partially invariant solutions of the considered system of integro-differential equations are studied.

A finite difference scheme for the equilibrium equations of elastic bodies

NASA Technical Reports Server (NTRS)

Phillips, T. N.; Rose, M. E.

1984-01-01

A compact difference scheme is described for treating the first-order system of partial differential equations which describe the equilibrium equations of an elastic body. An algebraic simplification enables the solution to be obtained by standard direct or iterative techniques.

Optical systolic solutions of linear algebraic equations

NASA Technical Reports Server (NTRS)

Neuman, C. P.; Casasent, D.

1984-01-01

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Conformational statistics of stiff macromolecules as solutions to partial differential equations on the rotation and motion groups

PubMed

Chirikjian; Wang

2000-07-01

Partial differential equations (PDE's) for the probability density function (PDF) of the position and orientation of the distal end of a stiff macromolecule relative to its proximal end are derived and solved. The Kratky-Porod wormlike chain, the Yamakawa helical wormlike chain, and the original and revised Marko-Siggia models are examples of stiffness models to which the present formulation is applied. The solution technique uses harmonic analysis on the rotation and motion groups to convert PDE's governing the PDF's of interest into linear algebraic equations which have mathematically elegant solutions.

Double diffusive conjugate heat transfer: Part III

NASA Astrophysics Data System (ADS)

Soudagar, Manzoor Elahi M.; Azeem

2018-05-01

The placement of a small solid wall towards cold surface of square porous cavity affects the heat transfer behavior of porous region due to restriction of fluid motion in the region occupied by solid wall. An investigation of heat transfer is carried out to understand the fluid flow and heat transfer behavior in porous cavity by solving the governing partial differential equations. Galerkin's approach is used to convert the partial differential equations into algebraic form of equations by applying finite element method. The heat transfer increases for solid towards right surface as compared to the case of solid at center of cavity.

An efficient numerical scheme for the study of equal width equation

NASA Astrophysics Data System (ADS)

Ghafoor, Abdul; Haq, Sirajul

2018-06-01

In this work a new numerical scheme is proposed in which Haar wavelet method is coupled with finite difference scheme for the solution of a nonlinear partial differential equation. The scheme transforms the partial differential equation to a system of algebraic equations which can be solved easily. The technique is applied to equal width equation in order to study the behaviour of one, two, three solitary waves, undular bore and soliton collision. For efficiency and accuracy of the scheme, L2 and L∞ norms and invariants are computed. The results obtained are compared with already existing results in literature.

Boundary-fitted coordinate systems for numerical solution of partial differential equations - A review

NASA Technical Reports Server (NTRS)

Thompson, J. F.; Warsi, Z. U. A.; Mastin, C. W.

1982-01-01

A comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given. Some general mathematical framework and error analysis common to such coordinate systems is also included. The general categories of generating systems are those based on conformal mapping, orthogonal systems, nearly orthogonal systems, systems produced as the solution of elliptic and hyperbolic partial differential equations, and systems generated algebraically by interpolation among the boundaries. Also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation. Dynamically adaptive coordinate systems, coupled with the physical solution, and time-dependent systems that follow moving boundaries are treated. References reporting experience using such coordinate systems are reviewed as well as those covering the system development.

Local uncontrollability for affine control systems with jumps

NASA Astrophysics Data System (ADS)

Treanţă, Savin

2017-09-01

This paper investigates affine control systems with jumps for which the ideal If(g1, …, gm) generated by the drift vector field f in the Lie algebra L(f, g1, …, gm) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.

Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.

2005-09-01

We present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. Numerical applications of how to use these methods are described. Numerical results obtained compare favorably with those of the analytical solutions. Accurate double ultraspherical spectral approximations for Poisson's and Helmholtz's equations are also noted. Numerical experiments show that spectral approximation based on Chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials.

Toward the classification of differential calculi on κ-Minkowski space and related field theories

NASA Astrophysics Data System (ADS)

Jurić, Tajron; Meljanac, Stjepan; Pikutić, Danijel; Štrajn, Rina

2015-07-01

Classification of differential forms on κ-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the κ-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to κ-Poincaré Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications.

Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence.

PubMed

Ozaktas, Haldun M; Yüksel, Serdar; Kutay, M Alper

2002-08-01

A linear algebraic theory of partial coherence is presented that allows precise mathematical definitions of concepts such as coherence and incoherence. This not only provides new perspectives and insights but also allows us to employ the conceptual and algebraic tools of linear algebra in applications. We define several scalar measures of the degree of partial coherence of an optical field that are zero for full incoherence and unity for full coherence. The mathematical definitions are related to our physical understanding of the corresponding concepts by considering them in the context of Young's experiment.

Time domain convergence properties of Lyapunov stable penalty methods

NASA Technical Reports Server (NTRS)

Kurdila, A. J.; Sunkel, John

1991-01-01

Linear hyperbolic partial differential equations are analyzed using standard techniques to show that a sequence of solutions generated by the Liapunov stable penalty equations approaches the solution of the differential-algebraic equations governing the dynamics of multibody problems arising in linear vibrations. The analysis does not require that the system be conservative and does not impose any specific integration scheme. Variational statements are derived which bound the error in approximation by the norm of the constraint violation obtained in the approximate solutions.

Application of numerical grid generation for improved CFD analysis of multiphase screw machines

NASA Astrophysics Data System (ADS)

Rane, S.; Kovačević, A.

2017-08-01

Algebraic grid generation is widely used for discretization of the working domain of twin screw machines. Algebraic grid generation is fast and has good control over the placement of grid nodes. However, the desired qualities of grid which should be able to handle multiphase flows such as oil injection, may be difficult to achieve at times. In order to obtain fast solution of multiphase screw machines, it is important to further improve the quality and robustness of the computational grid. In this paper, a deforming grid of a twin screw machine is generated using algebraic transfinite interpolation to produce initial mesh upon which an elliptic partial differential equations (PDE) of the Poisson’s form is solved numerically to produce smooth final computational mesh. The quality of numerical cells and their distribution obtained by the differential method is greatly improved. In addition, a similar procedure was introduced to fully smoothen the transition of the partitioning rack curve between the rotors thus improving continuous movement of grid nodes and in turn improve robustness and speed of the Computational Fluid Dynamic (CFD) solver. Analysis of an oil injected twin screw compressor is presented to compare the improvements in grid quality factors in the regions of importance such as interlobe space, radial tip and the core of the rotor. The proposed method that combines algebraic and differential grid generation offer significant improvement in grid quality and robustness of numerical solution.

Schwarz maps of algebraic linear ordinary differential equations

NASA Astrophysics Data System (ADS)

Sanabria Malagón, Camilo

2017-12-01

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.

Realization of preconditioned Lanczos and conjugate gradient algorithms on optical linear algebra processors.

PubMed

Ghosh, A

1988-08-01

Lanczos and conjugate gradient algorithms are important in computational linear algebra. In this paper, a parallel pipelined realization of these algorithms on a ring of optical linear algebra processors is described. The flow of data is designed to minimize the idle times of the optical multiprocessor and the redundancy of computations. The effects of optical round-off errors on the solutions obtained by the optical Lanczos and conjugate gradient algorithms are analyzed, and it is shown that optical preconditioning can improve the accuracy of these algorithms substantially. Algorithms for optical preconditioning and results of numerical experiments on solving linear systems of equations arising from partial differential equations are discussed. Since the Lanczos algorithm is used mostly with sparse matrices, a folded storage scheme to represent sparse matrices on spatial light modulators is also described.

Final Report: Subcontract B623868 Algebraic Multigrid solvers for coupled PDE systems

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

Brannick, J.

The Pennsylvania State University (“Subcontractor”) continued to work on the design of algebraic multigrid solvers for coupled systems of partial differential equations (PDEs) arising in numerical modeling of various applications, with a main focus on solving the Dirac equation arising in Quantum Chromodynamics (QCD). The goal of the proposed work was to develop combined geometric and algebraic multilevel solvers that are robust and lend themselves to efficient implementation on massively parallel heterogeneous computers for these QCD systems. The research in these areas built on previous works, focusing on the following three topics: (1) the development of parallel full-multigrid (PFMG) andmore » non-Galerkin coarsening techniques in this frame work for solving the Wilson Dirac system; (2) the use of these same Wilson MG solvers for preconditioning the Overlap and Domain Wall formulations of the Dirac equation; and (3) the design and analysis of algebraic coarsening algorithms for coupled PDE systems including Stokes equation, Maxwell equation and linear elasticity.« less

From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets

NASA Astrophysics Data System (ADS)

Jurčo, Branislav

2012-12-01

Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, such that each Gn is simply connected. We use the 1-jet of the classifying space W¯ G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a differential graded Lie algebra. The differential and the brackets are explicitly described in terms (of a part) of the corresponding k-hypercrossed complex structure of Ng. The result can be seen as a geometric interpretation of Quillen's (purely algebraic) construction of the adjunction between simplicial Lie algebras and dg-Lie algebras.

Learning partial differential equations via data discovery and sparse optimization

NASA Astrophysics Data System (ADS)

Schaeffer, Hayden

2017-01-01

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

Learning partial differential equations via data discovery and sparse optimization.

PubMed

Schaeffer, Hayden

2017-01-01

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

Learning partial differential equations via data discovery and sparse optimization

PubMed Central

2017-01-01

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection. PMID:28265183

Generalized intermediate long-wave hierarchy in zero-curvature representation with noncommutative spectral parameter

NASA Astrophysics Data System (ADS)

Degasperis, A.; Lebedev, D.; Olshanetsky, M.; Pakuliak, S.; Perelomov, A.; Santini, P. M.

1992-11-01

The simplest generalization of the intermediate long-wave hierarchy (ILW) is considered to show how to extend the Zakharov-Shabat dressing method to nonlocal, i.e., integro-partial differential, equations. The purpose is to give a procedure of constructing the zero-curvature representation of this class of equations. This result obtains by combining the Drinfeld-Sokolov formalism together with the introduction of an operator-valued spectral parameter, namely, a spectral parameter that does not commute with the space variable x. This extension provides a connection between the ILWk hierarchy and the Saveliev-Vershik continuum graded Lie algebras. In the case of ILW2 the Fairlie-Zachos sinh-algebra was found.

Continuum Model for River Networks

NASA Astrophysics Data System (ADS)

Giacometti, Achille; Maritan, Amos; Banavar, Jayanth R.

1995-07-01

The effects of erosion, avalanching, and random precipitation are captured in a simple stochastic partial differential equation for modeling the evolution of river networks. Our model leads to a self-organized structured landscape and to abstraction and piracy of the smaller tributaries as the evolution proceeds. An algebraic distribution of the average basin areas and a power law relationship between the drainage basin area and the river length are found.

A calculus based on a q-deformed Heisenberg algebra

DOE PAGES

Cerchiai, B. L.; Hinterding, R.; Madore, J.; ...

1999-04-27

We show how one can construct a differential calculus over an algebra where position variables $x$ and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by cursive Greek chi and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on thismore » derivative differential forms and an exterior differential calculus can be constructed.« less

D{sub {infinity}}-differential E{sub {infinity}}-algebras and spectral sequences of fibrations

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

Lapin, Sergei V

2007-10-31

The notion of an E{sub {infinity}}-algebra with a filtration is introduced. The connections are established between E{sub {infinity}}-algebras with filtrations and the theory of D{sub {infinity}}-differential E{sub {infinity}}-algebras over fields. Based on the technique of D{sub {infinity}}-differential E{sub {infinity}}-algebras, the apparatus of spectral sequences is developed for E{sub {infinity}}-algebras with filtrations, and applications of this apparatus to the multiplicative cohomology spectral sequences of fibrations are given. Bibliography: 21 titles.

Differential calculus and gauge transformations on a deformed space

NASA Astrophysics Data System (ADS)

Wess, Julius

2007-08-01

We consider a formalism by which gauge theories can be constructed on noncommutative space time structures. The coordinates are supposed to form an algebra, restricted by certain requirements that allow us to realise the algebra in terms of star products. In this formulation it is useful to define derivatives and to extend the algebra of coordinates by these derivatives. The elements of this extended algebra are deformed differential operators. We then show that there is a morphism between these deformed differential operators and the usual higher order differential operators acting on functions of commuting coordinates. In this way we obtain deformed gauge transformations and a deformed version of the algebra of diffeomorphisms. The deformation of these algebras can be clearly seen in the category of Hopf algebras. The comultiplication will be twisted. These twisted algebras can be realised on noncommutative spaces and allow the construction of deformed gauge theories and deformed gravity theory.

Partial differential equation-based localization of a monopole source from a circular array.

PubMed

Ando, Shigeru; Nara, Takaaki; Levy, Tsukassa

2013-10-01

Wave source localization from a sensor array has long been the most active research topics in both theory and application. In this paper, an explicit and time-domain inversion method for the direction and distance of a monopole source from a circular array is proposed. The approach is based on a mathematical technique, the weighted integral method, for signal/source parameter estimation. It begins with an exact form of the source-constraint partial differential equation that describes the unilateral propagation of wide-band waves from a single source, and leads to exact algebraic equations that include circular Fourier coefficients (phase mode measurements) as their coefficients. From them, nearly closed-form, single-shot and multishot algorithms are obtained that is suitable for use with band-pass/differential filter banks. Numerical evaluation and several experimental results obtained using a 16-element circular microphone array are presented to verify the validity of the proposed method.

On differential operators generating iterative systems of linear ODEs of maximal symmetry algebra

NASA Astrophysics Data System (ADS)

Ndogmo, J. C.

2017-06-01

Although every iterative scalar linear ordinary differential equation is of maximal symmetry algebra, the situation is different and far more complex for systems of linear ordinary differential equations, and an iterative system of linear equations need not be of maximal symmetry algebra. We illustrate these facts by examples and derive families of vector differential operators whose iterations are all linear systems of equations of maximal symmetry algebra. Some consequences of these results are also discussed.

Generalized EMV-Effect Algebras

NASA Astrophysics Data System (ADS)

Borzooei, R. A.; Dvurečenskij, A.; Sharafi, A. H.

2018-04-01

Recently in Dvurečenskij and Zahiri (2017), new algebraic structures, called EMV-algebras which generalize both MV-algebras and generalized Boolean algebras, were introduced. We present equivalent conditions for EMV-algebras. In addition, we define a partial algebraic structure, called a generalized EMV-effect algebra, which is close to generalized MV-effect algebras. Finally, we show that every generalized EMV-effect algebra is either an MV-effect algebra or can be embedded into an MV-effect algebra as a maximal ideal.

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

NASA Astrophysics Data System (ADS)

Saveliev, M. V.; Vershik, A. M.

1989-12-01

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations of Z-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras “continuum Lie algebras.” The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.

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

Mozrzymas, Marek; Horodecki, Michał; Studziński, Michał

We consider the structure of algebra of operators, acting in n-fold tensor product space, which are partially transposed on the last term. Using purely algebraical methods we show that this algebra is semi-simple and then, considering its regular representation, we derive basic properties of the algebra. In particular, we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations. It appears that there are two kinds of irreducible representations of the algebra. The first one is strictly connected with the representations of the group S(n − 1) induced by irreduciblemore » representations of the group S(n − 2). The second kind is structurally connected with irreducible representations of the group S(n − 1)« less

Algebra and topology for applications to physics

NASA Technical Reports Server (NTRS)

Rozhkov, S. S.

1987-01-01

The principal concepts of algebra and topology are examined with emphasis on applications to physics. In particular, attention is given to sets and mapping; topological spaces and continuous mapping; manifolds; and topological groups and Lie groups. The discussion also covers the tangential spaces of the differential manifolds, including Lie algebras, vector fields, and differential forms, properties of differential forms, mapping of tangential spaces, and integration of differential forms.

Partially Flipped Linear Algebra: A Team-Based Approach

ERIC Educational Resources Information Center

Carney, Debra; Ormes, Nicholas; Swanson, Rebecca

2015-01-01

In this article we describe a partially flipped Introductory Linear Algebra course developed by three faculty members at two different universities. We give motivation for our partially flipped design and describe our implementation in detail. Two main features of our course design are team-developed preview videos and related in-class activities.…

Structure of Lie point and variational symmetry algebras for a class of odes

NASA Astrophysics Data System (ADS)

Ndogmo, J. C.

2018-04-01

It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation y(n)=0. For arbitrary systems of ordinary differential equations of order n ≥ 3 reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.

Hidden role of Maxwell superalgebras in the free differential algebras of D = 4 and D = 11 supergravity

NASA Astrophysics Data System (ADS)

Ravera, Lucrezia

2018-03-01

The purpose of this paper is to show that the so-called Maxwell superalgebra in four dimensions, which naturally involves the presence of a nilpotent fermionic generator, can be interpreted as a hidden superalgebra underlying N=1, {D}=4 supergravity extended to include a 2-form gauge potential associated to a 2-index antisymmetric tensor. In this scenario, the theory is appropriately discussed in the context of Free Differential Algebras (an extension of the Maurer-Cartan equations to involve higher-degree differential forms). The study is then extended to the Free Differential Algebra describing D = 11 supergravity, showing that, also in this case, there exists a super-Maxwell algebra underlying the theory. The same extra spinors dual to the nilpotent fermionic generators whose presence is crucial for writing a supersymmetric extension of the Maxwell algebras, both in the D = 4 and in the D = 11 case, turn out to be fundamental ingredients also to reproduce the D = 4 and D = 11 Free Differential Algebras on ordinary superspace, whose basis is given by the supervielbein. The analysis of the gauge structure of the supersymmetric Free Differential Algebras is carried on taking into account the gauge transformations from the hidden supergroup-manifold associated with the Maxwell superalgebras.

Analysis on singular spaces: Lie manifolds and operator algebras

NASA Astrophysics Data System (ADS)

Nistor, Victor

2016-07-01

We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference Noncommutative geometry and applications, Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds-called ;Lie manifolds; -that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here-work that spans over close to two decades-was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.

Three-dimensional vibration analysis of a uniform beam with offset inertial masses at the ends

NASA Technical Reports Server (NTRS)

Robertson, D. K.

1985-01-01

Analysis of a flexible beam with displaced end-located inertial masses is presented. The resulting three-dimensional mode shape is shown to consist of two one-plane bending modes and one torsional mode. These three components of the mode shapes are shown to be linear combinations of trigonometric and hyperbolic sine and cosine functions. Boundary conditions are derived to obtain nonlinear algebraic equations through kinematic coupling of the general solutions of the three governing partial differential equations. A method of solution which takes these boundary conditions into account is also presented. A computer program has been written to obtain unique solutions to the resulting nonlinear algebraic equations. This program, which calculates natural frequencies and three-dimensional mode shapes for any number of modes, is presented and discussed.

D{sub {infinity}}-differential A{sub {infinity}}-algebras and spectral sequences

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

Lapin, S V

2002-02-28

In the present paper the construction of a D{sub {infinity}}-differential A{sub {infinity}}-(co)algebra is introduced and basic homotopy properties of this construction are studied. The connection between D{sub {infinity}}-differential A{sub {infinity}}-(co)algebras and spectral sequences is established, which enables us to construct the structure of an A{sub {infinity}} -coalgebra on the Milnor coalgebra directly from the differentials of the Adams spectral sequence.

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

Suh, Uhi Rinn, E-mail: uhrisu1@math.snu.ac.kr

We introduce a classical BRST complex (See Definition 3.2.) and show that one can construct a classical affine W-algebra via the complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible λ-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we describe explicit forms ofmore » free generators and compute λ-brackets between them. Provided some assumptions on a classical affine fractional W-algebra, we find an infinite sequence of integrable systems related to the algebra, using the generalized Drinfel’d and Sokolov reduction.« less

A Loomis-Sikorski theorem and functional calculus for a generalized Hermitian algebra

NASA Astrophysics Data System (ADS)

Foulis, David J.; Jenčová, Anna; Pulmannová, Sylvia

2017-10-01

A generalized Hermitian (GH-) algebra is a generalization of the partially ordered Jordan algebra of all Hermitian operators on a Hilbert space. We introduce the notion of a gh-tribe, which is a commutative GH-algebra of functions on a nonempty set X with pointwise partial order and operations, and we prove that every commutative GH-algebra is the image of a gh-tribe under a surjective GH-morphism. Using this result, we prove that each element a of a GH-algebra A corresponds to a real observable ξa on the σ-orthomodular lattice of projections in A and that ξa determines the spectral resolution of a. Also, if f is a continuous function defined on the spectrum of a, we formulate a definition of f (a), thus obtaining a continuous functional calculus for A.

A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations

NASA Astrophysics Data System (ADS)

Whiteley, J. P.

2017-10-01

Large, incompressible elastic deformations are governed by a system of nonlinear partial differential equations. The finite element discretisation of these partial differential equations yields a system of nonlinear algebraic equations that are usually solved using Newton's method. On each iteration of Newton's method, a linear system must be solved. We exploit the structure of the Jacobian matrix to propose a preconditioner, comprising two steps. The first step is the solution of a relatively small, symmetric, positive definite linear system using the preconditioned conjugate gradient method. This is followed by a small number of multigrid V-cycles for a larger linear system. Through the use of exemplar elastic deformations, the preconditioner is demonstrated to facilitate the iterative solution of the linear systems arising. The number of GMRES iterations required has only a very weak dependence on the number of degrees of freedom of the linear systems.

A simplified formalism of the algebra of partially transposed permutation operators with applications

NASA Astrophysics Data System (ADS)

Mozrzymas, Marek; Studziński, Michał; Horodecki, Michał

2018-03-01

Herein we continue the study of the representation theory of the algebra of permutation operators acting on the n -fold tensor product space, partially transposed on the last subsystem. We develop the concept of partially reduced irreducible representations, which allows us to significantly simplify previously proved theorems and, most importantly, derive new results for irreducible representations of the mentioned algebra. In our analysis we are able to reduce the complexity of the central expressions by getting rid of sums over all permutations from the symmetric group, obtaining equations which are much more handy in practical applications. We also find relatively simple matrix representations for the generators of the underlying algebra. The obtained simplifications and developments are applied to derive the characteristics of a deterministic port-based teleportation scheme written purely in terms of irreducible representations of the studied algebra. We solve an eigenproblem for the generators of the algebra, which is the first step towards a hybrid port-based teleportation scheme and gives us new proofs of the asymptotic behaviour of teleportation fidelity. We also show a connection between the density operator characterising port-based teleportation and a particular matrix composed of an irreducible representation of the symmetric group, which encodes properties of the investigated algebra.

A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras

NASA Astrophysics Data System (ADS)

Alshammari, Fahad; Isaac, Phillip S.; Marquette, Ian

2018-02-01

We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and (2) the Schrödinger Lie algebras. We find that an abstract form of dimensional analysis assists us in our algorithm, and greatly reduces the complexity of the problem.

Inverse Scattering and Applications. Proceedings of Conference on Inverse Scattering on the Line, Held in Amherst, Massachusetts on June 7 - 13, 1990

DTIC Science & Technology

1990-01-01

J. Laurie Snell S. A. Amitsur, D. J. Saltman, and 2 Proceedings of the conference on G. B. Seligman , Editors integration, topology, and geometry in...Rational constructions of modules 17 Nonlinear partial differential equations. for simple Lie algebras, George B. Joel A. Smoller, Editor Seligman 18...number theory, Michael R. Stein and Linda Keen, Editor R. Keith Dennis, Editors 65 Logic and combinatorics, Stephen G. 84 Partition problems in

Extended gauge theory and gauged free differential algebras

NASA Astrophysics Data System (ADS)

Salgado, P.; Salgado, S.

2018-01-01

Recently, Antoniadis, Konitopoulos and Savvidy introduced, in the context of the so-called extended gauge theory, a procedure to construct background-free gauge invariants, using non-abelian gauge potentials described by higher degree forms. In this article it is shown that the extended invariants found by Antoniadis, Konitopoulos and Savvidy can be constructed from an algebraic structure known as free differential algebra. In other words, we show that the above mentioned non-abelian gauge theory, where the gauge fields are described by p-forms with p ≥ 2, can be obtained by gauging free differential algebras.

A numerical method to solve the 1D and the 2D reaction diffusion equation based on Bessel functions and Jacobian free Newton-Krylov subspace methods

NASA Astrophysics Data System (ADS)

Parand, K.; Nikarya, M.

2017-11-01

In this paper a novel method will be introduced to solve a nonlinear partial differential equation (PDE). In the proposed method, we use the spectral collocation method based on Bessel functions of the first kind and the Jacobian free Newton-generalized minimum residual (JFNGMRes) method with adaptive preconditioner. In this work a nonlinear PDE has been converted to a nonlinear system of algebraic equations using the collocation method based on Bessel functions without any linearization, discretization or getting the help of any other methods. Finally, by using JFNGMRes, the solution of the nonlinear algebraic system is achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the famous Fisher equation. We compare our results with other methods.

Finite-analytic numerical solution of heat transfer in two-dimensional cavity flow

NASA Technical Reports Server (NTRS)

Chen, C.-J.; Naseri-Neshat, H.; Ho, K.-S.

1981-01-01

Heat transfer in cavity flow is numerically analyzed by a new numerical method called the finite-analytic method. The basic idea of the finite-analytic method is the incorporation of local analytic solutions in the numerical solutions of linear or nonlinear partial differential equations. In the present investigation, the local analytic solutions for temperature, stream function, and vorticity distributions are derived. When the local analytic solution is evaluated at a given nodal point, it gives an algebraic relationship between a nodal value in a subregion and its neighboring nodal points. A system of algebraic equations is solved to provide the numerical solution of the problem. The finite-analytic method is used to solve heat transfer in the cavity flow at high Reynolds number (1000) for Prandtl numbers of 0.1, 1, and 10.

Low dose reconstruction algorithm for differential phase contrast imaging.

PubMed

Wang, Zhentian; Huang, Zhifeng; Zhang, Li; Chen, Zhiqiang; Kang, Kejun; Yin, Hongxia; Wang, Zhenchang; Marco, Stampanoni

2011-01-01

Differential phase contrast imaging computed tomography (DPCI-CT) is a novel x-ray inspection method to reconstruct the distribution of refraction index rather than the attenuation coefficient in weakly absorbing samples. In this paper, we propose an iterative reconstruction algorithm for DPCI-CT which benefits from the new compressed sensing theory. We first realize a differential algebraic reconstruction technique (DART) by discretizing the projection process of the differential phase contrast imaging into a linear partial derivative matrix. In this way the compressed sensing reconstruction problem of DPCI reconstruction can be transformed to a resolved problem in the transmission imaging CT. Our algorithm has the potential to reconstruct the refraction index distribution of the sample from highly undersampled projection data. Thus it can significantly reduce the dose and inspection time. The proposed algorithm has been validated by numerical simulations and actual experiments.

Analysis of backward differentiation formula for nonlinear differential-algebraic equations with 2 delays.

PubMed

Sun, Leping

2016-01-01

This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true.

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

DTIC Science & Technology

2014-11-01

linear hybrid systems by linear algebraic methods. In SAS, volume 6337 of LNCS, pages 373–389. Springer, 2010. [19] E. W. Mayr. Membership in polynomial...383–394, 2009. [31] A. Tarski. A decision method for elementary algebra and geometry. Bull. Amer. Math. Soc., 59, 1951. [32] A. Tiwari. Abstractions...A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 November 2014 CMU

Realization of Uq(sp(2n)) within the Differential Algebra on Quantum Symplectic Space

NASA Astrophysics Data System (ADS)

Zhang, Jiao; Hu, Naihong

2017-10-01

We realize the Hopf algebra U_q({sp}_{2n}) as an algebra of quantum differential operators on the quantum symplectic space X(f_s;R) and prove that X(f_s;R) is a U_q({sp}_{2n})-module algebra whose irreducible summands are just its homogeneous subspaces. We give a coherence realization for all the positive root vectors under the actions of Lusztig's braid automorphisms of U_q({sp}_{2n}).

Super-Laplacians and their symmetries

NASA Astrophysics Data System (ADS)

Howe, P. S.; Lindström, U.

2017-05-01

A super-Laplacian is a set of differential operators in superspace whose highestdimensional component is given by the spacetime Laplacian. Symmetries of super-Laplacians are given by linear differential operators of arbitrary finite degree and are determined by superconformal Killing tensors. We investigate these in flat superspaces. The differential operators determining the symmetries give rise to algebras which can be identified in many cases with the tensor algebras of the relevant superconformal Lie algebras modulo certain ideals. They have applications to Higher Spin theories.

Using trees to compute approximate solutions to ordinary differential equations exactly

NASA Technical Reports Server (NTRS)

Grossman, Robert

1991-01-01

Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.

Labeled trees and the efficient computation of derivations

NASA Technical Reports Server (NTRS)

Grossman, Robert; Larson, Richard G.

1989-01-01

The effective parallel symbolic computation of operators under composition is discussed. Examples include differential operators under composition and vector fields under the Lie bracket. Data structures consisting of formal linear combinations of rooted labeled trees are discussed. A multiplication on rooted labeled trees is defined, thereby making the set of these data structures into an associative algebra. An algebra homomorphism is defined from the original algebra of operators into this algebra of trees. An algebra homomorphism from the algebra of trees into the algebra of differential operators is then described. The cancellation which occurs when noncommuting operators are expressed in terms of commuting ones occurs naturally when the operators are represented using this data structure. This leads to an algorithm which, for operators which are derivations, speeds up the computation exponentially in the degree of the operator. It is shown that the algebra of trees leads naturally to a parallel version of the algorithm.

TDIGG - TWO-DIMENSIONAL, INTERACTIVE GRID GENERATION CODE

NASA Technical Reports Server (NTRS)

Vu, B. T.

1994-01-01

TDIGG is a fast and versatile program for generating two-dimensional computational grids for use with finite-difference flow-solvers. Both algebraic and elliptic grid generation systems are included. The method for grid generation by algebraic transformation is based on an interpolation algorithm and the elliptic grid generation is established by solving the partial differential equation (PDE). Non-uniform grid distributions are carried out using a hyperbolic tangent stretching function. For algebraic grid systems, interpolations in one direction (univariate) and two directions (bivariate) are considered. These interpolations are associated with linear or cubic Lagrangian/Hermite/Bezier polynomial functions. The algebraic grids can subsequently be smoothed using an elliptic solver. For elliptic grid systems, the PDE can be in the form of Laplace (zero forcing function) or Poisson. The forcing functions in the Poisson equation come from the boundary or the entire domain of the initial algebraic grids. A graphics interface procedure using the Silicon Graphics (GL) Library is included to allow users to visualize the grid variations at each iteration. This will allow users to interactively modify the grid to match their applications. TDIGG is written in FORTRAN 77 for Silicon Graphics IRIS series computers running IRIX. This package requires either MIT's X Window System, Version 11 Revision 4 or SGI (Motif) Window System. A sample executable is provided on the distribution medium. It requires 148K of RAM for execution. The standard distribution medium is a .25 inch streaming magnetic IRIX tape cartridge in UNIX tar format. This program was developed in 1992.

On Fock-space representations of quantized enveloping algebras related to noncommutative differential geometry

NASA Astrophysics Data System (ADS)

Jurčo, B.; Schlieker, M.

1995-07-01

In this paper explicitly natural (from the geometrical point of view) Fock-space representations (contragradient Verma modules) of the quantized enveloping algebras are constructed. In order to do so, one starts from the Gauss decomposition of the quantum group and introduces the differential operators on the corresponding q-deformed flag manifold (assumed as a left comodule for the quantum group) by a projection to it of the right action of the quantized enveloping algebra on the quantum group. Finally, the representatives of the elements of the quantized enveloping algebra corresponding to the left-invariant vector fields on the quantum group are expressed as first-order differential operators on the q-deformed flag manifold.

Moving frames and prolongation algebras

NASA Technical Reports Server (NTRS)

Estabrook, F. B.

1982-01-01

Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.

A least-squares finite element method for 3D incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Lin, T. L.; Hou, Lin-Jun; Povinelli, Louis A.

1993-01-01

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

Acoustooptic linear algebra processors - Architectures, algorithms, and applications

NASA Technical Reports Server (NTRS)

Casasent, D.

1984-01-01

Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.

Lie algebras and linear differential equations.

NASA Technical Reports Server (NTRS)

Brockett, R. W.; Rahimi, A.

1972-01-01

Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

On the computation of steady Hopper flows. II: von Mises materials in various geometries

NASA Astrophysics Data System (ADS)

Gremaud, Pierre A.; Matthews, John V.; O'Malley, Meghan

2004-11-01

Similarity solutions are constructed for the flow of granular materials through hoppers. Unlike previous work, the present approach applies to nonaxisymmetric containers. The model involves ten unknowns (stresses, velocity, and plasticity function) determined by nine nonlinear first order partial differential equations together with a quadratic algebraic constraint (yield condition). A pseudospectral discretization is applied; the resulting problem is solved with a trust region method. The important role of the hopper geometry on the flow is illustrated by several numerical experiments of industrial relevance.

Three-dimensional analytic model of the magnetic field for the Chalk River Superconducting Cyclotron

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

Davies, W.G.; Lee-Whiting, G.E.; Douglas, S.R.

1994-07-01

A three-dimensional analytic model of the magnetic field for the TASCC cyclotron that satisfies Maxwell`s equations exactly has been constructed for use with the new differential-algebra orbit-dynamics code. The model includes: (1) the superconducting coils; (2) the saturated iron poles; (3) the partially saturated yoke; (4) the saturated-iron trim rods. Lines of dipole density along the edges of the hills account for the non-uniformities and edge effects and along with three yoke constants constitute the only free parameters.

Heat transfer in a conical porous medium due to inner and top surface heating: Effect of radius ratio

NASA Astrophysics Data System (ADS)

Ahamad, N. Ameer; Khan, T. M. Yunus

2018-05-01

The present study investigates the effect of radius ratio and Rayleigh number on beat transfer characteristics of an annular cone subjected to two side heating and one side cooling. Finite element method is used to convert the partial differential equations into algebraic equations. The resulting equations are solved with the help of in-house computer code developed for specific purpose of heat transfer in conical porous medium. The results are discussed with respect to the radius ratio and Rayleigh number.

Diagnosing students' misconceptions in algebra: results from an experimental pilot study.

PubMed

Russell, Michael; O'Dwyer, Laura M; Miranda, Helena

2009-05-01

Computer-based diagnostic assessment systems hold potential to help teachers identify sources of poor performance and to connect teachers and students to learning activities designed to help advance students' conceptual understandings. The present article presents findings from a study that examined how students' performance in algebra and their overcoming of common algebraic misconceptions were affected by the use of a diagnostic assessment system that focused on important algebra concepts. This study used a four-group randomized cluster trial design in which teachers were assigned randomly to one of four groups: a "business as usual" control group, a partial intervention group that was provided with access to diagnostic tests results, a partial intervention group that was provided with access to the learning activities, and a full intervention group that was given access to the test results and learning activities. Data were collected from 905 students (6th-12th grade) nested within 44 teachers. We used hierarchical linear modeling techniques to compare the effects of full, partial, and no (control) intervention on students' algebraic ability and misconceptions. The analyses indicate that full intervention had a net positive effect on ability and misconception measures.

Derive Workshop Matrix Algebra and Linear Algebra.

ERIC Educational Resources Information Center

Townsley Kulich, Lisa; Victor, Barbara

This document presents the course content for a workshop that integrates the use of the computer algebra system Derive with topics in matrix and linear algebra. The first section is a guide to using Derive that provides information on how to write algebraic expressions, make graphs, save files, edit, define functions, differentiate expressions,…

A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian decomposition method.

PubMed

Benhammouda, Brahim

2016-01-01

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

Mathematical model for Dengue with three states of infection

NASA Astrophysics Data System (ADS)

Hincapie, Doracelly; Ospina, Juan

2012-06-01

A mathematical model for dengue with three states of infection is proposed and analyzed. The model consists in a system of differential equations. The three states of infection are respectively asymptomatic, partially asymptomatic and fully asymptomatic. The model is analyzed using computer algebra software, specifically Maple, and the corresponding basic reproductive number and the epidemic threshold are computed. The resulting basic reproductive number is an algebraic synthesis of all epidemic parameters and it makes clear the possible control measures. The microscopic structure of the epidemic parameters is established using the quantum theory of the interactions between the atoms and radiation. In such approximation, the human individual is represented by an atom and the mosquitoes are represented by radiation. The force of infection from the mosquitoes to the humans is considered as the transition probability from the fundamental state of atom to excited states. The combination of computer algebra software and quantum theory provides a very complete formula for the basic reproductive number and the possible control measures tending to stop the propagation of the disease. It is claimed that such result may be important in military medicine and the proposed method can be applied to other vector-borne diseases.

FAST TRACK COMMUNICATION: On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

NASA Astrophysics Data System (ADS)

Man, Yiu-Kwong

2010-10-01

In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

NASA Astrophysics Data System (ADS)

Abdelaziz, Y.; Maillard, J.-M.

2017-05-01

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same {}_2F1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus {}_2F1 hypergeometric function example. We then focus on identities relating the same {}_2F1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to {}_3F2 , hypergeometric functions, and show that one just reduces to the previous {}_2F1 cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a _2F1 hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or {}_2F1 hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

Modified non-Abelian Toda field equations and twisted quasigraded Lie algebras

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

Skrypnyk, T.

We construct a new family of quasigraded Lie algebras that admit the Kostant-Adler scheme. They coincide with special quasigraded deformations of twisted subalgebras of the loop algebras. Using them we obtain new hierarchies of integrable equations in partial derivatives which we call 'modified' non-Abelian Toda field hierarchies.

Block algebra in two-component BKP and D type Drinfeld-Sokolov hierarchies

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

Li, Chuanzhong, E-mail: lichuanzhong@nbu.edu.cn; He, Jingsong, E-mail: hejingsong@nbu.edu.cn

We construct generalized additional symmetries of a two-component BKP hierarchy defined by two pseudo-differential Lax operators. These additional symmetry flows form a Block type algebra with some modified (or additional) terms because of a B type reduction condition of this integrable hierarchy. Further we show that the D type Drinfeld-Sokolov hierarchy, which is a reduction of the two-component BKP hierarchy, possess a complete Block type additional symmetry algebra. That D type Drinfeld-Sokolov hierarchy has a similar algebraic structure as the bigraded Toda hierarchy which is a differential-discrete integrable system.

Hopf-algebraic structure of combinatorial objects and differential operators

NASA Technical Reports Server (NTRS)

Grossman, Robert; Larson, Richard G.

1989-01-01

A Hopf-algebraic structure on a vector space which has as basis a family of trees is described. Some applications of this structure to combinatorics and to differential operators are surveyed. Some possible future directions for this work are indicated.

A Textbook for a First Course in Computational Fluid Dynamics

NASA Technical Reports Server (NTRS)

Zingg, D. W.; Pulliam, T. H.; Nixon, David (Technical Monitor)

1999-01-01

This paper describes and discusses the textbook, Fundamentals of Computational Fluid Dynamics by Lomax, Pulliam, and Zingg, which is intended for a graduate level first course in computational fluid dynamics. This textbook emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. Its underlying philosophy is that the theory of linear algebra and the attendant eigenanalysis of linear systems provides a mathematical framework to describe and unify most numerical methods in common use in the field of fluid dynamics. Two linear model equations, the linear convection and diffusion equations, are used to illustrate concepts throughout. Emphasis is on the semi-discrete approach, in which the governing partial differential equations (PDE's) are reduced to systems of ordinary differential equations (ODE's) through a discretization of the spatial derivatives. The ordinary differential equations are then reduced to ordinary difference equations (O(Delta)E's) using a time-marching method. This methodology, using the progression from PDE through ODE's to O(Delta)E's, together with the use of the eigensystems of tridiagonal matrices and the theory of O(Delta)E's, gives the book its distinctiveness and provides a sound basis for a deep understanding of fundamental concepts in computational fluid dynamics.

Contractions from grading

NASA Astrophysics Data System (ADS)

Krishnan, Chethan; Raju, Avinash

2018-04-01

We note that large classes of contractions of algebras that arise in physics can be understood purely algebraically via identifying appropriate Zm-gradings (and their generalizations) on the parent algebra. This includes various types of flat space/Carroll limits of finite and infinite dimensional (A)dS algebras, as well as Galilean and Galilean conformal algebras. Our observations can be regarded as providing a natural context for the Grassmann approach of Krishnan et al. [J. High Energy Phys. 2014(3), 36]. We also introduce a related notion, which we call partial grading, that arises naturally in this context.

Algebraic invariant curves of plane polynomial differential systems

NASA Astrophysics Data System (ADS)

Tsygvintsev, Alexei

2001-01-01

We consider a plane polynomial vector field P(x,y) dx + Q(x,y) dy of degree m>1. With each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential ω = dx/P = dy/Q. The asymptotic estimate of the degree of an arbitrary algebraic invariant curve is found. In the smooth case this estimate has already been found by Cerveau and Lins Neto in a different way.

Computer Algebra Systems in Undergraduate Instruction.

ERIC Educational Resources Information Center

Small, Don; And Others

1986-01-01

Computer algebra systems (such as MACSYMA and muMath) can carry out many of the operations of calculus, linear algebra, and differential equations. Use of them with sketching graphs of rational functions and with other topics is discussed. (MNS)

An abbreviated Reynolds stress turbulence model for airfoil flows

NASA Technical Reports Server (NTRS)

Gaffney, R. L., Jr.; Hassan, H. A.; Salas, M. D.

1990-01-01

An abbreviated Reynolds stress turbulence model is presented for solving turbulent flow over airfoils. The model consists of two partial differential equations, one for the Reynolds shear stress and the other for the turbulent kinetic energy. The normal stresses and the dissipation rate of turbulent kinetic energy are computed from algebraic relationships having the correct asymptotic near wall behavior. This allows the model to be integrated all the way to the wall without the use of wall functions. Results for a flat plate at zero angle of attack, a NACA 0012 airfoil and a RAE 2822 airfoil are presented.

Higher symmetries and exact solutions of linear and nonlinear Schr{umlt o}dinger equation

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

Fushchych, W.I.; Nikitin, A.G.

1997-11-01

A new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries. Higher symmetries of the Schr{umlt o}dinger equation with an arbitrary potential are investigated. Nonlinear determining equations for potentials are solved using reductions to Weierstrass, Painlev{acute e}, and Riccati forms. Algebraic properties of higher order symmetry operators are analyzed. Combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear Schr{umlt o}dinger equations. {copyright} {ital 1997 American Institute of Physics.}

Optimal control of first order distributed systems. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Johnson, T. L.

1972-01-01

The problem of characterizing optimal controls for a class of distributed-parameter systems is considered. The system dynamics are characterized mathematically by a finite number of coupled partial differential equations involving first-order time and space derivatives of the state variables, which are constrained at the boundary by a finite number of algebraic relations. Multiple control inputs, extending over the entire spatial region occupied by the system ("distributed controls') are to be designed so that the response of the system is optimal. A major example involving boundary control of an unstable low-density plasma is developed from physical laws.

Symbolic computer vector analysis

NASA Technical Reports Server (NTRS)

Stoutemyer, D. R.

1977-01-01

A MACSYMA program is described which performs symbolic vector algebra and vector calculus. The program can combine and simplify symbolic expressions including dot products and cross products, together with the gradient, divergence, curl, and Laplacian operators. The distribution of these operators over sums or products is under user control, as are various other expansions, including expansion into components in any specific orthogonal coordinate system. There is also a capability for deriving the scalar or vector potential of a vector field. Examples include derivation of the partial differential equations describing fluid flow and magnetohydrodynamics, for 12 different classic orthogonal curvilinear coordinate systems.

The exact fundamental solution for the Benes tracking problem

NASA Astrophysics Data System (ADS)

Balaji, Bhashyam

2009-05-01

The universal continuous-discrete tracking problem requires the solution of a Fokker-Planck-Kolmogorov forward equation (FPKfe) for an arbitrary initial condition. Using results from quantum mechanics, the exact fundamental solution for the FPKfe is derived for the state model of arbitrary dimension with Benes drift that requires only the computation of elementary transcendental functions and standard linear algebra techniques- no ordinary or partial differential equations need to be solved. The measurement process may be an arbitrary, discrete-time nonlinear stochastic process, and the time step size can be arbitrary. Numerical examples are included, demonstrating its utility in practical implementation.

On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

NASA Technical Reports Server (NTRS)

Gunzburger, M. D.; Nicolaides, R. A.

1986-01-01

Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.

Classical Affine W-Superalgebras via Generalized Drinfeld-Sokolov Reductions and Related Integrable Systems

NASA Astrophysics Data System (ADS)

Suh, Uhi Rinn

2018-02-01

The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel'd-Sokolov (D-S) reduction associated to a Lie superalgebra g and its even nilpotent element f, and we find a new definition of the classical affine W-superalgebra W(g,f,k) via the D-S reduction. This new construction allows us to find free generators of W(g,f,k), as a differential superalgebra, and two independent Lie brackets on W(g,f,k)/partial W(g,f,k). Moreover, we describe super-Hamiltonian systems with the Poisson vertex algebras theory. A W-superalgebra with certain properties can be understood as an underlying differential superalgebra of a series of integrable super-Hamiltonian systems.

a Triangular Deformation of the Two-Dimensional POINCARÉ Algebra

NASA Astrophysics Data System (ADS)

Khorrami, M.; Shariati, A.; Abolhassani, M. R.; Aghamohammadi, A.

Contracting the h-deformation of SL(2, ℝ), we construct a new deformation of two-dimensional Poincaré's algebra, the algebra of functions on its group and its differential structure. It is seen that these dual Hopf algebras are isomorphic to each other. It is also shown that the Hopf algebra is triangular, and its universal R-matrix is also constructed explicitly. We then find a deformation map for the universal enveloping algebra, and at the end, give the deformed mass shells and Lorentz transformation.

Bisimulation equivalence of differential-algebraic systems

NASA Astrophysics Data System (ADS)

Megawati, Noorma Yulia; Schaft, Arjan van der

2018-01-01

In this paper, the notion of bisimulation relation for linear input-state-output systems is extended to general linear differential-algebraic (DAE) systems. Geometric control theory is used to derive a linear-algebraic characterisation of bisimulation relations, and an algorithm for computing the maximal bisimulation relation between two linear DAE systems. The general definition is specialised to the case where the matrix pencil sE - A is regular. Furthermore, by developing a one-sided version of bisimulation, characterisations of simulation and abstraction are obtained.

A representation of solution of stochastic differential equations

NASA Astrophysics Data System (ADS)

Kim, Yoon Tae; Jeon, Jong Woo

2006-03-01

We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.

Differential calculus on quantized simple lie groups

NASA Astrophysics Data System (ADS)

Jurčo, Branislav

1991-07-01

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q ∈ ℝ are also discussed.

Using Computer Symbolic Algebra to Solve Differential Equations.

ERIC Educational Resources Information Center

Mathews, John H.

1989-01-01

This article illustrates that mathematical theory can be incorporated into the process to solve differential equations by a computer algebra system, muMATH. After an introduction to functions of muMATH, several short programs for enhancing the capabilities of the system are discussed. Listed are six references. (YP)

Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces

NASA Astrophysics Data System (ADS)

Meljanac, Stjepan; Krešić–Jurić, Saša; Martinić, Tea

2017-07-01

This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra g0, we construct a Lie superalgebra g =g0⊕g1 containing noncommutative coordinates and one-forms. We show that g can be extended by a set of generators TAB whose action on the enveloping algebra U (g ) gives the commutation relations between monomials in U (g0 ) and one-forms. Realizations of noncommutative coordinates, one-forms, and the generators TAB as formal power series in a semicompleted Weyl superalgebra are found. In the special case dim(g0 ) =dim(g1 ) , we also find a realization of the exterior derivative on U (g0 ) . The realizations of these geometric objects yield a bicovariant differential calculus on U (g0 ) as a deformation of the standard calculus on the Euclidean space.

Natural differential operations on manifolds: an algebraic approach

NASA Astrophysics Data System (ADS)

Katsylo, P. I.; Timashev, D. A.

2008-10-01

Natural algebraic differential operations on geometric quantities on smooth manifolds are considered. A method for the investigation and classification of such operations is described, the method of IT-reduction. With it the investigation of natural operations reduces to the analysis of rational maps between k-jet spaces, which are equivariant with respect to certain algebraic groups. On the basis of the method of IT-reduction a finite generation theorem is proved: for tensor bundles \\mathscr{V},\\mathscr{W}\\to M all the natural differential operations D\\colon\\Gamma(\\mathscr{V})\\to\\Gamma(\\mathscr{W}) of degree at most d can be algebraically constructed from some finite set of such operations. Conceptual proofs of known results on the classification of natural linear operations on arbitrary and symplectic manifolds are presented. A non-existence theorem is proved for natural deformation quantizations on Poisson manifolds and symplectic manifolds.Bibliography: 21 titles.

Grid generation for the solution of partial differential equations

NASA Technical Reports Server (NTRS)

Eiseman, Peter R.; Erlebacher, Gordon

1989-01-01

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given.

Grid generation for the solution of partial differential equations

NASA Technical Reports Server (NTRS)

Eiseman, Peter R.; Erlebacher, Gordon

1987-01-01

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given.

State-of-charge estimation in lithium-ion batteries: A particle filter approach

NASA Astrophysics Data System (ADS)

Tulsyan, Aditya; Tsai, Yiting; Gopaluni, R. Bhushan; Braatz, Richard D.

2016-11-01

The dynamics of lithium-ion batteries are complex and are often approximated by models consisting of partial differential equations (PDEs) relating the internal ionic concentrations and potentials. The Pseudo two-dimensional model (P2D) is one model that performs sufficiently accurately under various operating conditions and battery chemistries. Despite its widespread use for prediction, this model is too complex for standard estimation and control applications. This article presents an original algorithm for state-of-charge estimation using the P2D model. Partial differential equations are discretized using implicit stable algorithms and reformulated into a nonlinear state-space model. This discrete, high-dimensional model (consisting of tens to hundreds of states) contains implicit, nonlinear algebraic equations. The uncertainty in the model is characterized by additive Gaussian noise. By exploiting the special structure of the pseudo two-dimensional model, a novel particle filter algorithm that sweeps in time and spatial coordinates independently is developed. This algorithm circumvents the degeneracy problems associated with high-dimensional state estimation and avoids the repetitive solution of implicit equations by defining a 'tether' particle. The approach is illustrated through extensive simulations.

Tensor calculus in polar coordinates using Jacobi polynomials

NASA Astrophysics Data System (ADS)

Vasil, Geoffrey M.; Burns, Keaton J.; Lecoanet, Daniel; Olver, Sheehan; Brown, Benjamin P.; Oishi, Jeffrey S.

2016-11-01

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.

Combinatorial Formulas for Characteristic Classes, and Localization of Secondary Topological Invariants.

NASA Astrophysics Data System (ADS)

Smirnov, Mikhail

1995-01-01

The problems solved in this thesis originated from combinatorial formulas for characteristic classes. This thesis deals with Chern-Simons classes, their generalizations and related algebraic and analytic problems. (1) In this thesis, I describe a new class of algebras whose elements contain Chern and generalized Chern -Simons classes. There is a Poisson bracket in these algebras, similar to the bracket in Kontsevich's noncommutative symplectic geometry (Kon). I prove that the Poisson bracket gives rise to a graded Lie algebra containing differential forms representing Chern and Chern-Simons classes. This is a new result. I describe algebraic analogs of the dilogarithm and higher polylogarithms in the algebra corresponding to Chern-Simons classes. (2) I study the properties of this bracket. It is possible to write the exterior differential and other operations in the algebra using this bracket. The bracket of any two Chern classes is zero and the bracket of a Chern class and a Chern-Simons class is d-closed. The construction developed here easily gives explicit formulas for known secondary classes and makes it possible to construct new ones. (3) I develop an algebraic model for the action of the gauge group and describe how elements of algebra corresponding to the secondary characteristic classes change under this action (see theorem 3 page xi). (4) It is possible give new explicit formulas for cocycles on a gauge group of a bundle and for the corresponding cocycles on the Lie algebra of the gauge group. I use formulas for secondary characteristic classes and an algebraic approach developed in chapter 1. I also use the work of Faddeev, Reiman and Semyonov-Tian-Shanskii (FRS) on cocycles as quantum anomalies. (5) I apply the methods of differential geometry of formal power series to construct universal characteristic and secondary characteristic classes. Given a pair of gauge equivalent connections using local formulas I obtain dilogarithmic and trilogarithmic analogs of Chern-Simons classes.

Lie algebra of conformal Killing-Yano forms

NASA Astrophysics Data System (ADS)

Ertem, Ümit

2016-06-01

We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms. A new Lie bracket for conformal Killing-Yano forms that corresponds to slightly modified Schouten-Nijenhuis bracket of differential forms is proposed. We show that conformal Killing-Yano forms satisfy a graded Lie algebra in constant curvature manifolds. It is also proven that normal conformal Killing-Yano forms in Einstein manifolds also satisfy a graded Lie algebra. The constructed graded Lie algebras reduce to the graded Lie algebra of Killing-Yano forms and the Lie algebras of conformal Killing and Killing vector fields in special cases.

Coherent orthogonal polynomials

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

Celeghini, E., E-mail: celeghini@fi.infn.it; Olmo, M.A. del, E-mail: olmo@fta.uva.es

2013-08-15

We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis (|x〉), for an alternative countable basis (|n〉). The matrix elements that relatemore » these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinite-dimensional irreducible representation of a non-compact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the L{sup 2} functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L{sup 2} and, in particular, generalized coherent polynomials are thus obtained. -- Highlights: •Fundamental characteristic of orthogonal polynomials (OP): existence of a Lie algebra. •Differential recurrence relations of OP determine a unitary representation of a non-compact Lie group. •2nd order Casimir originates a 2nd order differential equation that defines the corresponding OP family. •Generalized coherent polynomials are obtained from OP.« less

Assessing non-uniqueness: An algebraic approach

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

Vasco, Don W.

Geophysical inverse problems are endowed with a rich mathematical structure. When discretized, most differential and integral equations of interest are algebraic (polynomial) in form. Techniques from algebraic geometry and computational algebra provide a means to address questions of existence and uniqueness for both linear and non-linear inverse problem. In a sense, the methods extend ideas which have proven fruitful in treating linear inverse problems.

Quasi-generalized variables

NASA Technical Reports Server (NTRS)

Baumgarten, J.; Ostermeyer, G. P.

1986-01-01

The numerical solution of a system of differential and algebraic equations is difficult, due to the appearance of numerical instabilities. A method is presented here which permits numerical solutions of such a system to be obtained which satisfy the algebraic constraint equations exactly without reducing the order of the differential equations. The method is demonstrated using examples from mechanics.

Exploring Students' Understanding of Ordinary Differential Equations Using Computer Algebraic System (CAS)

ERIC Educational Resources Information Center

Maat, Siti Mistima; Zakaria, Effandi

2011-01-01

Ordinary differential equations (ODEs) are one of the important topics in engineering mathematics that lead to the understanding of technical concepts among students. This study was conducted to explore the students' understanding of ODEs when they solve ODE questions using a traditional method as well as a computer algebraic system, particularly…

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

Miyadera, Takayuki; Imai, Hideki; Graduate School of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551

This paper discusses the no-cloning theorem in a logicoalgebraic approach. In this approach, an orthoalgebra is considered as a general structure for propositions in a physical theory. We proved that an orthoalgebra admits cloning operation if and only if it is a Boolean algebra. That is, only classical theory admits the cloning of states. If unsharp propositions are to be included in the theory, then a notion of effect algebra is considered. We proved that an atomic Archimedean effect algebra admitting cloning operation is a Boolean algebra. This paper also presents a partial result, indicating a relation between the cloningmore » on effect algebras and hidden variables.« less

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2006-10-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2011-03-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

NASA Astrophysics Data System (ADS)

Campoamor-Stursberg, R.

2018-03-01

A procedure for the construction of nonlinear realizations of Lie algebras in the context of Vessiot-Guldberg-Lie algebras of first-order systems of ordinary differential equations (ODEs) is proposed. The method is based on the reduction of invariants and projection of lowest-dimensional (irreducible) representations of Lie algebras. Applications to the description of parameterized first-order systems of ODEs related by contraction of Lie algebras are given. In particular, the kinematical Lie algebras in (2 + 1)- and (3 + 1)-dimensions are realized simultaneously as Vessiot-Guldberg-Lie algebras of parameterized nonlinear systems in R3 and R4, respectively.

Optimal linear-quadratic control of coupled parabolic-hyperbolic PDEs

NASA Astrophysics Data System (ADS)

Aksikas, I.; Moghadam, A. Alizadeh; Forbes, J. F.

2017-10-01

This paper focuses on the optimal control design for a system of coupled parabolic-hypebolic partial differential equations by using the infinite-dimensional state-space description and the corresponding operator Riccati equation. Some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear-quadratic (LQ)-optimal control problem. A state LQ-feedback operator is computed by solving the operator Riccati equation, which is converted into a set of algebraic and differential Riccati equations, thanks to the eigenvalues and the eigenvectors of the parabolic operator. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ-optimal controller designed in the early portion of the paper is implemented for the original nonlinear model. Numerical simulations are performed to show the controller performances.

On conforming mixed finite element methods for incompressible viscous flow problems

NASA Technical Reports Server (NTRS)

Gunzburger, M. D; Nicolaides, R. A.; Peterson, J. S.

1982-01-01

The application of conforming mixed finite element methods to obtain approximate solutions of linearized Navier-Stokes equations is examined. Attention is given to the convergence rates of various finite element approximations of the pressure and the velocity field. The optimality of the convergence rates are addressed in terms of comparisons of the approximation convergence to a smooth solution in relation to the best approximation available for the finite element space used. Consideration is also devoted to techniques for efficient use of a Gaussian elimination algorithm to obtain a solution to a system of linear algebraic equations derived by finite element discretizations of linear partial differential equations.

A note on the regularity of solutions of infinite dimensional Riccati equations

NASA Technical Reports Server (NTRS)

Burns, John A.; King, Belinda B.

1994-01-01

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR and LQG control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smoothes solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.

Synchronization of Reaction-Diffusion Neural Networks With Dirichlet Boundary Conditions and Infinite Delays.

PubMed

Sheng, Yin; Zhang, Hao; Zeng, Zhigang

2017-10-01

This paper is concerned with synchronization for a class of reaction-diffusion neural networks with Dirichlet boundary conditions and infinite discrete time-varying delays. By utilizing theories of partial differential equations, Green's formula, inequality techniques, and the concept of comparison, algebraic criteria are presented to guarantee master-slave synchronization of the underlying reaction-diffusion neural networks via a designed controller. Additionally, sufficient conditions on exponential synchronization of reaction-diffusion neural networks with finite time-varying delays are established. The proposed criteria herein enhance and generalize some published ones. Three numerical examples are presented to substantiate the validity and merits of the obtained theoretical results.

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients.

PubMed

Boyko, Vyacheslav M; Popovych, Roman O; Shapoval, Nataliya M

2013-01-01

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients

PubMed Central

Boyko, Vyacheslav M.; Popovych, Roman O.; Shapoval, Nataliya M.

2013-01-01

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach. PMID:23564972

Exceptional quantum geometry and particle physics

NASA Astrophysics Data System (ADS)

Dubois-Violette, Michel

2016-11-01

Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group SU (3) and on the existence of 3 generations, we develop an argumentation suggesting that the "finite quantum space" corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space C ⊕C3 is associated to the quark-lepton symmetry (one complex for the lepton and 3 for the corresponding quark). More generally it is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of "the algebra of real functions" on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.

On the construction of unitary quantum group differential calculus

NASA Astrophysics Data System (ADS)

Pyatov, Pavel

2016-10-01

We develop a construction of the unitary type anti-involution for the quantized differential calculus over {{GL}}q(n) in the case | q| =1. To this end, we consider a joint associative algebra of quantized functions, differential forms and Lie derivatives over {{GL}}q(n)/{{SL}}q(n), which is bicovariant with respect to {{GL}}q(n)/{{SL}}q(n) coactions. We define a specific non-central spectral extension of this algebra by the spectral variables of three matrices of the algebra generators. In the spectrally expended algebra, we construct a three-parametric family of its inner automorphisms. These automorphisms are used for the construction of the unitary anti-involution for the (spectrally extended) calculus over {{GL}}q(n). This work has been funded by the Russian Academic Excellence Project ‘5-100’. The results of section 5 (propositions 5.2, 5.3 and theorem 5.5) have been obtained under support of the RSF grant No.16-11-10160.

Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation

NASA Astrophysics Data System (ADS)

Bokhari, Ashfaque H.; Mahomed, F. M.; Zaman, F. D.

2010-05-01

The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.

Deriving Differential Equations from Process Algebra Models in Reagent-Centric Style

NASA Astrophysics Data System (ADS)

Hillston, Jane; Duguid, Adam

The reagent-centric style of modeling allows stochastic process algebra models of biochemical signaling pathways to be developed in an intuitive way. Furthermore, once constructed, the models are amenable to analysis by a number of different mathematical approaches including both stochastic simulation and coupled ordinary differential equations. In this chapter, we give a tutorial introduction to the reagent-centric style, in PEPA and Bio-PEPA, and the way in which such models can be used to generate systems of ordinary differential equations.

Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov-Kuznetsov equations

NASA Astrophysics Data System (ADS)

Huang, Ding-jiang; Ivanova, Nataliya M.

2016-02-01

In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov-Kuznetsov (GZK) equations of form ut +(F (u)) xxx +(G (u)) xyy +(H (u)) x = 0. As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov-Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs.

Differential invariants in nonclassical models of hydrodynamics

NASA Astrophysics Data System (ADS)

Bublik, Vasily V.

2017-10-01

In this paper, differential invariants are used to construct solutions for equations of the dynamics of a viscous heat-conducting gas and the dynamics of a viscous incompressible fluid modified by nanopowder inoculators. To describe the dynamics of a viscous heat-conducting gas, we use the complete system of Navier—Stokes equations with allowance for heat fluxes. Mathematical description of the dynamics of liquid metals under high-energy external influences (laser radiation or plasma flow) includes, in addition to the Navier—Stokes system of an incompressible viscous fluid, also heat fluxes and processes of nonequilibrium crystallization of a deformable fluid. Differentially invariant solutions are a generalization of partially invariant solutions, and their active study for various models of continuous medium mechanics is just beginning. Differentially invariant solutions can also be considered as solutions with differential constraints; therefore, when developing them, the approaches and methods developed by the science schools of academicians N. N. Yanenko and A. F. Sidorov will be actively used. In the construction of partially invariant and differentially invariant solutions, there are overdetermined systems of differential equations that require a compatibility analysis. The algorithms for reducing such systems to involution in a finite number of steps are described by Cartan, Finikov, Kuranishi, and other authors. However, the difficultly foreseeable volume of intermediate calculations complicates their practical application. Therefore, the methods of computer algebra are actively used here, which largely helps in solving this difficult problem. It is proposed to use the constructed exact solutions as tests for formulas, algorithms and their software implementations when developing and creating numerical methods and computational program complexes. This combination of effective numerical methods, capable of solving a wide class of problems, with analytical methods makes it possible to make the results of mathematical modeling more accurate and reliable.

Using Example Problems to Improve Student Learning in Algebra: Differentiating between Correct and Incorrect Examples

ERIC Educational Resources Information Center

Booth, Julie L.; Lange, Karin E.; Koedinger, Kenneth R.; Newton, Kristie J.

2013-01-01

In a series of two "in vivo" experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly…

Using Example Problems to Improve Student Learning in Algebra: Differentiating between Correct and Incorrect Examples

ERIC Educational Resources Information Center

Booth, Julie L.; Lange, Karin E.; Koedinger, Kenneth R.; Newton, Kristie J.

2013-01-01

In a series of two in vivo experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly assigned…

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

NASA Astrophysics Data System (ADS)

Plestenjak, Bor; Gheorghiu, Călin I.; Hochstenbach, Michiel E.

2015-10-01

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lamé's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems.

An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher-KPP equation

NASA Astrophysics Data System (ADS)

Shapovalov, A. V.; Trifonov, A. Yu.

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

A Second Order Semi-Discrete Cosserat Rod Model Suitable for Dynamic Simulations in Real Time

NASA Astrophysics Data System (ADS)

Lang, Holger; Linn, Joachim

2009-09-01

We present an alternative approach for a semi-discrete viscoelastic Cosserat rod model that allows both fast dynamic computations within milliseconds and accurate results compared to detailed finite element solutions. The model is able to represent extension, shearing, bending and torsion. For inner dissipation, a consistent damping potential from Antman is chosen. The continuous equations of motion, which consist a system of nonlinear hyperbolic partial differential algebraic equations, are derived from a two dimensional variational principle. The semi-discrete balance equations are obtained by spatial finite difference schemes on a staggered grid and standard index reduction techniques. The right-hand side of the model and its Jacobian can be chosen free of higher algebraic (e.g. root) or transcendent (e.g. trigonometric or exponential) functions and is therefore extremely cheap to evaluate numerically. For the time integration of the system, we use well established stiff solvers. As our model yields computational times within milliseconds, it is suitable for interactive manipulation. It reflects structural mechanics solutions sufficiently correct, as comparison with detailed finite element results shows.

Conference on Complex Turbulent Flows: Comparison of Computation and Experiment, Stanford University, Stanford, CA, September 14-18, 1981, Proceedings. Volume 2 - Taxonomies, reporters' summaries, evaluation, and conclusions

NASA Technical Reports Server (NTRS)

Kline, S. J. (Editor); Cantwell, B. J. (Editor); Lilley, G. M.

1982-01-01

Computational techniques for simulating turbulent flows were explored, together with the results of experimental investigations. Particular attention was devoted to the possibility of defining a universal closure model, applicable for all turbulence situations; however, conclusions were drawn that zonal models, describing localized structures, were the most promising techniques to date. The taxonomy of turbulent flows was summarized, as were algebraic, differential, integral, and partial differential methods for numerical depiction of turbulent flows. Numerous comparisons of theoretically predicted and experimentally obtained data for wall pressure distributions, velocity profiles, turbulent kinetic energy profiles, Reynolds shear stress profiles, and flows around transonic airfoils were presented. Simplifying techniques for reducing the necessary computational time for modeling complex flowfields were surveyed, together with the industrial requirements and applications of computational fluid dynamics techniques.

Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model

NASA Astrophysics Data System (ADS)

Dehghan, Mehdi; Mohammadi, Vahid

2017-03-01

As is said in [27], the tumor-growth model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models [27]. Simulations of this practical model using numerical methods can be applied for evaluating it. The present paper investigates the solution of the tumor growth model with meshless techniques. Meshless methods are applied based on the collocation technique which employ multiquadrics (MQ) radial basis function (RBFs) and generalized moving least squares (GMLS) procedures. The main advantages of these choices come back to the natural behavior of meshless approaches. As well as, a method based on meshless approach can be applied easily for finding the solution of partial differential equations in high-dimension using any distributions of points on regular and irregular domains. The present paper involves a time-dependent system of partial differential equations that describes four-species tumor growth model. To overcome the time variable, two procedures will be used. One of them is a semi-implicit finite difference method based on Crank-Nicolson scheme and another one is based on explicit Runge-Kutta time integration. The first case gives a linear system of algebraic equations which will be solved at each time-step. The second case will be efficient but conditionally stable. The obtained numerical results are reported to confirm the ability of these techniques for solving the two and three-dimensional tumor-growth equations.

Unification of the general non-linear sigma model and the Virasoro master equation

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

Boer, J. de; Halpern, M.B.

1997-06-01

The Virasoro master equation describes a large set of conformal field theories known as the affine-Virasoro constructions, in the operator algebra (affinie Lie algebra) of the WZW model, while the einstein equations of the general non-linear sigma model describe another large set of conformal field theories. This talk summarizes recent work which unifies these two sets of conformal field theories, together with a presumable large class of new conformal field theories. The basic idea is to consider spin-two operators of the form L{sub ij}{partial_derivative}x{sup i}{partial_derivative}x{sup j} in the background of a general sigma model. The requirement that these operators satisfymore » the Virasoro algebra leads to a set of equations called the unified Einstein-Virasoro master equation, in which the spin-two spacetime field L{sub ij} cuples to the usual spacetime fields of the sigma model. The one-loop form of this unified system is presented, and some of its algebraic and geometric properties are discussed.« less

Fem Formulation for Heat and Mass Transfer in Porous Medium

NASA Astrophysics Data System (ADS)

Azeem; Soudagar, Manzoor Elahi M.; Salman Ahmed, N. J.; Anjum Badruddin, Irfan

2017-08-01

Heat and mass transfer in porous medium can be modelled using three partial differential equations namely, momentum equation, energy equation and mass diffusion. These three equations are coupled to each other by some common terms that turn the whole phenomenon into a complex problem with inter-dependable variables. The current article describes the finite element formulation of heat and mass transfer in porous medium with respect to Cartesian coordinates. The problem under study is formulated into algebraic form of equations by using Galerkin's method with the help of two-node linear triangular element having three nodes. The domain is meshed with smaller sized elements near the wall region and bigger size away from walls.

Collector modulation in high-voltage bipolar transistor in the saturation mode: Analytical approach

NASA Astrophysics Data System (ADS)

Dmitriev, A. P.; Gert, A. V.; Levinshtein, M. E.; Yuferev, V. S.

2018-04-01

A simple analytical model is developed, capable of replacing the numerical solution of a system of nonlinear partial differential equations by solving a simple algebraic equation when analyzing the collector resistance modulation of a bipolar transistor in the saturation mode. In this approach, the leakage of the base current into the emitter and the recombination of non-equilibrium carriers in the base are taken into account. The data obtained are in good agreement with the results of numerical calculations and make it possible to describe both the motion of the front of the minority carriers and the steady state distribution of minority carriers across the collector in the saturation mode.

Noncommutative Differential Geometry of Generalized Weyl Algebras

NASA Astrophysics Data System (ADS)

Brzeziński, Tomasz

2016-06-01

Elements of noncommutative differential geometry of Z-graded generalized Weyl algebras A(p;q) over the ring of polynomials in two variables and their zero-degree subalgebras B(p;q), which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particular, three classes of skew derivations of A(p;q) are constructed, and three-dimensional first-order differential calculi induced by these derivations are described. The associated integrals are computed and it is shown that the dimension of the integral space coincides with the order of the defining polynomial p(z). It is proven that the restriction of these first-order differential calculi to the calculi on B(p;q) is isomorphic to the direct sum of degree 2 and degree -2 components of A(p;q). A Dirac operator for B(p;q) is constructed from a (strong) connection with respect to this differential calculus on the (free) spinor bimodule defined as the direct sum of degree 1 and degree -1 components of A(p;q). The real structure of KO-dimension two for this Dirac operator is also described.

Algebraic Riccati equations in zero-sum differential games

NASA Technical Reports Server (NTRS)

Johnson, T. L.; Chao, A.

1974-01-01

The procedure for finding the closed-loop Nash equilibrium solution of two-player zero-sum linear time-invariant differential games with quadratic performance criteria and classical information pattern may be reduced in most cases to the solution of an algebraic Riccati equation. Based on the results obtained by Willems, necessary and sufficient conditions for existence of solutions to these equations are derived, and explicit conditions for a scalar example are given.

Dolan Grady relations and noncommutative quasi-exactly solvable systems

NASA Astrophysics Data System (ADS)

Klishevich, Sergey M.; Plyushchay, Mikhail S.

2003-11-01

We investigate a U(1) gauge invariant quantum mechanical system on a 2D noncommutative space with coordinates generating a generalized deformed oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge covariant derivatives obeying the nonlinear Dolan-Grady relations. This restricts the structure function of the deformed oscillator algebra to a quadratic polynomial. The cases when the coordinates form the {\\mathfrak{su}}(2) and {\\mathfrak{sl}}(2,{\\bb {R}}) algebras are investigated in detail. Reducing the Hamiltonian to 1D finite-difference quasi-exactly solvable operators, we demonstrate partial algebraization of the spectrum of the corresponding systems on the fuzzy sphere and noncommutative hyperbolic plane. A completely covariant method based on the notion of intrinsic algebra is proposed to deal with the spectral problem of such systems.

Manifolds, Tensors, and Forms

NASA Astrophysics Data System (ADS)

Renteln, Paul

2013-11-01

Preface; 1. Linear algebra; 2. Multilinear algebra; 3. Differentiation on manifolds; 4. Homotopy and de Rham cohomology; 5. Elementary homology theory; 6. Integration on manifolds; 7. Vector bundles; 8. Geometric manifolds; 9. The degree of a smooth map; Appendixes; References; Index.

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.

[Feature extraction for breast cancer data based on geometric algebra theory and feature selection using differential evolution].

PubMed

Li, Jing; Hong, Wenxue

2014-12-01

The feature extraction and feature selection are the important issues in pattern recognition. Based on the geometric algebra representation of vector, a new feature extraction method using blade coefficient of geometric algebra was proposed in this study. At the same time, an improved differential evolution (DE) feature selection method was proposed to solve the elevated high dimension issue. The simple linear discriminant analysis was used as the classifier. The result of the 10-fold cross-validation (10 CV) classification of public breast cancer biomedical dataset was more than 96% and proved superior to that of the original features and traditional feature extraction method.

Algebraic features of some generalizations of the Lotka-Volterra system

NASA Astrophysics Data System (ADS)

Bibik, Yu. V.; Sarancha, D. A.

2010-10-01

For generalizations of the Lotka-Volterra system, an integration method is proposed based on the nontrivial algebraic structure of these generalizations. The method makes use of an auxiliary first-order differential equation derived from the phase curve equation with the help of this algebraic structure. Based on this equation, a Hamiltonian approach can be developed and canonical variables (moreover, action-angle variables) can be constructed.

Team Teaching and Cooperative Groups in Abstract Algebra: Nurturing a New Generation of Confident Mathematics Teachers

ERIC Educational Resources Information Center

Grassl, R.; Mingus, T. T. Y.

2007-01-01

Experiences in designing and teaching a reformed abstract algebra course are described. This effort was partially a result of a five year statewide National Science Foundation (NSF) grant entitled the Rocky Mountain Teacher Enhancement Collaborative. The major thrust of this grant was to implement reform in core mathematics courses that would…

Deriving Laws from Ordering Relations

NASA Technical Reports Server (NTRS)

Knuth, Kevin H.

2003-01-01

It took much effort in the early days of non-Euclidean geometry to break away from the mindset that all spaces are flat and that two distinct parallel lines do not cross. Up to that point, all that was known was Euclidean geometry, and it was difficult to imagine anything else. We have suffered a similar handicap brought on by the enormous relevance of Boolean algebra to the problems of our age-logic and set theory. Previously, I demonstrated that the algebra of questions is not Boolean, but rather is described by the free distributive algebra. To get to this stage took much effort, as many obstacles-most self-placed-had to be overcome. As Boolean algebras were all I had ever known, it was almost impossible for me to imagine working with an algebra where elements do not have complements. With this realization, it became very clear that the sum and product rules of probability theory at the most basic level had absolutely nothing to do with the Boolean algebra of logical statements. Instead, a measure of degree of inclusion can be invented for many different partially ordered sets, and the sum and product rules fall out of the associativity and distributivity of the algebra. To reinforce this very important idea, this paper will go over how these constructions are made, while focusing on the underlying assumptions. I will derive the sum and product rules for a distributive lattice in general and demonstrate how this leads to probability theory on the Boolean lattice and is related to the calculus of quantum mechanical amplitudes on the partially ordered set of experimental setups. I will also discuss the rules that can be derived from modular lattices and their relevance to the cross-ratio of projective geometry.

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)

Artificial Neural Networks Equivalent to Fuzzy Algebra T-Norm Conjunction Operators

NASA Astrophysics Data System (ADS)

Iliadis, L. S.; Spartalis, S. I.

2007-12-01

This paper describes the construction of three Artificial Neural Networks with fuzzy input and output, imitating the performance of fuzzy algebra conjunction operators. More specifically, it is applied over the results of a previous research effort that used T-Norms in order to produce a characteristic torrential risk index that unified the partial risk indices for the area of Xanthi. Each one of the three networks substitutes a T-Norm and consequently they can be used as equivalent operators. This means that ANN performing Fuzzy Algebra operations can be designed and developed.

A gradual update method for simulating the steady-state solution of stiff differential equations in metabolic circuits.

PubMed

Shiraishi, Emi; Maeda, Kazuhiro; Kurata, Hiroyuki

2009-02-01

Numerical simulation of differential equation systems plays a major role in the understanding of how metabolic network models generate particular cellular functions. On the other hand, the classical and technical problems for stiff differential equations still remain to be solved, while many elegant algorithms have been presented. To relax the stiffness problem, we propose new practical methods: the gradual update of differential-algebraic equations based on gradual application of the steady-state approximation to stiff differential equations, and the gradual update of the initial values in differential-algebraic equations. These empirical methods show a high efficiency for simulating the steady-state solutions for the stiff differential equations that existing solvers alone cannot solve. They are effective in extending the applicability of dynamic simulation to biochemical network models.

The algebraic criteria for the stability of control systems

NASA Technical Reports Server (NTRS)

Cremer, H.; Effertz, F. H.

1986-01-01

This paper critically examines the standard algebraic criteria for the stability of linear control systems and their proofs, reveals important previously unnoticed connections, and presents new representations. Algebraic stability criteria have also acquired significance for stability studies of non-linear differential equation systems by the Krylov-Bogoljubov-Magnus Method, and allow realization conditions to be determined for classes of broken rational functions as frequency characteristics of electrical network.

Initial Orbit Determination Based on Propagation of Admissible Regions with Differential Algebra

DTIC Science & Technology

2017-01-19

Asteroid close encounter characterization using differential algebra: the case of aphophis. Celestial Mechanics and Dynamical Astronomy , 107(4), 2010...Mechanics and Dynamical Astronomy , 112 (3):331–352, 2012. ISSN 09232958. doi: 10.1007/s10569-012-9400-8. Roberto Armellin, Pierluigi Di Lizia, and Renato... Astronomy , 90(1-2):59–87, 2004. ISSN 09232958. doi: 10.1007/s10569-004-6593-5. 50 DISTRIBUTION A. Approved for public release: distribution unlimited

Dual algebraic formulation of differential GPS

NASA Astrophysics Data System (ADS)

Lannes, A.; Dur, S.

2003-05-01

A new approach to differential GPS is presented. The corresponding theoretical framework calls on elementary concepts of algebraic graph theory. The notion of double difference, which is related to that of closure in the sense of Kirchhoff, is revisited in this context. The Moore-Penrose pseudo-inverse of the closure operator plays a key role in the corresponding dual formulation. This approach, which is very attractive from a conceptual point of view, sheds a new light on the Teunissen formulation.

C*-algebras associated with reversible extensions of logistic maps

NASA Astrophysics Data System (ADS)

Kwaśniewski, Bartosz K.

2012-10-01

The construction of reversible extensions of dynamical systems presented in a previous paper by the author and A.V. Lebedev is enhanced, so that it applies to arbitrary mappings (not necessarily with open range). It is based on calculating the maximal ideal space of C*-algebras that extends endomorphisms to partial automorphisms via partial isometric representations, and involves a new set of 'parameters' (the role of parameters is played by chosen sets or ideals). As model examples, we give a thorough description of reversible extensions of logistic maps and a classification of systems associated with compression of unitaries generating homeomorphisms of the circle. Bibliography: 34 titles.

Fem Formulation of Heat Transfer in Cylindrical Porous Medium

NASA Astrophysics Data System (ADS)

Azeem; Khaleed, H. M. T.; Soudagar, Manzoor Elahi M.

2017-08-01

Heat transfer in porous medium can be derived from the fundamental laws of flow in porous region ass given by Henry Darcy. The fluid flow and energy transport inside the porous medium can be described with the help of momentum and energy equations. The heat transfer in cylindrical porous medium differs from its counterpart in radial and axial coordinates. The present work is focused to discuss the finite element formulation of heat transfer in cylindrical porous medium. The basic partial differential equations are derived using Darcy law which is the converted into a set of algebraic equations with the help of finite element method. The resulting equations are solved by matrix method for two solution variables involved in the coupled equations.

Determining "small parameters" for quasi-steady state

NASA Astrophysics Data System (ADS)

Goeke, Alexandra; Walcher, Sebastian; Zerz, Eva

2015-08-01

For a parameter-dependent system of ordinary differential equations we present a systematic approach to the determination of parameter values near which singular perturbation scenarios (in the sense of Tikhonov and Fenichel) arise. We call these special values Tikhonov-Fenichel parameter values. The principal application we intend is to equations that describe chemical reactions, in the context of quasi-steady state (or partial equilibrium) settings. Such equations have rational (or even polynomial) right-hand side. We determine the structure of the set of Tikhonov-Fenichel parameter values as a semi-algebraic set, and present an algorithmic approach to their explicit determination, using Groebner bases. Examples and applications (which include the irreversible and reversible Michaelis-Menten systems) illustrate that the approach is rather easy to implement.

User's guide to PMESH: A grid-generation program for single-rotation and counterrotation advanced turboprops

NASA Technical Reports Server (NTRS)

Warsi, Saif A.

1989-01-01

A detailed operating manual is presented for a grid generating program that produces 3-D meshes for advanced turboprops. The code uses both algebraic and elliptic partial differential equation methods to generate single rotation and counterrotation, H or C type meshes for the z - r planes and H type for the z - theta planes. The code allows easy specification of geometrical constraints (such as blade angle, location of bounding surfaces, etc.), mesh control parameters (point distribution near blades and nacelle, number of grid points desired, etc.), and it has good runtime diagnostics. An overview is provided of the mesh generation procedure, sample input dataset with detailed explanation of all input, and example meshes.

Automated airplane surface generation

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

Smith, R.E.; Cordero, Y.; Jones, W.

1996-12-31

An efficient methodology and software axe presented for defining a class of airplane configurations. A small set of engineering design parameters and grid control parameters govern the process. The general airplane configuration has wing, fuselage, vertical tall, horizontal tail, and canard components. Wing, canard, and tail surface grids axe manifested by solving a fourth-order partial differential equation subject to Dirichlet and Neumann boundary conditions. The design variables are incorporated into the boundary conditions, and the solution is expressed as a Fourier series. The fuselage is described by an algebraic function with four design parameters. The computed surface grids are suitablemore » for a wide range of Computational Fluid Dynamics simulation and configuration optimizations. Both batch and interactive software are discussed for applying the methodology.« less

Ionospheric-thermospheric UV tomography: 1. Image space reconstruction algorithms

NASA Astrophysics Data System (ADS)

Dymond, K. F.; Budzien, S. A.; Hei, M. A.

2017-03-01

We present and discuss two algorithms of the class known as Image Space Reconstruction Algorithms (ISRAs) that we are applying to the solution of large-scale ionospheric tomography problems. ISRAs have several desirable features that make them useful for ionospheric tomography. In addition to producing nonnegative solutions, ISRAs are amenable to sparse-matrix formulations and are fast, stable, and robust. We present the results of our studies of two types of ISRA: the Least Squares Positive Definite and the Richardson-Lucy algorithms. We compare their performance to the Multiplicative Algebraic Reconstruction and Conjugate Gradient Least Squares algorithms. We then discuss the use of regularization in these algorithms and present our new approach based on regularization to a partial differential equation.

Grid adaption for hypersonic flow

NASA Technical Reports Server (NTRS)

Abolhassani, Jamshid S.; Tiwari, Surendra N.; Smith, Robert E.

1987-01-01

The methods of grid adaption are reviewed and a method is developed with the capability of adaption to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. Also the method has been formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose body. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.

Algebraic Methods to Design Signals

DTIC Science & Technology

2015-08-27

sequence pairs with optimal correlation values. 5. K.T. Arasu, Pradeep Bansal , Cody Watson, Partially balanced incomplete block designs with two...IEEE Transactions Information Theory, Volume: 58, Issue: 11, Nov 2012, Page(s): 6968 – 6978 5. K.T. Arasu, Pradeep Bansal , Cody Watson, Partially

Three-dimensional polarization algebra for all polarization sensitive optical systems.

PubMed

Li, Yahong; Fu, Yuegang; Liu, Zhiying; Zhou, Jianhong; Bryanston-Cross, P J; Li, Yan; He, Wenjun

2018-05-28

Using three-dimensional (3D) coherency vector (9 × 1), we develop a new 3D polarization algebra to calculate the polarization properties of all polarization sensitive optical systems, especially when the incident optical field is partially polarized or un-polarized. The polarization properties of a high numerical aperture (NA) microscope objective (NA = 1.25 immersed in oil) are analyzed based on the proposed 3D polarization algebra. Correspondingly, the polarization simulation of this high NA optical system is performed by the commercial software VirtualLAB Fusion. By comparing the theoretical calculations with polarization simulations, a perfect matching relation is obtained, which demonstrates that this 3D polarization algebra is valid to quantify the 3D polarization properties for all polarization sensitive optical systems.

Residus de 2-formes differentielles sur les surfaces algebriques et applications aux codes correcteurs d'erreurs

NASA Astrophysics Data System (ADS)

Couvreur, A.

2009-05-01

The theory of algebraic-geometric codes has been developed in the beginning of the 80's after a paper of V.D. Goppa. Given a smooth projective algebraic curve X over a finite field, there are two different constructions of error-correcting codes. The first one, called "functional", uses some rational functions on X and the second one, called "differential", involves some rational 1-forms on this curve. Hundreds of papers are devoted to the study of such codes. In addition, a generalization of the functional construction for algebraic varieties of arbitrary dimension is given by Y. Manin in an article of 1984. A few papers about such codes has been published, but nothing has been done concerning a generalization of the differential construction to the higher-dimensional case. In this thesis, we propose a differential construction of codes on algebraic surfaces. Afterwards, we study the properties of these codes and particularly their relations with functional codes. A pretty surprising fact is that a main difference with the case of curves appears. Indeed, if in the case of curves, a differential code is always the orthogonal of a functional one, this assertion generally fails for surfaces. Last observation motivates the study of codes which are the orthogonal of some functional code on a surface. Therefore, we prove that, under some condition on the surface, these codes can be realized as sums of differential codes. Moreover, we show that some answers to some open problems "a la Bertini" could give very interesting informations on the parameters of these codes.

Computational approach to compact Riemann surfaces

NASA Astrophysics Data System (ADS)

Frauendiener, Jörg; Klein, Christian

2017-01-01

A purely numerical approach to compact Riemann surfaces starting from plane algebraic curves is presented. The critical points of the algebraic curve are computed via a two-dimensional Newton iteration. The starting values for this iteration are obtained from the resultants with respect to both coordinates of the algebraic curve and a suitable pairing of their zeros. A set of generators of the fundamental group for the complement of these critical points in the complex plane is constructed from circles around these points and connecting lines obtained from a minimal spanning tree. The monodromies are computed by solving the defining equation of the algebraic curve on collocation points along these contours and by analytically continuing the roots. The collocation points are chosen to correspond to Chebychev collocation points for an ensuing Clenshaw-Curtis integration of the holomorphic differentials which gives the periods of the Riemann surface with spectral accuracy. At the singularities of the algebraic curve, Puiseux expansions computed by contour integration on the circles around the singularities are used to identify the holomorphic differentials. The Abel map is also computed with the Clenshaw-Curtis algorithm and contour integrals. As an application of the code, solutions to the Kadomtsev-Petviashvili equation are computed on non-hyperelliptic Riemann surfaces.

The Boundary Function Method. Fundamentals

NASA Astrophysics Data System (ADS)

Kot, V. A.

2017-03-01

The boundary function method is proposed for solving applied problems of mathematical physics in the region defined by a partial differential equation of the general form involving constant or variable coefficients with a Dirichlet, Neumann, or Robin boundary condition. In this method, the desired function is defined by a power polynomial, and a boundary function represented in the form of the desired function or its derivative at one of the boundary points is introduced. Different sequences of boundary equations have been set up with the use of differential operators. Systems of linear algebraic equations constructed on the basis of these sequences allow one to determine the coefficients of a power polynomial. Constitutive equations have been derived for initial boundary-value problems of all the main types. With these equations, an initial boundary-value problem is transformed into the Cauchy problem for the boundary function. The determination of the boundary function by its derivative with respect to the time coordinate completes the solution of the problem.

Rapid Airplane Parametric Input Design(RAPID)

NASA Technical Reports Server (NTRS)

Smith, Robert E.; Bloor, Malcolm I. G.; Wilson, Michael J.; Thomas, Almuttil M.

2004-01-01

An efficient methodology is presented for defining a class of airplane configurations. Inclusive in this definition are surface grids, volume grids, and grid sensitivity. A small set of design parameters and grid control parameters govern the process. The general airplane configuration has wing, fuselage, vertical tail, horizontal tail, and canard components. The wing, tail, and canard components are manifested by solving a fourth-order partial differential equation subject to Dirichlet and Neumann boundary conditions. The design variables are incorporated into the boundary conditions, and the solution is expressed as a Fourier series. The fuselage has circular cross section, and the radius is an algebraic function of four design parameters and an independent computational variable. Volume grids are obtained through an application of the Control Point Form method. Grid sensitivity is obtained by applying the automatic differentiation precompiler ADIFOR to software for the grid generation. The computed surface grids, volume grids, and sensitivity derivatives are suitable for a wide range of Computational Fluid Dynamics simulation and configuration optimizations.

Reactive solute transport in streams: 1. Development of an equilibrium- based model

USGS Publications Warehouse

Runkel, Robert L.; Bencala, Kenneth E.; Broshears, Robert E.; Chapra, Steven C.

1996-01-01

An equilibrium-based solute transport model is developed for the simulation of trace metal fate and transport in streams. The model is formed by coupling a solute transport model with a chemical equilibrium submodel based on MINTEQ. The solute transport model considers the physical processes of advection, dispersion, lateral inflow, and transient storage, while the equilibrium submodel considers the speciation and complexation of aqueous species, precipitation/dissolution and sorption. Within the model, reactions in the water column may result in the formation of solid phases (precipitates and sorbed species) that are subject to downstream transport and settling processes. Solid phases on the streambed may also interact with the water column through dissolution and sorption/desorption reactions. Consideration of both mobile (water-borne) and immobile (streambed) solid phases requires a unique set of governing differential equations and solution techniques that are developed herein. The partial differential equations describing physical transport and the algebraic equations describing chemical equilibria are coupled using the sequential iteration approach.

Orders on Intervals Over Partially Ordered Sets: Extending Allen's Algebra and Interval Graph Results

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

Zapata, Francisco; Kreinovich, Vladik; Joslyn, Cliff A.

2013-08-01

To make a decision, we need to compare the values of quantities. In many practical situations, we know the values with interval uncertainty. In such situations, we need to compare intervals. Allen’s algebra describes all possible relations between intervals on the real line, and ordering relations between such intervals are well studied. In this paper, we extend this description to intervals in an arbitrary partially ordered set (poset). In particular, we explicitly describe ordering relations between intervals that generalize relation between points. As auxiliary results, we provide a logical interpretation of the relation between intervals, and extend the results aboutmore » interval graphs to intervals over posets.« less

Effects of Computer Algebra System (CAS) with Metacognitive Training on Mathematical Reasoning.

ERIC Educational Resources Information Center

Kramarski, Bracha; Hirsch, Chaya

2003-01-01

Describes a study that investigated the differential effects of Computer Algebra Systems (CAS) and metacognitive training (META) on mathematical reasoning. Participants were 83 Israeli eighth-grade students. Results showed that CAS embedded within META significantly outperformed the META and CAS alone conditions, which in turn significantly…

Algebraic Structure of tt * Equations for Calabi-Yau Sigma Models

NASA Astrophysics Data System (ADS)

Alim, Murad

2017-08-01

The tt * equations define a flat connection on the moduli spaces of {2d, \\mathcal{N}=2} quantum field theories. For conformal theories with c = 3 d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. We show that the non-holomorphic content of the tt * equations, restricted to the conformal directions, in the cases d = 1, 2, 3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with the generators of a Lie algebra. This Lie algebra replaces the non-holomorphic derivatives of tt * and provides these with a finer and algebraic meaning. For sigma models into lattice polarized K3 manifolds, the differential ring of special functions on the moduli space is constructed, extending known structures for d = 1 and 3. The generators of the differential rings of special functions are given by quasi-modular forms for d = 1 and their generalizations in d = 2, 3. Some explicit examples are worked out including the case of the mirror of the quartic in {\\mathbbm{P}^3}, where due to further algebraic constraints, the differential ring coincides with quasi modular forms.

Gauss-Manin Connection in Disguise: Calabi-Yau Threefolds

NASA Astrophysics Data System (ADS)

Alim, Murad; Movasati, Hossein; Scheidegger, Emanuel; Yau, Shing-Tung

2016-06-01

We describe a Lie Algebra on the moduli space of non-rigid compact Calabi-Yau threefolds enhanced with differential forms and its relation to the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation. In particular, we describe algebraic topological string partition functions {{F}g^alg, g ≥ 1}, which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck's algebraic de Rham cohomology and on the algebraic Gauss-Manin connection. In this way, we recover a result of Yamaguchi-Yau and Alim-Länge in an algebraic context. Our proofs use the fact that the special polynomial generators defined using the special geometry of deformation spaces of Calabi-Yau threefolds correspond to coordinates on such a moduli space. We discuss the mirror quintic as an example.

Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions [Algebraic multigrid preconditioners for multiphase flow in porous media with phase transitions

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

Bui, Quan M.; Wang, Lu; Osei-Kuffuor, Daniel

Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the partial differential equations modeling the flow. In the case of multiphase multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are taken into account. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem (NCP). Unlike in the primary variable switchingmore » technique, the set of primary variables in this approach is fixed even when there is phase transition. Not only does this improve the robustness of the nonlinear solver, it opens up the possibility to use multigrid methods to solve the resulting linear system. The disadvantage of the complementarity approach, however, is that when a phase disappears, the linear system has the structure of a saddle point problem and becomes indefinite, and current algebraic multigrid (AMG) algorithms cannot be applied directly. In this study, we explore the effectiveness of a new multilevel strategy, based on the multigrid reduction technique, to deal with problems of this type. We demonstrate the effectiveness of the method through numerical results for the case of two-phase, two-component flow with phase appearance/disappearance. In conclusion, we also show that the strategy is efficient and scales optimally with problem size.« less

Aspects géométriques et intégrables des modèles de matrices aléatoires

NASA Astrophysics Data System (ADS)

Marchal, Olivier

2010-12-01

This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of integrable systems. The variety of these applications shows why matrix models are important from a mathematical point of view. First, the thesis will focus on the study of the merging of two intervals of the eigenvalues density near a singular point. Specifically, we will show why this special limit gives universal equations from the Painlevé II hierarchy of integrable systems theory. Then, following the approach of (bi) orthogonal polynomials introduced by Mehta to compute partition functions, we will find Riemann-Hilbert and isomonodromic problems connected to matrix models, making the link with the theory of Jimbo, Miwa and Ueno. In particular, we will describe how the hermitian two-matrix models provide a degenerate case of Jimbo-Miwa-Ueno's theory that we will generalize in this context. Furthermore, the loop equations method, with its central notions of spectral curve and topological expansion, will lead to the symplectic invariants of algebraic geometry recently proposed by Eynard and Orantin. This last point will be generalized to the case of non-hermitian matrix models (arbitrary beta) paving the way to "quantum algebraic geometry" and to the generalization of symplectic invariants to "quantum curves". Finally, this set up will be applied to combinatorics in the context of topological string theory, with the explicit computation of an hermitian random matrix model enumerating the Gromov-Witten invariants of a toric Calabi-Yau threefold.

Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions [Algebraic multigrid preconditioners for multiphase flow in porous media with phase transitions

DOE PAGES

Bui, Quan M.; Wang, Lu; Osei-Kuffuor, Daniel

2018-02-06

Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the partial differential equations modeling the flow. In the case of multiphase multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are taken into account. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem (NCP). Unlike in the primary variable switchingmore » technique, the set of primary variables in this approach is fixed even when there is phase transition. Not only does this improve the robustness of the nonlinear solver, it opens up the possibility to use multigrid methods to solve the resulting linear system. The disadvantage of the complementarity approach, however, is that when a phase disappears, the linear system has the structure of a saddle point problem and becomes indefinite, and current algebraic multigrid (AMG) algorithms cannot be applied directly. In this study, we explore the effectiveness of a new multilevel strategy, based on the multigrid reduction technique, to deal with problems of this type. We demonstrate the effectiveness of the method through numerical results for the case of two-phase, two-component flow with phase appearance/disappearance. In conclusion, we also show that the strategy is efficient and scales optimally with problem size.« less

Heat and Mass Transfer in an L Shaped Porous Medium

NASA Astrophysics Data System (ADS)

Salman Ahmed, N. J.; Azeem; Yunus Khan, T. M.

2017-08-01

This article is an extension to the heat transfer in L-shaped porous medium by including the mass diffusion. The heat and mass transfer in the porous domain is represented by three coupled partial differential equations representing the fluid movement, energy transport and mass transport. The equations are converted into algebraic form of equations by the application of finite element method that can be conveniently solved by matrix method. An iterative approach is adopted to solve the coupled equations by setting suitable convergence criterion. The results are discussed in terms of heat transfer characteristics influenced by physical parameters such as buoyancy ratio, Lewis number, Rayleigh number etc. It is found that these physical parameters have significant effect on heat and mass transfer behavior of L-shaped porous medium.

Workshop on Engineering Turbulence Modeling

NASA Technical Reports Server (NTRS)

Povinelli, Louis A. (Editor); Liou, W. W. (Editor); Shabbir, A. (Editor); Shih, T.-H. (Editor)

1992-01-01

Discussed here is the future direction of various levels of engineering turbulence modeling related to computational fluid dynamics (CFD) computations for propulsion. For each level of computation, there are a few turbulence models which represent the state-of-the-art for that level. However, it is important to know their capabilities as well as their deficiencies in order to help engineers select and implement the appropriate models in their real world engineering calculations. This will also help turbulence modelers perceive the future directions for improving turbulence models. The focus is on one-point closure models (i.e., from algebraic models to higher order moment closure schemes and partial differential equation methods) which can be applied to CFD computations. However, other schemes helpful in developing one-point closure models, are also discussed.

Nonlinear Solver Approaches for the Diffusive Wave Approximation to the Shallow Water Equations

NASA Astrophysics Data System (ADS)

Collier, N.; Knepley, M.

2015-12-01

The diffusive wave approximation to the shallow water equations (DSW) is a doubly-degenerate, nonlinear, parabolic partial differential equation used to model overland flows. Despite its challenges, the DSW equation has been extensively used to model the overland flow component of various integrated surface/subsurface models. The equation's complications become increasingly problematic when ponding occurs, a feature which becomes pervasive when solving on large domains with realistic terrain. In this talk I discuss the various forms and regularizations of the DSW equation and highlight their effect on the solvability of the nonlinear system. In addition to this analysis, I present results of a numerical study which tests the applicability of a class of composable nonlinear algebraic solvers recently added to the Portable, Extensible, Toolkit for Scientific Computation (PETSc).

Unsteady MHD free convection flow of casson fluid over an inclined vertical plate embedded in a porous media

NASA Astrophysics Data System (ADS)

Manideep, P.; Raju, R. Srinivasa; Rao, T. Siva Nageswar; Reddy, G. Jithender

2018-05-01

This paper deals, an unsteady magnetohydrodynamic heat transfer natural convection flow of non-Newtonian Casson fluid over an inclined vertical plate embedded in a porous media with the presence of boundary conditions such as oscillating velocity, constant wall temperature. The governing dimensionless boundary layer partial differential equations are reduced to simultaneous algebraic linear equation for velocity, temperature of Casson fluid through finite element method. Those equations are solved by Thomas algorithm after imposing the boundary conditions through MATLAB for analyzing the behavior of Casson fluid velocity and temperature with various physical parameters. Also analyzed the local skin-friction and rate of heat transfer. Compared the present results with earlier reported studies, the results are comprehensively authenticated and robust FEM.

Grid adaption for bluff bodies

NASA Technical Reports Server (NTRS)

Abolhassani, Jamshid S.; Tiwari, Surendra N.

1986-01-01

Methods of grid adaptation are reviewed and a method is developed with the capability of adaptation to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. Also the method was formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method and a movie was produced which shows simulataneously the transient behavior of the solution and the grid adaptation. The results are compared with the experimental and other numerical results.

Support for Struggling Students in Algebra: Contributions of Incorrect Worked Examples

ERIC Educational Resources Information Center

Barbieri, Christina; Booth, Julie L.

2016-01-01

Middle school algebra students (N = 125) randomly assigned within classroom to a Problem-solving control group, a Correct worked examples control group, or an Incorrect worked examples group, completed an experimental classroom study to assess the differential effects of incorrect examples versus the two control groups on students' algebra…

Importance of Grades and Placement for Math Attainment

ERIC Educational Resources Information Center

Tyson, Will; Roksa, Josipa

2017-01-01

Research on high school math course taking documents the advantages of starting high school at or beyond Algebra 1. Fewer studies examine differentiation into remedial, general, and honors Algebra 1 course types by course rigor. This study examines how course grades and course rigor are associated with math attainment among students with similar…

Calabi's conjecture and some new results in algebraic geometry

PubMed Central

Yau, Shing-Tung

1977-01-01

We announce a proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold and then apply it to prove some new results in algebraic geometry and differential geometry. For example, we prove that the only Kähler structure on a complex projective space is the standard one. PMID:16592394

Automorphisms of Order Structures of Abelian Parts of Operator Algebras and Their Role in Quantum Theory

NASA Astrophysics Data System (ADS)

Hamhalter, Jan; Turilova, Ekaterina

2014-10-01

It is shown that any order isomorphism between the structures of unital associative JB subalgebras of JB algebras is given naturally by a partially linear Jordan isomorphism. The same holds for nonunital subalgebras and order isomorphisms preserving the unital subalgebra. Finally, we recover usual action of time evolution group on a von Neumann factor from group of automorphisms of the structure of Abelian subalgebras.

C-semiring Frameworks for Minimum Spanning Tree Problems

NASA Astrophysics Data System (ADS)

Bistarelli, Stefano; Santini, Francesco

In this paper we define general algebraic frameworks for the Minimum Spanning Tree problem based on the structure of c-semirings. We propose general algorithms that can compute such trees by following different cost criteria, which must be all specific instantiation of c-semirings. Our algorithms are extensions of well-known procedures, as Prim or Kruskal, and show the expressivity of these algebraic structures. They can deal also with partially-ordered costs on the edges.

A nonlocal continuum model for the biaxial buckling analysis of composite nanoplates with shape memory alloy nanowires

NASA Astrophysics Data System (ADS)

Farajpour, M. R.; Shahidi, A. R.; Farajpour, A.

2018-03-01

In this study, the buckling behavior of a three-layered composite nanoplate reinforced with shape memory alloy (SMA) nanowires is examined. Whereas the upper and lower layers are reinforced with typical nanowires, SMA nanoscale wires are used to strengthen the middle layer of the system. The composite nanoplate is assumed to be under the action of biaxial compressive loading. A scale-dependent mathematical model is presented with the consideration of size effects within the context of the Eringen’s nonlocal continuum mechanics. Using the one-dimensional Brinson’s theory and the Kirchhoff theory of plates, the governing partial differential equations of SMA nanowire-reinforced hybrid nanoplates are derived. Both lateral and longitudinal deflections are taken into consideration in the theoretical formulation and method of solution. In order to reduce the governing differential equations to their corresponding algebraic equations, a discretization approach based on the differential quadrature method is employed. The critical buckling loads of the hybrid nanosystem with various boundary conditions are obtained with the use of a standard eigenvalue solver. It is found that the stability response of SMA composite nanoplates is strongly sensitive to the small scale effect.

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.

Vector 33: A reduce program for vector algebra and calculus in orthogonal curvilinear coordinates

NASA Astrophysics Data System (ADS)

Harper, David

1989-06-01

This paper describes a package with enables REDUCE 3.3 to perform algebra and calculus operations upon vectors. Basic algebraic operations between vectors and between scalars and vectors are provided, including scalar (dot) product and vector (cross) product. The vector differential operators curl, divergence, gradient and Laplacian are also defined, and are valid in any orthogonal curvilinear coordinate system. The package is written in RLISP to allow algebra and calculus to be performed using notation identical to that for operations. Scalars and vectors can be mixed quite freely in the same expression. The package will be of interest to mathematicians, engineers and scientists who need to perform vector calculations in orthogonal curvilinear coordinates.

Institution Morphisms

NASA Technical Reports Server (NTRS)

Goguen, Joseph; Rosu, Grigore; Norvig, Peter (Technical Monitor)

2001-01-01

Institutions formalize the intuitive notion of logical system, including both syntax and semantics. A surprising number of different notions of morphisim have been suggested for forming categories with institutions as objects, and a surprising variety of names have been proposed for them. One goal of this paper is to suggest a terminology that is both uniform and informative to replace the current rather chaotic nomenclature. Another goal is to investigate the properties and interrelations of these notions. Following brief expositions of indexed categories, twisted relations, and Kan extensions, we demonstrate and then exploit the duality between institution morphisms in the original sense of Goguen and Burstall, and the 'plain maps' of Meseguer, obtaining simple uniform proofs of completeness and cocompleteness for both resulting categories; because of this duality, we prefer the name 'comorphism' over 'plain map.' We next consider 'theoroidal' morphisms and comorphisims, which generalize signatures to theories, finding that the 'maps' of Meseguer are theoroidal comorphisms, while theoroidal morphisms are a new concept. We then introduce 'forward' and 'semi-natural' morphisms, and appendices discuss institutions for hidden algebra, universal algebra, partial equational logic, and a variant of order sorted algebra supporting partiality.

An algebraic iterative reconstruction technique for differential X-ray phase-contrast computed tomography.

PubMed

Fu, Jian; Schleede, Simone; Tan, Renbo; Chen, Liyuan; Bech, Martin; Achterhold, Klaus; Gifford, Martin; Loewen, Rod; Ruth, Ronald; Pfeiffer, Franz

2013-09-01

Iterative reconstruction has a wide spectrum of proven advantages in the field of conventional X-ray absorption-based computed tomography (CT). In this paper, we report on an algebraic iterative reconstruction technique for grating-based differential phase-contrast CT (DPC-CT). Due to the differential nature of DPC-CT projections, a differential operator and a smoothing operator are added to the iterative reconstruction, compared to the one commonly used for absorption-based CT data. This work comprises a numerical study of the algorithm and its experimental verification using a dataset measured at a two-grating interferometer setup. Since the algorithm is easy to implement and allows for the extension to various regularization possibilities, we expect a significant impact of the method for improving future medical and industrial DPC-CT applications. Copyright © 2012. Published by Elsevier GmbH.

Algebraic integrability: a survey.

PubMed

Vanhaecke, Pol

2008-03-28

We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems.

A Solution Space for a System of Null-State Partial Differential Equations: Part 3

NASA Astrophysics Data System (ADS)

Flores, Steven M.; Kleban, Peter

2015-01-01

This article is the third of four that completely and rigorously characterize a solution space for a homogeneous system of 2 N + 3 linear partial differential equations (PDEs) in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE κ ). The system comprises 2 N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2 N one-leg boundary operators. In the first two articles (Flores and Kleban, in Commun Math Phys, arXiv:1212.2301, 2012; Commun Math Phys, arXiv:1404.0035, 2014), we use methods of analysis and linear algebra to prove that dim , with C N the Nth Catalan number. Extending these results, we prove in this article that dim and entirely consists of (real-valued) solutions constructed with the CFT Coulomb gas (contour integral) formalism. In order to prove this claim, we show that a certain set of C N such solutions is linearly independent. Because the formulas for these solutions are complicated, we prove linear independence indirectly. We use the linear injective map of Lemma 15 in Flores and Kleban (Commun Math Phys, arXiv:1212.2301, 2012) to send each solution of the mentioned set to a vector in , whose components we find as inner products of elements in a Temperley-Lieb algebra. We gather these vectors together as columns of a symmetric matrix, with the form of a meander matrix. If the determinant of this matrix does not vanish, then the set of C N Coulomb gas solutions is linearly independent. And if this determinant does vanish, then we construct an alternative set of C N Coulomb gas solutions and follow a similar procedure to show that this set is linearly independent. The latter situation is closely related to CFT minimal models. We emphasize that, although the system of PDEs arises in CFT in away that is typically non-rigorous, our treatment of this system here and in Flores and Kleban (Commun Math Phys, arXiv:1212.2301, 2012; Commun Math Phys, arXiv:1404.0035, 2014; Commun Math Phys, arXiv:1405.2747, 2014) is completely rigorous.

Generating a New Higher-Dimensional Coupled Integrable Dispersionless System: Algebraic Structures, Bäcklund Transformation and Hidden Structural Symmetries

NASA Astrophysics Data System (ADS)

Souleymanou, Abbagari; Thomas, B. Bouetou; Timoleon, C. Kofane

2013-08-01

The prolongation structure methodologies of Wahlquist—Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 (1975) 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Bäcklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new (2 + 1)-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.

Large-scale computation of incompressible viscous flow by least-squares finite element method

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Lin, T. L.; Povinelli, Louis A.

1993-01-01

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations and results in symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli function-vorticity formulation for incompressible viscous flows is also tested. For three-dimensional cases, an additional compatibility equation, i.e., the divergence of the vorticity vector should be zero, is included to make the first-order system elliptic. The simple substitution of the Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this scheme for large-scale computation, we give numerical results for 2D driven cavity problem at Re = 10000 with 408 x 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1,000 with 50 x 50 x 50 trilinear elements. The Taylor-Goertler-like vortices are observed for Re = 1,000.

Effect of visual experience on structural organization of the human brain: a voxel based morphometric study using DARTEL.

PubMed

Modi, Shilpi; Bhattacharya, Manisha; Singh, Namita; Tripathi, Rajendra Prasad; Khushu, Subash

2012-10-01

To investigate structural reorganization in the brain with differential visual experience using Voxel-Based Morphometry with Diffeomorphic Anatomic Registration Through Exponentiated Lie algebra algorithm (DARTEL) approach. High resolution structural MR images were taken in fifteen normal sighted healthy controls, thirteen totally blind subjects and six partial blind subjects. The analysis was carried out using SPM8 software on MATLAB 7.6.0 platform. VBM study revealed gray matter volume atrophy in the cerebellum and left inferior parietal cortex in total blind subjects and in left inferior parietal cortex, right caudate nucleus, and left primary visual cortex in partial blind subjects as compared to controls. White matter volume loss was found in calcarine gyrus in total blind subjects and Thlamus-somatosensory region in partially blind subjects as compared to controls. Besides, an increase in Gray Matter volume was also found in left middle occipital and middle frontal gyrus and right entorhinal cortex, and an increase in White Matter volume was found in superior frontal gyrus, left middle temporal gyrus and right Heschl's gyrus in totally blind subjects as compared to controls. Comparison between total and partial blind subjects revealed a greater Gray Matter volume in left cerebellum of partial blinds and left Brodmann area 18 of total blind subjects. Results suggest that, loss of vision at an early age can induce significant structural reorganization on account of the loss of visual input. These plastic changes are different in early onset of total blindness as compared to partial blindness. Copyright © 2011 Elsevier Ireland Ltd. All rights reserved.

Differential invariants for spherical flows of a viscid fluid

NASA Astrophysics Data System (ADS)

Duyunova, Anna; Lychagin, Valentin; Tychkov, Sergey

2018-01-01

Symmetries and the corresponding algebras of differential invariants of viscid fluids on a sphere are given. Their dependence on thermodynamical states of media is studied, and a classification of thermodynamical states is given.

Fast template matching with polynomials.

PubMed

Omachi, Shinichiro; Omachi, Masako

2007-08-01

Template matching is widely used for many applications in image and signal processing. This paper proposes a novel template matching algorithm, called algebraic template matching. Given a template and an input image, algebraic template matching efficiently calculates similarities between the template and the partial images of the input image, for various widths and heights. The partial image most similar to the template image is detected from the input image for any location, width, and height. In the proposed algorithm, a polynomial that approximates the template image is used to match the input image instead of the template image. The proposed algorithm is effective especially when the width and height of the template image differ from the partial image to be matched. An algorithm using the Legendre polynomial is proposed for efficient approximation of the template image. This algorithm not only reduces computational costs, but also improves the quality of the approximated image. It is shown theoretically and experimentally that the computational cost of the proposed algorithm is much smaller than the existing methods.

a Non-Overlapping Discretization Method for Partial Differential Equations

NASA Astrophysics Data System (ADS)

Rosas-Medina, A.; Herrera, I.

2013-05-01

Mathematical models of many systems of interest, including very important continuous systems of Engineering and Science, lead to a great variety of partial differential equations whose solution methods are based on the computational processing of large-scale algebraic systems. Furthermore, the incredible expansion experienced by the existing computational hardware and software has made amenable to effective treatment problems of an ever increasing diversity and complexity, posed by engineering and scientific applications. The emergence of parallel computing prompted on the part of the computational-modeling community a continued and systematic effort with the purpose of harnessing it for the endeavor of solving boundary-value problems (BVPs) of partial differential equations. Very early after such an effort began, it was recognized that domain decomposition methods (DDM) were the most effective technique for applying parallel computing to the solution of partial differential equations, since such an approach drastically simplifies the coordination of the many processors that carry out the different tasks and also reduces very much the requirements of information-transmission between them. Ideally, DDMs intend producing algorithms that fulfill the DDM-paradigm; i.e., such that "the global solution is obtained by solving local problems defined separately in each subdomain of the coarse-mesh -or domain-decomposition-". Stated in a simplistic manner, the basic idea is that, when the DDM-paradigm is satisfied, full parallelization can be achieved by assigning each subdomain to a different processor. When intensive DDM research began much attention was given to overlapping DDMs, but soon after attention shifted to non-overlapping DDMs. This evolution seems natural when the DDM-paradigm is taken into account: it is easier to uncouple the local problems when the subdomains are separated. However, an important limitation of non-overlapping domain decompositions, as that concept is usually understood today, is that interface nodes are shared by two or more subdomains of the coarse-mesh and, therefore, even non-overlapping DDMs are actually overlapping when seen from the perspective of the nodes used in the discretization. In this talk we present and discuss a discretization method in which the nodes used are non-overlapping, in the sense that each one of them belongs to one and only one subdomain of the coarse-mesh.

Rings of h-deformed differential operators

NASA Astrophysics Data System (ADS)

Ogievetsky, O. V.; Herlemont, B.

2017-08-01

We describe the center of the ring Diff h ( n) of h- deformed differential operators of type A. We establish an isomorphism between certain localizations of Diff h ( n) and the Weyl algebra W n , extended by n indeterminates.

Linearized gravity in terms of differential forms

NASA Astrophysics Data System (ADS)

Baykal, Ahmet; Dereli, Tekin

2017-01-01

A technique to linearize gravitational field equations is developed in which the perturbation metric coefficients are treated as second rank, symmetric, 1-form fields belonging to the Minkowski background spacetime by using the exterior algebra of differential forms.

The Automation of Stochastization Algorithm with Use of SymPy Computer Algebra Library

NASA Astrophysics Data System (ADS)

Demidova, Anastasya; Gevorkyan, Migran; Kulyabov, Dmitry; Korolkova, Anna; Sevastianov, Leonid

2018-02-01

SymPy computer algebra library is used for automatic generation of ordinary and stochastic systems of differential equations from the schemes of kinetic interaction. Schemes of this type are used not only in chemical kinetics but also in biological, ecological and technical models. This paper describes the automatic generation algorithm with an emphasis on application details.

Final Report, DE-FG01-06ER25718 Domain Decomposition and Parallel Computing

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

Widlund, Olof B.

2015-06-09

The goal of this project is to develop and improve domain decomposition algorithms for a variety of partial differential equations such as those of linear elasticity and electro-magnetics.These iterative methods are designed for massively parallel computing systems and allow the fast solution of the very large systems of algebraic equations that arise in large scale and complicated simulations. A special emphasis is placed on problems arising from Maxwell's equation. The approximate solvers, the preconditioners, are combined with the conjugate gradient method and must always include a solver of a coarse model in order to have a performance which is independentmore » of the number of processors used in the computer simulation. A recent development allows for an adaptive construction of this coarse component of the preconditioner.« less

On the ``Matrix Approach'' to Interacting Particle Systems

NASA Astrophysics Data System (ADS)

de Sanctis, L.; Isopi, M.

2004-04-01

Derrida et al. and Schütz and Stinchcombe gave algebraic formulas for the correlation functions of the partially asymmetric simple exclusion process. Here we give a fairly general recipe of how to get these formulas and extend them to the whole time evolution (starting from the generator of the process), for a certain class of interacting systems. We then analyze the algebraic relations obtained to show that the matrix approach does not work with some models such as the voter and the contact processes.

The numerical-analytical implementation of the cross-sections method to the open waveguide transition of the "horn" type

NASA Astrophysics Data System (ADS)

Divakov, Dmitriy; Malykh, Mikhail; Sevastianov, Leonid; Sevastianov, Anton; Tiutiunnik, Anastasiia

2017-04-01

In the paper we construct a method for approximate solution of the waveguide problem for guided modes of an open irregular waveguide transition. The method is based on straightening of the curved waveguide boundaries by introducing new variables and applying the Kantorovich method to the problem formulated in the new variables to get a system of ordinary second-order differential equations. In the method, the boundary conditions are formulated by analogy with the partial radiation conditions in the similar problem for closed waveguide transitions. The method is implemented in the symbolic-numeric form using the Maple computer algebra system. The coefficient matrices of the system of differential equations and boundary conditions are calculated symbolically, and then the obtained boundary-value problem is solved numerically using the finite difference method. The chosen coordinate functions of Kantorovich expansions provide good conditionality of the coefficient matrices. The numerical experiment simulating the propagation of guided modes in the open waveguide transition confirms the validity of the method proposed to solve the problem.

Numerical Solution of Systems of Loaded Ordinary Differential Equations with Multipoint Conditions

NASA Astrophysics Data System (ADS)

Assanova, A. T.; Imanchiyev, A. E.; Kadirbayeva, Zh. M.

2018-04-01

A system of loaded ordinary differential equations with multipoint conditions is considered. The problem under study is reduced to an equivalent boundary value problem for a system of ordinary differential equations with parameters. A system of linear algebraic equations for the parameters is constructed using the matrices of the loaded terms and the multipoint condition. The conditions for the unique solvability and well-posedness of the original problem are established in terms of the matrix made up of the coefficients of the system of linear algebraic equations. The coefficients and the righthand side of the constructed system are determined by solving Cauchy problems for linear ordinary differential equations. The solutions of the system are found in terms of the values of the desired function at the initial points of subintervals. The parametrization method is numerically implemented using the fourth-order accurate Runge-Kutta method as applied to the Cauchy problems for ordinary differential equations. The performance of the constructed numerical algorithms is illustrated by examples.

Finding higher symmetries of differential equations using the MAPLE package DESOLVII

NASA Astrophysics Data System (ADS)

Vu, K. T.; Jefferson, G. F.; Carminati, J.

2012-04-01

We present and describe, with illustrative examples, the MAPLE computer algebra package DESOLVII, which is a major upgrade of DESOLV. DESOLVII now includes new routines allowing the determination of higher symmetries (contact and Lie-Bäcklund) for systems of both ordinary and partial differential equations. Catalogue identifier: ADYZ_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYZ_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 10 858 No. of bytes in distributed program, including test data, etc.: 112 515 Distribution format: tar.gz Programming language: MAPLE internal language Computer: PCs and workstations Operating system: Linux, Windows XP and Windows 7 RAM: Depends on the type of problem and the complexity of the system (small ≈ MB, large ≈ GB) Classification: 4.3, 5 Catalogue identifier of previous version: ADYZ_v1_0 Journal reference of previous version: Comput. Phys. Comm. 176 (2007) 682 Does the new version supersede the previous version?: Yes Nature of problem: There are a number of approaches one may use to find solutions to systems of differential equations. These include numerical, perturbative, and algebraic methods. Unfortunately, approximate or numerical solution methods may be inappropriate in many cases or even impossible due to the nature of the system and hence exact methods are important. In their own right, exact solutions are valuable not only as a yardstick for approximate/numerical solutions but also as a means of elucidating the physical meaning of fundamental quantities in systems. One particular method of finding special exact solutions is afforded by the work of Sophus Lie and the use of continuous transformation groups. The power of Lie's group theoretic method lies in its ability to unify a number of ad hoc integration methods through the use of symmetries, that is, continuous groups of transformations which leave the differential system “unchanged”. These symmetry groups may then be used to find special solutions. Solutions found in this manner are called similarity or invariant solutions. The method of finding symmetry transformations initially requires the generation of a large overdetermined system of linear, homogeneous, coupled PDEs. The integration of this system is usually reasonably straightforward requiring the (often elementary) integration of equations by splitting the system according to dependency on different orders and degrees of the dependent variable/s. Unfortunately, in the case of contact and Lie-Bäcklund symmetries, the integration of the determining system becomes increasingly more difficult as the order of the symmetry is increased. This is because the symmetry generating functions become dependent on higher orders of the derivatives of the dependent variables and this diminishes the overall resulting “separable” differential conditions derived from the main determining system. Furthermore, typical determining systems consist of tens to hundreds of equations and this, combined with standard mechanical solution methods, makes the process well suited to automation using computer algebra systems. The new MAPLE package DESOLVII, which is a major upgrade of DESOLV, now includes routines allowing the determination of higher symmetries (contact and Lie-Bäcklund) for systems of both ordinary and partial differential equations. In addition, significant improvements have been implemented to the algorithm for PDE solution. Finally, we have made some improvements in the overall automated process so as to improve user friendliness by reducing user intervention where possible. Solution method: See “Nature of problem” above. Reasons for new version: New and improved functionality. New functionality - can now compute generalised symmetries. Much improved efficiency (speed and memory use) of existing routines. Restrictions: Sufficient memory may be required for complex systems. Running time: Depends on the type of problem and the complexity of the system (small ≈ seconds, large ≈ hours).

On the optimal systems of subalgebras for the equations of hydrodynamic stability analysis of smooth shear flows and their group-invariant solutions

NASA Astrophysics Data System (ADS)

Hau, Jan-Niklas; Oberlack, Martin; Chagelishvili, George

2017-04-01

We present a unifying solution framework for the linearized compressible equations for two-dimensional linearly sheared unbounded flows using the Lie symmetry analysis. The full set of symmetries that are admitted by the underlying system of equations is employed to systematically derive the one- and two-dimensional optimal systems of subalgebras, whose connected group reductions lead to three distinct invariant ansatz functions for the governing sets of partial differential equations (PDEs). The purpose of this analysis is threefold and explicitly we show that (i) there are three invariant solutions that stem from the optimal system. These include a general ansatz function with two free parameters, as well as the ansatz functions of the Kelvin mode and the modal approach. Specifically, the first approach unifies these well-known ansatz functions. By considering two limiting cases of the free parameters and related algebraic transformations, the general ansatz function is reduced to either of them. This fact also proves the existence of a link between the Kelvin mode and modal ansatz functions, as these appear to be the limiting cases of the general one. (ii) The Lie algebra associated with the Lie group admitted by the PDEs governing the compressible dynamics is a subalgebra associated with the group admitted by the equations governing the incompressible dynamics, which allows an additional (scaling) symmetry. Hence, any consequences drawn from the compressible case equally hold for the incompressible counterpart. (iii) In any of the systems of ordinary differential equations, derived by the three ansatz functions in the compressible case, the linearized potential vorticity is a conserved quantity that allows us to analyze vortex and wave mode perturbations separately.

The Laguerre finite difference one-way equation solver

NASA Astrophysics Data System (ADS)

Terekhov, Andrew V.

2017-05-01

This paper presents a new finite difference algorithm for solving the 2D one-way wave equation with a preliminary approximation of a pseudo-differential operator by a system of partial differential equations. As opposed to the existing approaches, the integral Laguerre transform instead of Fourier transform is used. After carrying out the approximation of spatial variables it is possible to obtain systems of linear algebraic equations with better computing properties and to reduce computer costs for their solution. High accuracy of calculations is attained at the expense of employing finite difference approximations of higher accuracy order that are based on the dispersion-relationship-preserving method and the Richardson extrapolation in the downward continuation direction. The numerical experiments have verified that as compared to the spectral difference method based on Fourier transform, the new algorithm allows one to calculate wave fields with a higher degree of accuracy and a lower level of numerical noise and artifacts including those for non-smooth velocity models. In the context of solving the geophysical problem the post-stack migration for velocity models of the types Syncline and Sigsbee2A has been carried out. It is shown that the images obtained contain lesser noise and are considerably better focused as compared to those obtained by the known Fourier Finite Difference and Phase-Shift Plus Interpolation methods. There is an opinion that purely finite difference approaches do not allow carrying out the seismic migration procedure with sufficient accuracy, however the results obtained disprove this statement. For the supercomputer implementation it is proposed to use the parallel dichotomy algorithm when solving systems of linear algebraic equations with block-tridiagonal matrices.

Operator pencil passing through a given operator

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

Biggs, A., E-mail: khudian@manchester.ac.uk, E-mail: adam.biggs@student.manchester.ac.uk; Khudaverdian, H. M., E-mail: khudian@manchester.ac.uk, E-mail: adam.biggs@student.manchester.ac.uk

Let Δ be a linear differential operator acting on the space of densities of a given weight λ{sub 0} on a manifold M. One can consider a pencil of operators Π-circumflex(Δ)=(Δ{sub λ}) passing through the operator Δ such that any Δ{sub λ} is a linear differential operator acting on densities of weight λ. This pencil can be identified with a linear differential operator Δ-circumflex acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e., pencils of self-adjoint and anti-self-adjoint operators. We studymore » lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular, we analyze the relation between these two concepts, and apply it to the study of diff (M)-equivariant liftings. Finally, we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and describe all regular self-adjoint and anti-self-adjoint liftings. Our constructions can be considered as a generalisation of equivariant quantisation.« less

The 6th International Conference on Computer Science and Computational Mathematics (ICCSCM 2017)

NASA Astrophysics Data System (ADS)

2017-09-01

The ICCSCM 2017 (The 6th International Conference on Computer Science and Computational Mathematics) has aimed to provide a platform to discuss computer science and mathematics related issues including Algebraic Geometry, Algebraic Topology, Approximation Theory, Calculus of Variations, Category Theory; Homological Algebra, Coding Theory, Combinatorics, Control Theory, Cryptology, Geometry, Difference and Functional Equations, Discrete Mathematics, Dynamical Systems and Ergodic Theory, Field Theory and Polynomials, Fluid Mechanics and Solid Mechanics, Fourier Analysis, Functional Analysis, Functions of a Complex Variable, Fuzzy Mathematics, Game Theory, General Algebraic Systems, Graph Theory, Group Theory and Generalizations, Image Processing, Signal Processing and Tomography, Information Fusion, Integral Equations, Lattices, Algebraic Structures, Linear and Multilinear Algebra; Matrix Theory, Mathematical Biology and Other Natural Sciences, Mathematical Economics and Financial Mathematics, Mathematical Physics, Measure Theory and Integration, Neutrosophic Mathematics, Number Theory, Numerical Analysis, Operations Research, Optimization, Operator Theory, Ordinary and Partial Differential Equations, Potential Theory, Real Functions, Rings and Algebras, Statistical Mechanics, Structure Of Matter, Topological Groups, Wavelets and Wavelet Transforms, 3G/4G Network Evolutions, Ad-Hoc, Mobile, Wireless Networks and Mobile Computing, Agent Computing & Multi-Agents Systems, All topics related Image/Signal Processing, Any topics related Computer Networks, Any topics related ISO SC-27 and SC- 17 standards, Any topics related PKI(Public Key Intrastructures), Artifial Intelligences(A.I.) & Pattern/Image Recognitions, Authentication/Authorization Issues, Biometric authentication and algorithms, CDMA/GSM Communication Protocols, Combinatorics, Graph Theory, and Analysis of Algorithms, Cryptography and Foundation of Computer Security, Data Base(D.B.) Management & Information Retrievals, Data Mining, Web Image Mining, & Applications, Defining Spectrum Rights and Open Spectrum Solutions, E-Comerce, Ubiquitous, RFID, Applications, Fingerprint/Hand/Biometrics Recognitions and Technologies, Foundations of High-performance Computing, IC-card Security, OTP, and Key Management Issues, IDS/Firewall, Anti-Spam mail, Anti-virus issues, Mobile Computing for E-Commerce, Network Security Applications, Neural Networks and Biomedical Simulations, Quality of Services and Communication Protocols, Quantum Computing, Coding, and Error Controls, Satellite and Optical Communication Systems, Theory of Parallel Processing and Distributed Computing, Virtual Visions, 3-D Object Retrievals, & Virtual Simulations, Wireless Access Security, etc. The success of ICCSCM 2017 is reflected in the received papers from authors around the world from several countries which allows a highly multinational and multicultural idea and experience exchange. The accepted papers of ICCSCM 2017 are published in this Book. Please check http://www.iccscm.com for further news. A conference such as ICCSCM 2017 can only become successful using a team effort, so herewith we want to thank the International Technical Committee and the Reviewers for their efforts in the review process as well as their valuable advices. We are thankful to all those who contributed to the success of ICCSCM 2017. The Secretary

Fast multipole method using Cartesian tensor in beam dynamic simulation

DOE PAGES

Zhang, He; Huang, He; Li, Rui; ...

2017-03-06

Here, the fast multipole method (FMM) using traceless totally symmetric Cartesian tensor to calculate the Coulomb interaction between charged particles will be presented. The Cartesian tensor-based FMM can be generalized to treat other non-oscillating interactions with the help of the differential algebra or the truncated power series algebra. Issues on implementation of the FMM in beam dynamic simulations are also discussed.

Conformal Galilei algebras, symmetric polynomials and singular vectors

NASA Astrophysics Data System (ADS)

Křižka, Libor; Somberg, Petr

2018-01-01

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras cga_ℓ (d,C) with d=1 for any integer value ℓ \\in N. The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.

On superintegrable monopole systems

NASA Astrophysics Data System (ADS)

Fazlul Hoque, Md; Marquette, Ian; Zhang, Yao-Zhong

2018-02-01

Superintegrable systems with monopole interactions in flat and curved spaces have attracted much attention. For example, models in spaces with a Taub-NUT metric are well-known to admit the Kepler-type symmetries and provide non-trivial generalizations of the usual Kepler problems. In this paper, we overview new families of superintegrable Kepler, MIC-harmonic oscillator and deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and curved Taub-NUT spaces. We present their higher-order, algebraically independent integrals of motion via the direct and constructive approaches which prove the superintegrability of the models. The integrals form symmetry polynomial algebras of the systems with structure constants involving Casimir operators of certain Lie algebras. Such algebraic approaches provide a deeper understanding to the degeneracies of the energy spectra and connection between wave functions and differential equations and geometry.

Combinatorial operad actions on cochains

NASA Astrophysics Data System (ADS)

Berger, Clemens; Fresse, Benoit

2004-07-01

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this paper is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.

Optimal control in a model of malaria with differential susceptibility

NASA Astrophysics Data System (ADS)

Hincapié, Doracelly; Ospina, Juan

2014-06-01

A malaria model with differential susceptibility is analyzed using the optimal control technique. In the model the human population is classified as susceptible, infected and recovered. Susceptibility is assumed dependent on genetic, physiological, or social characteristics that vary between individuals. The model is described by a system of differential equations that relate the human and vector populations, so that the infection is transmitted to humans by vectors, and the infection is transmitted to vectors by humans. The model considered is analyzed using the optimal control method when the control consists in using of insecticide-treated nets and educational campaigns; and the optimality criterion is to minimize the number of infected humans, while keeping the cost as low as is possible. One first goal is to determine the effects of differential susceptibility in the proposed control mechanism; and the second goal is to determine the algebraic form of the basic reproductive number of the model. All computations are performed using computer algebra, specifically Maple. It is claimed that the analytical results obtained are important for the design and implementation of control measures for malaria. It is suggested some future investigations such as the application of the method to other vector-borne diseases such as dengue or yellow fever; and also it is suggested the possible application of free software of computer algebra like Maxima.

Differential morphology and image processing.

PubMed

Maragos, P

1996-01-01

Image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain. We provide a unified view and analytic tools for morphological image processing that is based on ideas from differential calculus and dynamical systems. This includes ideas on using partial differential or difference equations (PDEs) to model distance propagation or nonlinear multiscale processes in images. We briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics. We also review some nonlinear PDEs that model the evolution of multiscale morphological operators and use morphological derivatives. Among the new ideas presented, we develop some general 2-D max/min-sum difference equations that model the space dynamics of 2-D morphological systems (including the distance computations) and some nonlinear signal transforms, called slope transforms, that can analyze these systems in a transform domain in ways conceptually similar to the application of Fourier transforms to linear systems. Thus, distance transforms are shown to be bandpass slope filters. We view the analysis of the multiscale morphological PDEs and of the eikonal PDE solved via weighted distance transforms as a unified area in nonlinear image processing, which we call differential morphology, and briefly discuss its potential applications to image processing and computer vision.

Real-Time Algebraic Derivative Estimations Using a Novel Low-Cost Architecture Based on Reconfigurable Logic

PubMed Central

Morales, Rafael; Rincón, Fernando; Gazzano, Julio Dondo; López, Juan Carlos

2014-01-01

Time derivative estimation of signals plays a very important role in several fields, such as signal processing and control engineering, just to name a few of them. For that purpose, a non-asymptotic algebraic procedure for the approximate estimation of the system states is used in this work. The method is based on results from differential algebra and furnishes some general formulae for the time derivatives of a measurable signal in which two algebraic derivative estimators run simultaneously, but in an overlapping fashion. The algebraic derivative algorithm presented in this paper is computed online and in real-time, offering high robustness properties with regard to corrupting noises, versatility and ease of implementation. Besides, in this work, we introduce a novel architecture to accelerate this algebraic derivative estimator using reconfigurable logic. The core of the algorithm is implemented in an FPGA, improving the speed of the system and achieving real-time performance. Finally, this work proposes a low-cost platform for the integration of hardware in the loop in MATLAB. PMID:24859033

Lie-Hamilton systems on the plane: Properties, classification and applications

NASA Astrophysics Data System (ADS)

Ballesteros, A.; Blasco, A.; Herranz, F. J.; de Lucas, J.; Sardón, C.

2015-04-01

We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in González-López, Kamran, and Olver (1992) [23] and we interpret their results as a local classification of Lie systems. By determining which of these real Lie algebras consist of Hamiltonian vector fields relative to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. We also analyse biomathematical models, the Milne-Pinney equations, second-order Kummer-Schwarz equations, complex Riccati equations and Buchdahl equations.

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.

Feedback Controller Design for the Synchronization of Boolean Control Networks.

PubMed

Liu, Yang; Sun, Liangjie; Lu, Jianquan; Liang, Jinling

2016-09-01

This brief investigates the partial and complete synchronization of two Boolean control networks (BCNs). Necessary and sufficient conditions for partial and complete synchronization are established by the algebraic representations of logical dynamics. An algorithm is obtained to construct the feedback controller that guarantees the synchronization of master and slave BCNs. Two biological examples are provided to illustrate the effectiveness of the obtained results.

Little strings, quasi-topological sigma model on loop group, and toroidal Lie algebras

NASA Astrophysics Data System (ADS)

Ashwinkumar, Meer; Cao, Jingnan; Luo, Yuan; Tan, Meng-Chwan; Zhao, Qin

2018-03-01

We study the ground states and left-excited states of the Ak-1 N = (2 , 0) little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on CP1 with target space the based loop group of SU (k). The ground states, described by L2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.

The Matrix Pencil and its Applications to Speech Processing

DTIC Science & Technology

2007-03-01

Elementary Linear Algebra ” 8th edition, pp. 278, 2000 John Wiley & Sons, Inc., New York [37] Wai C. Chu, “Speech Coding Algorithms”, New Jeresy: John...Ben; Daniel, James W.; “Applied Linear Algebra ”, pp. 342-345, 1988 Prentice Hall, Englewood Cliffs, NJ [35] Haykin, Simon “Applied Linear Adaptive...ABSTRACT Matrix Pencils facilitate the study of differential equations resulting from oscillating systems. Certain problems in linear ordinary

Linear Ordinary Differential Equations with Constant Coefficients. Revisiting the Impulsive Response Method Using Factorization

ERIC Educational Resources Information Center

Camporesi, Roberto

2011-01-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of…

Monotonically improving approximate answers to relational algebra queries

NASA Technical Reports Server (NTRS)

Smith, Kenneth P.; Liu, J. W. S.

1989-01-01

We present here a query processing method that produces approximate answers to queries posed in standard relational algebra. This method is monotone in the sense that the accuracy of the approximate result improves with the amount of time spent producing the result. This strategy enables us to trade the time to produce the result for the accuracy of the result. An approximate relational model that characterizes appromimate relations and a partial order for comparing them is developed. Relational operators which operate on and return approximate relations are defined.

An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations

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

Prokopenko, Andrey; Tuminaro, Raymond S.

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Speci cally, we investigate a Q 2-Q 1 mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches lever- aging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocitymore » dof relationships of the Q 2-Q 1 discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the nest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.« less

Semi-automatic sparse preconditioners for high-order finite element methods on non-uniform meshes

NASA Astrophysics Data System (ADS)

Austin, Travis M.; Brezina, Marian; Jamroz, Ben; Jhurani, Chetan; Manteuffel, Thomas A.; Ruge, John

2012-05-01

High-order finite elements often have a higher accuracy per degree of freedom than the classical low-order finite elements. However, in the context of implicit time-stepping methods, high-order finite elements present challenges to the construction of efficient simulations due to the high cost of inverting the denser finite element matrix. There are many cases where simulations are limited by the memory required to store the matrix and/or the algorithmic components of the linear solver. We are particularly interested in preconditioned Krylov methods for linear systems generated by discretization of elliptic partial differential equations with high-order finite elements. Using a preconditioner like Algebraic Multigrid can be costly in terms of memory due to the need to store matrix information at the various levels. We present a novel method for defining a preconditioner for systems generated by high-order finite elements that is based on a much sparser system than the original high-order finite element system. We investigate the performance for non-uniform meshes on a cube and a cubed sphere mesh, showing that the sparser preconditioner is more efficient and uses significantly less memory. Finally, we explore new methods to construct the sparse preconditioner and examine their effectiveness for non-uniform meshes. We compare results to a direct use of Algebraic Multigrid as a preconditioner and to a two-level additive Schwarz method.

An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations

DOE PAGES

Prokopenko, Andrey; Tuminaro, Raymond S.

2016-07-01

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Speci cally, we investigate a Q 2-Q 1 mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches lever- aging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocitymore » dof relationships of the Q 2-Q 1 discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the nest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.« less

Solving multi-customer FPR model with quality assurance and discontinuous deliveries using a two-phase algebraic approach.

PubMed

Chiu, Yuan-Shyi Peter; Chou, Chung-Li; Chang, Huei-Hsin; Chiu, Singa Wang

2016-01-01

A multi-customer finite production rate (FPR) model with quality assurance and discontinuous delivery policy was investigated in a recent paper (Chiu et al. in J Appl Res Technol 12(1):5-13, 2014) using differential calculus approach. This study employs mathematical modeling along with a two-phase algebraic method to resolve such a specific multi-customer FPR model. As a result, the optimal replenishment lot size and number of shipments can be derived without using the differential calculus. Such a straightforward method may assist practitioners who with insufficient knowledge of calculus in learning and managing the real multi-customer FPR systems more effectively.

Semistrict higher gauge theory

NASA Astrophysics Data System (ADS)

Jurčo, Branislav; Sämann, Christian; Wolf, Martin

2015-04-01

We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Čech groupoid to weak Lie 2-groups. As is demonstrated, some of these Lie 2-groups can be differentiated to semistrict Lie 2-algebras by a method due to Ševera. We further derive the full description of connective structures on semistrict principal 2-bundles including the non-linear gauge transformations. As an application, we use a twistor construction to derive superconformal constraint equations in six dimensions for a non-Abelian tensor multiplet taking values in a semistrict Lie 2-algebra.

Quantum dressing orbits on compact groups

NASA Astrophysics Data System (ADS)

Jurčo, Branislav; Šťovíček, Pavel

1993-02-01

The quantum double is shown to imply the dressing transformation on quantum compact groups and the quantum Iwasawa decompositon in the general case. Quantum dressing orbits are described explicitly as *-algebras. The dual coalgebras consisting of differential operators are related to the quantum Weyl elements. Besides, the differential geometry on a quantum leaf allows a remarkably simple construction of irreducible *-representations of the algebras of quantum functions. Representation spaces then consist of analytic functions on classical phase spaces. These representations are also interpreted in the framework of quantization in the spirit of Berezin applied to symplectic leaves on classical compact groups. Convenient “coherent states” are introduced and a correspondence between classical and quantum observables is given.

An Algebraic Construction of the First Integrals of the Stationary KdV Hierarchy

NASA Astrophysics Data System (ADS)

Matsushima, Masatomo; Ohmiya, Mayumi

2009-09-01

The stationary KdV hierarchy is constructed using a kind of recursion operator called Λ-operator. The notion of the maximal solution of the n-th stationary KdV equation is introduced. Using this maximal solution, a specific differential polynomial with the auxiliary spectral parameter called the spectral M-function is constructed as the quadratic form of the fundamental system of the eigenvalue problem for the 2-nd order linear ordinary differential equation which is related to the linearizing operator of the hierarchy. By calculating a perfect square condition of the quadratic form by an elementary algebraic method, the complete set of first integrals of this hierarchy is constructed.

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.

Partitioned coupling of advection-diffusion-reaction systems and Brinkman flows

NASA Astrophysics Data System (ADS)

Lenarda, Pietro; Paggi, Marco; Ruiz Baier, Ricardo

2017-09-01

We present a partitioned algorithm aimed at extending the capabilities of existing solvers for the simulation of coupled advection-diffusion-reaction systems and incompressible, viscous flow. The space discretisation of the governing equations is based on mixed finite element methods defined on unstructured meshes, whereas the time integration hinges on an operator splitting strategy that exploits the differences in scales between the reaction, advection, and diffusion processes, considering the global system as a number of sequentially linked sets of partial differential, and algebraic equations. The flow solver presents the advantage that all unknowns in the system (here vorticity, velocity, and pressure) can be fully decoupled and thus turn the overall scheme very attractive from the computational perspective. The robustness of the proposed method is illustrated with a series of numerical tests in 2D and 3D, relevant in the modelling of bacterial bioconvection and Boussinesq systems.

Metrics for Labeled Markov Systems

NASA Technical Reports Server (NTRS)

Desharnais, Josee; Jagadeesan, Radha; Gupta, Vineet; Panangaden, Prakash

1999-01-01

Partial Labeled Markov Chains are simultaneously generalizations of process algebra and of traditional Markov chains. They provide a foundation for interacting discrete probabilistic systems, the interaction being synchronization on labels as in process algebra. Existing notions of process equivalence are too sensitive to the exact probabilities of various transitions. This paper addresses contextual reasoning principles for reasoning about more robust notions of "approximate" equivalence between concurrent interacting probabilistic systems. The present results indicate that:We develop a family of metrics between partial labeled Markov chains to formalize the notion of distance between processes. We show that processes at distance zero are bisimilar. We describe a decision procedure to compute the distance between two processes. We show that reasoning about approximate equivalence can be done compositionally by showing that process combinators do not increase distance. We introduce an asymptotic metric to capture asymptotic properties of Markov chains; and show that parallel composition does not increase asymptotic distance.

GENERAL: Application of Symplectic Algebraic Dynamics Algorithm to Circular Restricted Three-Body Problem

NASA Astrophysics Data System (ADS)

Lu, Wei-Tao; Zhang, Hua; Wang, Shun-Jin

2008-07-01

Symplectic algebraic dynamics algorithm (SADA) for ordinary differential equations is applied to solve numerically the circular restricted three-body problem (CR3BP) in dynamical astronomy for both stable motion and chaotic motion. The result is compared with those of Runge-Kutta algorithm and symplectic algorithm under the fourth order, which shows that SADA has higher accuracy than the others in the long-term calculations of the CR3BP.

Electrokinetics Models for Micro and Nano Fluidic Impedance Sensors

DTIC Science & Technology

2010-11-01

primitive Differential-Algebraic Equations (DAEs), used to process and interpret the experimentally measured electrical impedance data (Sun and Morgan...field, and species respectively. A second-order scheme was used to calculate the ionic species distribution. The linearized algebraic equations were...is governed by the Poisson equation 2 0 0 r i i i F z cε ε φ∇ + =∑ where ε0 and εr are, respectively, the electrical permittivity in the vacuum

An effective automatic procedure for testing parameter identifiability of HIV/AIDS models.

PubMed

Saccomani, Maria Pia

2011-08-01

Realistic HIV models tend to be rather complex and many recent models proposed in the literature could not yet be analyzed by traditional identifiability testing techniques. In this paper, we check a priori global identifiability of some of these nonlinear HIV models taken from the recent literature, by using a differential algebra algorithm based on previous work of the author. The algorithm is implemented in a software tool, called DAISY (Differential Algebra for Identifiability of SYstems), which has been recently released (DAISY is freely available on the web site http://www.dei.unipd.it/~pia/ ). The software can be used to automatically check global identifiability of (linear and) nonlinear models described by polynomial or rational differential equations, thus providing a general and reliable tool to test global identifiability of several HIV models proposed in the literature. It can be used by researchers with a minimum of mathematical background.

On an example of a system of differential equations that are integrated in Abelian functions

NASA Astrophysics Data System (ADS)

Malykh, M. D.; Sevastianov, L. A.

2017-12-01

The short review of the theory of Abelian functions and its applications in mechanics and analytical theory of differential equations is given. We think that Abelian functions are the natural generalization of commonly used functions because if the general solution of the 2nd order differential equation depends algebraically on the constants of integration, then integrating this equation does not lead out of the realm of commonly used functions complemented by the Abelian functions (Painlevé theorem). We present a relatively simple example of a dynamical system that is integrated in Abelian integrals by “pairing” two copies of a hyperelliptic curve. Unfortunately, initially simple formulas unfold into very long ones. Apparently the theory of Abelian functions hasn’t been finished in the last century because without computer algebra systems it was impossible to complete the calculations to the end. All calculations presented in our report are performed in Sage.

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…

A Fresh Look at Linear Ordinary Differential Equations with Constant Coefficients. Revisiting the Impulsive Response Method Using Factorization

ERIC Educational Resources Information Center

Camporesi, Roberto

2016-01-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as…

High-Order Automatic Differentiation of Unmodified Linear Algebra Routines via Nilpotent Matrices

NASA Astrophysics Data System (ADS)

Dunham, Benjamin Z.

This work presents a new automatic differentiation method, Nilpotent Matrix Differentiation (NMD), capable of propagating any order of mixed or univariate derivative through common linear algebra functions--most notably third-party sparse solvers and decomposition routines, in addition to basic matrix arithmetic operations and power series--without changing data-type or modifying code line by line; this allows differentiation across sequences of arbitrarily many such functions with minimal implementation effort. NMD works by enlarging the matrices and vectors passed to the routines, replacing each original scalar with a matrix block augmented by derivative data; these blocks are constructed with special sparsity structures, termed "stencils," each designed to be isomorphic to a particular multidimensional hypercomplex algebra. The algebras are in turn designed such that Taylor expansions of hypercomplex function evaluations are finite in length and thus exactly track derivatives without approximation error. Although this use of the method in the "forward mode" is unique in its own right, it is also possible to apply it to existing implementations of the (first-order) discrete adjoint method to find high-order derivatives with lowered cost complexity; for example, for a problem with N inputs and an adjoint solver whose cost is independent of N--i.e., O(1)--the N x N Hessian can be found in O(N) time, which is comparable to existing second-order adjoint methods that require far more problem-specific implementation effort. Higher derivatives are likewise less expensive--e.g., a N x N x N rank-three tensor can be found in O(N2). Alternatively, a Hessian-vector product can be found in O(1) time, which may open up many matrix-based simulations to a range of existing optimization or surrogate modeling approaches. As a final corollary in parallel to the NMD-adjoint hybrid method, the existing complex-step differentiation (CD) technique is also shown to be capable of finding the Hessian-vector product. All variants are implemented on a stochastic diffusion problem and compared in-depth with various cost and accuracy metrics.

Decomposition of algebraic sets and applications to weak centers of cubic systems

NASA Astrophysics Data System (ADS)

Chen, Xingwu; Zhang, Weinian

2009-10-01

There are many methods such as Gröbner basis, characteristic set and resultant, in computing an algebraic set of a system of multivariate polynomials. The common difficulties come from the complexity of computation, singularity of the corresponding matrices and some unnecessary factors in successive computation. In this paper, we decompose algebraic sets, stratum by stratum, into a union of constructible sets with Sylvester resultants, so as to simplify the procedure of elimination. Applying this decomposition to systems of multivariate polynomials resulted from period constants of reversible cubic differential systems which possess a quadratic isochronous center, we determine the order of weak centers and discuss the bifurcation of critical periods.

Laplace-Runge-Lenz vector for arbitrary spin

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

Nikitin, A. G.

2013-12-15

A countable set of superintegrable quantum mechanical systems is presented which admit the dynamical symmetry with respect to algebra so(4). This algebra is generated by the Laplace-Runge-Lenz vector generalized to the case of arbitrary spin. The presented systems describe neutral particles with non-trivial multipole momenta. Their spectra can be found algebraically like in the case of hydrogen atom. Solutions for the systems with spins 1/2 and 1 are presented explicitly, solutions for spin 3/2 can be expressed via solutions of an ordinary differential equation of first order. A more extended version of this paper including detailed calculations is published asmore » an e-print arXiv:1308.4279.« less

Introducing DAE Systems in Undergraduate and Graduate Chemical Engineering Curriculum

ERIC Educational Resources Information Center

Mandela, Ravi Kumar; Sridhar, L. N.; Rengaswamy, Raghunathan

2010-01-01

Models play an important role in understanding chemical engineering systems. While differential equation models are taught in standard modeling and control courses, Differential Algebraic Equation (DAE) system models are not usually introduced. These models appear naturally in several chemical engineering problems. In this paper, the introduction…

Exploring the Phase Space of a System of Differential Equations: Different Mathematical Registers

ERIC Educational Resources Information Center

Dana-Picard, Thierry; Kidron, Ivy

2008-01-01

We describe and analyze a situation involving symbolic representation and graphical visualization of the solution of a system of two linear differential equations, using a computer algebra system. Symbolic solution and graphical representation complement each other. Graphical representation helps to understand the behavior of the symbolic…

Advanced Placement Mathematics Calculus, Grade 12 Curriculum Guide.

ERIC Educational Resources Information Center

Scharf, John; And Others

This document is a guide to the advanced placement program in calculus for grade 12 in the city schools in Warren, Ohio. The program covers analytic geometry, differential and integral calculus of algebraic functions, elementary transcendental functions, and applications of differentiation and integration. The philosophy and aims of the program…

DIFFERENTIAL AND INTEGRAL CALCULUS. A TENTATIVE CURRICULUM GUIDE.

ERIC Educational Resources Information Center

BRANT, VINCENT; GERARDI, WILLIAM

A GUIDE FOR A 1-YEAR COURSE IN DIFFERENTIAL AND INTEGRAL CALCULUS PREREQUISITED KNOWLEDGE IN ALGEBRA, ANALYTIC TRIGONOMETRY, AND ELEMENTARY ANALYSIS. EACH ASSIGNMENT CONTAINED BOTH NEW AND REVIEW WORK TO REINFORCE THE NEW WORK. THERE WERE ELEVEN UNITS OF STUDY USING THE FOLLOWING FOUR BOOKS--"CALCULUS AND ANALYTIC GEOMETRY, THIRD…

Teachers' Reactions to Pre-Differentiated and Enriched Mathematics Curricula

ERIC Educational Resources Information Center

Rubenstein, Lisa DaVia; Gilson, Cindy M.; Bruce-Davis, Micah N.; Gubbins, E. Jean

2015-01-01

Modern classrooms are often comprised of a heterogeneous student population with varying abilities. To address this variance, third-grade teachers implemented researcher-designed, pre-differentiated, and enriched math curricula in algebra, geometry and measurement, and graphing and data analysis. The goal of the curricula was to provide academic…

Constructing Hopf bifurcation lines for the stability of nonlinear systems with two time delays

NASA Astrophysics Data System (ADS)

Nguimdo, Romain Modeste

2018-03-01

Although the plethora real-life systems modeled by nonlinear systems with two independent time delays, the algebraic expressions for determining the stability of their fixed points remain the Achilles' heel. Typically, the approach for studying the stability of delay systems consists in finding the bifurcation lines separating the stable and unstable parameter regions. This work deals with the parametric construction of algebraic expressions and their use for the determination of the stability boundaries of fixed points in nonlinear systems with two independent time delays. In particular, we concentrate on the cases for which the stability of the fixed points can be ascertained from a characteristic equation corresponding to that of scalar two-delay differential equations, one-component dual-delay feedback, or nonscalar differential equations with two delays for which the characteristic equation for the stability analysis can be reduced to that of a scalar case. Then, we apply our obtained algebraic expressions to identify either the parameter regions of stable microwaves generated by dual-delay optoelectronic oscillators or the regions of amplitude death in identical coupled oscillators.

Sensitivity Analysis and Optimization of Aerodynamic Configurations with Blend Surfaces

NASA Technical Reports Server (NTRS)

Thomas, A. M.; Tiwari, S. N.

1997-01-01

A novel (geometrical) parametrization procedure using solutions to a suitably chosen fourth order partial differential equation is used to define a class of airplane configurations. Inclusive in this definition are surface grids, volume grids, and grid sensitivity. The general airplane configuration has wing, fuselage, vertical tail and horizontal tail. The design variables are incorporated into the boundary conditions, and the solution is expressed as a Fourier series. The fuselage has circular cross section, and the radius is an algebraic function of four design parameters and an independent computational variable. Volume grids are obtained through an application of the Control Point Form method. A graphic interface software is developed which dynamically changes the surface of the airplane configuration with the change in input design variable. The software is made user friendly and is targeted towards the initial conceptual development of any aerodynamic configurations. Grid sensitivity with respect to surface design parameters and aerodynamic sensitivity coefficients based on potential flow is obtained using an Automatic Differentiation precompiler software tool ADIFOR. Aerodynamic shape optimization of the complete aircraft with twenty four design variables is performed. Unstructured and structured volume grids and Euler solutions are obtained with standard software to demonstrate the feasibility of the new surface definition.

Generalized Cartan Calculus in general dimension

DOE PAGES

Wang, Yi -Nan

2015-07-22

We develop the generalized Cartan Calculus for the groups G = SL(2,R) × R +, SL(5,R) and SO(5,5). They are the underlying algebraic structures of d=9,7,6 exceptional field theory, respectively. These algebraic identities are needed for the "tensor hierarchy" structure in exceptional field theory. The validity of Poincar\\'e lemmas in this new differential geometry is also discussed. Lastly, we explore some possible extension of the generalized Cartan calculus beyond the exceptional series.

Multiple shooting algorithms for jump-discontinuous problems in optimal control and estimation

NASA Technical Reports Server (NTRS)

Mook, D. J.; Lew, Jiann-Shiun

1991-01-01

Multiple shooting algorithms are developed for jump-discontinuous two-point boundary value problems arising in optimal control and optimal estimation. Examples illustrating the origin of such problems are given to motivate the development of the solution algorithms. The algorithms convert the necessary conditions, consisting of differential equations and transversality conditions, into algebraic equations. The solution of the algebraic equations provides exact solutions for linear problems. The existence and uniqueness of the solution are proved.

Symmetries of the Space of Linear Symplectic Connections

NASA Astrophysics Data System (ADS)

Fox, Daniel J. F.

2017-01-01

There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle polynomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their! linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term.

3D algebraic iterative reconstruction for cone-beam x-ray differential phase-contrast computed tomography.

PubMed

Fu, Jian; Hu, Xinhua; Velroyen, Astrid; Bech, Martin; Jiang, Ming; Pfeiffer, Franz

2015-01-01

Due to the potential of compact imaging systems with magnified spatial resolution and contrast, cone-beam x-ray differential phase-contrast computed tomography (DPC-CT) has attracted significant interest. The current proposed FDK reconstruction algorithm with the Hilbert imaginary filter will induce severe cone-beam artifacts when the cone-beam angle becomes large. In this paper, we propose an algebraic iterative reconstruction (AIR) method for cone-beam DPC-CT and report its experiment results. This approach considers the reconstruction process as the optimization of a discrete representation of the object function to satisfy a system of equations that describes the cone-beam DPC-CT imaging modality. Unlike the conventional iterative algorithms for absorption-based CT, it involves the derivative operation to the forward projections of the reconstructed intermediate image to take into account the differential nature of the DPC projections. This method is based on the algebraic reconstruction technique, reconstructs the image ray by ray, and is expected to provide better derivative estimates in iterations. This work comprises a numerical study of the algorithm and its experimental verification using a dataset measured with a three-grating interferometer and a mini-focus x-ray tube source. It is shown that the proposed method can reduce the cone-beam artifacts and performs better than FDK under large cone-beam angles. This algorithm is of interest for future cone-beam DPC-CT applications.

The Mathlet Toolkit: Creating Dynamic Applets for Differential Equations and Dynamical Systems

ERIC Educational Resources Information Center

Decker, Robert

2011-01-01

Dynamic/interactive graphing applets can be used to supplement standard computer algebra systems such as Maple, Mathematica, Derive, or TI calculators, in courses such as Calculus, Differential Equations, and Dynamical Systems. The addition of this type of software can lead to discovery learning, with students developing their own conjectures, and…

The algebraic theory of latent projectors in lambda matrices

NASA Technical Reports Server (NTRS)

Denman, E. D.; Leyva-Ramos, J.; Jeon, G. J.

1981-01-01

Multivariable systems such as a finite-element model of vibrating structures, control systems, and large-scale systems are often formulated in terms of differential equations which give rise to lambda matrices. The present investigation is concerned with the formulation of the algebraic theory of lambda matrices and the relationship of latent roots, latent vectors, and latent projectors to the eigenvalues, eigenvectors, and eigenprojectors of the companion form. The chain rule for latent projectors and eigenprojectors for the repeated latent root or eigenvalues is given.

A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank

NASA Astrophysics Data System (ADS)

Fu, Yuchen; Shelley-Abrahamson, Seth

2016-06-01

We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using R-matrices for U_q(sl_N). Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.

A Comprehensive Analytical Model of Rotorcraft Aerodynamics and Dynamics. Part 1. Analysis Development

DTIC Science & Technology

1980-06-01

sufficient. Dropping the time lag terms, the equations for Xu, Xx’, and X reduce to linear algebraic equations.Y Hence in the quasistatic case the...quasistatic variables now are not described by differential equations but rather by linear algebraic equations. The solution for x0 then is simply -365...matrices for two-bladed rotor 414 7. LINEAR SYSTEM ANALYSIS 425 7,1 State Variable Form 425 7.2 Constant Coefficient System 426 7.2. 1 Eigen-analysis 426

Geometric algebra description of polarization mode dispersion, polarization-dependent loss, and Stokes tensor transformations.

PubMed

Soliman, George; Yevick, David; Jessop, Paul

2014-09-01

This paper demonstrates that numerous calculations involving polarization transformations can be condensed by employing suitable geometric algebra formalism. For example, to describe polarization mode dispersion and polarization-dependent loss, both the material birefringence and differential loss enter as bivectors and can be combined into a single symmetric quantity. Their frequency and distance evolution, as well as that of the Stokes vector through an optical system, can then each be expressed as a single compact expression, in contrast to the corresponding Mueller matrix formulations. The intrinsic advantage of the geometric algebra framework is further demonstrated by presenting a simplified derivation of generalized Stokes parameters that include the electric field phase. This procedure simultaneously establishes the tensor transformation properties of these parameters.

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

Structural identifiability analyses of candidate models for in vitro Pitavastatin hepatic uptake.

PubMed

Grandjean, Thomas R B; Chappell, Michael J; Yates, James W T; Evans, Neil D

2014-05-01

In this paper a review of the application of four different techniques (a version of the similarity transformation approach for autonomous uncontrolled systems, a non-differential input/output observable normal form approach, the characteristic set differential algebra and a recent algebraic input/output relationship approach) to determine the structural identifiability of certain in vitro nonlinear pharmacokinetic models is provided. The Organic Anion Transporting Polypeptide (OATP) substrate, Pitavastatin, is used as a probe on freshly isolated animal and human hepatocytes. Candidate pharmacokinetic non-linear compartmental models have been derived to characterise the uptake process of Pitavastatin. As a prerequisite to parameter estimation, structural identifiability analyses are performed to establish that all unknown parameters can be identified from the experimental observations available. Copyright © 2013. Published by Elsevier Ireland Ltd.

A comparison of different methods to implement higher order derivatives of density functionals

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

van Dam, Hubertus J.J.

Density functional theory is the dominant approach in electronic structure methods today. To calculate properties higher order derivatives of the density functionals are required. These derivatives might be implemented manually,by automatic differentiation, or by symbolic algebra programs. Different authors have cited different reasons for using the particular method of their choice. This paper presents work where all three approaches were used and the strengths and weaknesses of each approach are considered. It is found that all three methods produce code that is suffficiently performanted for practical applications, despite the fact that our symbolic algebra generated code and our automatic differentiationmore » code still have scope for significant optimization. The automatic differentiation approach is the best option for producing readable and maintainable code.« less

Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions

NASA Astrophysics Data System (ADS)

Bui, Quan M.; Wang, Lu; Osei-Kuffuor, Daniel

2018-04-01

Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the partial differential equations modeling the flow. In the case of multiphase multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are taken into account. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem (NCP). Unlike in the primary variable switching technique, the set of primary variables in this approach is fixed even when there is phase transition. Not only does this improve the robustness of the nonlinear solver, it opens up the possibility to use multigrid methods to solve the resulting linear system. The disadvantage of the complementarity approach, however, is that when a phase disappears, the linear system has the structure of a saddle point problem and becomes indefinite, and current algebraic multigrid (AMG) algorithms cannot be applied directly. In this study, we explore the effectiveness of a new multilevel strategy, based on the multigrid reduction technique, to deal with problems of this type. We demonstrate the effectiveness of the method through numerical results for the case of two-phase, two-component flow with phase appearance/disappearance. We also show that the strategy is efficient and scales optimally with problem size.

The use of spectral methods in bidomain studies.

PubMed

Trayanova, N; Pilkington, T

1992-01-01

A Fourier transform method is developed for solving the bidomain coupled differential equations governing the intracellular and extracellular potentials on a finite sheet of cardiac cells undergoing stimulation. The spectral formulation converts the system of differential equations into a "diagonal" system of algebraic equations. Solving the algebraic equations directly and taking the inverse transform of the potentials proved numerically less expensive than solving the coupled differential equations by means of traditional numerical techniques, such as finite differences; the comparison between the computer execution times showed that the Fourier transform method was about 40 times faster than the finite difference method. By application of the Fourier transform method, transmembrane potential distributions in the two-dimensional myocardial slice were calculated. For a tissue characterized by a ratio of the intra- to extracellular conductivities that is different in all principal directions, the transmembrane potential distribution exhibits a rather complicated geometrical pattern. The influence of the different anisotropy ratios, the finite tissue size, and the stimuli configuration on the pattern of membrane polarization is investigated.

Multiple symbol partially coherent detection of MPSK

NASA Technical Reports Server (NTRS)

Simon, M. K.; Divsalar, D.

1992-01-01

It is shown that by using the known (or estimated) value of carrier tracking loop signal to noise ratio (SNR) in the decision metric, it is possible to improve the error probability performance of a partially coherent multiple phase-shift-keying (MPSK) system relative to that corresponding to the commonly used ideal coherent decision rule. Using a maximum-likeihood approach, an optimum decision metric is derived and shown to take the form of a weighted sum of the ideal coherent decision metric (i.e., correlation) and the noncoherent decision metric which is optimum for differential detection of MPSK. The performance of a receiver based on this optimum decision rule is derived and shown to provide continued improvement with increasing length of observation interval (data symbol sequence length). Unfortunately, increasing the observation length does not eliminate the error floor associated with the finite loop SNR. Nevertheless, in the limit of infinite observation length, the average error probability performance approaches the algebraic sum of the error floor and the performance of ideal coherent detection, i.e., at any error probability above the error floor, there is no degradation due to the partial coherence. It is shown that this limiting behavior is virtually achievable with practical size observation lengths. Furthermore, the performance is quite insensitive to mismatch between the estimate of loop SNR (e.g., obtained from measurement) fed to the decision metric and its true value. These results may be of use in low-cost Earth-orbiting or deep-space missions employing coded modulations.

Stable isotope ratios of carbon and nitrogen and mercury concentrations in 13 toothed whale species taken from the western Pacific Ocean off Japan.

PubMed

Endo, Tetsuya; Hisamichi, Yohsuke; Kimura, Osamu; Haraguchi, Koichi; Lavery, Shane; Dalebout, Merel L; Funahashi, Naoko; Baker, C Scott

2010-04-01

Stable isotope ratios of carbon (partial differential(13)C) and nitrogen (partial differential(15)N) and total mercury (T-Hg) concentrations were measured in red meat samples from 11 odontocete species (toothed whales, dolphins, and porpoises) sold in Japan (n = 96) and in muscle samples from stranded killer whales (n = 6) and melon-headed whales (n = 15), and the analytical data for these species were classified into three regions (northern, central, and southern Japan) depending on the locations in which they were caught or stranded. The partial differential(15)N in the samples from southern Japan tended to be lower than that in samples from the north, whereas both partial differential(13)C and T-Hg concentrations in samples from the south tended to higher than those in samples from northern Japan. Negative correlations were found between the partial differential(13)C and partial differential(15)N values and between the partial differential(15)N value and T-Hg concentrations in the combined samples all three regions (gamma= -0.238, n = 117, P < 0.01). The partial differential(13)C, partial differential(15)N, and T-Hg concentrations in the samples varied more by habitat than by species. Spatial variations in partial differential(13)C, partial differential(15)N, and T-Hg concentrations in the ocean may be the cause of these phenomena.

Computational algebraic geometry of epidemic models

NASA Astrophysics Data System (ADS)

Rodríguez Vega, Martín.

2014-06-01

Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.

COED Transactions, Vol. IX, No. 2, February 1977. Prism: An Educational Aide to Symbolic Differentiation and Simplification of Algebraic Expressions.

ERIC Educational Resources Information Center

Marcovitz, Alan B., Ed.

A computer program for numeric and symbolic manipulation and the methodology underlying its development are presented. Some features of the program are: an option for implied multiplication; computation of higher-order derivatives; differentiation of 26 different trigonometric, hyperbolic, inverse trigonometric, and inverse hyperbolic functions;…

Control of nonlinear systems with applications to constrained robots and spacecraft attitude stabilization

NASA Technical Reports Server (NTRS)

Krishnan, Hariharan

1993-01-01

This thesis is organized in two parts. In Part 1, control systems described by a class of nonlinear differential and algebraic equations are introduced. A procedure for local stabilization based on a local state realization is developed. An alternative approach to local stabilization is developed based on a classical linearization of the nonlinear differential-algebraic equations. A theoretical framework is established for solving a tracking problem associated with the differential-algebraic system. First, a simple procedure is developed for the design of a feedback control law which ensures, at least locally, that the tracking error in the closed loop system lies within any given bound if the reference inputs are sufficiently slowly varying. Next, by imposing additional assumptions, a procedure is developed for the design of a feedback control law which ensures that the tracking error in the closed loop system approaches zero exponentially for reference inputs which are not necessarily slowly varying. The control design methodologies are used for simultaneous force and position control in constrained robot systems. The differential-algebraic equations are shown to characterize the slow dynamics of a certain nonlinear control system in nonstandard singularly perturbed form. In Part 2, the attitude stabilization (reorientation) of a rigid spacecraft using only two control torques is considered. First, the case of momentum wheel actuators is considered. The complete spacecraft dynamics are not controllable. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but a discontinuous feedback control strategy is constructed. Next, the case of gas jet actuators is considered. If the uncontrolled principal axis is not an axis of symmetry, the complete spacecraft dynamics are small time locally controllable. However, the spacecraft attitude cannot be asymptotically stabilized using continuous feedback, but a discontinuous stabilizing feedback control strategy is constructed. If the uncontrolled principal axis is an axis of symmetry, the complete spacecraft dynamics cannot be stabilized. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but again a discontinuous feedback control strategy is constructed.

Static Analysis of Large-Scale Multibody System Using Joint Coordinates and Spatial Algebra Operator

PubMed Central

Omar, Mohamed A.

2014-01-01

Initial transient oscillations inhibited in the dynamic simulations responses of multibody systems can lead to inaccurate results, unrealistic load prediction, or simulation failure. These transients could result from incompatible initial conditions, initial constraints violation, and inadequate kinematic assembly. Performing static equilibrium analysis before the dynamic simulation can eliminate these transients and lead to stable simulation. Most exiting multibody formulations determine the static equilibrium position by minimizing the system potential energy. This paper presents a new general purpose approach for solving the static equilibrium in large-scale articulated multibody. The proposed approach introduces an energy drainage mechanism based on Baumgarte constraint stabilization approach to determine the static equilibrium position. The spatial algebra operator is used to express the kinematic and dynamic equations of the closed-loop multibody system. The proposed multibody system formulation utilizes the joint coordinates and modal elastic coordinates as the system generalized coordinates. The recursive nonlinear equations of motion are formulated using the Cartesian coordinates and the joint coordinates to form an augmented set of differential algebraic equations. Then system connectivity matrix is derived from the system topological relations and used to project the Cartesian quantities into the joint subspace leading to minimum set of differential equations. PMID:25045732

Remediating Number Combination and Word Problem Deficits Among Students With Mathematics Difficulties: A Randomized Control Trial

PubMed Central

Fuchs, Lynn S.; Powell, Sarah R.; Seethaler, Pamela M.; Cirino, Paul T.; Fletcher, Jack M.; Fuchs, Douglas; Hamlett, Carol L.; Zumeta, Rebecca O.

2009-01-01

The purposes of this study were to assess the efficacy of remedial tutoring for 3rd graders with mathematics difficulty, to investigate whether tutoring is differentially efficacious depending on students’ math difficulty status (mathematics difficulty alone vs. mathematics plus reading difficulty), to explore transfer from number combination (NC) remediation, and to examine the transportability of the tutoring protocols. At 2 sites, 133 students were stratified on mathematics difficulty status and site and then randomly assigned to 3 conditions: control (no tutoring), tutoring on automatic retrieval of NCs (i.e., Math Flash), or tutoring on word problems with attention to the foundational skills of NCs, procedural calculations, and algebra (i.e., Pirate Math). Tutoring occurred for 16 weeks, 3 sessions per week and 20–30 min per session. Math Flash enhanced fluency with NCs with transfer to procedural computation but without transfer to algebra or word problems. Pirate Math enhanced word problem skill as well as fluency with NCs, procedural computation, and algebra. Tutoring was not differentially efficacious as a function of students’ mathematics difficulty status. The tutoring protocols proved transportable across sites. PMID:19865600

Static analysis of large-scale multibody system using joint coordinates and spatial algebra operator.

PubMed

Omar, Mohamed A

2014-01-01

Initial transient oscillations inhibited in the dynamic simulations responses of multibody systems can lead to inaccurate results, unrealistic load prediction, or simulation failure. These transients could result from incompatible initial conditions, initial constraints violation, and inadequate kinematic assembly. Performing static equilibrium analysis before the dynamic simulation can eliminate these transients and lead to stable simulation. Most exiting multibody formulations determine the static equilibrium position by minimizing the system potential energy. This paper presents a new general purpose approach for solving the static equilibrium in large-scale articulated multibody. The proposed approach introduces an energy drainage mechanism based on Baumgarte constraint stabilization approach to determine the static equilibrium position. The spatial algebra operator is used to express the kinematic and dynamic equations of the closed-loop multibody system. The proposed multibody system formulation utilizes the joint coordinates and modal elastic coordinates as the system generalized coordinates. The recursive nonlinear equations of motion are formulated using the Cartesian coordinates and the joint coordinates to form an augmented set of differential algebraic equations. Then system connectivity matrix is derived from the system topological relations and used to project the Cartesian quantities into the joint subspace leading to minimum set of differential equations.

Dual number algebra method for Green's function derivatives in 3D magneto-electro-elasticity

NASA Astrophysics Data System (ADS)

Dziatkiewicz, Grzegorz

2018-01-01

The Green functions are the basic elements of the boundary element method. To obtain the boundary integral formulation the Green function and its derivative should be known for the considered differential operator. Today the interesting group of materials are electronic composites. The special case of the electronic composite is the magnetoelectroelastic continuum. The mentioned continuum is a model of the piezoelectric-piezomagnetic composites. The anisotropy of their physical properties makes the problem of Green's function determination very difficult. For that reason Green's functions for the magnetoelectroelastic continuum are not known in the closed form and numerical methods should be applied to determine such Green's functions. These means that the problem of the accurate and simply determination of Green's function derivatives is even harder. Therefore in the present work the dual number algebra method is applied to calculate numerically the derivatives of 3D Green's functions for the magnetoelectroelastic materials. The introduced method is independent on the step size and it can be treated as a special case of the automatic differentiation method. Therefore, the dual number algebra method can be applied as a tool for checking the accuracy of the well-known finite difference schemes.

New Actions Upon Old Objects: A New Ontological Perspective on Functions.

ERIC Educational Resources Information Center

Schwarz, Baruch; Dreyfus, Tommy

1995-01-01

A computer microworld called Triple Representation Model uses graphical, tabular, and algebraic representations to influence conceptions of function. A majority of students were able to cope with partial data, recognize invariants while coordinating actions among representations, and recognize invariants while creating and comparing different…

Radioactivity Calculations

ERIC Educational Resources Information Center

Onega, Ronald J.

1969-01-01

Three problems in radioactive buildup and decay are presented and solved. Matrix algebra is used to solve the second problem. The third problem deals with flux depression and is solved by the use of differential equations. (LC)

A multivariate variational objective analysis-assimilation method. Part 1: Development of the basic model

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.; Ochs, Harry T., III

1988-01-01

The variational method of undetermined multipliers is used to derive a multivariate model for objective analysis. The model is intended for the assimilation of 3-D fields of rawinsonde height, temperature and wind, and mean level temperature observed by satellite into a dynamically consistent data set. Relative measurement errors are taken into account. The dynamic equations are the two nonlinear horizontal momentum equations, the hydrostatic equation, and an integrated continuity equation. The model Euler-Lagrange equations are eleven linear and/or nonlinear partial differential and/or algebraic equations. A cyclical solution sequence is described. Other model features include a nonlinear terrain-following vertical coordinate that eliminates truncation error in the pressure gradient terms of the horizontal momentum equations and easily accommodates satellite observed mean layer temperatures in the middle and upper troposphere. A projection of the pressure gradient onto equivalent pressure surfaces removes most of the adverse impacts of the lower coordinate surface on the variational adjustment.

Exhaustive Classification of the Invariant Solutions for a Specific Nonlinear Model Describing Near Planar and Marginally Long-Wave Unstable Interfaces for Phase Transition

NASA Astrophysics Data System (ADS)

Ahangari, Fatemeh

2018-05-01

Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dynamics can be modeled by an interface motion. This paper is devoted to exhaustive analysis of the invariant solutions for a modified Kuramoto-Sivashinsky equation in two spatial and one temporal dimensions is presented. This nonlinear partial differential equation asymptotically characterizes near planar interfaces, which are marginally long-wave unstable. For this purpose, by applying the classical symmetry method for this model the classical symmetry operators are attained. Moreover, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras, which yields the preliminary classification of group invariant solutions is constructed. Mainly, the Lie invariants corresponding to the infinitesimal symmetry generators as well as associated similarity reduced equations are also pointed out. Furthermore, the nonclassical symmetries of this nonlinear PDE are also comprehensively investigated.

Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics

NASA Astrophysics Data System (ADS)

Kholodenko, Arkady L.; Kauffman, Louis H.

2018-03-01

Huygens triviality - a concept invented by Jacques Hadamard - describes an equivalence class connecting those 2nd order partial differential equations which are transformable into the wave equation. In this work it is demonstrated, that the Schrödinger equation with the time-independent Hamiltonian belongs to such an equivalence class. The wave equation is the equation for which Huygens' principle (HP) holds. The HP was a subject of confusion in both physics and mathematics literature for a long time. Not surprisingly, the role of this principle was obscured from the beginnings of quantum mechanics causing some theoretical and experimental misunderstandings. The purpose of this work is to bring the full clarity into this topic. By doing so, we obtained a large amount of new results related to uses of Lie sphere geometry, of twistors, of Dupin cyclides, of null electromagnetic fields, of AdS-CFT correspondence, of Penrose limits, of geometric algebra, etc. in physical problems ranging from the atomic to high energy physics and cosmology.

On nonlinear Tollmien-Schlichting/vortex interaction in three-dimensional boundary layers

NASA Technical Reports Server (NTRS)

Davis, Dominic A. R.; Smith, Frank T.

1993-01-01

The instability of an incompressible three-dimensional boundary layer (that is, one with cross-flow) is considered theoretically and computationally in the context of vortex/wave interactions. Specifically the work centers on two low amplitude, lower-branch Tollmien-Schlichting waves which mutually interact to induce a weak longitudinal vortex flow; the vortex motion, in turn, gives rise to significant wave-modulation via wall-shear forcing. The characteristic Reynolds number is taken as a large parameter and, as a consequence, the waves' and the vortex motion are governed primarily by triple-deck theory. The nonlinear interaction is captured by a viscous partial-differential system for the vortex coupled with a pair of amplitude equations for each wave pressure. Three distinct possibilities were found to emerge for the nonlinear behavior of the flow solution downstream - an algebraic finite-distance singularity, far downstream saturation or far-downstream wave-decay (leaving pure vortex flow) - depending on the input conditions, the wave angles, and the size of the cross-flow.

Computational manipulation of a radiative MHD flow with Hall current and chemical reaction in the presence of rotating fluid

NASA Astrophysics Data System (ADS)

Alias Suba, Subbu; Muthucumaraswamy, R.

2018-04-01

A numerical analysis of transient radiative MHD(MagnetoHydroDynamic) natural convective flow of a viscous, incompressible, electrically conducting and rotating fluid along a semi-infinite isothermal vertical plate is carried out taking into consideration Hall current, rotation and first order chemical reaction.The coupled non-linear partial differential equations are expressed in difference form using implicit finite difference scheme. The difference equations are then reduced to a system of linear algebraic equations with a tri-diagonal structure which is solved by Thomas Algorithm. The primary and secondary velocity profiles, temperature profile, concentration profile, skin friction, Nusselt number and Sherwood Number are depicted graphically for a range of values of rotation parameter, Hall parameter,magnetic parameter, chemical reaction parameter, radiation parameter, Prandtl number and Schmidt number.It is recognized that rate of heat transfer and rate of mass transfer decrease with increase in time but they increase with increasing values of radiation parameter and Schmidt number respectively.

An automatic multigrid method for the solution of sparse linear systems

NASA Technical Reports Server (NTRS)

Shapira, Yair; Israeli, Moshe; Sidi, Avram

1993-01-01

An automatic version of the multigrid method for the solution of linear systems arising from the discretization of elliptic PDE's is presented. This version is based on the structure of the algebraic system solely, and does not use the original partial differential operator. Numerical experiments show that for the Poisson equation the rate of convergence of our method is equal to that of classical multigrid methods. Moreover, the method is robust in the sense that its high rate of convergence is conserved for other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. No double discretization or special treatment of sub-domains (e.g. boundaries) is needed. When supplemented with a vector extrapolation method, high rates of convergence are achieved also for anisotropic and discontinuous problems and also for indefinite Helmholtz equations. A new double discretization strategy is proposed for finite and spectral element schemes and is found better than known strategies.

Helmholtz-Smoluchowski velocity for viscoelastic electroosmotic flows.

PubMed

Park, H M; Lee, W M

2008-01-15

Many biofluids such as blood and DNA solutions are viscoelastic and exhibit extraordinary flow behaviors, not existing in Newtonian fluids. Adopting appropriate constitutive equations these exotic flow behaviors can be modeled and predicted reasonably using various numerical methods. However, the governing equations for viscoelastic flows are not easily solvable, especially for electroosmotic flows where the streamwise velocity varies rapidly from zero at the wall to a nearly uniform velocity at the outside of the very thin electric double layer. In the present investigation, we have devised a simple method to find the volumetric flow rate of viscoelastic electroosmotic flows through microchannels. It is based on the concept of the Helmholtz-Smoluchowski velocity which is widely adopted in the electroosmotic flows of Newtonian fluids. It is shown that the Helmholtz-Smoluchowski velocity for viscoelastic fluids can be found by solving a simple cubic algebraic equation. The volumetric flow rate obtained using this Helmholtz-Smoluchowski velocity is found to be almost the same as that obtained by solving the governing partial differential equations for various viscoelastic fluids.

Lumped Model Generation and Evaluation: Sensitivity and Lie Algebraic Techniques with Applications to Combustion

DTIC Science & Technology

1989-03-03

address global parameter space mapping issues for first order differential equations. The rigorous criteria for the existence of exact lumping by linear projective transformations was also established.

Microscopic approach based on a multiscale algebraic version of the resonating group model for radiative capture reactions

NASA Astrophysics Data System (ADS)

Solovyev, Alexander S.; Igashov, Sergey Yu.

2017-12-01

A microscopic approach to description of radiative capture reactions based on a multiscale algebraic version of the resonating group model is developed. The main idea of the approach is to expand wave functions of discrete spectrum and continuum for a nuclear system over different bases of the algebraic version of the resonating group model. These bases differ from each other by values of oscillator radius playing a role of scale parameter. This allows us in a unified way to calculate total and partial cross sections (astrophysical S factors) as well as branching ratio for the radiative capture reaction, to describe phase shifts for the colliding nuclei in the initial channel of the reaction, and at the same time to reproduce breakup thresholds of the final nucleus. The approach is applied to the theoretical study of the mirror 3H(α ,γ )7Li and 3He(α ,γ )7Be reactions, which are of great interest to nuclear astrophysics. The calculated results are compared with existing experimental data and with our previous calculations in the framework of the single-scale algebraic version of the resonating group model.

Eisenstein type series for Calabi-Yau varieties

NASA Astrophysics Data System (ADS)

Movasati, Hossein

2011-06-01

In this article we introduce an ordinary differential equation associated to the one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions written in the q-expansion form and the Yukawa coupling turns out to be rational in these functions. We prove that these functions are algebraically independent over the field of complex numbers, and hence, the algebra generated by such functions can be interpreted as the theory of (quasi) modular forms attached to the one parameter family of Calabi-Yau varieties. Our result is a reformulation and realization of a problem of Griffiths around seventies on the existence of automorphic functions for the moduli of polarized Hodge structures. It is a generalization of the Ramanujan differential equation satisfied by three Eisenstein series.

Quantum Koszul formula on quantum spacetime

NASA Astrophysics Data System (ADS)

Majid, Shahn; Williams, Liam

2018-07-01

Noncommutative or quantum Riemannian geometry has been proposed as an effective theory for aspects of quantum gravity. Here the metric is an invertible bimodule map Ω1⊗AΩ1 → A where A is a possibly noncommutative or 'quantum' spacetime coordinate algebra and (Ω1 , d) is a specified bimodule of 1-forms or 'differential calculus' over it. In this paper we explore the proposal of a 'quantum Koszul formula' in Majid [12] with initial data a degree - 2 bilinear map ⊥ on the full exterior algebra Ω obeying the 4-term relations

An Algorithm for Deriving Fiscal Equity Indicators of a School Financing Structure: Application to Nebraska.

ERIC Educational Resources Information Center

Hayden, F. Gregory

1980-01-01

By algebraically defining a school finance structure as a total budget system, partial derivatives can be used to find the kinds of rewards, incentives, and distributions the structure defines for individual districts and among districts. Equity concerns can also be answered. (Author/IRT)

On the geometry of inhomogeneous quantum groups

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

Aschieri, Paolo

1998-01-01

The author gives a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case. He further analyzes the relation between differential calculus and quantum Lie algebra of left (right) invariant vectorfields. Equivalent definitions of bicovariant differential calculus are studied and their geometrical interpretation is explained. From these data he constructs and analyzes the space of vectorfields, and naturally introduces a contraction operator and a Lie derivative. Their properties are discussed.

Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: An eigenvalue analysis

NASA Technical Reports Server (NTRS)

Warming, Robert F.; Beam, Richard M.

1986-01-01

A hyperbolic initial-boundary-value problem can be approximated by a system of ordinary differential equations (ODEs) by replacing the spatial derivatives by finite-difference approximations. The resulting system of ODEs is called a semidiscrete approximation. A complication is the fact that more boundary conditions are required for the spatially discrete approximation than are specified for the partial differential equation. Consequently, additional numerical boundary conditions are required and improper treatment of these additional conditions can lead to instability. For a linear initial-boundary-value problem (IBVP) with homogeneous analytical boundary conditions, the semidiscrete approximation results in a system of ODEs of the form du/dt = Au whose solution can be written as u(t) = exp(At)u(O). Lax-Richtmyer stability requires that the matrix norm of exp(At) be uniformly bounded for O less than or = t less than or = T independent of the spatial mesh size. Although the classical Lax-Richtmyer stability definition involves a conventional vector norm, there is no known algebraic test for the uniform boundedness of the matrix norm of exp(At) for hyperbolic IBVPs. An alternative but more complicated stability definition is used in the theory developed by Gustafsson, Kreiss, and Sundstrom (GKS). The two methods are compared.

Some applications of mathematics in theoretical physics - A review

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

Bora, Kalpana

2016-06-21

Mathematics is a very beautiful subject−very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like−differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical toolsmore » are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.« less

Some applications of mathematics in theoretical physics - A review

NASA Astrophysics Data System (ADS)

Bora, Kalpana

2016-06-01

Mathematics is a very beautiful subject-very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like-differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical tools are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.

GHM method for obtaining rationalsolutions of nonlinear differential equations.

PubMed

Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo

2015-01-01

In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.

A brief survey of constrained mechanics and variational problems in terms of differential forms

NASA Technical Reports Server (NTRS)

Hermann, Robert

1994-01-01

There has been considerable interest recently in constrained mechanics and variational problems. This is in part due to applied interests (such as 'non-holonomic mechanics in robotics') and in other part due to the fact that several schools of 'pure' mathematics have found that this classical subject is of importance for what they are trying to do. I have made various attempts at developing these subjects since my Lincoln lab days of the late 1950's. In this Chapter, I will sketch a Unified point of view, using Cartan's approach with differential forms. This has the advantage from the C-O-R viewpoint being developed in this Volume that the extension from 'smooth' to 'generalized' data is very systematic and algebraic. (I will only deal with the 'smooth' point of view in this Chapter; I will develop the 'generalized function' material at a later point.) The material presented briefly here about Variational Calculus and Constrained Mechanics can be found in more detail in my books, 'Differential Geometry and the Calculus of Variations', 'Lie Algebras and Quantum Mechanics', and 'Geometry, Physics and Systems'.

On method of solving third-order ordinary differential equations directly using Bernstein polynomials

NASA Astrophysics Data System (ADS)

Khataybeh, S. N.; Hashim, I.

2018-04-01

In this paper, we propose for the first time a method based on Bernstein polynomials for solving directly a class of third-order ordinary differential equations (ODEs). This method gives a numerical solution by converting the equation into a system of algebraic equations which is solved directly. Some numerical examples are given to show the applicability of the method.

Accessible Information Without Disturbing Partially Known Quantum States on a von Neumann Algebra

NASA Astrophysics Data System (ADS)

Kuramochi, Yui

2018-04-01

This paper addresses the problem of how much information we can extract without disturbing a statistical experiment, which is a family of partially known normal states on a von Neumann algebra. We define the classical part of a statistical experiment as the restriction of the equivalent minimal sufficient statistical experiment to the center of the outcome space, which, in the case of density operators on a Hilbert space, corresponds to the classical probability distributions appearing in the maximal decomposition by Koashi and Imoto (Phys. Rev. A 66, 022,318 2002). We show that we can access by a Schwarz or completely positive channel at most the classical part of a statistical experiment if we do not disturb the states. We apply this result to the broadcasting problem of a statistical experiment. We also show that the classical part of the direct product of statistical experiments is the direct product of the classical parts of the statistical experiments. The proof of the latter result is based on the theorem that the direct product of minimal sufficient statistical experiments is also minimal sufficient.

SH c realization of minimal model CFT: triality, poset and Burge condition

NASA Astrophysics Data System (ADS)

Fukuda, M.; Nakamura, S.; Matsuo, Y.; Zhu, R.-D.

2015-11-01

Recently an orthogonal basis of {{W}}_N -algebra (AFLT basis) labeled by N-tuple Young diagrams was found in the context of 4D/2D duality. Recursion relations among the basis are summarized in the form of an algebra SH c which is universal for any N. We show that it has an {{S}}_3 automorphism which is referred to as triality. We study the level-rank duality between minimal models, which is a special example of the automorphism. It is shown that the nonvanishing states in both systems are described by N or M Young diagrams with the rows of boxes appropriately shuffled. The reshuffling of rows implies there exists partial ordering of the set which labels them. For the simplest example, one can compute the partition functions for the partially ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan identities. We also study the description of minimal models by SH c . Simple analysis reproduces some known properties of minimal models, the structure of singular vectors and the N-Burge condition in the Hilbert space.

On 3-gauge transformations, 3-curvatures, and Gray-categories

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

Wang, Wei, E-mail: wwang@zju.edu.cn

In the 3-gauge theory, a 3-connection is given by a 1-form A valued in the Lie algebra g, a 2-form B valued in the Lie algebra h, and a 3-form C valued in the Lie algebra l, where (g,h,l) constitutes a differential 2-crossed module. We give the 3-gauge transformations from one 3-connection to another, and show the transformation formulae of the 1-curvature 2-form, the 2-curvature 3-form, and the 3-curvature 4-form. The gauge configurations can be interpreted as smooth Gray-functors between two Gray 3-groupoids: the path 3-groupoid P{sub 3}(X) and the 3-gauge group G{sup L} associated to the 2-crossed module L,more » whose differential is (g,h,l). The derivatives of Gray-functors are 3-connections, and the derivatives of lax-natural transformations between two such Gray-functors are 3-gauge transformations. We give the 3-dimensional holonomy, the lattice version of the 3-curvature, whose derivative gives the 3-curvature 4-form. The covariance of 3-curvatures easily follows from this construction. This Gray-categorical construction explains why 3-gauge transformations and 3-curvatures have the given forms. The interchanging 3-arrows are responsible for the appearance of terms with the Peiffer commutator (, )« less

Lectures on Kähler Geometry - Series: London Mathematical Society Student Texts (No. 69)

NASA Astrophysics Data System (ADS)

Moroianu, Andrei

2004-03-01

Kähler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. Kähler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous Kähler identities. The final part of the text studies several aspects of compact Kähler manifolds: the Calabi conjecture, Weitzenböck techniques, Calabi Yau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory. The first graduate-level text on Kähler geometry, providing a concise introduction for both mathematicians and physicists with a basic knowledge of calculus in several variables and linear algebra Over 130 exercises and worked examples Self-contained and presents varying viewpoints including Riemannian, complex and algebraic

On Asymptotically Good Ramp Secret Sharing Schemes

NASA Astrophysics Data System (ADS)

Geil, Olav; Martin, Stefano; Martínez-Peñas, Umberto; Matsumoto, Ryutaroh; Ruano, Diego

Asymptotically good sequences of linear ramp secret sharing schemes have been intensively studied by Cramer et al. in terms of sequences of pairs of nested algebraic geometric codes. In those works the focus is on full privacy and full reconstruction. In this paper we analyze additional parameters describing the asymptotic behavior of partial information leakage and possibly also partial reconstruction giving a more complete picture of the access structure for sequences of linear ramp secret sharing schemes. Our study involves a detailed treatment of the (relative) generalized Hamming weights of the considered codes.

More on the hidden symmetries of 11D supergravity

NASA Astrophysics Data System (ADS)

Andrianopoli, Laura; D'Auria, Riccardo; Ravera, Lucrezia

2017-09-01

In this paper we clarify the relations occurring among the osp (1 | 32) algebra, the M-algebra and the hidden superalgebra underlying the Free Differential Algebra of D=11 supergravity (to which we will refer as DF-algebra) that was introduced in the literature by D'Auria and Frè in 1981 and is actually a (Lorentz valued) central extension of the M-algebra including a nilpotent spinor generator, Q‧. We focus in particular on the 4-form cohomology in 11D superspace of the supergravity theory, strictly related to the presence in the theory of a 3-form A (3). Once formulated in terms of its hidden superalgebra of 1-forms, we find that A (3) can be decomposed into the sum of two parts having different group-theoretical meaning: One of them allows to reproduce the FDA of the 11D Supergravity due to non-trivial contributions to the 4-form cohomology in superspace, while the second one does not contribute to the 4-form cohomology, being a closed 3-form in the vacuum, defining however a one parameter family of trilinear forms invariant under a symmetry algebra related to osp (1 | 32) by redefining the spin connection and adding a new Maurer-Cartan equation. We further discuss about the crucial role played by the 1-form spinor η (dual to the nilpotent generator Q‧) for the 4-form cohomology of the eleven dimensional theory on superspace.

Predicting course performance in freshman and sophomore physics courses: Women are more predictable than men

NASA Astrophysics Data System (ADS)

McCammon, Susan; Golden, Jeannie; Wuensch, Karl L.

This study investigated the extent to which thinking skills and mathematical competency would predict the course performance of freshman and sophomore science majors enrolled in physics courses. Multiple-regression equations revealed that algebra and critical thinking skills were the best overall predictors across several physics courses. Although arithmetic skills, math anxiety, and primary mental abilities scores also correlated with performance, they were redundant with the algebra and critical thinking. The most surprising finding of the study was the differential validity by sex; predictor variables were successful in predicting course performance for women but not for men.

Algebraic multigrid methods applied to problems in computational structural mechanics

NASA Technical Reports Server (NTRS)

Mccormick, Steve; Ruge, John

1989-01-01

The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.

Novel symmetries in Christ-Lee model

NASA Astrophysics Data System (ADS)

Kumar, R.; Shukla, A.

2016-07-01

We demonstrate that the gauge-fixed Lagrangian of the Christ-Lee model respects four fermionic symmetries, namely; (anti-)BRST symmetries, (anti-)co-BRST symmetries within the framework of BRST formalism. The appropriate anticommutators amongst the fermionic symmetries lead to a unique bosonic symmetry. It turns out that the algebra obeyed by the symmetry transformations (and their corresponding conserved charges) is reminiscent of the algebra satisfied by the de Rham cohomological operators of differential geometry. We also provide the physical realizations of the cohomological operators in terms of the symmetry properties. Thus, the present model provides a simple model for the Hodge theory.

Hamiltonian approach to continuum dynamics

NASA Astrophysics Data System (ADS)

Isaev, A. A.; Kovalevskii, M. Yu.; Peletminskii, S. V.

1995-02-01

A study is made of the problem of obtaining the Poisson-bracket algebra of the dynamical variables of continuous media on the basis of specification of the kinematic part of the Lagrangian in terms of generalized coordinates and momenta. Within this algebra, subalgebras of variables corresponding to the description of elastic media, the hydrodynamics of ordinary liquids, and the dynamics of some phases of liquid crystals are identified. The differential conservation laws associated with the symmetries of the Hamiltonian of the system are studied. The dynamics of nematics is considered, and features of the dynamics of the cholesteric, smectic, and discotic phases are noted.

Transfer Functions Via Laplace- And Fourier-Borel Transforms

NASA Technical Reports Server (NTRS)

Can, Sumer; Unal, Aynur

1991-01-01

Approach to solution of nonlinear ordinary differential equations involves transfer functions based on recently-introduced Laplace-Borel and Fourier-Borel transforms. Main theorem gives transform of response of nonlinear system as Cauchy product of transfer function and transform of input function of system, together with memory effects. Used to determine responses of electrical circuits containing variable inductances or resistances. Also possibility of doing all noncommutative algebra on computers in such symbolic programming languages as Macsyma, Reduce, PL1, or Lisp. Process of solution organized and possibly simplified by algebraic manipulations reducing integrals in solutions to known or tabulated forms.

General purpose nonlinear system solver based on Newton-Krylov method.

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

2013-12-01

KINSOL is part of a software family called SUNDIALS: SUite of Nonlinear and Differential/Algebraic equation Solvers [1]. KINSOL is a general-purpose nonlinear system solver based on Newton-Krylov and fixed-point solver technologies [2].

The BRST complex of homological Poisson reduction

NASA Astrophysics Data System (ADS)

Müller-Lennert, Martin

2017-02-01

BRST complexes are differential graded Poisson algebras. They are associated with a coisotropic ideal J of a Poisson algebra P and provide a description of the Poisson algebra (P/J)^J as their cohomology in degree zero. Using the notion of stable equivalence introduced in Felder and Kazhdan (Contemporary Mathematics 610, Perspectives in representation theory, 2014), we prove that any two BRST complexes associated with the same coisotropic ideal are quasi-isomorphic in the case P = R[V] where V is a finite-dimensional symplectic vector space and the bracket on P is induced by the symplectic structure on V. As a corollary, the cohomology of the BRST complexes is canonically associated with the coisotropic ideal J in the symplectic case. We do not require any regularity assumptions on the constraints generating the ideal J. We finally quantize the BRST complex rigorously in the presence of infinitely many ghost variables and discuss the uniqueness of the quantization procedure.

More on quantum groups from the quantization point of view

NASA Astrophysics Data System (ADS)

Jurčo, Branislav

1994-12-01

Star products on the classical double group of a simple Lie group and on corresponding symplectic groupoids are given so that the quantum double and the “quantized tangent bundle” are obtained in the deformation description. “Complex” quantum groups and bicovariant quantum Lie algebras are discussed from this point of view. Further we discuss the quantization of the Poisson structure on the symmetric algebra S(g) leading to the quantized enveloping algebra U h (g) as an example of biquantization in the sense of Turaev. Description of U h (g) in terms of the generators of the bicovariant differential calculus on F(G q ) is very convenient for this purpose. Finaly we interpret in the deformation framework some well known properties of compact quantum groups as simple consequences of corresponding properties of classical compact Lie groups. An analogue of the classical Kirillov's universal character formula is given for the unitary irreducble representation in the compact case.

Linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

NASA Astrophysics Data System (ADS)

Camporesi, Roberto

2011-06-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and the variation of constants method. The approach presented here can be used in a first course on differential equations for science and engineering majors.

Teaching Modeling with Partial Differential Equations: Several Successful Approaches

ERIC Educational Resources Information Center

Myers, Joseph; Trubatch, David; Winkel, Brian

2008-01-01

We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation…

Towards Direct Simulations of Counterflow Flames with Consistent Differential-Algebraic Boundary Conditions

DTIC Science & Technology

2015-01-05

Mechanical and Civil Engineering Dept., California Institute of Technology, Pasadena, CA 91125, USA Josette Bellan † Jet Propulsion Laboratory...91125, USA Kenneth Harstad ‡ Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA A new approach for the formulation of...BDF (Backward Differentiation Formula ), in fixed-leading-coefficient form where the or- der of the method varies between 1 and 5. The BDF method can

The ε-form of the differential equations for Feynman integrals in the elliptic case

NASA Astrophysics Data System (ADS)

Adams, Luise; Weinzierl, Stefan

2018-06-01

Feynman integrals are easily solved if their system of differential equations is in ε-form. In this letter we show by the explicit example of the kite integral family that an ε-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The ε-form is obtained by a (non-algebraic) change of basis for the master integrals.

Software Reviews.

ERIC Educational Resources Information Center

Teles, Elizabeth, Ed.; And Others

1990-01-01

Reviewed are two computer software packages for Macintosh microcomputers including "Phase Portraits," an exploratory graphics tool for studying first-order planar systems; and "MacMath," a set of programs for exploring differential equations, linear algebra, and other mathematical topics. Features, ease of use, cost, availability, and hardware…

The method of Ritz applied to the equation of Hamilton. [for pendulum systems

NASA Technical Reports Server (NTRS)

Bailey, C. D.

1976-01-01

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

Lie group classification of first-order delay ordinary differential equations

NASA Astrophysics Data System (ADS)

Dorodnitsyn, Vladimir A.; Kozlov, Roman; Meleshko, Sergey V.; Winternitz, Pavel

2018-05-01

A group classification of first-order delay ordinary differential equations (DODEs) accompanied by an equation for the delay parameter (delay relation) is presented. A subset of such systems (delay ordinary differential systems or DODSs), which consists of linear DODEs and solution-independent delay relations, have infinite-dimensional symmetry algebras—as do nonlinear ones that are linearizable by an invertible transformation of variables. Genuinely nonlinear DODSs have symmetry algebras of dimension n, . It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction.

Global differential geometry: An introduction for control engineers

NASA Technical Reports Server (NTRS)

Doolin, B. F.; Martin, C. F.

1982-01-01

The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.

An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations

NASA Astrophysics Data System (ADS)

Alam Khan, Najeeb; Razzaq, Oyoon Abdul

2016-03-01

In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.

Maslov indices, Poisson brackets, and singular differential forms

NASA Astrophysics Data System (ADS)

Esterlis, I.; Haggard, H. M.; Hedeman, A.; Littlejohn, R. G.

2014-06-01

Maslov indices are integers that appear in semiclassical wave functions and quantization conditions. They are often notoriously difficult to compute. We present methods of computing the Maslov index that rely only on typically elementary Poisson brackets and simple linear algebra. We also present a singular differential form, whose integral along a curve gives the Maslov index of that curve. The form is closed but not exact, and transforms by an exact differential under canonical transformations. We illustrate the method with the 6j-symbol, which is important in angular-momentum theory and in quantum gravity.

Finite Difference Schemes as Algebraic Correspondences between Layers

NASA Astrophysics Data System (ADS)

Malykh, Mikhail; Sevastianov, Leonid

2018-02-01

For some differential equations, especially for Riccati equation, new finite difference schemes are suggested. These schemes define protective correspondences between the layers. Calculation using these schemes can be extended to the area beyond movable singularities of exact solution without any error accumulation.

Abel's theorem in the noncommutative case

NASA Astrophysics Data System (ADS)

Leitenberger, Frank

2004-03-01

We define noncommutative binary forms. Using the typical representation of Hermite we prove the fundamental theorem of algebra and we derive a noncommutative Cardano formula for cubic forms. We define quantized elliptic and hyperelliptic differentials of the first kind. Following Abel we prove Abel's theorem.

Application of the Group Foliation Method to the Complex Monge-Ampère Equation

NASA Astrophysics Data System (ADS)

Nutku, Y.; Sheftel, M. B.

2001-04-01

We apply the method of group foliation to the complex Monge-Ampère equation ( CMA 2) to establish a regular framework for finding its non-invariant solutions. We employ an infinite symmetry subgroup of CMA 2 to produce a foliation of the solution space into orbits of solutions with respect to this group and a corresponding splitting of CMA 2 into an automorphic system and a resolvent system. We propose a new approach to group foliation which is based on the commutator algebra of operators of invariant differentiation. This algebra together with its Jacobi identities provides the commutator representation of the resolvent system.

Legendre-Tau approximation for functional differential equations. Part 3: Eigenvalue approximations and uniform stability

NASA Technical Reports Server (NTRS)

Ito, K.

1984-01-01

The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.

Sex Differences in Mathematics: An Investigation of Sex Differentiated Attitudes Toward Mathematics and Sex-Differentiated Achievement in Mathematics on the Ninth Grade Level in Eight Schools in New Jersey.

ERIC Educational Resources Information Center

Merkel-Keller, Claudia

This study was undertaken: (1) to determine achievement in mathematics of ninth graders (general mathematics students and algebra students) in eight public schools in New Jersey as measured by an achievement test developed and tested by the author; (2) to determine attitudes toward mathematics of the students as measured by an attitude scale…

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

Kwasniewski, Bartosz K

The construction of reversible extensions of dynamical systems presented in a previous paper by the author and A.V. Lebedev is enhanced, so that it applies to arbitrary mappings (not necessarily with open range). It is based on calculating the maximal ideal space of C*-algebras that extends endomorphisms to partial automorphisms via partial isometric representations, and involves a new set of 'parameters' (the role of parameters is played by chosen sets or ideals). As model examples, we give a thorough description of reversible extensions of logistic maps and a classification of systems associated with compression of unitaries generating homeomorphisms of themore » circle. Bibliography: 34 titles.« less

A round trip from Caldirola to Bateman systems

NASA Astrophysics Data System (ADS)

Guerrero, J.; López-Ruiz, F. F.; Aldaya, V.; Cossío, F.

2011-03-01

For the quantum Caldirola-Kanai Hamiltonian, describing a quantum damped harmonic oscillator, a couple of constant of motion operators generating the Heisenberg algebra can be found. The inclusion in this algebra, in a unitary manner, of the standard time evolution generator , which is not a constant of motion, requires a non-trivial extension of this basic algebra and the physical system itself, which now includes a new dual particle. This enlarged algebra, when exponentiated, leads to a group, named the Bateman group, which admits unitary representations with support in the Hilbert space of functions satisfying the Schrodinger equation associated with the quantum Bateman Hamiltonian, either as a second order differential operator as well as a first order one. The classical Bateman Hamiltonian describes a dual system of a damped (losing energy) particle and a dual (gaining energy) particle. The classical Bateman system has a solution submanifold containing the trajectories of the original system as a submanifold. When restricted to this submanifold, the Bateman dual classical Hamiltonian leads to the Caldirola-Kanai Hamiltonian for a single damped particle. This construction can also be done at the quantum level, and the Caldirola-Kanai Hamiltonian operator can be derived from the Bateman Hamiltonian operator when appropriate constraints are imposed.

Generalizations of the classical Yang-Baxter equation and O-operators

NASA Astrophysics Data System (ADS)

Bai, Chengming; Guo, Li; Ni, Xiang

2011-06-01

Tensor solutions (r-matrices) of the classical Yang-Baxter equation (CYBE) in a Lie algebra, obtained as the classical limit of the R-matrix solution of the quantum Yang-Baxter equation, is an important structure appearing in different areas such as integrable systems, symplectic geometry, quantum groups, and quantum field theory. Further study of CYBE led to its interpretation as certain operators, giving rise to the concept of {O}-operators. The O-operators were in turn interpreted as tensor solutions of CYBE by enlarging the Lie algebra [Bai, C., "A unified algebraic approach to the classical Yang-Baxter equation," J. Phys. A: Math. Theor. 40, 11073 (2007)], 10.1088/1751-8113/40/36/007. The purpose of this paper is to extend this study to a more general class of operators that were recently introduced [Bai, C., Guo, L., and Ni, X., "Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras," Commun. Math. Phys. 297, 553 (2010)], 10.1007/s00220-010-0998-7 in the study of Lax pairs in integrable systems. Relations between O-operators, relative differential operators, and Rota-Baxter operators are also discussed.

Non-geometric fluxes, quasi-Hopf twist deformations, and nonassociative quantum mechanics

NASA Astrophysics Data System (ADS)

Mylonas, Dionysios; Schupp, Peter; Szabo, Richard J.

2014-12-01

We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting, nonassociativity is characterised by the associator 3-cocycle which controls non-coassociativity of the quasi-Hopf algebra. We use abelian 2-cocycle twists to construct maps between the dynamical nonassociative star product and a family of associative star products parametrized by constant momentum surfaces in phase space. We define a suitable integration on these nonassociative spaces and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. Using this star product quantization on phase space together with 3-cyclicity, we formulate a consistent version of nonassociative quantum mechanics, in which we calculate the expectation values of area and volume operators, and find coarse-graining of the string background due to the R-flux.

Phase noise in oscillators as differential-algebraic systems with colored noise sources

NASA Astrophysics Data System (ADS)

Demir, Alper

2004-05-01

Oscillators are key components of many kinds of systems, particularly electronic and opto-electronic systems. Undesired perturbations, i.e. noise, in practical systems adversely affect the spectral and timing properties of the signals generated by oscillators resulting in phase noise and timing jitter, which are key performance limiting factors, being major contributors to bit-error-rate (BER) of RF and possibly optical communication systems, and creating synchronization problems in clocked and sampled-data electronic systems. In this paper, we review our work on the theory and numerical methods for nonlinear perturbation and noise analysis of oscillators described by a system of differential-algebraic equations (DAEs) with white and colored noise sources. The bulk of the work reviewed in this paper first appeared in [1], then in [2] and [3]. Prior to the work mentioned above, we developed a theory and numerical methods for nonlinear perturbation and noise analysis of oscillators described by a system of ordinary differential equations (ODEs) with white noise sources only [4, 5]. In this paper, we also discuss some open problems and issues in the modeling and analysis of phase noise both in free running oscillators and in phase/injection-locked ones.

Bifurcations in two-image photometric stereo for orthogonal illuminations

NASA Astrophysics Data System (ADS)

Kozera, R.; Prokopenya, A.; Noakes, L.; Śluzek, A.

2017-07-01

This paper discusses the ambiguous shape recovery in two-image photometric stereo for a Lambertian surface. The current uniqueness analysis refers to linearly independent light-source directions p = (0, 0, -1) and q arbitrary. For this case necessary and sufficient condition determining ambiguous reconstruction is governed by a second-order linear partial differential equation with constant coefficients. In contrast, a general position of both non-colinear illumination directions p and q leads to a highly non-linear PDE which raises a number of technical difficulties. As recently shown, the latter can also be handled for another family of orthogonal illuminations parallel to the OXZ-plane. For the special case of p = (0, 0, -1) a potential ambiguity stems also from the possible bifurcations of sub-local solutions glued together along a curve defined by an algebraic equation in terms of the data. This paper discusses the occurrence of similar bifurcations for such configurations of orthogonal light-source directions. The discussion to follow is supplemented with examples based on continuous reflectance map model and generated synthetic images.

Modal Structures in flow past a cylinder

NASA Astrophysics Data System (ADS)

Murshed, Mohammad

2017-11-01

With the advent of data, there have been opportunities to apply formalism to detect patterns or simple relations. For instance, a phenomenon can be defined through a partial differential equation which may not be very useful right away, whereas a formula for the evolution of a primary variable may be interpreted quite easily. Having access to data is not enough to move on since doing advanced linear algebra can put strain on the way computations are being done. A canonical problem in the field of aerodynamics is the transient flow past a cylinder where the viscosity can be adjusted to set the Reynolds number (Re). We observe the effect of the critical Re on the certain modes of behavior in time scale. A 2D-velocity field works as an input to analyze the modal structure of the flow using the Proper Orthogonal Decomposition and Koopman Mode/Dynamic Mode Decomposition. This will enable prediction of the solution further in time (taking into account the dependence on Re) and help us evaluate and discuss the associated error in the mechanism.

Finite-difference model to simulate the areal flow of saltwater and fresh water separated by an interface

USGS Publications Warehouse

Mercer, James W.; Larson, S.P.; Faust, Charles R.

1980-01-01

Model documentation is presented for a two-dimensional (areal) model capable of simulating ground-water flow of salt water and fresh water separated by an interface. The partial differential equations are integrated over the thicknesses of fresh water and salt water resulting in two equations describing the flow characteristics in the areal domain. These equations are approximated using finite-difference techniques and the resulting algebraic equations are solved for the dependent variables, fresh water head and salt water head. An iterative solution method was found to be most appropriate. The program is designed to simulate time-dependent problems such as those associated with the development of coastal aquifers, and can treat water-table conditions or confined conditions with steady-state leakage of fresh water. The program will generally be most applicable to the analysis of regional aquifer problems in which the zone between salt water and fresh water can be considered a surface (sharp interface). Example problems and a listing of the computer code are included. (USGS).

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).

An efficient method for solving the steady Euler equations

NASA Technical Reports Server (NTRS)

Liou, M. S.

1986-01-01

An efficient numerical procedure for solving a set of nonlinear partial differential equations is given, specifically for the steady Euler equations. Solutions of the equations were obtained by Newton's linearization procedure, commonly used to solve the roots of nonlinear algebraic equations. In application of the same procedure for solving a set of differential equations we give a theorem showing that a quadratic convergence rate can be achieved. While the domain of quadratic convergence depends on the problems studied and is unknown a priori, we show that firstand second-order derivatives of flux vectors determine whether the condition for quadratic convergence is satisfied. The first derivatives enter as an implicit operator for yielding new iterates and the second derivatives indicates smoothness of the flows considered. Consequently flows involving shocks are expected to require larger number of iterations. First-order upwind discretization in conjunction with the Steger-Warming flux-vector splitting is employed on the implicit operator and a diagonal dominant matrix results. However the explicit operator is represented by first- and seond-order upwind differencings, using both Steger-Warming's and van Leer's splittings. We discuss treatment of boundary conditions and solution procedures for solving the resulting block matrix system. With a set of test problems for one- and two-dimensional flows, we show detailed study as to the efficiency, accuracy, and convergence of the present method.

The convergence of the order sequence and the solution function sequence on fractional partial differential equation

NASA Astrophysics Data System (ADS)

Rusyaman, E.; Parmikanti, K.; Chaerani, D.; Asefan; Irianingsih, I.

2018-03-01

One of the application of fractional ordinary differential equation is related to the viscoelasticity, i.e., a correlation between the viscosity of fluids and the elasticity of solids. If the solution function develops into function with two or more variables, then its differential equation must be changed into fractional partial differential equation. As the preliminary study for two variables viscoelasticity problem, this paper discusses about convergence analysis of function sequence which is the solution of the homogenous fractional partial differential equation. The method used to solve the problem is Homotopy Analysis Method. The results show that if given two real number sequences (αn) and (βn) which converge to α and β respectively, then the solution function sequences of fractional partial differential equation with order (αn, βn) will also converge to the solution function of fractional partial differential equation with order (α, β).

A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

NASA Astrophysics Data System (ADS)

Camporesi, Roberto

2016-01-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.

On differential transformations between Cartesian and curvilinear (geodetic) coordinates

NASA Technical Reports Server (NTRS)

Soler, T.

1976-01-01

Differential transformations are developed between Cartesian and curvilinear orthogonal coordinates. Only matrix algebra is used for the presentation of the basic concepts. After defining the reference systems used the rotation (R), metric (H), and Jacobian (J) matrices of the transformations between cartesian and curvilinear coordinate systems are introduced. A value of R as a function of H and J is presented. Likewise an analytical expression for J(-1) as a function of H(-2) and R is obtained. Emphasis is placed on showing that differential equations are equivalent to conventional similarity transformations. Scaling methods are discussed along with ellipsoidal coordinates. Differential transformations between elipsoidal and geodetic coordinates are established.

A Brief Historical Introduction to Determinants with Applications

ERIC Educational Resources Information Center

Debnath, L.

2013-01-01

This article deals with a short historical introduction to determinants with applications to the theory of equations, geometry, multiple integrals, differential equations and linear algebra. Included are some properties of determinants with proofs, eigenvalues, eigenvectors and characteristic equations with examples of applications to simple…

Conditional Independence in Applied Probability.

ERIC Educational Resources Information Center

Pfeiffer, Paul E.

This material assumes the user has the background provided by a good undergraduate course in applied probability. It is felt that introductory courses in calculus, linear algebra, and perhaps some differential equations should provide the requisite experience and proficiency with mathematical concepts, notation, and argument. The document is…

GROUNDWATER MASS TRANSPORT AND EQUILIBRIUM CHEMISTRY MODEL FOR MULTICOMPONENT SYSTEMS

EPA Science Inventory

A mass transport model, TRANQL, for a multicomponent solution system has been developed. The equilibrium interaction chemistry is posed independently of the mass transport equations which leads to a set of algebraic equations for the chemistry coupled to a set of differential equ...

Exploration of the Factors That Support Learning: Web-Based Activity and Testing Systems in Community College Algebra

ERIC Educational Resources Information Center

Hauk, Shandy; Matlen, Bryan

2016-01-01

A variety of computerized interactive learning platforms exist. Most include instructional supports in the form of problem sets. Feedback to users ranges from a single word like "Correct!" to offers of hints and partially to fully worked examples. Behind-the-scenes design of such systems varies as well --from static dictionaries of…

Exploration of the Factors That Support Learning: Web-Based Activity and Testing Systems in Community College Algebra [Contributed Report

ERIC Educational Resources Information Center

Hauk, Shandy; Matlen, Bryan; Thomas, Larry

2017-01-01

A variety of computerized interactive learning platforms exist. Most include instructional supports in the form of problem sets. Feedback to users ranges from a single word like "Correct!" to offers of hints and partially- to fully-worked examples. Behind-the-scenes design of systems varies as well--from static dictionaries of problems…

Factors Influencing Students' Propensity for Semantic and Syntactic Reasoning in Proof Writing: A Case Study

ERIC Educational Resources Information Center

Mejía-Ramos, Juan Pablo; Weber, Keith; Fuller, Evan

2015-01-01

In this paper we present a case study of an individual student who consistently used semantic reasoning to construct proofs in calculus but infrequently used semantic reasoning to produce proofs in linear algebra. We hypothesize that the differences in these reasoning styles can be partially attributed to this student's familiarity with the…

Stellar Mixing: I. Formalism

NASA Technical Reports Server (NTRS)

Canuto, V .M.

2011-01-01

In this paper we use the Reynolds stress models (RSM) to derive algebraic expressions for the following variables: a) heat fluxes; b) J.l fluxes; and c) momentum fluxes. These relations, which are fully 3D, include: 1) stable and unstable stratification, represented by the Brunt-Vaislila frequency, N(exp 2) =-g/H(sub p_(del - del(sub ad))(1 - RI(sub mu)); 2) double diffusion, salt-fingers, and semi-convection, represented by the density ratio R(sub mu) = del(sub mu)/(del - del(sub ad)); 3) shear (differential rotation), represented by the mean squared shear Sigma(exp 2) or by the Richardson number, Ri =N(exp 2)Sigma(exp -2); 4) radiative losses represented by a Peclet number, Pe; 5) a complete analytical solution of the ID version of the model. In general, the model requires the solution of two differential equations for the eddy kinetic energy K and its rate of dissipation, epsilon. In the local and stationary cases, when production equals dissipation, the model equations are all algebraic.

BFV-BRST quantization of two-dimensional supergravity

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

Fujiwara, T.; Igarashi, Y.; Kuriki, R.

1996-01-01

Two-dimensional supergravity theory is quantized as an anomalous gauge theory. In the Batalin-Fradkin (BF) formalism, the anomaly-canceling super-Liouville fields are introduced to identify the original second-class constrained system with a gauge-fixed version of a first-class system. The BFV-BRST quantization applies to formulate the theory in the most general class of gauges. A local effective action constructed in the configuration space contains two super-Liouville actions; one is a noncovariant but local functional written only in terms of two-dimensional supergravity fields, and the other contains the super-Liouville fields canceling the super-Weyl anomaly. Auxiliary fields for the Liouville and the gravity supermultiplets aremore » introduced to make the BRST algebra close off-shell. Inclusion of them turns out to be essentially important especially in the super-light-cone gauge fixing, where the supercurvature equations ({partial_derivative}{sup 3}{sub {minus}}{ital g}{sub +}{sub +}={partial_derivative}{sup 2}{sub {minus}}{chi}{sub +}{sub +}=0) are obtained as a result of BRST invariance of the theory. Our approach reveals the origin of the OSp(1,2) current algebra symmetry in a transparent manner. {copyright} {ital 1996 The American Physical Society.}« less

Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics

NASA Astrophysics Data System (ADS)

Rangan, Aaditya V.; Cai, David; Tao, Louis

2007-02-01

Recently developed kinetic theory and related closures for neuronal network dynamics have been demonstrated to be a powerful theoretical framework for investigating coarse-grained dynamical properties of neuronal networks. The moment equations arising from the kinetic theory are a system of (1 + 1)-dimensional nonlinear partial differential equations (PDE) on a bounded domain with nonlinear boundary conditions. The PDEs themselves are self-consistently specified by parameters which are functions of the boundary values of the solution. The moment equations can be stiff in space and time. Numerical methods are presented here for efficiently and accurately solving these moment equations. The essential ingredients in our numerical methods include: (i) the system is discretized in time with an implicit Euler method within a spectral deferred correction framework, therefore, the PDEs of the kinetic theory are reduced to a sequence, in time, of boundary value problems (BVPs) with nonlinear boundary conditions; (ii) a set of auxiliary parameters is introduced to recast the original BVP with nonlinear boundary conditions as BVPs with linear boundary conditions - with additional algebraic constraints on the auxiliary parameters; (iii) a careful combination of two Newton's iterates for the nonlinear BVP with linear boundary condition, interlaced with a Newton's iterate for solving the associated algebraic constraints is constructed to achieve quadratic convergence for obtaining the solutions with self-consistent parameters. It is shown that a simple fixed-point iteration can only achieve a linear convergence for the self-consistent parameters. The practicability and efficiency of our numerical methods for solving the moment equations of the kinetic theory are illustrated with numerical examples. It is further demonstrated that the moment equations derived from the kinetic theory of neuronal network dynamics can very well capture the coarse-grained dynamical properties of integrate-and-fire neuronal networks.

Numerical algebraic geometry for model selection and its application to the life sciences

PubMed Central

Gross, Elizabeth; Davis, Brent; Ho, Kenneth L.; Bates, Daniel J.

2016-01-01

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are available. Here, we consider polynomial models (e.g. mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometrical structures relating models and data, and we demonstrate its utility on examples from cell signalling, synthetic biology and epidemiology. PMID:27733697

Hermeneutics of differential calculus in eighteenth-century northern Germany.

PubMed

Blanco, Mónica

2008-01-01

This paper applies comparative textbook analysis to studying the mathematical development of differential calculus in northern German states during the eighteenth century. It begins with describing how the four textbooks analyzed presented the foundations of calculus and continues with assessing the influence each of these foundational approaches exerted on the resolution of problems, such as the determination of tangents and extreme values, and even on the choice of coordinates for both algebraic and transcendental curves.

Mathematical Model for Dengue Epidemics with Differential Susceptibility and Asymptomatic Patients Using Computer Algebra

NASA Astrophysics Data System (ADS)

Saldarriaga Vargas, Clarita

When there are diseases affecting large populations where the social, economic and cultural diversity is significant within the same region, the biological parameters that determine the behavior of the dispersion disease analysis are affected by the selection of different individuals. Therefore and because of the variety and magnitude of the communities at risk of contracting dengue disease around all over the world, suggest defining differentiated populations with individual contributions in the results of the dispersion dengue disease analysis. In this paper those conditions were taken in account when several epidemiologic models were analyzed. Initially a stability analysis was done for a SEIR mathematical model of Dengue disease without differential susceptibility. Both free disease and endemic equilibrium states were found in terms of the basic reproduction number and were defined in the Theorem (3.1). Then a DSEIR model was solved when a new susceptible group was introduced to consider the effects of important biological parameters of non-homogeneous populations in the spreading analysis. The results were compiled in the Theorem (3.2). Finally Theorems (3.3) and (3.4) resumed the basic reproduction numbers for three and n different susceptible groups respectively, giving an idea of how differential susceptibility affects the equilibrium states. The computations were done using an algorithmic method implemented in Maple 11, a general-purpose computer algebra system.

Incorporating Writing in an Integrated Calculus, Linear Algebra, and Differential Equations Sequence.

ERIC Educational Resources Information Center

Kelly, Susan E.; LeDocq, Rebecca Lewin

2001-01-01

Describes the specific courses in a sequence along with how the writing has been implemented in each course. Provides ideas for how to efficiently handle the additional paper load so students receive the necessary feedback while keeping the grading time reasonable. (Author/ASK)

Differentiation from First Principles Using Spreadsheets

ERIC Educational Resources Information Center

Lim, Kieran F.

2008-01-01

In the teaching of calculus, the algebraic derivation of the derivative (gradient function) enables the student to obtain an analytic "global" gradient function. However, to the best of this author's knowledge, all current technology-based approaches require the student to obtain the derivative (gradient) at a single point by…

The roles of prefrontal and posterior parietal cortex in algebra problem solving: a case of using cognitive modeling to inform neuroimaging data.

PubMed

Danker, Jared F; Anderson, John R

2007-04-15

In naturalistic algebra problem solving, the cognitive processes of representation and retrieval are typically confounded, in that transformations of the equations typically require retrieval of mathematical facts. Previous work using cognitive modeling has associated activity in the prefrontal cortex with the retrieval demands of algebra problems and activity in the posterior parietal cortex with the transformational demands of algebra problems, but these regions tend to behave similarly in response to task manipulations (Anderson, J.R., Qin, Y., Sohn, M.-H., Stenger, V.A., Carter, C.S., 2003. An information-processing model of the BOLD response in symbol manipulation tasks. Psychon. Bull. Rev. 10, 241-261; Qin, Y., Carter, C.S., Silk, E.M., Stenger, A., Fissell, K., Goode, A., Anderson, J.R., 2004. The change of brain activation patterns as children learn algebra equation solving. Proc. Natl. Acad. Sci. 101, 5686-5691). With this study we attempt to isolate activity in these two regions by using a multi-step algebra task in which transformation (parietal) is manipulated in the first step and retrieval (prefrontal) is manipulated in the second step. Counter to our initial predictions, both brain regions were differentially active during both steps. We designed two cognitive models, one encompassing our initial assumptions and one in which both processes were engaged during both steps. The first model provided a poor fit to the behavioral and neural data, while the second model fit both well. This simultaneously emphasizes the strong relationship between retrieval and representation in mathematical reasoning and demonstrates that cognitive modeling can serve as a useful tool for understanding task manipulations in neuroimaging experiments.

Thermodynamics. [algebraic structure

NASA Technical Reports Server (NTRS)

Zeleznik, F. J.

1976-01-01

The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.

Super-Lie n-algebra extensions, higher WZW models and super-p-branes with tensor multiplet fields

NASA Astrophysics Data System (ADS)

Fiorenza, Domenico; Sati, Hisham; Schreiber, Urs

2015-12-01

We formalize higher-dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L∞-extension theory and how the resulting "extended (super-)space-times" formalize spacetimes containing σ-model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super-p-brane spectrum of superstring/M-theory is realized this way, including the pure σ-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional (11D) spacetime with an M2-brane condensate turns out to be the "M-theory super-Lie algebra". We also observe that in this formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11D sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string theory. Finally we give the non-perturbative description of all this by higher WZW-type σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology.

A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type,

DTIC Science & Technology

NONLINEAR DIFFERENTIAL EQUATIONS, INTEGRATION), (*PARTIAL DIFFERENTIAL EQUATIONS, BOUNDARY VALUE PROBLEMS), BANACH SPACE , MAPPING (TRANSFORMATIONS), SET THEORY, TOPOLOGY, ITERATIONS, STABILITY, THEOREMS

Explorations in fuzzy physics and non-commutative geometry

NASA Astrophysics Data System (ADS)

Kurkcuoglu, Seckin

Fuzzy spaces arise as discrete approximations to continuum manifolds. They are usually obtained through quantizing coadjoint orbits of compact Lie groups and they can be described in terms of finite-dimensional matrix algebras, which for large matrix sizes approximate the algebra of functions of the limiting continuum manifold. Their ability to exactly preserve the symmetries of their parent manifolds is especially appealing for physical applications. Quantum Field Theories are built over them as finite-dimensional matrix models preserving almost all the symmetries of their respective continuum models. In this dissertation, we first focus our attention to the study of fuzzy supersymmetric spaces. In this regard, we obtain the fuzzy supersphere S2,2F through quantizing the supersphere, and demonstrate that it has exact supersymmetry. We derive a finite series formula for the *-product of functions over S2,2F and analyze the differential geometric information encoded in this formula. Subsequently, we show that quantum field theories on S2,2F are realized as finite-dimensional supermatrix models, and in particular we obtain the non-linear sigma model over the fuzzy supersphere by constructing the fuzzy supersymmetric extensions of a certain class of projectors. We show that this model too, is realized as a finite-dimensional supermatrix model with exact supersymmetry. Next, we show that fuzzy spaces have a generalized Hopf algebra structure. By focusing on the fuzzy sphere, we establish that there is a *-homomorphism from the group algebra SU(2)* of SU(2) to the fuzzy sphere. Using this and the canonical Hopf algebra structure of SU(2)* we show that both the fuzzy sphere and their direct sum are Hopf algebras. Using these results, we discuss processes in which a fuzzy sphere with angular momenta J splits into fuzzy spheres with angular momenta K and L. Finally, we study the formulation of Chern-Simons (CS) theory on an infinite strip of the non-commutative plane. We develop a finite-dimensional matrix model, whose large size limit approximates the CS theory on the infinite strip, and show that there are edge observables in this model obeying a finite-dimensional Lie algebra, that resembles the Kac-Moody algebra.

Intelligent Machines in the 21st Century: Automating the Processes of Inference and Inquiry

NASA Technical Reports Server (NTRS)

Knuth, Kevin H.

2003-01-01

The last century saw the application of Boolean algebra toward the construction of computing machines, which work by applying logical transformations to information contained in their memory. The development of information theory and the generalization of Boolean algebra to Bayesian inference have enabled these computing machines. in the last quarter of the twentieth century, to be endowed with the ability to learn by making inferences from data. This revolution is just beginning as new computational techniques continue to make difficult problems more accessible. However, modern intelligent machines work by inferring knowledge using only their pre-programmed prior knowledge and the data provided. They lack the ability to ask questions, or request data that would aid their inferences. Recent advances in understanding the foundations of probability theory have revealed implications for areas other than logic. Of relevance to intelligent machines, we identified the algebra of questions as the free distributive algebra, which now allows us to work with questions in a way analogous to that which Boolean algebra enables us to work with logical statements. In this paper we describe this logic of inference and inquiry using the mathematics of partially ordered sets and the scaffolding of lattice theory, discuss the far-reaching implications of the methodology, and demonstrate its application with current examples in machine learning. Automation of both inference and inquiry promises to allow robots to perform science in the far reaches of our solar system and in other star systems by enabling them to not only make inferences from data, but also decide which question to ask, experiment to perform, or measurement to take given what they have learned and what they are designed to understand.

Applications of algebraic topology to compatible spatial discretizations.

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

Bochev, Pavel Blagoveston; Hyman, James M.

We provide a common framework for compatible discretizations using algebraic topology to guide our analysis. The main concept is the natural inner product on cochains, which induces a combinatorial Hodge theory. The framework comprises of mutually consistent operations of differentiation and integration, has a discrete Stokes theorem, and preserves the invariants of the DeRham cohomology groups. The latter allows for an elementary calculation of the kernel of the discrete Laplacian. Our framework provides an abstraction that includes examples of compatible finite element, finite volume and finite difference methods. We describe how these methods result from the choice of a reconstructionmore » operator and when they are equivalent.« less

Exploration of the Factors That Support Learning: Web-Based Activity and Testing Systems in Community College Algebra [Conference Long Paper

ERIC Educational Resources Information Center

Hauk, Shandy; Matlen, Bryan J.

2017-01-01

This is an extended conference proceedings report [Long Paper] based on a shorter contributed report at the same conference. A variety of computerized learning platforms exist. In mathematics, most include sets of problems to complete. Feedback to users ranges from a single word like "Correct!" to offers of hints and partially- to…

Integration of CAI into a Freshmen Liberal Arts Math Course in the Community College.

ERIC Educational Resources Information Center

McCall, Michael B.; Holton, Jean L.

1982-01-01

Discusses four computer-assisted-instruction programs used in a college-level mathematics course to introduce computer literacy and improve mathematical skills. The BASIC programs include polynomial functions, trigonometric functions, matrix algebra, and differential calculus. Each program discusses mathematics theory and introduces programming…

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…

A heuristic way of obtaining the Kerr metric

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

Enderlein, J.

1997-09-01

An intuitive, straightforward way of finding the metric of a rotating black hole is presented, based on the algebra of differential forms. The representation obtained for the metric displays a simplicity which is not obvious in the usual Boyer{endash}Lindquist coordinates. {copyright} {ital 1997 American Association of Physics Teachers.}

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…

Hermann-Bernoulli-Laplace-Hamilton-Runge-Lenz Vector.

ERIC Educational Resources Information Center

Subramanian, P. R.; And Others

1991-01-01

A way for students to refresh and use their knowledge in both mathematics and physics is presented. By the study of the properties of the "Runge-Lenz" vector the subjects of algebra, analytical geometry, calculus, classical mechanics, differential equations, matrices, quantum mechanics, trigonometry, and vector analysis can be reviewed. (KR)

Controlling Population with Pollution

ERIC Educational Resources Information Center

Browne, Joseph

2010-01-01

Population models are often discussed in algebra, calculus, and differential equations courses. In this article we will use the human population of the world as our application. After quick looks at two common models we'll investigate more deeply a model which incorporates the negative effect that accumulated pollution may have on population.

The approach to engineering tasks composition on knowledge portals

NASA Astrophysics Data System (ADS)

Novogrudska, Rina; Globa, Larysa; Schill, Alexsander; Romaniuk, Ryszard; Wójcik, Waldemar; Karnakova, Gaini; Kalizhanova, Aliya

2017-08-01

The paper presents an approach to engineering tasks composition on engineering knowledge portals. The specific features of engineering tasks are highlighted, their analysis makes the basis for partial engineering tasks integration. The formal algebraic system for engineering tasks composition is proposed, allowing to set the context-independent formal structures for engineering tasks elements' description. The method of engineering tasks composition is developed that allows to integrate partial calculation tasks into general calculation tasks on engineering portals, performed on user request demand. The real world scenario «Calculation of the strength for the power components of magnetic systems» is represented, approving the applicability and efficiency of proposed approach.

The ATOMFT integrator - Using Taylor series to solve ordinary differential equations

NASA Technical Reports Server (NTRS)

Berryman, Kenneth W.; Stanford, Richard H.; Breckheimer, Peter J.

1988-01-01

This paper discusses the application of ATOMFT, an integration package based on Taylor series solution with a sophisticated user interface. ATOMFT has the capabilities to allow the implementation of user defined functions and the solution of stiff and algebraic equations. Detailed examples, including the solutions to several astrodynamics problems, are presented. Comparisons with its predecessor ATOMCC and other modern integrators indicate that ATOMFT is a fast, accurate, and easy method to use to solve many differential equation problems.

Workshop on Advances in Scientific Computation and Differential Equations 󈨡 (SCADE)

DTIC Science & Technology

1994-07-18

STATEMENT ~~’"j’’ Approved for public release; distribution unlimited. I ABSTRACT (MAMMU 200WOMW 94 808 1 64 4.L SUBIECT TERMS Ii11URE Of PAGES 12 16...called differential algebraic ODEs (DAES). (Some important early research on this topic was by L. Petzold.) Both theoretically and in terms of...completely specify the solution. In many physical systems, especially those in biology, or other large scale slowly responding systems, the inclusion of some

On deformation of complex continuum immersed in a plane space

NASA Astrophysics Data System (ADS)

Kovalev, V. A.; Murashkin, E. V.; Radayev, Y. N.

2018-05-01

The present paper is devoted to mathematical modelling of complex continua deformations considered as immersed in an external plane space. The complex continuum is defined as a differential manifold supplied with metrics induced by the external space. A systematic derivation of strain tensors by notion of isometric immersion of the complex continuum into a plane space of a higher dimension is proposed. Problem of establishing complete systems of irreducible objective strain and extrastrain tensors for complex continuum immersed in an external plane space is resolved. The solution to the problem is obtained by methods of the field theory and the theory of rational algebraic invariants. Strain tensors of the complex continuum are derived as irreducible algebraic invariants of contravariant vectors of the external space emerging as functional arguments in the complex continuum action density. Present analysis is restricted to rational algebraic invariants. Completeness of the considered systems of rational algebraic invariants is established for micropolar elastic continua. Rational syzygies for non-quadratic invariants are discussed. Objective strain tensors (indifferent to frame rotations in the external plane space) for micropolar continuum are alternatively obtained by properly combining multipliers of polar decompositions of deformation and extra-deformation gradients. The latter is realized only for continua immersed in a plane space of the equal mathematical dimension.

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems

NASA Astrophysics Data System (ADS)

Fathizadeh, Farzad; Gabriel, Olivier

2016-02-01

The analog of the Chern-Gauss-Bonnet theorem is studied for a C^*-dynamical system consisting of a C^*-algebra A equipped with an ergodic action of a compact Lie group G. The structure of the Lie algebra g of G is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra A subset A as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique G-invariant state on A, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on A and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

On Special Functions in the Context of Clifford Analysis

NASA Astrophysics Data System (ADS)

Malonek, H. R.; Falcão, M. I.

2010-09-01

Considering the foundation of Quaternionic Analysis by R. Fueter and his collaborators in the beginning of the 1930s as starting point of Clifford Analysis, we can look back to 80 years of work in this field. However the interest in multivariate analysis using Clifford algebras only started to grow significantly in the 70s. Since then a great amount of papers on Clifford Analysis referring different classes of Special Functions have appeared. This situation may have been triggered by a more systematic treatment of monogenic functions by their multiple series development derived from Gegenbauer or associated Legendre polynomials (and not only by their integral representation). Also approaches to Special Functions by means of algebraic methods, either Lie algebras or through Lie groups and symmetric spaces gained by that time importance and influenced their treatment in Clifford Analysis. In our talk we will rely on the generalization of the classical approach to Special Functions through differential equations with respect to the hypercomplex derivative, which is a more recently developed tool in Clifford Analysis. In this context special attention will be payed to the role of Special Functions as intermediator between continuous and discrete mathematics. This corresponds to a more recent trend in combinatorics, since it has been revealed that many algebraic structures have hidden combinatorial underpinnings.

Graphical construction of a local perspective on differentiation and integration

NASA Astrophysics Data System (ADS)

Hong, Ye Yoon; Thomas, Michael O. J.

2015-06-01

Recent studies of the transition from school to university mathematics have identified a number of epistemological gaps, including the need to change from an emphasis on equality to that of inequality. Another crucial epistemological change during this transition involves the movement from the pointwise and global perspectives of functions usually established through the school curriculum to a view of function that includes a local, or interval, perspective. This is necessary for study of concepts such as continuity and limit that underpin calculus and analysis at university. In this study, a first-year university calculus course in Korea was constructed that integrated use of digital technology and considered the epistemic value of the associated techniques. The aim was to encourage versatile thinking about functions, especially in relation to properties arising from a graphical investigation of differentiation and integration. In this paper, the results of this approach for the learning of derivative and antiderivative, based on integrated technology use, are presented. They show the persistence of what Tall ( Mathematics Education Research Journal, 20(2), 5-24, 2008) describes as symbolic world algebraic thinking on the part of a significant minority of students, who feel the need to introduce algebraic methods, in spite of its disadvantages, even when no explicit algebra is provided. However, the results also demonstrate the ability of many of the students to use technology mediation to build local or interval conceptual thinking about derivative and antiderivative functions.

On Monoids in the Category of Sets and Relations

NASA Astrophysics Data System (ADS)

Jenčová, Anna; Jenča, Gejza

2017-12-01

The category R e l is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, R e l is a monoidal category. Moreover, R e l is a locally posetal 2-category, since every homset R e l( A, B) is a poset with respect to inclusion. We examine the 2-category of monoids R e l M o n in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.

On the hierarchy of partially invariant submodels of differential equations

NASA Astrophysics Data System (ADS)

Golovin, Sergey V.

2008-07-01

It is noted that the partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PISs of the higher rank. This introduces a hierarchic structure in the set of all PISs of a given system of differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. The hierarchy simplifies the process of enumeration and analysis of partially invariant submodels to the given system of differential equations. In this framework, the complete classification of regular partially invariant solutions of ideal MHD equations is given.

Optimal moving grids for time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Wathen, A. J.

1989-01-01

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of partial differential equation solutions in the least squares norm.

Solution of differential equations by application of transformation groups

NASA Technical Reports Server (NTRS)

Driskell, C. N., Jr.; Gallaher, L. J.; Martin, R. H., Jr.

1968-01-01

Report applies transformation groups to the solution of systems of ordinary differential equations and partial differential equations. Lies theorem finds an integrating factor for appropriate invariance group or groups can be found and can be extended to partial differential equations.

Evolutionary optimization with data collocation for reverse engineering of biological networks.

PubMed

Tsai, Kuan-Yao; Wang, Feng-Sheng

2005-04-01

Modern experimental biology is moving away from analyses of single elements to whole-organism measurements. Such measured time-course data contain a wealth of information about the structure and dynamic of the pathway or network. The dynamic modeling of the whole systems is formulated as a reverse problem that requires a well-suited mathematical model and a very efficient computational method to identify the model structure and parameters. Numerical integration for differential equations and finding global parameter values are still two major challenges in this field of the parameter estimation of nonlinear dynamic biological systems. We compare three techniques of parameter estimation for nonlinear dynamic biological systems. In the proposed scheme, the modified collocation method is applied to convert the differential equations to the system of algebraic equations. The observed time-course data are then substituted into the algebraic system equations to decouple system interactions in order to obtain the approximate model profiles. Hybrid differential evolution (HDE) with population size of five is able to find a global solution. The method is not only suited for parameter estimation but also can be applied for structure identification. The solution obtained by HDE is then used as the starting point for a local search method to yield the refined estimates.

On Finding and Using Identifiable Parameter Combinations in Nonlinear Dynamic Systems Biology Models and COMBOS: A Novel Web Implementation

PubMed Central

DiStefano, Joseph

2014-01-01

Parameter identifiability problems can plague biomodelers when they reach the quantification stage of development, even for relatively simple models. Structural identifiability (SI) is the primary question, usually understood as knowing which of P unknown biomodel parameters p 1,…, pi,…, pP are-and which are not-quantifiable in principle from particular input-output (I-O) biodata. It is not widely appreciated that the same database also can provide quantitative information about the structurally unidentifiable (not quantifiable) subset, in the form of explicit algebraic relationships among unidentifiable pi. Importantly, this is a first step toward finding what else is needed to quantify particular unidentifiable parameters of interest from new I–O experiments. We further develop, implement and exemplify novel algorithms that address and solve the SI problem for a practical class of ordinary differential equation (ODE) systems biology models, as a user-friendly and universally-accessible web application (app)–COMBOS. Users provide the structural ODE and output measurement models in one of two standard forms to a remote server via their web browser. COMBOS provides a list of uniquely and non-uniquely SI model parameters, and–importantly-the combinations of parameters not individually SI. If non-uniquely SI, it also provides the maximum number of different solutions, with important practical implications. The behind-the-scenes symbolic differential algebra algorithms are based on computing Gröbner bases of model attributes established after some algebraic transformations, using the computer-algebra system Maxima. COMBOS was developed for facile instructional and research use as well as modeling. We use it in the classroom to illustrate SI analysis; and have simplified complex models of tumor suppressor p53 and hormone regulation, based on explicit computation of parameter combinations. It’s illustrated and validated here for models of moderate complexity, with and without initial conditions. Built-in examples include unidentifiable 2 to 4-compartment and HIV dynamics models. PMID:25350289

From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations

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

Angstmann, C.N.; Donnelly, I.C.; Henry, B.I., E-mail: B.Henry@unsw.edu.au

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also showmore » that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.« less

Reciprocal links among differential parenting, perceived partiality, and self-worth: a three-wave longitudinal study.

PubMed

Shebloski, Barbara; Conger, Katherine J; Widaman, Keith F

2005-12-01

This study examined reciprocal links between parental differential treatment, siblings' perception of partiality, and self-worth with 3 waves of data from 384 adolescent sibling dyads. Results suggest that birth-order status was significantly associated with self-worth and perception of maternal and paternal differential treatment. There was a consistent across-time effect of self-worth on perception of parental partiality for later born siblings, but not earlier born siblings, and a consistent effect of differential treatment on perception of partiality for earlier born but not later born siblings. The results contribute new insight into the associations between perception of differential parenting and adolescents' adjustment and the role of birth order. Copyright 2006 APA, all rights reserved).

Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids

PubMed Central

Eisenberg, Bob; Hyon, YunKyong; Liu, Chun

2010-01-01

Ionic solutions are mixtures of interacting anions and cations. They hardly resemble dilute gases of uncharged noninteracting point particles described in elementary textbooks. Biological and electrochemical solutions have many components that interact strongly as they flow in concentrated environments near electrodes, ion channels, or active sites of enzymes. Interactions in concentrated environments help determine the characteristic properties of electrodes, enzymes, and ion channels. Flows are driven by a combination of electrical and chemical potentials that depend on the charges, concentrations, and sizes of all ions, not just the same type of ion. We use a variational method EnVarA (energy variational analysis) that combines Hamilton’s least action and Rayleigh’s dissipation principles to create a variational field theory that includes flow, friction, and complex structure with physical boundary conditions. EnVarA optimizes both the action integral functional of classical mechanics and the dissipation functional. These functionals can include entropy and dissipation as well as potential energy. The stationary point of the action is determined with respect to the trajectory of particles. The stationary point of the dissipation is determined with respect to rate functions (such as velocity). Both variations are written in one Eulerian (laboratory) framework. In variational analysis, an “extra layer” of mathematics is used to derive partial differential equations. Energies and dissipations of different components are combined in EnVarA and Euler–Lagrange equations are then derived. These partial differential equations are the unique consequence of the contributions of individual components. The form and parameters of the partial differential equations are determined by algebra without additional physical content or assumptions. The partial differential equations of mixtures automatically combine physical properties of individual (unmixed) components. If a new component is added to the energy or dissipation, the Euler–Lagrange equations change form and interaction terms appear without additional adjustable parameters. EnVarA has previously been used to compute properties of liquid crystals, polymer fluids, and electrorheological fluids containing solid balls and charged oil droplets that fission and fuse. Here we apply EnVarA to the primitive model of electrolytes in which ions are spheres in a frictional dielectric. The resulting Euler–Lagrange equations include electrostatics and diffusion and friction. They are a time dependent generalization of the Poisson–Nernst–Planck equations of semiconductors, electrochemistry, and molecular biophysics. They include the finite diameter of ions. The EnVarA treatment is applied to ions next to a charged wall, where layering is observed. Applied to an ion channel, EnVarA calculates a quick transient pile-up of electric charge, transient and steady flow through the channel, stationary “binding” in the channel, and the eventual accumulation of salts in “unstirred layers” near channels. EnVarA treats electrolytes in a unified way as complex rather than simple fluids. Ad hoc descriptions of interactions and flow have been used in many areas of science to deal with the nonideal properties of electrolytes. It seems likely that the variational treatment can simplify, unify, and perhaps derive and improve those descriptions. PMID:20849161

Steps Forward and Back in Adult Numeracy Teacher Professional Development: A Reflection on a Teacher Workshop Experience

ERIC Educational Resources Information Center

Saliga, Linda Marie; Daviso, Al; Stuart, Denise; Pachnowski, Lynne

2015-01-01

In this project, a university team of teacher education and mathematics professors conducted eight professional development sessions for General Educational Development (GED) teachers in the area of mathematics teaching. Topics included concretely modeling mathematics concepts in algebra, number sense, geometry, and differentiating instruction in…

The Effect of Strategy on Problem Solving: An FMRI Study

ERIC Educational Resources Information Center

Newman, Sharlene D.; Pruce, Benjamin; Rusia, Akash; Burns, Thomas, Jr.

2010-01-01

fMRI was used to examine the differential effect of two problem-solving strategies. Participants were trained to use both a pictorial/spatial and a symbolic/algebraic strategy to solve word problems. While these two strategies activated similar cortical regions, a number of differences were noted in the level of activation. These differences…

Proceedings: Summer Conference for College Teachers on Applied Mathematics, University of Missouri-Rolla, 1971.

ERIC Educational Resources Information Center

Committee on the Undergraduate Program in Mathematics, Berkeley, CA.

Proceedings from four sessions of the Summer Conference for College Teachers on Applied Mathematics are presented. The four sessions were: (1) Applications of Elementary Calculus, (2) Applications of Linear Algebra, (3) Applications of Elementary Differential Equations, and (4) Applications of Probability and Statistics. Nine lectures were given…

Modelling the Landing of a Plane in a Calculus Lab

ERIC Educational Resources Information Center

Morante, Antonio; Vallejo, Jose A.

2012-01-01

We exhibit a simple model of a plane landing that involves only basic concepts of differential calculus, so it is suitable for a first-year calculus lab. We use the computer algebra system Maxima and the interactive geometry software GeoGebra to do the computations and graphics. (Contains 5 figures and 1 note.)

Inverse dynamics of underactuated mechanical systems: A simple case study and experimental verification

NASA Astrophysics Data System (ADS)

Blajer, W.; Dziewiecki, K.; Kołodziejczyk, K.; Mazur, Z.

2011-05-01

Underactuated systems are featured by fewer control inputs than the degrees-of-freedom, m < n. The determination of an input control strategy that forces such a system to complete a set of m specified motion tasks is a challenging task, and the explicit solution existence is conditioned to differential flatness of the problem. The flatness-based solution denotes that all the 2 n states and m control inputs can be algebraically expressed in terms of the m specified outputs and their time derivatives up to a certain order, which is in practice attainable only for simple systems. In this contribution the problem is posed in a more practical way as a set of index-three differential-algebraic equations, and the solution is obtained numerically. The formulation is then illustrated by a two-degree-of-freedom underactuated system composed of two rotating discs connected by a torsional spring, in which the pre-specified motion of one of the discs is actuated by the torque applied to the other disc, n = 2 and m = 1. Experimental verification of the inverse simulation control methodology is reported.

Image model: new perspective for image processing and computer vision

NASA Astrophysics Data System (ADS)

Ziou, Djemel; Allili, Madjid

2004-05-01

We propose a new image model in which the image support and image quantities are modeled using algebraic topology concepts. The image support is viewed as a collection of chains encoding combination of pixels grouped by dimension and linking different dimensions with the boundary operators. Image quantities are encoded using the notion of cochain which associates values for pixels of given dimension that can be scalar, vector, or tensor depending on the problem that is considered. This allows obtaining algebraic equations directly from the physical laws. The coboundary and codual operators, which are generic operations on cochains allow to formulate the classical differential operators as applied for field functions and differential forms in both global and local forms. This image model makes the association between the image support and the image quantities explicit which results in several advantages: it allows the derivation of efficient algorithms that operate in any dimension and the unification of mathematics and physics to solve classical problems in image processing and computer vision. We show the effectiveness of this model by considering the isotropic diffusion.

Fluid Flow and Heat Transfer Analysis of a Nanofluid Containing Motile Gyrotactic Micro-Organisms Passing a Nonlinear Stretching Vertical Sheet in the Presence of a Non-Uniform Magnetic Field; Numerical Approach

PubMed Central

M. Mehryan, S. A.; Moradi Kashkooli, Farshad; Soltani, M.; Raahemifar, Kaamran

2016-01-01

The behavior of a water-based nanofluid containing motile gyrotactic micro-organisms passing an isothermal nonlinear stretching sheet in the presence of a non-uniform magnetic field is studied numerically. The governing partial differential equations including continuity, momentums, energy, concentration of the nanoparticles, and density of motile micro-organisms are converted into a system of the ordinary differential equations via a set of similarity transformations. New set of equations are discretized using the finite difference method and have been linearized by employing the Newton’s linearization technique. The tri-diagonal system of algebraic equations from discretization is solved using the well-known Thomas algorithm. The numerical results for profiles of velocity, temperature, nanoparticles concentration and density of motile micro-organisms as well as the local skin friction coefficient Cfx, the local Nusselt number Nux, the local Sherwood number Shx and the local density number of the motile microorganism Nnx are expressed graphically and described in detail. This investigation shows the density number of the motile micro-organisms enhances with rise of M, Gr/Re2, Pe and Ω but it decreases with augment of Rb and n. Also, Sherwood number augments with an increase of M and Gr/Re2, while decreases with n, Rb, Nb and Nr. To show the validity of the current results, a comparison between the present results and the existing literature has been carried out. PMID:27322536

One-Dimensional Transport with Equilibrium Chemistry (OTEQ) - A Reactive Transport Model for Streams and Rivers

USGS Publications Warehouse

Runkel, Robert L.

2010-01-01

OTEQ is a mathematical simulation model used to characterize the fate and transport of waterborne solutes in streams and rivers. The model is formed by coupling a solute transport model with a chemical equilibrium submodel. The solute transport model is based on OTIS, a model that considers the physical processes of advection, dispersion, lateral inflow, and transient storage. The equilibrium submodel is based on MINTEQ, a model that considers the speciation and complexation of aqueous species, acid-base reactions, precipitation/dissolution, and sorption. Within OTEQ, reactions in the water column may result in the formation of solid phases (precipitates and sorbed species) that are subject to downstream transport and settling processes. Solid phases on the streambed may also interact with the water column through dissolution and sorption/desorption reactions. Consideration of both mobile (waterborne) and immobile (streambed) solid phases requires a unique set of governing differential equations and solution techniques that are developed herein. The partial differential equations describing physical transport and the algebraic equations describing chemical equilibria are coupled using the sequential iteration approach. The model's ability to simulate pH, precipitation/dissolution, and pH-dependent sorption provides a means of evaluating the complex interactions between instream chemistry and hydrologic transport at the field scale. This report details the development and application of OTEQ. Sections of the report describe model theory, input/output specifications, model applications, and installation instructions. OTEQ may be obtained over the Internet at http://water.usgs.gov/software/OTEQ.

Perturbed Coulomb Potentials in the Klein-Gordon Equation: Quasi-Exact Solution

NASA Astrophysics Data System (ADS)

Baradaran, M.; Panahi, H.

2018-05-01

Using the Lie algebraic approach, we present the quasi-exact solutions of the relativistic Klein-Gordon equation for perturbed Coulomb potentials namely the Cornell potential, the Kratzer potential and the Killingbeck potential. We calculate the general exact expressions for the energies, corresponding wave functions and the allowed values of the parameters of the potential within the representation space of sl(2) Lie algebra. In addition, we show that the considered equations can be transformed into the Heun's differential equations and then we reproduce the results using the associated special functions. Also, we study the special case of the Coulomb potential and show that in the non-relativistic limit, the solution of the Klein-Gordon equation converges to that of Schrödinger equation.

Symbolic-numeric interface: A review

NASA Technical Reports Server (NTRS)

Ng, E. W.

1980-01-01

A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach.

Dirac delta representation by exact parametric equations.. Application to impulsive vibration systems

NASA Astrophysics Data System (ADS)

Chicurel-Uziel, Enrique

2007-08-01

A pair of closed parametric equations are proposed to represent the Heaviside unit step function. Differentiating the step equations results in two additional parametric equations, that are also hereby proposed, to represent the Dirac delta function. These equations are expressed in algebraic terms and are handled by means of elementary algebra and elementary calculus. The proposed delta representation complies exactly with the values of the definition. It complies also with the sifting property and the requisite unit area and its Laplace transform coincides with the most general form given in the tables. Furthermore, it leads to a very simple method of solution of impulsive vibrating systems either linear or belonging to a large class of nonlinear problems. Two example solutions are presented.

Polynomial algebra reveals diverging roles of the unfolded protein response in endothelial cells during ischemia-reperfusion injury.

PubMed

Le Pape, Sylvain; Dimitrova, Elena; Hannaert, Patrick; Konovalov, Alexander; Volmer, Romain; Ron, David; Thuillier, Raphaël; Hauet, Thierry

2014-08-25

The unfolded protein response (UPR)--the endoplasmic reticulum stress response--is found in various pathologies including ischemia-reperfusion injury (IRI). However, its role during IRI is still unclear. Here, by combining two different bioinformatical methods--a method based on ordinary differential equations (Time Series Network Inference) and an algebraic method (probabilistic polynomial dynamical systems)--we identified the IRE1α-XBP1 and the ATF6 pathways as the main UPR effectors involved in cell's adaptation to IRI. We validated these findings experimentally by assessing the impact of their knock-out and knock-down on cell survival during IRI. Copyright © 2014 Federation of European Biochemical Societies. Published by Elsevier B.V. All rights reserved.

Twist for Snyder space

NASA Astrophysics Data System (ADS)

Meljanac, Daniel; Meljanac, Stjepan; Mignemi, Salvatore; Pikutić, Danijel; Štrajn, Rina

2018-03-01

We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincaré symmetry is described by a non-associative Hopf algebra, while the twisted Lorentz symmetry is described by the undeformed Hopf algebra. This new twist might be important in the construction of different types of field theories on Snyder space.

Gauge Theories of Vector Particles

DOE R&D Accomplishments Database

Glashow, S. L.; Gell-Mann, M.

1961-04-24

The possibility of generalizing the Yang-Mills trick is examined. Thus we seek theories of vector bosons invariant under continuous groups of coordinate-dependent linear transformations. All such theories may be expressed as superpositions of certain "simple" theories; we show that each "simple theory is associated with a simple Lie algebra. We may introduce mass terms for the vector bosons at the price of destroying the gauge-invariance for coordinate-dependent gauge functions. The theories corresponding to three particular simple Lie algebras - those which admit precisely two commuting quantum numbers - are examined in some detail as examples. One of them might play a role in the physics of the strong interactions if there is an underlying super-symmetry, transcending charge independence, that is badly broken. The intermediate vector boson theory of weak interactions is discussed also. The so-called "schizon" model cannot be made to conform to the requirements of partial gauge-invariance.

Spatial complexity of solutions of higher order partial differential equations

NASA Astrophysics Data System (ADS)

Kukavica, Igor

2004-03-01

We address spatial oscillation properties of solutions of higher order parabolic partial differential equations. In the case of the Kuramoto-Sivashinsky equation ut + uxxxx + uxx + u ux = 0, we prove that for solutions u on the global attractor, the quantity card {x epsi [0, L]:u(x, t) = lgr}, where L > 0 is the spatial period, can be bounded by a polynomial function of L for all \\lambda\\in{\\Bbb R} . A similar property is proven for a general higher order partial differential equation u_t+(-1)^{s}\\partial_x^{2s}u+ \\sum_{k=0}^{2s-1}v_k(x,t)\\partial_x^k u =0 .

Efficient model reduction of parametrized systems by matrix discrete empirical interpolation

NASA Astrophysics Data System (ADS)

Negri, Federico; Manzoni, Andrea; Amsallem, David

2015-12-01

In this work, we apply a Matrix version of the so-called Discrete Empirical Interpolation (MDEIM) for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations. Dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced-order models (ROMs). However, in many cases such an affine decomposition is not readily available, and must be recovered through (often) intrusive procedures, such as the empirical interpolation method (EIM) and its discrete variant DEIM. In this paper we show that MDEIM represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non-intrusive, efficient and purely algebraic way. We propose different strategies to combine MDEIM with a state approximation resulting either from a reduced basis greedy approach or Proper Orthogonal Decomposition. A posteriori error estimates accounting for the MDEIM error are also developed in the case of parametrized elliptic and parabolic equations. Finally, the capability of MDEIM to generate accurate and efficient ROMs is demonstrated on the solution of two computationally-intensive classes of problems occurring in engineering contexts, namely PDE-constrained shape optimization and parametrized coupled problems.

Transforming a fourth year modern optics course using a deliberate practice framework

NASA Astrophysics Data System (ADS)

Jones, David J.; Madison, Kirk W.; Wieman, Carl E.

2015-12-01

[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present a study of active learning pedagogies in an upper-division physics course. This work was guided by the principle of deliberate practice for the development of expertise, and this principle was used in the design of the materials and the orchestration of the classroom activities of the students. We present our process for efficiently converting a traditional lecture course based on instructor notes into activities for such a course with active learning methods. Ninety percent of the same material was covered and scores on common exam problems showed a 15% improvement with an effect size greater than 1 after the transformation. We observe that the improvement and the associated effect size is sustained after handing off the materials to a second instructor. Because the improvement on exam questions was independent of specific problem topics and because the material tested was so mathematically advanced and broad (including linear algebra, Fourier transforms, partial differential equations, and vector calculus), we expect the transformation process could be applied to most upper-division physics courses having a similar mathematical base.

[Differential gene expression profile in ischemic myocardium of Wistar rats with acute myocardial infarction: the study on gene construction, identification and function].

PubMed

Guo, Chun Yu; Yin, Hui Jun; Jiang, Yue Rong; Xue, Mei; Zhang, Lu; Shi, Da Zhuo

2008-06-18

To construct the differential genes expressed profile in the ischemic myocardium tissue reduced from acute myocardial infarction(AMI), and determine the biological functions of target genes. AMI model was generated by ligation of the left anterior descending coronary artery in Wistar rats. Total RNA was extracted from the normal and the ischemic heart tissues under the ligation point 7 days after the operation. Differential gene expression profiles of the two samples were constructed using Long Serial Analysis of Gene Expression(LongSAGE). Real time fluorescence quantitative PCR was used to verify gene expression profile and to identify the expression of 2 functional genes. The activities of enzymes from functional genes were determined by histochemistry. A total of 15,966 tags were screened from the normal and the ischemic LongSAGE maps. The similarities of the sequences were compared using the BLAST algebra in NCBI and 7,665 novel tags were found. In the ischemic tissue 142 genes were significantly changed compared with those in the normal tissue (P<0.05). These differentially expressed genes represented the proteins which might play important roles in the pathways of oxidation and phosphorylation, ATP synthesis and glycolysis. The partial genes identified by LongSAGE were confirmed using real time fluorescence quantitative PCR. Two genes related to energy metabolism, COX5a and ATP5e, were screened and quantified. Expression of two functional genes down-regulated at their mRNA levels and the activities of correlative functional enzymes decreased compared with those in the normal tissue. AMI causes a series of changes in gene expression, in which the abnormal expression of genes related to energy metabolism could be one of the molecular mechanisms of AMI. The intervention of the expressions of COX5a and ATP5e may be a new target for AMI therapy.

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.

On the Solution of Elliptic Partial Differential Equations on Regions with Corners

DTIC Science & Technology

2015-07-09

In this report we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations . We observe...that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of...efficient numerical algorithms. The results are illustrated by a number of numerical examples. On the solution of elliptic partial differential equations on

Modelling, Simulation, Animation, and Real-Time Control (Mosart) for a Class of Electromechanical Systems: A System-Theoretic Approach

ERIC Educational Resources Information Center

Rodriguez, Armando A.; Metzger, Richard P.; Cifdaloz, Oguzhan; Dhirasakdanon, Thanate; Welfert, Bruno

2004-01-01

This paper describes an interactive modelling, simulation, animation, and real-time control (MoSART) environment for a class of 'cart-pendulum' electromechanical systems that may be used to enhance learning within differential equations and linear algebra classes. The environment is useful for conveying fundamental mathematical/systems concepts…

Armed Services Vocational Aptitude Battery (ASVAB): The Equity of ASVAB Form 14 in the Prediction of High School Course Grades

DTIC Science & Technology

1990-08-01

differential validity 612 ASVAB test constru~tion, 16 .PRIC’E CODE bias 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION jig. SECURITY CLAS...15 Juniors 1985-86 .............................. 15 Seniors 1984-85 .............................. 16 Algebra... 16 Freshmen 1984-85 ............................. 16 Freshmen 1 85-86 ............................. 16 Sophomores 1984-85

Examining Gender DIF on a Multiple-Choice Test of Mathematics: A Confirmatory Approach.

ERIC Educational Resources Information Center

Ryan, Katherine E.; Fan, Meichu

1996-01-01

Results for 3,244 female and 3,033 male junior high school students from the Second International Mathematics Study show that applied items in algebra, geometry, and computation were easier for males but arithmetic items were differentially easier for females. Implications of these findings for assessment and instruction are discussed. (SLD)

Towards Direct Simulations of Counterflow Flames with Consistent Numerical Differential-Algebraic Boundary Conditions

DTIC Science & Technology

2015-05-18

First, the gov - erning equations of the problem are presented. A detailed discussion on the construction of the initial profile of the flow follows...time from the DoD HPCMP Open Research Systems and JPL/ NASA is gratefully acknowledged. References [1] H. Tsuji, Prog. Energ. Combust.8(2) (1982) 93-119

JOURNAL SCOPE GUIDELINES: Paper classification scheme

NASA Astrophysics Data System (ADS)

2005-06-01

This scheme is used to clarify the journal's scope and enable authors and readers to more easily locate the appropriate section for their work. For each of the sections listed in the scope statement we suggest some more detailed subject areas which help define that subject area. These lists are by no means exhaustive and are intended only as a guide to the type of papers we envisage appearing in each section. We acknowledge that no classification scheme can be perfect and that there are some papers which might be placed in more than one section. We are happy to provide further advice on paper classification to authors upon request (please email jphysa@iop.org). 1. Statistical physics numerical and computational methods statistical mechanics, phase transitions and critical phenomena quantum condensed matter theory Bose-Einstein condensation strongly correlated electron systems exactly solvable models in statistical mechanics lattice models, random walks and combinatorics field-theoretical models in statistical mechanics disordered systems, spin glasses and neural networks nonequilibrium systems network theory 2. Chaotic and complex systems nonlinear dynamics and classical chaos fractals and multifractals quantum chaos classical and quantum transport cellular automata granular systems and self-organization pattern formation biophysical models 3. Mathematical physics combinatorics algebraic structures and number theory matrix theory classical and quantum groups, symmetry and representation theory Lie algebras, special functions and orthogonal polynomials ordinary and partial differential equations difference and functional equations integrable systems soliton theory functional analysis and operator theory inverse problems geometry, differential geometry and topology numerical approximation and analysis geometric integration computational methods 4. Quantum mechanics and quantum information theory coherent states eigenvalue problems supersymmetric quantum mechanics scattering theory relativistic quantum mechanics semiclassical approximations foundations of quantum mechanics and measurement theory entanglement and quantum nonlocality geometric phases and quantum tomography quantum tunnelling decoherence and open systems quantum cryptography, communication and computation theoretical quantum optics 5. Classical and quantum field theory quantum field theory gauge and conformal field theory quantum electrodynamics and quantum chromodynamics Casimir effect integrable field theory random matrix theory applications in field theory string theory and its developments classical field theory and electromagnetism metamaterials 6. Fluid and plasma theory turbulence fundamental plasma physics kinetic theory magnetohydrodynamics and multifluid descriptions strongly coupled plasmas one-component plasmas non-neutral plasmas astrophysical and dusty plasmas

A Semianalytical Analysis of Compressible Electrophoretic Cake Formation

NASA Astrophysics Data System (ADS)

Kambham, Kiran K. R.; Tuncay, Kagan; Corapcioglu, M. Yavuz

1995-05-01

Leaks in geomembrane liners of waste landfills and liquid impoundments cause chemical contaminants to leak into the subsurface environment. A mathematical model is presented to simulate electrophoretic sealing of impoundment leaks. The model describes the formation of a compressible clay cake because of electrical and gravitational forces. The model includes mass balance equations for the solid particles and liquid phase, modified Darcy's law in an electrical field, and Terzaghi's definition of effective stress. The formulation is presented in the Eulerian coordinates. The resulting second-order, nonlinear partial differential equation and the lower boundary condition are linearized to obtain an analytical solution for time-dependent settlement. After discretizing in time the analytical solution is applied to simulate compression of an accreting sediment. In the simulation of an accreting sediment, solid fluxes on either side of suspension/sediment interface are coupled using a no-jump condition. The velocity of a discrete particle in the suspension zone is assumed to be equal to the algebraic sum of electrophoretic and Stoke's settling velocities. An empirical relationship available in the literature is used to account for the effect of concentration on the velocity of solid particles in the suspension zone. The validity of the semianalytical approach is partially verified using an exact steady state solution for self-weight consolidation. The simulation results obtained for a set of material parameters are presented graphically. It is noted that the electrokinetic consolidation of sediment continues even after the completion of electrophoretic settling of all clay particles. An analysis reveals that the electrophoretic cake formation process is quite sensitive to voltage gradient and the coefficient of compressibility.

Intelligent machines in the twenty-first century: foundations of inference and inquiry.

PubMed

Knuth, Kevin H

2003-12-15

The last century saw the application of Boolean algebra to the construction of computing machines, which work by applying logical transformations to information contained in their memory. The development of information theory and the generalization of Boolean algebra to Bayesian inference have enabled these computing machines, in the last quarter of the twentieth century, to be endowed with the ability to learn by making inferences from data. This revolution is just beginning as new computational techniques continue to make difficult problems more accessible. Recent advances in our understanding of the foundations of probability theory have revealed implications for areas other than logic. Of relevance to intelligent machines, we recently identified the algebra of questions as the free distributive algebra, which will now allow us to work with questions in a way analogous to that which Boolean algebra enables us to work with logical statements. In this paper, we examine the foundations of inference and inquiry. We begin with a history of inferential reasoning, highlighting key concepts that have led to the automation of inference in modern machine-learning systems. We then discuss the foundations of inference in more detail using a modern viewpoint that relies on the mathematics of partially ordered sets and the scaffolding of lattice theory. This new viewpoint allows us to develop the logic of inquiry and introduce a measure describing the relevance of a proposed question to an unresolved issue. Last, we will demonstrate the automation of inference, and discuss how this new logic of inquiry will enable intelligent machines to ask questions. Automation of both inference and inquiry promises to allow robots to perform science in the far reaches of our solar system and in other star systems by enabling them not only to make inferences from data, but also to decide which question to ask, which experiment to perform, or which measurement to take given what they have learned and what they are designed to understand.

Intelligent machines in the twenty-first century: foundations of inference and inquiry

NASA Technical Reports Server (NTRS)

Knuth, Kevin H.

2003-01-01

The last century saw the application of Boolean algebra to the construction of computing machines, which work by applying logical transformations to information contained in their memory. The development of information theory and the generalization of Boolean algebra to Bayesian inference have enabled these computing machines, in the last quarter of the twentieth century, to be endowed with the ability to learn by making inferences from data. This revolution is just beginning as new computational techniques continue to make difficult problems more accessible. Recent advances in our understanding of the foundations of probability theory have revealed implications for areas other than logic. Of relevance to intelligent machines, we recently identified the algebra of questions as the free distributive algebra, which will now allow us to work with questions in a way analogous to that which Boolean algebra enables us to work with logical statements. In this paper, we examine the foundations of inference and inquiry. We begin with a history of inferential reasoning, highlighting key concepts that have led to the automation of inference in modern machine-learning systems. We then discuss the foundations of inference in more detail using a modern viewpoint that relies on the mathematics of partially ordered sets and the scaffolding of lattice theory. This new viewpoint allows us to develop the logic of inquiry and introduce a measure describing the relevance of a proposed question to an unresolved issue. Last, we will demonstrate the automation of inference, and discuss how this new logic of inquiry will enable intelligent machines to ask questions. Automation of both inference and inquiry promises to allow robots to perform science in the far reaches of our solar system and in other star systems by enabling them not only to make inferences from data, but also to decide which question to ask, which experiment to perform, or which measurement to take given what they have learned and what they are designed to understand.

Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces

NASA Technical Reports Server (NTRS)

Ito, Kazufumi

1987-01-01

The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems.

New developments in FeynCalc 9.0

NASA Astrophysics Data System (ADS)

Shtabovenko, Vladyslav; Mertig, Rolf; Orellana, Frederik

2016-10-01

In this note we report on the new version of FEYNCALC, a MATHEMATICA package for symbolic semi-automatic evaluation of Feynman diagrams and algebraic expressions in quantum field theory. The main features of version 9.0 are: improved tensor reduction and partial fractioning of loop integrals, new functions for using FEYNCALC together with tools for reduction of scalar loop integrals using integration-by-parts (IBP) identities, better interface to FEYNARTS and support for SU(N) generators with explicit fundamental indices.

Olson Order of Quantum Observables

NASA Astrophysics Data System (ADS)

Dvurečenskij, Anatolij

2016-11-01

M.P. Olson, Proc. Am. Math. Soc. 28, 537-544 (1971) showed that the system of effect operators of the Hilbert space can be ordered by the so-called spectral order such that the system of effect operators is a complete lattice. Using his ideas, we introduce a partial order, called the Olson order, on the set of bounded observables of a complete lattice effect algebra. We show that the set of bounded observables is a Dedekind complete lattice.

Using Productive Disposition to Differentiate between Students' Level of Precision When Critiquing a Peer's Work

ERIC Educational Resources Information Center

Gilbert, Melissa C.

2014-01-01

This study examined the productive disposition of pre-algebra students who demonstrated similar knowledge of the focal content but varied in other academic behaviors expected in the Common Core State Standards for Mathematics (CCSSM). Specifically, the study considered students' attention to precision when critiquing a peer's work. The…

[Differential and ontogenetic meaningfulness of iconic, linguistic and formal codes on cognitive development: new questions].

PubMed

Wittwer, J

1990-01-01

Man is a semiotic functioning animal, i.e. civilizations are units of symbolic (architectural), iconic, linguistic, formal, etc...) organizations. These units can only initially develop when enabled--but not necessarily produced--by a material base of a bio-physical nature, namely the central nervous system. In short, taking but three more academic factors, images, texts, and algebra, for example, are grasped by this material base. However, it is clear that the effects produced on children (and on adults, for that matter) are not equal. Scholastic goals, however, emphasize "fables" and "equations" whereas social mediatization emphasizes "images" and economic mediatization "equations". Hence the problems of appropriation of linguistic codes. To show the danger of an imbalance in these appropriations, the concept of differential semanticization is called upon: images are over-semanticized, with identification at risk; algebra is under-semanticized, at risks of obsessionalization. Texts, themselves, call upon the imagination and not on an imaginary structure imposed by a multivocal iconic pressure nor an imaginary structure rarefied by the prevalence of systems with univocal elements. Hence the importance of reading and writing for maintaining a nondepersonalizing semiotic balance.

Free vibration of functionally graded beams and frameworks using the dynamic stiffness method

NASA Astrophysics Data System (ADS)

Banerjee, J. R.; Ananthapuvirajah, A.

2018-05-01

The free vibration analysis of functionally graded beams (FGBs) and frameworks containing FGBs is carried out by applying the dynamic stiffness method and deriving the elements of the dynamic stiffness matrix in explicit algebraic form. The usually adopted rule that the material properties of the FGB vary continuously through the thickness according to a power law forms the fundamental basis of the governing differential equations of motion in free vibration. The differential equations are solved in closed analytical form when the free vibratory motion is harmonic. The dynamic stiffness matrix is then formulated by relating the amplitudes of forces to those of the displacements at the two ends of the beam. Next, the explicit algebraic expressions for the dynamic stiffness elements are derived with the help of symbolic computation. Finally the Wittrick-Williams algorithm is applied as solution technique to solve the free vibration problems of FGBs with uniform cross-section, stepped FGBs and frameworks consisting of FGBs. Some numerical results are validated against published results, but in the absence of published results for frameworks containing FGBs, consistency checks on the reliability of results are performed. The paper closes with discussion of results and conclusions.

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations

NASA Astrophysics Data System (ADS)

Husa, Sascha; Hinder, Ian; Lechner, Christiane

2006-06-01

We present a suite of Mathematica-based computer-algebra packages, termed "Kranc", which comprise a toolbox to convert certain (tensorial) systems of partial differential evolution equations to parallelized C or Fortran code for solving initial boundary value problems. Kranc can be used as a "rapid prototyping" system for physicists or mathematicians handling very complicated systems of partial differential equations, but through integration into the Cactus computational toolkit we can also produce efficient parallelized production codes. Our work is motivated by the field of numerical relativity, where Kranc is used as a research tool by the authors. In this paper we describe the design and implementation of both the Mathematica packages and the resulting code, we discuss some example applications, and provide results on the performance of an example numerical code for the Einstein equations. Program summaryTitle of program: Kranc Catalogue identifier: ADXS_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXS_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Distribution format: tar.gz Computer for which the program is designed and others on which it has been tested: General computers which run Mathematica (for code generation) and Cactus (for numerical simulations), tested under Linux Programming language used: Mathematica, C, Fortran 90 Memory required to execute with typical data: This depends on the number of variables and gridsize, the included ADM example requires 4308 KB Has the code been vectorized or parallelized: The code is parallelized based on the Cactus framework. Number of bytes in distributed program, including test data, etc.: 1 578 142 Number of lines in distributed program, including test data, etc.: 11 711 Nature of physical problem: Solution of partial differential equations in three space dimensions, which are formulated as an initial value problem. In particular, the program is geared towards handling very complex tensorial equations as they appear, e.g., in numerical relativity. The worked out examples comprise the Klein-Gordon equations, the Maxwell equations, and the ADM formulation of the Einstein equations. Method of solution: The method of numerical solution is finite differencing and method of lines time integration, the numerical code is generated through a high level Mathematica interface. Restrictions on the complexity of the program: Typical numerical relativity applications will contain up to several dozen evolution variables and thousands of source terms, Cactus applications have shown scaling up to several thousand processors and grid sizes exceeding 500 3. Typical running time: This depends on the number of variables and the grid size: the included ADM example takes approximately 100 seconds on a 1600 MHz Intel Pentium M processor. Unusual features of the program: based on Mathematica and Cactus

Pion and Kaon Lab Frame Differential Cross Sections for Intermediate Energy Nucleus-Nucleus Collisions

NASA Technical Reports Server (NTRS)

Norbury, John W.; Blattnig, Steve R.

2008-01-01

Space radiation transport codes require accurate models for hadron production in intermediate energy nucleus-nucleus collisions. Codes require cross sections to be written in terms of lab frame variables and it is important to be able to verify models against experimental data in the lab frame. Several models are compared to lab frame data. It is found that models based on algebraic parameterizations are unable to describe intermediate energy differential cross section data. However, simple thermal model parameterizations, when appropriately transformed from the center of momentum to the lab frame, are able to account for the data.

The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations.

PubMed

Khader, M M

2013-10-01

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

The conceptual basis of mathematics in cardiology: (II). Calculus and differential equations.

PubMed

Bates, Jason H T; Sobel, Burton E

2003-04-01

This is the second in a series of four articles developed for the readers of Coronary Artery Disease. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

Scheduled Relaxation Jacobi method: Improvements and applications

NASA Astrophysics Data System (ADS)

Adsuara, J. E.; Cordero-Carrión, I.; Cerdá-Durán, P.; Aloy, M. A.

2016-09-01

Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficiency in the reduction of the residual increases with the number of levels employed in the algorithm. Applying the original methodology to compute the algorithm parameters with more than 5 levels notably hinders obtaining optimal SRJ schemes, as the mixed (non-linear) algebraic-differential system of equations from which they result becomes notably stiff. Here we present a new methodology for obtaining the parameters of SRJ schemes that overcomes the limitations of the original algorithm and provide parameters for SRJ schemes with up to 15 levels and resolutions of up to 215 points per dimension, allowing for acceleration factors larger than several hundreds with respect to the Jacobi method for typical resolutions and, in some high resolution cases, close to 1000. Most of the success in finding SRJ optimal schemes with more than 10 levels is based on an analytic reduction of the complexity of the previously mentioned system of equations. Furthermore, we extend the original algorithm to apply it to certain systems of non-linear ePDEs.

The rational parameterization theorem for multisite post-translational modification systems.

PubMed

Thomson, Matthew; Gunawardena, Jeremy

2009-12-21

Post-translational modification of proteins plays a central role in cellular regulation but its study has been hampered by the exponential increase in substrate modification forms ("modforms") with increasing numbers of sites. We consider here biochemical networks arising from post-translational modification under mass-action kinetics, allowing for multiple substrates, having different types of modification (phosphorylation, methylation, acetylation, etc.) on multiple sites, acted upon by multiple forward and reverse enzymes (in total number L), using general enzymatic mechanisms. These assumptions are substantially more general than in previous studies. We show that the steady-state modform concentrations constitute an algebraic variety that can be parameterized by rational functions of the L free enzyme concentrations, with coefficients which are rational functions of the rate constants. The parameterization allows steady states to be calculated by solving L algebraic equations, a dramatic reduction compared to simulating an exponentially large number of differential equations. This complexity collapse enables analysis in contexts that were previously intractable and leads to biological predictions that we review. Our results lay a foundation for the systems biology of post-translational modification and suggest deeper connections between biochemical networks and algebraic geometry.

Serre duality, Abel's theorem, and Jacobi inversion for supercurves over a thick superpoint

NASA Astrophysics Data System (ADS)

Rothstein, Mitchell J.; Rabin, Jeffrey M.

2015-04-01

The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1 | q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincaré duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic groups and class fields, combined with an invariance result for the topology on the dual of the space of répartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov's characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf D of differential operators. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n.

A Simple Algebraic Grid Adaptation Scheme with Applications to Two- and Three-dimensional Flow Problems

NASA Technical Reports Server (NTRS)

Hsu, Andrew T.; Lytle, John K.

1989-01-01

An algebraic adaptive grid scheme based on the concept of arc equidistribution is presented. The scheme locally adjusts the grid density based on gradients of selected flow variables from either finite difference or finite volume calculations. A user-prescribed grid stretching can be specified such that control of the grid spacing can be maintained in areas of known flowfield behavior. For example, the grid can be clustered near a wall for boundary layer resolution and made coarse near the outer boundary of an external flow. A grid smoothing technique is incorporated into the adaptive grid routine, which is found to be more robust and efficient than the weight function filtering technique employed by other researchers. Since the present algebraic scheme requires no iteration or solution of differential equations, the computer time needed for grid adaptation is trivial, making the scheme useful for three-dimensional flow problems. Applications to two- and three-dimensional flow problems show that a considerable improvement in flowfield resolution can be achieved by using the proposed adaptive grid scheme. Although the scheme was developed with steady flow in mind, it is a good candidate for unsteady flow computations because of its efficiency.

Construction and accuracy of partial differential equation approximations to the chemical master equation.

PubMed

Grima, Ramon

2011-11-01

The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.

The discrete adjoint method for parameter identification in multibody system dynamics.

PubMed

Lauß, Thomas; Oberpeilsteiner, Stefan; Steiner, Wolfgang; Nachbagauer, Karin

2018-01-01

The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. An additional set of differential equations has to be solved to compute the adjoint variables, which are further used for the gradient computation. However, the accuracy of the numerical solution of the adjoint differential equation has a great impact on the gradient. Hence, an alternative approach is the discrete adjoint method , where the adjoint differential equations are replaced by algebraic equations. Therefore, a finite difference scheme is constructed for the adjoint system directly from the numerical time integration method. The method provides the exact gradient of the discretized cost function subjected to the discretized equations of motion.

Extended Riemannian geometry II: local heterotic double field theory

NASA Astrophysics Data System (ADS)

Deser, Andreas; Heller, Marc Andre; Sämann, Christian

2018-04-01

We continue our exploration of local Double Field Theory (DFT) in terms of symplectic graded manifolds carrying compatible derivations and study the case of heterotic DFT. We start by developing in detail the differential graded manifold that captures heterotic Generalized Geometry which leads to new observations on the generalized metric and its twists. We then give a symplectic pre-N Q-manifold that captures the symmetries and the geometry of local heterotic DFT. We derive a weakened form of the section condition, which arises algebraically from consistency of the symmetry Lie 2-algebra and its action on extended tensors. We also give appropriate notions of twists — which are required for global formulations — and of the torsion and Riemann tensors. Finally, we show how the observed α'-corrections are interpreted naturally in our framework.

Analysis of SI models with multiple interacting populations using subpopulations.

PubMed

Thomas, Evelyn K; Gurski, Katharine F; Hoffman, Kathleen A

2015-02-01

Computing endemic equilibria and basic reproductive numbers for systems of differential equations describing epidemiological systems with multiple connections between subpopulations is often algebraically intractable. We present an alternative method which deconstructs the larger system into smaller subsystems and captures the interactions between the smaller systems as external forces using an approximate model. We bound the basic reproductive numbers of the full system in terms of the basic reproductive numbers of the smaller systems and use the alternate model to provide approximations for the endemic equilibrium. In addition to creating algebraically tractable reproductive numbers and endemic equilibria, we can demonstrate the influence of the interactions between subpopulations on the basic reproductive number of the full system. The focus of this paper is to provide analytical tools to help guide public health decisions with limited intervention resources.

On B-type Open-Closed Landau-Ginzburg Theories Defined on Calabi-Yau Stein Manifolds

NASA Astrophysics Data System (ADS)

Babalic, Elena Mirela; Doryn, Dmitry; Lazaroiu, Calin Iuliu; Tavakol, Mehdi

2018-05-01

We consider the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair (X,W), where X is a non-compact Calabi-Yau manifold and W is a complex-valued holomorphic function. When X is a Stein manifold (but not restricted to be a domain of holomorphy), we extract equivalent descriptions of the bulk algebra and of the category of topological D-branes which are constructed using only the analytic space associated to X. In particular, we show that the D-brane category is described by projective factorizations defined over the ring of holomorphic functions of X. We also discuss simplifications of the analytic models which arise when X is holomorphically parallelizable and illustrate these in a few classes of examples.

Calculation of normal modes of the closed waveguides in general vector case

NASA Astrophysics Data System (ADS)

Malykh, M. D.; Sevastianov, L. A.; Tiutiunnik, A. A.

2018-04-01

The article is devoted to the calculation of normal modes of the closed waveguides with an arbitrary filling ɛ, μ in the system of computer algebra Sage. Maxwell equations in the cylinder are reduced to the system of two bounded Helmholtz equations, the notion of weak solution of this system is given and then this system is investigated as a system of ordinary differential equations. The normal modes of this system are an eigenvectors of a matrix pencil. We suggest to calculate the matrix elements approximately and to truncate the matrix by usual way but further to solve the truncated eigenvalue problem exactly in the field of algebraic numbers. This approach allows to keep the symmetry of the initial problem and in particular the multiplicity of the eigenvalues. In the work would be presented some results of calculations.

Stepwise Analysis of Differential Item Functioning Based on Multiple-Group Partial Credit Model.

ERIC Educational Resources Information Center

Muraki, Eiji

1999-01-01

Extended an Item Response Theory (IRT) method for detection of differential item functioning to the partial credit model and applied the method to simulated data using a stepwise procedure. Then applied the stepwise DIF analysis based on the multiple-group partial credit model to writing trend data from the National Assessment of Educational…

Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's. [ordinary differential equations

NASA Technical Reports Server (NTRS)

Geddes, K. O.

1977-01-01

If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.

Explicit Filtering Based Low-Dose Differential Phase Reconstruction Algorithm with the Grating Interferometry.

PubMed

Jiang, Xiaolei; Zhang, Li; Zhang, Ran; Yin, Hongxia; Wang, Zhenchang

2015-01-01

X-ray grating interferometry offers a novel framework for the study of weakly absorbing samples. Three kinds of information, that is, the attenuation, differential phase contrast (DPC), and dark-field images, can be obtained after a single scanning, providing additional and complementary information to the conventional attenuation image. Phase shifts of X-rays are measured by the DPC method; hence, DPC-CT reconstructs refraction indexes rather than attenuation coefficients. In this work, we propose an explicit filtering based low-dose differential phase reconstruction algorithm, which enables reconstruction from reduced scanning without artifacts. The algorithm adopts a differential algebraic reconstruction technique (DART) with the explicit filtering based sparse regularization rather than the commonly used total variation (TV) method. Both the numerical simulation and the biological sample experiment demonstrate the feasibility of the proposed algorithm.

Explicit Filtering Based Low-Dose Differential Phase Reconstruction Algorithm with the Grating Interferometry

PubMed Central

Zhang, Li; Zhang, Ran; Yin, Hongxia; Wang, Zhenchang

2015-01-01

X-ray grating interferometry offers a novel framework for the study of weakly absorbing samples. Three kinds of information, that is, the attenuation, differential phase contrast (DPC), and dark-field images, can be obtained after a single scanning, providing additional and complementary information to the conventional attenuation image. Phase shifts of X-rays are measured by the DPC method; hence, DPC-CT reconstructs refraction indexes rather than attenuation coefficients. In this work, we propose an explicit filtering based low-dose differential phase reconstruction algorithm, which enables reconstruction from reduced scanning without artifacts. The algorithm adopts a differential algebraic reconstruction technique (DART) with the explicit filtering based sparse regularization rather than the commonly used total variation (TV) method. Both the numerical simulation and the biological sample experiment demonstrate the feasibility of the proposed algorithm. PMID:26089971

Another short and elementary proof of strong subadditivity of quantum entropy

NASA Astrophysics Data System (ADS)

Ruskai, Mary Beth

2007-08-01

A short and elementary proof of the joint convexity of relative entropy is presented, using nothing beyond linear algebra. The key ingredients are an easily verified integral representation and the strategy used to prove the Cauchy-Schwarz inequality in elementary courses. Several consequences are proved in a way which allows an elementary proof of strong subadditivity in a few more lines. Some expository material on Schwarz inequalities for operators and the Holevo bound for partial measurements is also included.

US-Latin American Workshop on Molecular and Materials Sciences: Theoretical and Computational Aspects Held at the University of Florida, Gainesville, on February 8-10, 1994

DTIC Science & Technology

1994-08-09

Observables During a Collision Inst. de Fisica , Cuernavaca, Mexico Ruben D. Santiago Acosta An Algebraic Model for 3-dimensional Atom-Diatom Inst C...STRUCTURES. MOLECULAR DYNAMICS SIMULATION M. C .Donnamaria and J. R. Grigera Instituto de Fisica de Liquidos y Sistemas Biologicos (IFLYSIB),CONICET...Crybiology, 1981, 18, 631. ACKNOWLEDGMENTS This work has been partially funded by the Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET) of

Highly Productive Application Development with ViennaCL for Accelerators

NASA Astrophysics Data System (ADS)

Rupp, K.; Weinbub, J.; Rudolf, F.

2012-12-01

The use of graphics processing units (GPUs) for the acceleration of general purpose computations has become very attractive over the last years, and accelerators based on many integrated CPU cores are about to hit the market. However, there are discussions about the benefit of GPU computing when comparing the reduction of execution times with the increased development effort [1]. To counter these concerns, our open-source linear algebra library ViennaCL [2,3] uses modern programming techniques such as generic programming in order to provide a convenient access layer for accelerator and GPU computing. Other GPU-accelerated libraries are primarily tuned for performance, but less tailored to productivity and portability: MAGMA [4] provides dense linear algebra operations via a LAPACK-comparable interface, but no dedicated matrix and vector types. Cusp [5] is closest in functionality to ViennaCL for sparse matrices, but is based on CUDA and thus restricted to devices from NVIDIA. However, no convenience layer for dense linear algebra is provided with Cusp. ViennaCL is written in C++ and uses OpenCL to access the resources of accelerators, GPUs and multi-core CPUs in a unified way. On the one hand, the library provides iterative solvers from the family of Krylov methods, including various preconditioners, for the solution of linear systems typically obtained from the discretization of partial differential equations. On the other hand, dense linear algebra operations are supported, including algorithms such as QR factorization and singular value decomposition. The user application interface of ViennaCL is compatible to uBLAS [6], which is part of the peer-reviewed Boost C++ libraries [7]. This allows to port existing applications based on uBLAS with a minimum of effort to ViennaCL. Conversely, the interface compatibility allows to use the iterative solvers from ViennaCL with uBLAS types directly, thus enabling code reuse beyond CPU-GPU boundaries. Out-of-the-box support for types from the Eigen library [8] and MTL 4 [9] are provided as well, enabling a seamless transition from single-core CPU to GPU and multi-core CPU computations. Case studies from the numerical solution of PDEs are given and isolated performance benchmarks are discussed. Also, pitfalls in scientific computing with GPUs and accelerators are addressed, allowing for a first evaluation of whether these novel devices can be mapped well to certain applications. References: [1] R. Bordawekar et al., Technical Report, IBM, 2010 [2] ViennaCL library. Online: http://viennacl.sourceforge.net/ [3] K. Rupp et al., GPUScA, 2010 [4] MAGMA library. Online: http://icl.cs.utk.edu/magma/ [5] Cusp library. Online: http://code.google.com/p/cusp-library/ [6] uBLAS library. Online: http://www.boost.org/libs/numeric/ublas/ [7] Boost C++ Libraries. Online: http://www.boost.org/ [8] Eigen library. Online: http://eigen.tuxfamily.org/ [9] MTL 4 Library. Online: http://www.mtl4.org/

Entanglement criteria via the uncertainty relations in su(2) and su(1,1) algebras: Detection of non-Gaussian entangled states

NASA Astrophysics Data System (ADS)

Nha, Hyunchul; Kim, Jaewan

2006-07-01

We derive a class of inequalities, from the uncertainty relations of the su(1,1) and the su(2) algebra in conjunction with partial transposition, that must be satisfied by any separable two-mode states. These inequalities are presented in terms of the su(2) operators Jx=(a†b+ab†)/2 , Jy=(a†b-ab†)/2i , and the total photon number ⟨Na+Nb⟩ . They include as special cases the inequality derived by Hillery and Zubairy [Phys. Rev. Lett. 96, 050503 (2006)], and the one by Agarwal and Biswas [New J. Phys. 7, 211 (2005)]. In particular, optimization over the whole inequalities leads to the criterion obtained by Agarwal and Biswas. We show that this optimal criterion can detect entanglement for a broad class of non-Gaussian entangled states, i.e., the su(2) minimum-uncertainty states. Experimental schemes to test the optimal criterion are also discussed, especially the one using linear optical devices and photodetectors.

Domain Derivatives in Dielectric Rough Surface Scattering

DTIC Science & Technology

2015-01-01

and require the gradient of the objective function in the unknown model parameter vector at each stage of iteration. For large N, finite...differencing becomes numerically intensive, and an efficient alternative is domain differentiation in which the full gradient is obtained by solving a single...derivative calculation of the gradient for a locally perturbed dielectric interface. The method is non-variational, and algebraic in nature in that it

A Solution Space for a System of Null-State Partial Differential Equations: Part 4

NASA Astrophysics Data System (ADS)

Flores, Steven M.; Kleban, Peter

2015-01-01

This article is the last of four that completely and rigorously characterize a solution space for a homogeneous system of 2 N + 3 linear partial differential equations in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE). The system comprises 2 N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2 N one-leg boundary operators. In the first two articles (Flores and Kleban in Commun Math Phys, 2012; Flores and Kleban, in Commun Math Phys, 2014), we use methods of analysis and linear algebra to prove that dim , with C N the Nth Catalan number. Using these results in the third article (Flores and Kleban, in Commun Math Phys, 2013), we prove that dim and is spanned by (real-valued) solutions constructed with the Coulomb gas (contour integral) formalism of CFT. In this article, we use these results to prove some facts concerning the solution space . First, we show that each of its elements equals a sum of at most two distinct Frobenius series in powers of the difference between two adjacent points (unless is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion for one-leg boundary operators, assumed in CFT. We also identify particular elements of , which we call connectivity weights, and exploit their special properties to conjecture a formula for the probability that the curves of a multiple-SLE process join in a particular connectivity. This leads to new formulas for crossing probabilities of critical lattice models inside polygons with a free/fixed side-alternating boundary condition, which we derive in Flores et al. (Partition functions and crossing probabilities for critical systems inside polygons, in preparation). Finally, we propose a reason for why the exceptional speeds [certain values that appeared in the analysis of the Coulomb gas solutions in Flores and Kleban (Commun Math Phys, 2013)] and the minimal models of CFT are connected.

Time Parallel Solution of Linear Partial Differential Equations on the Intel Touchstone Delta Supercomputer

NASA Technical Reports Server (NTRS)

Toomarian, N.; Fijany, A.; Barhen, J.

1993-01-01

Evolutionary partial differential equations are usually solved by decretization in time and space, and by applying a marching in time procedure to data and algorithms potentially parallelized in the spatial domain.

Classification of symmetry-protected phases for interacting fermions in two dimensions

NASA Astrophysics Data System (ADS)

Cheng, Meng; Bi, Zhen; You, Yi-Zhuang; Gu, Zheng-Cheng

2018-05-01

Recently, it has been established that two-dimensional bosonic symmetry-protected topological (SPT) phases with on-site unitary symmetry G can be completely classified by the group cohomology H3( G ,U (1 ) ) . Later, group supercohomology was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the algebraic theory of symmetry and defects in two-dimensional topological phases. We reproduce the partial classifications given by group supercohomology, and we also show that with an additional H1(G ,Z2) structure, a complete classification of SPT phases for two-dimensional interacting fermion systems with a total symmetry group G ×Z2f is obtained. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.

ML 3.0 smoothed aggregation user's guide.

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

Sala, Marzio; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen

2004-05-01

ML is a multigrid preconditioning package intended to solve linear systems of equations Az = b where A is a user supplied n x n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. ML should be used on large sparse linear systems arising from partial differential equation (PDE) discretizations. While technically any linear system can be considered, ML should be used on linear systems that correspond to things that work well with multigrid methods (e.g. elliptic PDEs). ML can be used as a stand-alone package ormore » to generate preconditioners for a traditional iterative solver package (e.g. Krylov methods). We have supplied support for working with the AZTEC 2.1 and AZTECOO iterative package [15]. However, other solvers can be used by supplying a few functions. This document describes one specific algebraic multigrid approach: smoothed aggregation. This approach is used within several specialized multigrid methods: one for the eddy current formulation for Maxwell's equations, and a multilevel and domain decomposition method for symmetric and non-symmetric systems of equations (like elliptic equations, or compressible and incompressible fluid dynamics problems). Other methods exist within ML but are not described in this document. Examples are given illustrating the problem definition and exercising multigrid options.« less

ML 3.1 smoothed aggregation user's guide.

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

Sala, Marzio; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen

2004-10-01

ML is a multigrid preconditioning package intended to solve linear systems of equations Ax = b where A is a user supplied n x n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. ML should be used on large sparse linear systems arising from partial differential equation (PDE) discretizations. While technically any linear system can be considered, ML should be used on linear systems that correspond to things that work well with multigrid methods (e.g. elliptic PDEs). ML can be used as a stand-alone package ormore » to generate preconditioners for a traditional iterative solver package (e.g. Krylov methods). We have supplied support for working with the Aztec 2.1 and AztecOO iterative package [16]. However, other solvers can be used by supplying a few functions. This document describes one specific algebraic multigrid approach: smoothed aggregation. This approach is used within several specialized multigrid methods: one for the eddy current formulation for Maxwell's equations, and a multilevel and domain decomposition method for symmetric and nonsymmetric systems of equations (like elliptic equations, or compressible and incompressible fluid dynamics problems). Other methods exist within ML but are not described in this document. Examples are given illustrating the problem definition and exercising multigrid options.« less

Huygens-Fresnel picture for electron-molecule elastic scattering★

NASA Astrophysics Data System (ADS)

Baltenkov, Arkadiy S.; Msezane, Alfred Z.

2017-11-01

The elastic scattering cross sections for a slow electron by C2 and H2 molecules have been calculated within the framework of the non-overlapping atomic potential model. For the amplitudes of the multiple electron scattering by a target the wave function of the molecular continuum is represented as a combination of a plane wave and two spherical waves generated by the centers of atomic spheres. This wave function obeys the Huygens-Fresnel principle according to which the electron wave scattering by a system of two centers is accompanied by generation of two spherical waves; their interaction creates a diffraction pattern far from the target. Each of the Huygens waves, in turn, is a superposition of the partial spherical waves with different orbital angular momenta l and their projections m. The amplitudes of these partial waves are defined by the corresponding phases of electron elastic scattering by an isolated atomic potential. In numerical calculations the s- and p-phase shifts are taken into account. So the number of interfering electron waves is equal to eight: two of which are the s-type waves and the remaining six waves are of the p-type with different m values. The calculation of the scattering amplitudes in closed form (rather than in the form of S-matrix expansion) is reduced to solving a system of eight inhomogeneous algebraic equations. The differential and total cross sections of electron scattering by fixed-in-space molecules and randomly oriented ones have been calculated as well. We conclude by discussing the special features of the S-matrix method for the case of arbitrary non-spherical potentials. Contribution to the Topical Issue "Low energy positron and electron interactions", edited by James Sullivan, Ron White, Michael Bromley, Ilya Fabrikant, and David Cassidy.

In AppreciationThe Depth and Breadth of John Bell's Physics

NASA Astrophysics Data System (ADS)

Jackiw, Roman; Shimony, Abner

This essay surveys the work of John Stewart Bell, one of the great physicists of the twentieth century. Section 1 is a brief biography, tracing his career from working-class origins and undergraduate training in Belfast, Northern Ireland, to research in accelerator and nuclear physics in the British national laboratories at Harwell and Malvern, to his profound research on elementary particle physics as a member of the Theory Group at CERN and his equally profound ``hobby'' of investigating the foundations of quantum mechanics. Section 2 concerns this hobby, which began in his discontent with Bohr's and Heisenberg's analyses of the measurement process. He was attracted to the program of hidden variables interpretations, but he revolutionized the foundations of quantum mechanics by a powerful negative result: that no hidden variables theory that is ``local'' (in a clear and well-motivated sense) can agree with all the correlations predicted by quantum mechanics regarding well-separated systems. He further deepened the foundations of quantum mechanics by penetrating conceptual analyses of results concerning measurement theory of von Neumann, de Broglie and Bohm, Gleason, Jauch and Piron, Everett, and Ghirardi-Rimini-Weber. Bell's work in particle theory (Section 3) began with a proof of the CPT theorem in his doctoral dissertation, followed by investigations of the phenomenology of CP-violating experiments. At CERN Bell investigated the commutation relations in current algebras from various standpoints. The failure of current algebra combined with partially conserved current algebra to permit the experimentally observed decay of the neutral pi-meson into two photons stimulated the discovery by Bell and Jackiw of anomalous or quantal symmetry breaking, which has numerous implications for elementary particle phenomena. Other late investigations of Bell on elementary particle physics were bound states in quantum chromodynamics (in collaboration with Bertlmann) and estimates for the anomalous magnetic moment of the muon (in collaboration with de Rafael). Section 4 concerns accelerations, starting at Harwell with the algebra of strong focusing and the stability of orbits in linear accelerators and synchrotrons. At CERN he continued to contribute to accelerator physics, and with his wife Mary Bell he wrote on electron cooling and Beamstrahlung. A spectacular late achievement in accelerator physics was the demonstration (in collaboration with Leinaas) that the effective black-body radiation seen by an accelerated observer in an electromagnetic vacuum - the ``Unruh effect''- had already been observed experimentally in the partial depolarization of electrons traversing circular orbits.

Mathematical Modelling of Continuous Biotechnological Processes

ERIC Educational Resources Information Center

Pencheva, T.; Hristozov, I.; Shannon, A. G.

2003-01-01

Biotechnological processes (BTP) are characterized by a complicated structure of organization and interdependent characteristics. Partial differential equations or systems of partial differential equations are used for their behavioural description as objects with distributed parameters. Modelling of substrate without regard to dispersion…

Fault Tolerant Optimal Control.

DTIC Science & Technology

1982-08-01

subsystem is modelled by deterministic or stochastic finite-dimensional vector differential or difference equations. The parameters of these equations...is no partial differential equation that must be solved. Thus we can sidestep the inability to solve the Bellman equation for control problems with x...transition models and cost functionals can be reduced to the search for solutions of nonlinear partial differential equations using ’verification

Differential geometry techniques for sets of nonlinear partial differential equations

NASA Technical Reports Server (NTRS)

Estabrook, Frank B.

1990-01-01

An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.

A procedure to construct exact solutions of nonlinear fractional differential equations.

PubMed

Güner, Özkan; Cevikel, Adem C

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

NASA Astrophysics Data System (ADS)

Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.

2017-04-01

This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.

Linear or linearizable first-order delay ordinary differential equations and their Lie point symmetries

NASA Astrophysics Data System (ADS)

Dorodnitsyn, Vladimir A.; Kozlov, Roman; Meleshko, Sergey V.; Winternitz, Pavel

2018-05-01

A recent article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs, which have infinite-dimensional Lie point symmetry groups due to the linear superposition principle. Their symmetry algebra always contains a two-dimensional subalgebra realized by linearly connected vector fields. We identify all classes of linear first-order DODSs that have additional symmetries, not due to linearity alone, and we present representatives of each class. These additional symmetries are then used to construct exact analytical particular solutions using symmetry reduction.

How to differentiate collective variables in free energy codes: Computer-algebra code generation and automatic differentiation

NASA Astrophysics Data System (ADS)

Giorgino, Toni

2018-07-01

The proper choice of collective variables (CVs) is central to biased-sampling free energy reconstruction methods in molecular dynamics simulations. The PLUMED 2 library, for instance, provides several sophisticated CV choices, implemented in a C++ framework; however, developing new CVs is still time consuming due to the need to provide code for the analytical derivatives of all functions with respect to atomic coordinates. We present two solutions to this problem, namely (a) symbolic differentiation and code generation, and (b) automatic code differentiation, in both cases leveraging open-source libraries (SymPy and Stan Math, respectively). The two approaches are demonstrated and discussed in detail implementing a realistic example CV, the local radius of curvature of a polymer. Users may use the code as a template to streamline the implementation of their own CVs using high-level constructs and automatic gradient computation.

Proposed method to construct Boolean functions with maximum possible annihilator immunity

NASA Astrophysics Data System (ADS)

Goyal, Rajni; Panigrahi, Anupama; Bansal, Rohit

2017-07-01

Nonlinearity and Algebraic(annihilator) immunity are two core properties of a Boolean function because optimum values of Annihilator Immunity and nonlinearity are required to resist fast algebraic attack and differential cryptanalysis respectively. For a secure cypher system, Boolean function(S-Boxes) should resist maximum number of attacks. It is possible if a Boolean function has optimal trade-off among its properties. Before constructing Boolean functions, we fixed the criteria of our constructions based on its properties. In present work, our construction is based on annihilator immunity and nonlinearity. While keeping above facts in mind,, we have developed a multi-objective evolutionary approach based on NSGA-II and got the optimum value of annihilator immunity with good bound of nonlinearity. We have constructed balanced Boolean functions having the best trade-off among balancedness, Annihilator immunity and nonlinearity for 5, 6 and 7 variables by the proposed method.

Numerical stability in problems of linear algebra.

NASA Technical Reports Server (NTRS)

Babuska, I.

1972-01-01

Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.

Nonlinear analysis of 0-3 polarized PLZT microplate based on the new modified couple stress theory

NASA Astrophysics Data System (ADS)

Wang, Liming; Zheng, Shijie

2018-02-01

In this study, based on the new modified couple stress theory, the size- dependent model for nonlinear bending analysis of a pure 0-3 polarized PLZT plate is developed for the first time. The equilibrium equations are derived from a variational formulation based on the potential energy principle and the new modified couple stress theory. The Galerkin method is adopted to derive the nonlinear algebraic equations from governing differential equations. And then the nonlinear algebraic equations are solved by using Newton-Raphson method. After simplification, the new model includes only a material length scale parameter. In addition, numerical examples are carried out to study the effect of material length scale parameter on the nonlinear bending of a simply supported pure 0-3 polarized PLZT plate subjected to light illumination and uniform distributed load. The results indicate the new model is able to capture the size effect and geometric nonlinearity.

Optimal moving grids for time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Wathen, A. J.

1992-01-01

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of PDE solutions in the least-squares norm are reported.

Constructing general partial differential equations using polynomial and neural networks.

PubMed

Zjavka, Ladislav; Pedrycz, Witold

2016-01-01

Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems. Copyright © 2015 Elsevier Ltd. All rights reserved.

Differential characters and cohomology of the moduli of flat connections

NASA Astrophysics Data System (ADS)

Castrillón López, Marco; Ferreiro Pérez, Roberto

2018-05-01

Let π {:} P→ M be a principal bundle and p an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern-Simons differential characters is exploited to define a homology map χ k {:} H_{2r-k-1}(M)× Hk(F/G)→ R/Z , for k

A splitting algorithm for the wavelet transform of cubic splines on a nonuniform grid

NASA Astrophysics Data System (ADS)

Sulaimanov, Z. M.; Shumilov, B. M.

2017-10-01

For cubic splines with nonuniform nodes, splitting with respect to the even and odd nodes is used to obtain a wavelet expansion algorithm in the form of the solution to a three-diagonal system of linear algebraic equations for the coefficients. Computations by hand are used to investigate the application of this algorithm for numerical differentiation. The results are illustrated by solving a prediction problem.

On the huge Lie superalgebra of pseudo-superdifferential operators and super KP-hierarchies

NASA Astrophysics Data System (ADS)

Sedra, M. B.

1996-07-01

Lie superalgebraic methods are used to establish a connection between the huge Lie superalgebra Ξ of super- (pseudo-) differential operators and various super KP-hierarchies. We show in particular that Ξ splits into 5=2×2+1 graded algebras expected to correspond to five classes of super-KP-hierarchies generalizing the well-known Manin-Radul and Figueroa-Mas-Ramos supersymmetric KP-hierarchies.

Degrees of Freedom: Diversifying Math Requirements for College Readiness and Graduation (Report 1 of a 3-Part Series)

ERIC Educational Resources Information Center

Burdman, Pamela

2015-01-01

Since the mid-20th century, the standard U.S. high school and college math curriculum has been based on two years of algebra and a year of geometry, preparing students to take classes in pre-calculus followed by calculus. Students' math pursuits have been differentiated primarily by how far or how rapidly they proceed along a clearly defined…

Continuous analog of multiplicative algebraic reconstruction technique for computed tomography

NASA Astrophysics Data System (ADS)

Tateishi, Kiyoko; Yamaguchi, Yusaku; Abou Al-Ola, Omar M.; Kojima, Takeshi; Yoshinaga, Tetsuya

2016-03-01

We propose a hybrid dynamical system as a continuous analog to the block-iterative multiplicative algebraic reconstruction technique (BI-MART), which is a well-known iterative image reconstruction algorithm for computed tomography. The hybrid system is described by a switched nonlinear system with a piecewise smooth vector field or differential equation and, for consistent inverse problems, the convergence of non-negatively constrained solutions to a globally stable equilibrium is guaranteed by the Lyapunov theorem. Namely, we can prove theoretically that a weighted Kullback-Leibler divergence measure can be a common Lyapunov function for the switched system. We show that discretizing the differential equation by using the first-order approximation (Euler's method) based on the geometric multiplicative calculus leads to the same iterative formula of the BI-MART with the scaling parameter as a time-step of numerical discretization. The present paper is the first to reveal that a kind of iterative image reconstruction algorithm is constructed by the discretization of a continuous-time dynamical system for solving tomographic inverse problems. Iterative algorithms with not only the Euler method but also the Runge-Kutta methods of lower-orders applied for discretizing the continuous-time system can be used for image reconstruction. A numerical example showing the characteristics of the discretized iterative methods is presented.

Quasipolynomial generalization of Lotka-Volterra mappings

NASA Astrophysics Data System (ADS)

Hernández-Bermejo, Benito; Brenig, Léon

2002-07-01

In recent years, it has been shown that Lotka-Volterra mappings constitute a valuable tool from both the theoretical and the applied points of view, with developments in very diverse fields such as physics, population dynamics, chemistry and economy. The purpose of this work is to demonstrate that many of the most important ideas and algebraic methods that constitute the basis of the quasipolynomial formalism (originally conceived for the analysis of ordinary differential equations) can be extended into the mapping domain. The extension of the formalism into the discrete-time context is remarkable as far as the quasipolynomial methodology had never been shown to be applicable beyond the differential case. It will be demonstrated that Lotka-Volterra mappings play a central role in the quasipolynomial formalism for the discrete-time case. Moreover, the extension of the formalism into the discrete-time domain allows a significant generalization of Lotka-Volterra mappings as well as a whole transfer of algebraic methods into the discrete-time context. The result is a novel and more general conceptual framework for the understanding of Lotka-Volterra mappings as well as a new range of possibilities that become open not only for the theoretical analysis of Lotka-Volterra mappings and their generalizations, but also for the development of new applications.

On new classes of solutions of nonlinear partial differential equations in the form of convergent special series

NASA Astrophysics Data System (ADS)

Filimonov, M. Yu.

2017-12-01

The method of special series with recursively calculated coefficients is used to solve nonlinear partial differential equations. The recurrence of finding the coefficients of the series is achieved due to a special choice of functions, in powers of which the solution is expanded in a series. We obtain a sequence of linear partial differential equations to find the coefficients of the series constructed. In many cases, one can deal with a sequence of linear ordinary differential equations. We construct classes of solutions in the form of convergent series for a certain class of nonlinear evolution equations. A new class of solutions of generalized Boussinesque equation with an arbitrary function in the form of a convergent series is constructed.

A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations

PubMed Central

Güner, Özkan; Cevikel, Adem C.

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions. PMID:24737972

On implicit abstract neutral nonlinear differential equations

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

Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br; O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie

2016-04-15

In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.

Oscillation of certain higher-order neutral partial functional differential equations.

PubMed

Li, Wei Nian; Sheng, Weihong

2016-01-01

In this paper, we study the oscillation of certain higher-order neutral partial functional differential equations with the Robin boundary conditions. Some oscillation criteria are established. Two examples are given to illustrate the main results in the end of this paper.

On generalized Melvin solution for the Lie algebra E_6

NASA Astrophysics Data System (ADS)

Bolokhov, S. V.; Ivashchuk, V. D.

2017-10-01

A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra G is considered. The gravitational model in D dimensions, D ≥ 4, contains n 2-forms and l ≥ n scalar fields, where n is the rank of G. The solution is governed by a set of n functions H_s(z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). The polynomials H_s(z), s = 1,\\ldots ,6, for the Lie algebra E_6 are obtained and a corresponding solution for l = n = 6 is presented. The polynomials depend upon integration constants Q_s, s = 1,\\ldots ,6. They obey symmetry and duality identities. The latter ones are used in deriving asymptotic relations for solutions at large distances. The power-law asymptotic relations for E_6-polynomials at large z are governed by the integer-valued matrix ν = A^{-1} (I + P), where A^{-1} is the inverse Cartan matrix, I is the identity matrix and P is a permutation matrix, corresponding to a generator of the Z_2-group of symmetry of the Dynkin diagram. The 2-form fluxes Φ ^s, s = 1,\\ldots ,6, are calculated.

Newton's method: A link between continuous and discrete solutions of nonlinear problems

NASA Technical Reports Server (NTRS)

Thurston, G. A.

1980-01-01

Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions.

Pseudo almost periodic solutions to some systems of nonlinear hyperbolic second-order partial differential equations

NASA Astrophysics Data System (ADS)

Al-Islam, Najja Shakir

In this Dissertation, the existence of pseudo almost periodic solutions to some systems of nonlinear hyperbolic second-order partial differential equations is established. For that, (Al-Islam [4]) is first studied and then obtained under some suitable assumptions. That is, the existence of pseudo almost periodic solutions to a hyperbolic second-order partial differential equation with delay. The second-order partial differential equation (1) represents a mathematical model for the dynamics of gas absorption, given by uxt+a x,tux=Cx,t,u x,t , u0,t=4 t, 1 where a : [0, L] x RR , C : [0, L] x R x RR , and ϕ : RR are (jointly) continuous functions ( t being the greatest integer function) and L > 0. The results in this Dissertation generalize those of Poorkarimi and Wiener [22]. Secondly, a generalization of the above-mentioned system consisting of the non-linear hyperbolic second-order partial differential equation uxt+a x,tux+bx,t ut+cx,tu=f x,t,u, x∈ 0,L,t∈ R, 2 equipped with the boundary conditions ux,0 =40x, u0,t=u 0t, uxx,0=y 0x, x∈0,L, t∈R, 3 where a, b, c : [0, L ] x RR and f : [0, L] x R x RR are (jointly) continuous functions is studied. Under some suitable assumptions, the existence and uniqueness of pseudo almost periodic solutions to particular cases, as well as the general case of the second-order hyperbolic partial differential equation (2) are studied. The results of all studies contained within this text extend those obtained by Aziz and Meyers [6] in the periodic setting.

Reversible effects of oxygen partial pressure on genes associated with placental angiogenesis and differentiation in primary-term cytotrophoblast cell culture.

PubMed

Debiève, F; Depoix, C; Gruson, D; Hubinont, C

2013-09-01

Timely regulated changes in oxygen partial pressure are important for placental formation. Disturbances could be responsible for pregnancy-related diseases like preeclampsia and intrauterine growth restriction. We aimed to (i) determine the effect of oxygen partial pressure on cytotrophoblast differentiation; (ii) measure mRNA expression and protein secretion from genes associated with placental angiogenesis; and (iii) determine the reversibility of these effects at different oxygen partial pressures. Term cytotrophoblasts were incubated at 21% and 2.5% O2 for 96 hr, or were switched between the two oxygen concentrations after 48 hr. Real-time PCR and enzyme-linked immunosorbent assays (ELISAs) were used to evaluate cell fusion and differentiation, measuring transcript levels for those genes involved in cell fusion and placental angiogenesis, including VEGF, PlGF, VEGFR1, sVEGFR1, sENG, INHA, and GCM1. Cytotrophoblasts underwent fusion and differentiation in 2.5% O2 . PlGF expression was inhibited while sVEGFR1 expression increased. VEGF and sENG mRNA expressions increased in 2.5% compared to 21% O2 , but no protein was detected in the cell supernatants. Finally, GCM1 mRNA expression increased during trophoblast differentiation at 21% O2 , but was inhibited at 2.5% O2 . These mRNA expression effects were reversed by returning the cells to 21% O2 . Thus, low-oxygen partial pressure does not inhibit term-cytotrophoblast cell fusion and differentiation in vitro. Lowering the oxygen partial pressure from 21% to 2.5% caused normal-term trophoblasts to reversibly modify their expression of genes associated with placental angiogenesis. This suggests that modifications observed in pregnancy diseases such as preeclampsia or growth retardation are probably due to an extrinsic effect on trophoblasts. Copyright © 2013 Wiley Periodicals, Inc.

Lattice Boltzmann model for high-order nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Lattice Boltzmann model for high-order nonlinear partial differential equations.

PubMed

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

On the classification of scalar evolution equations with non-constant separant

NASA Astrophysics Data System (ADS)

Hümeyra Bilge, Ayşe; Mizrahi, Eti

2017-01-01

The ‘separant’ of the evolution equation u t   =  F, where F is some differentiable function of the derivatives of u up to order m, is the partial derivative \\partial F/\\partial {{u}m}, where {{u}m}={{\\partial}m}u/\\partial {{x}m} . As an integrability test, we use the formal symmetry method of Mikhailov-Shabat-Sokolov, which is based on the existence of a recursion operator as a formal series. The solvability of its coefficients in the class of local functions gives a sequence of conservation laws, called the ‘conserved densities’ {ρ(i)}, i=-1,1,2,3,\\ldots . We apply this method to the classification of scalar evolution equations of orders 3≤slant m≤slant 15 , for which {ρ(-1)}={≤ft[\\partial F/\\partial {{u}m}\\right]}-1/m} and {{ρ(1)} are non-trivial, i.e. they are not total derivatives and {ρ(-1)} is not linear in its highest order derivative. We obtain the ‘top level’ parts of these equations and their ‘top dependencies’ with respect to the ‘level grading’, that we defined in a previous paper, as a grading on the algebra of polynomials generated by the derivatives u b+i , over the ring of {{C}∞} functions of u,{{u}1},\\ldots,{{u}b} . In this setting b and i are called ‘base’ and ‘level’, respectively. We solve the conserved density conditions to show that if {ρ(-1)} depends on u,{{u}1},\\ldots,{{u}b}, then, these equations are level homogeneous polynomials in {{u}b+i},\\ldots,{{u}m} , i≥slant 1 . Furthermore, we prove that if {ρ(3)} is non-trivial, then {ρ(-1)}={≤ft(α ub2+β {{u}b}+γ \\right)}1/2} , with b≤slant 3 while if {{ρ(3)} is trivial, then {ρ(-1)}={≤ft(λ {{u}b}+μ \\right)}1/3} , where b≤slant 5 and α, β, γ, λ and μ are functions of u,\\ldots,{{u}b-1} . We show that the equations that we obtain form commuting flows and we construct their recursion operators that are respectively of orders 2 and 6 for non-trivial and trivial {{ρ(3)} respectively. Omitting lower order dependencies, we show that equations with non-trivial {ρ(3)} and b  =  3 are symmetries of the ‘essentially non-linear third order equation’ for trivial {ρ(3)} , the equations with b  =  5 are symmetries of a non-quasilinear fifth order equation obtained in previous work, while for b  =  3, 4 they are symmetries of quasilinear fifth order equations.

PREFACE: Symmetries and integrability of difference equations Symmetries and integrability of difference equations

NASA Astrophysics Data System (ADS)

Levi, Decio; Olver, Peter; Thomova, Zora; Winternitz, Pavel

2009-11-01

The concept of integrability was introduced in classical mechanics in the 19th century for finite dimensional continuous Hamiltonian systems. It was extended to certain classes of nonlinear differential equations in the second half of the 20th century with the discovery of the inverse scattering transform and the birth of soliton theory. Also at the end of the 19th century Lie group theory was invented as a powerful tool for obtaining exact analytical solutions of large classes of differential equations. Together, Lie group theory and integrability theory in its most general sense provide the main tools for solving nonlinear differential equations. Like differential equations, difference equations play an important role in physics and other sciences. They occur very naturally in the description of phenomena that are genuinely discrete. Indeed, they may actually be more fundamental than differential equations if space-time is actually discrete at very short distances. On the other hand, even when treating continuous phenomena described by differential equations it is very often necessary to resort to numerical methods. This involves a discretization of the differential equation, i.e. a replacement of the differential equation by a difference one. Given the well developed and understood techniques of symmetry and integrability for differential equations a natural question to ask is whether it is possible to develop similar techniques for difference equations. The aim is, on one hand, to obtain powerful methods for solving `integrable' difference equations and to establish practical integrability criteria, telling us when the methods are applicable. On the other hand, Lie group methods can be adapted to solve difference equations analytically. Finally, integrability and symmetry methods can be combined with numerical methods to obtain improved numerical solutions of differential equations. The origin of the SIDE meetings goes back to the early 1990s and the first meeting with the name `Symmetries and Integrability of Discrete Equations (SIDE)' was held in Estérel, Québec, Canada. This was organized by D Levi, P Winternitz and L Vinet. After the success of the first meeting the scientific community decided to hold bi-annual SIDE meetings. They were held in 1996 at the University of Kent (UK), 1998 in Sabaudia (Italy), 2000 at the University of Tokyo (Japan), 2002 in Giens (France), 2004 in Helsinki (Finland) and in 2006 at the University of Melbourne (Australia). In 2008 the SIDE 8 meeting was again organized near Montreal, in Ste-Adèle, Québec, Canada. The SIDE 8 International Advisory Committee (also the SIDE steering committee) consisted of Frank Nijhoff, Alexander Bobenko, Basil Grammaticos, Jarmo Hietarinta, Nalini Joshi, Decio Levi, Vassilis Papageorgiou, Junkichi Satsuma, Yuri Suris, Claude Vialet and Pavel Winternitz. The local organizing committee consisted of Pavel Winternitz, John Harnad, Véronique Hussin, Decio Levi, Peter Olver and Luc Vinet. Financial support came from the Centre de Recherches Mathématiques in Montreal and the National Science Foundation (through the University of Minnesota). Proceedings of the first three SIDE meetings were published in the LMS Lecture Note series. Since 2000 the emphasis has been on publishing selected refereed articles in response to a general call for papers issued after the conference. This allows for a wider author base, since the call for papers is not restricted to conference participants. The SIDE topics thus are represented in special issues of Journal of Physics A: Mathematical and General 34 (48) and Journal of Physics A: Mathematical and Theoretical, 40 (42) (SIDE 4 and SIDE 7, respectively), Journal of Nonlinear Mathematical Physics 10 (Suppl. 2) and 12 (Suppl. 2) (SIDE 5 and SIDE 6 respectively). The SIDE 8 meeting was organized around several topics and the contributions to this special issue reflect the diversity presented during the meeting. The papers presented at the SIDE 8 meeting were organized into the following special sessions: geometry of discrete and continuous Painlevé equations; continuous symmetries of discrete equations—theory and computational applications; algebraic aspects of discrete equations; singularity confinement, algebraic entropy and Nevanlinna theory; discrete differential geometry; discrete integrable systems and isomonodromy transformations; special functions as solutions of difference and q-difference equations. This special issue of the journal is organized along similar lines. The first three articles are topical review articles appearing in alphabetical order (by first author). The article by Doliwa and Nieszporski describes the Darboux transformations in a discrete setting, namely for the discrete second order linear problem. The article by Grammaticos, Halburd, Ramani and Viallet concentrates on the integrability of the discrete systems, in particular they describe integrability tests for difference equations such as singularity confinement, algebraic entropy (growth and complexity), and analytic and arithmetic approaches. The topical review by Konopelchenko explores the relationship between the discrete integrable systems and deformations of associative algebras. All other articles are presented in alphabetical order (by first author). The contributions were solicited from all participants as well as from the general scientific community. The contributions published in this special issue can be loosely grouped into several overlapping topics, namely: •Geometry of discrete and continuous Painlevé equations (articles by Spicer and Nijhoff and by Lobb and Nijhoff). •Continuous symmetries of discrete equations—theory and applications (articles by Dorodnitsyn and Kozlov; Levi, Petrera and Scimiterna; Scimiterna; Ste-Marie and Tremblay; Levi and Yamilov; Rebelo and Winternitz). •Yang--Baxter maps (article by Xenitidis and Papageorgiou). •Algebraic aspects of discrete equations (articles by Doliwa and Nieszporski; Konopelchenko; Tsarev and Wolf). •Singularity confinement, algebraic entropy and Nevanlinna theory (articles by Grammaticos, Halburd, Ramani and Viallet; Grammaticos, Ramani and Tamizhmani). •Discrete integrable systems and isomonodromy transformations (article by Dzhamay). •Special functions as solutions of difference and q-difference equations (articles by Atakishiyeva, Atakishiyev and Koornwinder; Bertola, Gekhtman and Szmigielski; Vinet and Zhedanov). •Other topics (articles by Atkinson; Grünbaum Nagai, Kametaka and Watanabe; Nagiyev, Guliyeva and Jafarov; Sahadevan and Uma Maheswari; Svinin; Tian and Hu; Yao, Liu and Zeng). This issue is the result of the collaboration of many individuals. We would like to thank the authors who contributed and everyone else involved in the preparation of this special issue.

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

Oxidation Behavior of Carbon Fiber-Reinforced Composites

NASA Technical Reports Server (NTRS)

Sullivan, Roy M.

2008-01-01

OXIMAP is a numerical (FEA-based) solution tool capable of calculating the carbon fiber and fiber coating oxidation patterns within any arbitrarily shaped carbon silicon carbide composite structure as a function of time, temperature, and the environmental oxygen partial pressure. The mathematical formulation is derived from the mechanics of the flow of ideal gases through a chemically reacting, porous solid. The result of the formulation is a set of two coupled, non-linear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations are solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of the spatial location and time. The local rate of carbon oxidation is determined at each time step using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The non-linear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual finite element method, allowing for the solution of the differential equations numerically.

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.

Differential phase measurements of D-region partial reflections

NASA Technical Reports Server (NTRS)

Wiersma, D. J.; Sechrist, C. F., Jr.

1972-01-01

Differential phase partial reflection measurements were used to deduce D region electron density profiles. The phase difference was measured by taking sums and differences of amplitudes received on an array of crossed dipoles. The reflection model used was derived from Fresnel reflection theory. Seven profiles obtained over the period from 13 October 1971 to 5 November 1971 are presented, along with the results from simultaneous measurements of differential absorption. Some possible sources of error and error propagation are discussed. A collision frequency profile was deduced from the electron concentration calculated from differential phase and differential absorption.

Quantized Detector Networks

NASA Astrophysics Data System (ADS)

Jaroszkiewicz, George

2017-12-01

Preface; Acronyms; 1. Introduction; 2. Questions and answers; 3. Classical bits; 4. Quantum bits; 5. Classical and quantum registers; 6. Classical register mechanics; 7. Quantum register dynamics; 8. Partial observations; 9. Mixed states and POVMs; 10. Double-slit experiments; 11. Modules; 12. Computerization and computer algebra; 13. Interferometers; 14. Quantum eraser experiments; 15. Particle decays; 16. Non-locality; 17. Bell inequalities; 18. Change and persistence; 19. Temporal correlations; 20. The Franson experiment; 21. Self-intervening networks; 22. Separability and entanglement; 23. Causal sets; 24. Oscillators; 25. Dynamical theory of observation; 26. Conclusions; Appendix; Index.