33 CFR 80.805 - Rock Island, FL to Cape San Blas, FL.

Code of Federal Regulations, 2014 CFR

2014-07-01

... 33 Navigation and Navigable Waters 1 2014-07-01 2014-07-01 false Rock Island, FL to Cape San Blas, FL. 80.805 Section 80.805 Navigation and Navigable Waters COAST GUARD, DEPARTMENT OF HOMELAND SECURITY INTERNATIONAL NAVIGATION RULES COLREGS DEMARCATION LINES Eighth District § 80.805 Rock Island, FL...

33 CFR 80.805 - Rock Island, FL to Cape San Blas, FL.

Code of Federal Regulations, 2010 CFR

2010-07-01

... 33 Navigation and Navigable Waters 1 2010-07-01 2010-07-01 false Rock Island, FL to Cape San Blas, FL. 80.805 Section 80.805 Navigation and Navigable Waters COAST GUARD, DEPARTMENT OF HOMELAND SECURITY INTERNATIONAL NAVIGATION RULES COLREGS DEMARCATION LINES Eighth District § 80.805 Rock Island, FL...

33 CFR 80.805 - Rock Island, FL to Cape San Blas, FL.

Code of Federal Regulations, 2012 CFR

2012-07-01

... 33 Navigation and Navigable Waters 1 2012-07-01 2012-07-01 false Rock Island, FL to Cape San Blas, FL. 80.805 Section 80.805 Navigation and Navigable Waters COAST GUARD, DEPARTMENT OF HOMELAND SECURITY INTERNATIONAL NAVIGATION RULES COLREGS DEMARCATION LINES Eighth District § 80.805 Rock Island, FL...

33 CFR 80.805 - Rock Island, FL to Cape San Blas, FL.

Code of Federal Regulations, 2013 CFR

2013-07-01

... 33 Navigation and Navigable Waters 1 2013-07-01 2013-07-01 false Rock Island, FL to Cape San Blas, FL. 80.805 Section 80.805 Navigation and Navigable Waters COAST GUARD, DEPARTMENT OF HOMELAND SECURITY INTERNATIONAL NAVIGATION RULES COLREGS DEMARCATION LINES Eighth District § 80.805 Rock Island, FL...

33 CFR 80.805 - Rock Island, FL to Cape San Blas, FL.

Code of Federal Regulations, 2011 CFR

2011-07-01

... 33 Navigation and Navigable Waters 1 2011-07-01 2011-07-01 false Rock Island, FL to Cape San Blas, FL. 80.805 Section 80.805 Navigation and Navigable Waters COAST GUARD, DEPARTMENT OF HOMELAND SECURITY INTERNATIONAL NAVIGATION RULES COLREGS DEMARCATION LINES Eighth District § 80.805 Rock Island, FL...

A VLT/MUSE galaxy survey towards QSO Q1410: looking for a WHIM traced by BLAs in inter-cluster filaments

NASA Astrophysics Data System (ADS)

Pessa, Ismael; Tejos, Nicolas; Barrientos, L. Felipe; Werk, Jessica; Bielby, Richard; Padilla, Nelson; Morris, Simon L.; Prochaska, J. Xavier; Lopez, Sebastian; Hummels, Cameron

2018-07-01

Cosmological simulations predict that a significant fraction of the low-z baryon budget resides in large-scale filaments in the form of a diffuse plasma at temperatures T ˜ 105 - 107 K. However, direct observation of this so-called warm-hot intergalactic medium (WHIM) has been elusive. In the Λcold dark matter paradigm, galaxy clusters correspond to the nodes of the cosmic web at the intersection of several large-scale filamentary threads. In previous work, we used HST/COS data to conduct the first survey of broad H I Lyα absorbers (BLAs) potentially produced by WHIM in inter-cluster filaments. We targeted a single QSO, namely Q1410, whose sightline intersects seven independent inter-cluster axes at impact parameters <3 Mpc (comoving), and found a tentative excess of a factor of ˜4 with respect to the field. Here, we further investigate the origin of these BLAs by performing a blind galaxy survey within the Q1410 field using VLT/MUSE. We identified 77 sources and obtained the redshifts for 52 of them. Out of the total sample of seven BLAs in inter-cluster axes, we found three without any galaxy counterpart to stringent luminosity limits (˜4 × 108 L⊙ ˜0.01 L*), providing further evidence that these BLAs may represent genuine WHIM detections. We combined this sample with other suitable BLAs from the literature and inferred the corresponding baryon mean density for these filaments in the range Ω ^fil_bar= 0.02-0.04. Our rough estimates are consistent with the predictions from numerical simulations but still subject to large systematic uncertainties, mostly from the adopted geometry, ionization corrections, and density profile.

33 CFR 334.670 - Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing...

Code of Federal Regulations, 2010 CFR

2010-07-01

... 33 Navigation and Navigable Waters 3 2010-07-01 2010-07-01 false Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing practice range, Tyndall Air Force Base, Fla. 334... Apalachicola, San Blas, and St. Joseph bays; air-to-air firing practice range, Tyndall Air Force Base, Fla. (a...

Extreme-Scale Algorithms & Software Resilience (EASIR) Architecture-Aware Algorithms for Scalable Performance and Resilience on Heterogeneous Architectures

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

Demmel, James W.

This project addresses both communication-avoiding algorithms, and reproducible floating-point computation. Communication, i.e. moving data, either between levels of memory or processors over a network, is much more expensive per operation than arithmetic (measured in time or energy), so we seek algorithms that greatly reduce communication. We developed many new algorithms for both dense and sparse, and both direct and iterative linear algebra, attaining new communication lower bounds, and getting large speedups in many cases. We also extended this work in several ways: (1) We minimize writes separately from reads, since writes may be much more expensive than reads on emergingmore » memory technologies, like Flash, sometimes doing asymptotically fewer writes than reads. (2) We extend the lower bounds and optimal algorithms to arbitrary algorithms that may be expressed as perfectly nested loops accessing arrays, where the array subscripts may be arbitrary affine functions of the loop indices (eg A(i), B(i,j+k, k+3*m-7, …) etc.). (3) We extend our communication-avoiding approach to some machine learning algorithms, such as support vector machines. This work has won a number of awards. We also address reproducible floating-point computation. We define reproducibility to mean getting bitwise identical results from multiple runs of the same program, perhaps with different hardware resources or other changes that should ideally not change the answer. Many users depend on reproducibility for debugging or correctness. However, dynamic scheduling of parallel computing resources, combined with nonassociativity of floating point addition, makes attaining reproducibility a challenge even for simple operations like summing a vector of numbers, or more complicated operations like the Basic Linear Algebra Subprograms (BLAS). We describe an algorithm that computes a reproducible sum of floating point numbers, independent of the order of summation. The algorithm depends only on

Libraries for Software Use on Peregrine | High-Performance Computing | NREL

Science.gov Websites

-specific libraries. Libraries List Name Description BLAS Basic Linear Algebra Subroutines, libraries only managing hierarchically structured data. LAPACK Standard Netlib offering for computational linear algebra

Parallelization of the FLAPW method and comparison with the PPW method

NASA Astrophysics Data System (ADS)

Canning, Andrew; Mannstadt, Wolfgang; Freeman, Arthur

2000-03-01

The FLAPW (full-potential linearized-augmented plane-wave) method is one of the most accurate first-principles methods for determining electronic and magnetic properties of crystals and surfaces. In the past the FLAPW method has been limited to systems of about a hundred atoms due to the lack of an efficient parallel implementation to exploit the power and memory of parallel computers. In this work we present an efficient parallelization of the method by division among the processors of the plane-wave components for each state. The code is also optimized for RISC (reduced instruction set computer) architectures, such as those found on most parallel computers, making full use of BLAS (basic linear algebra subprograms) wherever possible. Scaling results are presented for systems of up to 686 silicon atoms and 343 palladium atoms per unit cell running on up to 512 processors on a Cray T3E parallel supercomputer. Some results will also be presented on a comparison of the plane-wave pseudopotential method and the FLAPW method on large systems.

GENERAL PURPOSE ADA PACKAGES

NASA Technical Reports Server (NTRS)

Klumpp, A. R.

1994-01-01

Ten families of subprograms are bundled together for the General-Purpose Ada Packages. The families bring to Ada many features from HAL/S, PL/I, FORTRAN, and other languages. These families are: string subprograms (INDEX, TRIM, LOAD, etc.); scalar subprograms (MAX, MIN, REM, etc.); array subprograms (MAX, MIN, PROD, SUM, GET, and PUT); numerical subprograms (EXP, CUBIC, etc.); service subprograms (DATE_TIME function, etc.); Linear Algebra II; Runge-Kutta integrators; and three text I/O families of packages. In two cases, a family consists of a single non-generic package. In all other cases, a family comprises a generic package and its instances for a selected group of scalar types. All generic packages are designed to be easily instantiated for the types declared in the user facility. The linear algebra package is LINRAG2. This package includes subprograms supplementing those in NPO-17985, An Ada Linear Algebra Package Modeled After HAL/S (LINRAG). Please note that LINRAG2 cannot be compiled without LINRAG. Most packages have widespread applicability, although some are oriented for avionics applications. All are designed to facilitate writing new software in Ada. Several of the packages use conventions introduced by other programming languages. A package of string subprograms is based on HAL/S (a language designed for the avionics software in the Space Shuttle) and PL/I. Packages of scalar and array subprograms are taken from HAL/S or generalized current Ada subprograms. A package of Runge-Kutta integrators is patterned after a built-in MAC (MIT Algebraic Compiler) integrator. Those packages modeled after HAL/S make it easy to translate existing HAL/S software to Ada. The General-Purpose Ada Packages program source code is available on two 360K 5.25" MS-DOS format diskettes. The software was developed using VAX Ada v1.5 under DEC VMS v4.5. It should be portable to any validated Ada compiler and it should execute either interactively or in batch. The largest package

MATH77 - A LIBRARY OF MATHEMATICAL SUBPROGRAMS FOR FORTRAN 77, RELEASE 4.0

NASA Technical Reports Server (NTRS)

Lawson, C. L.

1994-01-01

MATH77 is a high quality library of ANSI FORTRAN 77 subprograms implementing contemporary algorithms for the basic computational processes of science and engineering. The portability of MATH77 meets the needs of present-day scientists and engineers who typically use a variety of computing environments. Release 4.0 of MATH77 contains 454 user-callable and 136 lower-level subprograms. Usage of the user-callable subprograms is described in 69 sections of the 416 page users' manual. The topics covered by MATH77 are indicated by the following list of chapter titles in the users' manual: Mathematical Functions, Pseudo-random Number Generation, Linear Systems of Equations and Linear Least Squares, Matrix Eigenvalues and Eigenvectors, Matrix Vector Utilities, Nonlinear Equation Solving, Curve Fitting, Table Look-Up and Interpolation, Definite Integrals (Quadrature), Ordinary Differential Equations, Minimization, Polynomial Rootfinding, Finite Fourier Transforms, Special Arithmetic , Sorting, Library Utilities, Character-based Graphics, and Statistics. Besides subprograms that are adaptations of public domain software, MATH77 contains a number of unique packages developed by the authors of MATH77. Instances of the latter type include (1) adaptive quadrature, allowing for exceptional generality in multidimensional cases, (2) the ordinary differential equations solver used in spacecraft trajectory computation for JPL missions, (3) univariate and multivariate table look-up and interpolation, allowing for "ragged" tables, and providing error estimates, and (4) univariate and multivariate derivative-propagation arithmetic. MATH77 release 4.0 is a subroutine library which has been carefully designed to be usable on any computer system that supports the full ANSI standard FORTRAN 77 language. It has been successfully implemented on a CRAY Y/MP computer running UNICOS, a UNISYS 1100 computer running EXEC 8, a DEC VAX series computer running VMS, a Sun4 series computer running Sun

Bleaching of reef coelenterates in the San Blas Islands, Panama

NASA Astrophysics Data System (ADS)

Lasker, Howard R.; Peters, Esther C.; Coffroth, Mary Alice

1984-12-01

Starting in June 1983, 25 species of hermatypic corals, gorgonians, hydrocorals, anemones and zoanthids in the San Blas Islands, Panama, began showing signs of a loss of colour leading in some cases to a white “bleached” appearance. Histologic examination of six coral species indicated that bleaching was associated with drastic reductions in the density of zooxanthellae and with the atrophy and necrosis of the animal tissue. The severity of the bleaching varied among species and many species were unaffected. The species most extensively affected were: Agaricia spp., which became completely bleached and frequently died; Montastraea annularis which bleached and continued to survive; and Millepora spp. which bleached white but quickly regained their colouration. Shallow reefs dominated by Agaricia spp. suffered the most extensive bleaching. At one site, Pico Feo, 99% of the Agaricia (32% of the living cover) was bleached. On fore reers, which were dominated by Agaricia spp. and M. annularis, the proportion of M. annularis bleached ranged from 18 to 100% and that of Agaricia spp. from 30 to 53%. Transects at Sail Rock and House Reef were surveyed in August 1983 and January 1984. At those sites, 53% of the Agaricia cover died between August and January. The remaining living cover of Agaricia and of all other species exhibited normal colouration in January. Salinity and temperature were monitored every second day at 4 m depth between May 10 and August 28, 1983 at one of the localities. Bleaching was first observed within two weeks of a 2 °C rise in temperature which occurred in late May 1983. Temperatures remained at or above 31.5 °C for the following 3 weeks and were at or above 30 °C for an additional 4 weeks. The bleaching of corals in the San Blas was most likely due to those elevanted temperatures.

GPU Linear Algebra Libraries and GPGPU Programming for Accelerating MOPAC Semiempirical Quantum Chemistry Calculations.

PubMed

Maia, Julio Daniel Carvalho; Urquiza Carvalho, Gabriel Aires; Mangueira, Carlos Peixoto; Santana, Sidney Ramos; Cabral, Lucidio Anjos Formiga; Rocha, Gerd B

2012-09-11

In this study, we present some modifications in the semiempirical quantum chemistry MOPAC2009 code that accelerate single-point energy calculations (1SCF) of medium-size (up to 2500 atoms) molecular systems using GPU coprocessors and multithreaded shared-memory CPUs. Our modifications consisted of using a combination of highly optimized linear algebra libraries for both CPU (LAPACK and BLAS from Intel MKL) and GPU (MAGMA and CUBLAS) to hasten time-consuming parts of MOPAC such as the pseudodiagonalization, full diagonalization, and density matrix assembling. We have shown that it is possible to obtain large speedups just by using CPU serial linear algebra libraries in the MOPAC code. As a special case, we show a speedup of up to 14 times for a methanol simulation box containing 2400 atoms and 4800 basis functions, with even greater gains in performance when using multithreaded CPUs (2.1 times in relation to the single-threaded CPU code using linear algebra libraries) and GPUs (3.8 times). This degree of acceleration opens new perspectives for modeling larger structures which appear in inorganic chemistry (such as zeolites and MOFs), biochemistry (such as polysaccharides, small proteins, and DNA fragments), and materials science (such as nanotubes and fullerenes). In addition, we believe that this parallel (GPU-GPU) MOPAC code will make it feasible to use semiempirical methods in lengthy molecular simulations using both hybrid QM/MM and QM/QM potentials.

Matrix Algebra for GPU and Multicore Architectures (MAGMA) for Large Petascale Systems

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

Dongarra, Jack J.; Tomov, Stanimire

2014-03-24

The goal of the MAGMA project is to create a new generation of linear algebra libraries that achieve the fastest possible time to an accurate solution on hybrid Multicore+GPU-based systems, using all the processing power that future high-end systems can make available within given energy constraints. Our efforts at the University of Tennessee achieved the goals set in all of the five areas identified in the proposal: 1. Communication optimal algorithms; 2. Autotuning for GPU and hybrid processors; 3. Scheduling and memory management techniques for heterogeneity and scale; 4. Fault tolerance and robustness for large scale systems; 5. Building energymore » efficiency into software foundations. The University of Tennessee’s main contributions, as proposed, were the research and software development of new algorithms for hybrid multi/many-core CPUs and GPUs, as related to two-sided factorizations and complete eigenproblem solvers, hybrid BLAS, and energy efficiency for dense, as well as sparse, operations. Furthermore, as proposed, we investigated and experimented with various techniques targeting the five main areas outlined.« less

Tropical Storm Blas off the Pacific Coast of Mexico

NASA Image and Video Library

2004-07-14

Tropical Storm Blas as observed by the Atmospheric Infrared Sounder AIRS onboard NASA Aqua in the year 2004. The major contribution to radiation (infrared light) that AIRS channels sense comes from different levels in the atmosphere, depending upon the channel wavelength. To create the movie, a set of AIRS channels were selected which probe the atmosphere at progressively deeper levels. If there were no clouds, the color in each frame would be nearly uniform until the Earth's surface is encountered. The tropospheric air temperature warms at a rate of 6 K (about 11 F) for each kilometer of descent toward the surface. Thus the colors would gradually change from cold to warm as the movie progresses. Clouds block the infrared radiation. Thus wherever there are clouds we can penetrate no deeper in infrared. The color remains fixed as the movie progresses, for that area of the image is "stuck" to the cloud top temperature. The coldest temperatures around 220 K (about -65 F) come from altitudes of about 10 miles. http://photojournal.jpl.nasa.gov/catalog/PIA00436

Automatic Blocking Of QR and LU Factorizations for Locality

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

Yi, Q; Kennedy, K; You, H

2004-03-26

QR and LU factorizations for dense matrices are important linear algebra computations that are widely used in scientific applications. To efficiently perform these computations on modern computers, the factorization algorithms need to be blocked when operating on large matrices to effectively exploit the deep cache hierarchy prevalent in today's computer memory systems. Because both QR (based on Householder transformations) and LU factorization algorithms contain complex loop structures, few compilers can fully automate the blocking of these algorithms. Though linear algebra libraries such as LAPACK provides manually blocked implementations of these algorithms, by automatically generating blocked versions of the computations, moremore » benefit can be gained such as automatic adaptation of different blocking strategies. This paper demonstrates how to apply an aggressive loop transformation technique, dependence hoisting, to produce efficient blockings for both QR and LU with partial pivoting. We present different blocking strategies that can be generated by our optimizer and compare the performance of auto-blocked versions with manually tuned versions in LAPACK, both using reference BLAS, ATLAS BLAS and native BLAS specially tuned for the underlying machine architectures.« less

33 CFR 334.670 - Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing...

Code of Federal Regulations, 2014 CFR

2014-07-01

... 33 Navigation and Navigable Waters 3 2014-07-01 2014-07-01 false Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing practice range, Tyndall Air Force Base, Fla. 334..., DEPARTMENT OF DEFENSE DANGER ZONE AND RESTRICTED AREA REGULATIONS § 334.670 Gulf of Mexico south and west of...

33 CFR 334.670 - Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing...

Code of Federal Regulations, 2011 CFR

2011-07-01

... 33 Navigation and Navigable Waters 3 2011-07-01 2011-07-01 false Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing practice range, Tyndall Air Force Base, Fla. 334..., DEPARTMENT OF DEFENSE DANGER ZONE AND RESTRICTED AREA REGULATIONS § 334.670 Gulf of Mexico south and west of...

33 CFR 334.670 - Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing...

Code of Federal Regulations, 2012 CFR

2012-07-01

... 33 Navigation and Navigable Waters 3 2012-07-01 2012-07-01 false Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing practice range, Tyndall Air Force Base, Fla. 334..., DEPARTMENT OF DEFENSE DANGER ZONE AND RESTRICTED AREA REGULATIONS § 334.670 Gulf of Mexico south and west of...

33 CFR 334.670 - Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing...

Code of Federal Regulations, 2013 CFR

2013-07-01

... 33 Navigation and Navigable Waters 3 2013-07-01 2013-07-01 false Gulf of Mexico south and west of Apalachicola, San Blas, and St. Joseph bays; air-to-air firing practice range, Tyndall Air Force Base, Fla. 334..., DEPARTMENT OF DEFENSE DANGER ZONE AND RESTRICTED AREA REGULATIONS § 334.670 Gulf of Mexico south and west of...

Quantum cluster algebras and quantum nilpotent algebras.

PubMed

Goodearl, Kenneth R; Yakimov, Milen T

2014-07-08

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein-Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405-455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337-397] for the case of symmetric Kac-Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1-52] associated with double Bruhat cells coincide with the corresponding cluster algebras.

Quantum cluster algebras and quantum nilpotent algebras

PubMed Central

Goodearl, Kenneth R.; Yakimov, Milen T.

2014-01-01

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197

Duncan F. Gregory, William Walton and the development of British algebra: 'algebraical geometry', 'geometrical algebra', abstraction.

PubMed

Verburgt, Lukas M

2016-01-01

This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.

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,…

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.

Banach Synaptic Algebras

NASA Astrophysics Data System (ADS)

Foulis, David J.; Pulmannov, Sylvia

2018-04-01

Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Størmer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C∗-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW∗-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.

The Strategic Technologies for Automation and Robotics (STEAR) program: Protection of materials in the space environment subprogram

NASA Technical Reports Server (NTRS)

Schmidt, Lorne R.; Francoeur, J.; Aguero, Alina; Wertheimer, Michael R.; Klemberg-Sapieha, J. E.; Martinu, L.; Blezius, J. W.; Oliver, M.; Singh, A.

1995-01-01

Three projects are currently underway for the development of new coatings for the protection of materials in the space environment. These coatings are based on vacuum deposition technologies. The projects will go as far as the proof-of-concept stage when the commercial potential for the technology will be demonstrated on pilot-scale fabrication facilities in 1996. These projects are part of a subprogram to develop supporting technologies for automation and robotics technologies being developed under the Canadian Space Agency's STEAR Program, part of the Canadian Space Station Program.

Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

NASA Astrophysics Data System (ADS)

Zhang, Tianjie; Gao, Xing; Guo, Li

2016-10-01

The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.

FPGA implementation of sparse matrix algorithm for information retrieval

NASA Astrophysics Data System (ADS)

Bojanic, Slobodan; Jevtic, Ruzica; Nieto-Taladriz, Octavio

2005-06-01

Information text data retrieval requires a tremendous amount of processing time because of the size of the data and the complexity of information retrieval algorithms. In this paper the solution to this problem is proposed via hardware supported information retrieval algorithms. Reconfigurable computing may adopt frequent hardware modifications through its tailorable hardware and exploits parallelism for a given application through reconfigurable and flexible hardware units. The degree of the parallelism can be tuned for data. In this work we implemented standard BLAS (basic linear algebra subprogram) sparse matrix algorithm named Compressed Sparse Row (CSR) that is showed to be more efficient in terms of storage space requirement and query-processing timing over the other sparse matrix algorithms for information retrieval application. Although inverted index algorithm is treated as the de facto standard for information retrieval for years, an alternative approach to store the index of text collection in a sparse matrix structure gains more attention. This approach performs query processing using sparse matrix-vector multiplication and due to parallelization achieves a substantial efficiency over the sequential inverted index. The parallel implementations of information retrieval kernel are presented in this work targeting the Virtex II Field Programmable Gate Arrays (FPGAs) board from Xilinx. A recent development in scientific applications is the use of FPGA to achieve high performance results. Computational results are compared to implementations on other platforms. The design achieves a high level of parallelism for the overall function while retaining highly optimised hardware within processing unit.

Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories

NASA Astrophysics Data System (ADS)

Cheng, Tao; Huang, Hua-Lin; Yang, Yuping

2016-01-01

By viewing Clifford algebras as algebras in some suitable symmetric Gr-categories, Albuquerque and Majid were able to give a new derivation of some well known results about Clifford algebras and to generalize them. Along the same line, Bulacu observed that Clifford algebras are weak Hopf algebras in the aforementioned categories and obtained other interesting properties. The aim of this paper is to study generalized Clifford algebras in a similar manner and extend the results of Albuquerque, Majid and Bulacu to the generalized setting. In particular, by taking full advantage of the gauge transformations in symmetric linear Gr-categories, we derive the decomposition theorem and provide categorical weak Hopf structures for generalized Clifford algebras in a conceptual and simpler manner.

Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction

ERIC Educational Resources Information Center

Wasserman, Nicholas H.

2016-01-01

This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…

Algebraic K-theory, K-regularity, and -duality of -stable C ∗-algebras

NASA Astrophysics Data System (ADS)

Mahanta, Snigdhayan

2015-12-01

We develop an algebraic formalism for topological -duality. More precisely, we show that topological -duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C ∗-algebra . Then we show that there is a functorial topological K-theory symmetric spectrum construction on the category of separable C ∗-algebras, such that is an algebra over this operad; moreover, is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C ∗-algebras. We also show that -stable C ∗-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of a x+ b-semigroup C ∗-algebras coming from number theory and that of -stabilized noncommutative tori.

Using Linear Algebra to Introduce Computer Algebra, Numerical Analysis, Data Structures and Algorithms (and To Teach Linear Algebra, Too).

ERIC Educational Resources Information Center

Gonzalez-Vega, Laureano

1999-01-01

Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)

Algebra for Everyone.

ERIC Educational Resources Information Center

Edwards, Edgar L., Jr., Ed.

The fundamentals of algebra and algebraic thinking should be a part of the background of all citizens in society. The vast increase in the use of technology requires that school mathematics ensure the teaching of algebraic thinking as well as its use at both the elementary and secondary school levels. Algebra is a universal theme that runs through…

Virasoro algebra in the KN algebra; Bosonic string with fermionic ghosts on Riemann surfaces

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

Koibuchi, H.

1991-10-10

In this paper the bosonic string model with fermionic ghosts is considered in the framework of the KN algebra. The authors' attentions are paid to representations of KN algebra and a Clifford algebra of the ghosts. The authors show that a Virasoro-like algebra is obtained from KN algebra when KN algebra has certain antilinear anti-involution, and that it is isomorphic to the usual Virasoro algebra. The authors show that there is an expected relation between a central charge of this Virasoro-like algebra and an anomaly of the combined system.

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.

Generalizing the bms3 and 2D-conformal algebras by expanding the Virasoro algebra

NASA Astrophysics Data System (ADS)

Caroca, Ricardo; Concha, Patrick; Rodríguez, Evelyn; Salgado-Rebolledo, Patricio

2018-03-01

By means of the Lie algebra expansion method, the centrally extended conformal algebra in two dimensions and the bms3 algebra are obtained from the Virasoro algebra. We extend this result to construct new families of expanded Virasoro algebras that turn out to be infinite-dimensional lifts of the so-called Bk, Ck and Dk algebras recently introduced in the literature in the context of (super)gravity. We also show how some of these new infinite-dimensional symmetries can be obtained from expanded Kač-Moody algebras using modified Sugawara constructions. Applications in the context of three-dimensional gravity are briefly discussed.

Quiver W-algebras

NASA Astrophysics Data System (ADS)

Kimura, Taro; Pestun, Vasily

2018-06-01

For a quiver with weighted arrows, we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al. and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds-Kac-Moody Lie algebras, their quantum affinizations and associated W-algebras.

Computer algebra and operators

NASA Technical Reports Server (NTRS)

Fateman, Richard; Grossman, Robert

1989-01-01

The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.

Abstract Algebra to Secondary School Algebra: Building Bridges

ERIC Educational Resources Information Center

Christy, Donna; Sparks, Rebecca

2015-01-01

The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…

Making Algebra Work: Instructional Strategies that Deepen Student Understanding, within and between Algebraic Representations

ERIC Educational Resources Information Center

Star, Jon R.; Rittle-Johnson, Bethany

2009-01-01

Competence in algebra is increasingly recognized as a critical milestone in students' middle and high school years. The transition from arithmetic to algebra is a notoriously difficult one, and improvements in algebra instruction are greatly needed (National Research Council, 2001). Algebra historically has represented students' first sustained…

A constrained regularization method for inverting data represented by linear algebraic or integral equations

NASA Astrophysics Data System (ADS)

Provencher, Stephen W.

1982-09-01

CONTIN is a portable Fortran IV package for inverting noisy linear operator equations. These problems occur in the analysis of data from a wide variety experiments. They are generally ill-posed problems, which means that errors in an unregularized inversion are unbounded. Instead, CONTIN seeks the optimal solution by incorporating parsimony and any statistical prior knowledge into the regularizor and absolute prior knowledge into equallity and inequality constraints. This can be greatly increase the resolution and accuracyh of the solution. CONTIN is very flexible, consisting of a core of about 50 subprograms plus 13 small "USER" subprograms, which the user can easily modify to specify special-purpose constraints, regularizors, operator equations, simulations, statistical weighting, etc. Specjial collections of USER subprograms are available for photon correlation spectroscopy, multicomponent spectra, and Fourier-Bessel, Fourier and Laplace transforms. Numerically stable algorithms are used throughout CONTIN. A fairly precise definition of information content in terms of degrees of freedom is given. The regularization parameter can be automatically chosen on the basis of an F-test and confidence region. The interpretation of the latter and of error estimates based on the covariance matrix of the constrained regularized solution are discussed. The strategies, methods and options in CONTIN are outlined. The program itself is described in the following paper.

Derivation in INK-algebras

NASA Astrophysics Data System (ADS)

Kaviyarasu, M.; Indhira, K.

2018-04-01

In 2017 we introduced a new notion of algebra called IKN-algebra. Motivated by some result on derivations (rightleft)-derivation and (leftright)- derivation in ring. In this paper we introduce derivation in INK-Algebras and investigate some important result.

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.

Color Algebras

NASA Technical Reports Server (NTRS)

Mulligan, Jeffrey B.

2017-01-01

A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.

Algebraic theory of molecules

NASA Technical Reports Server (NTRS)

Iachello, Franco

1995-01-01

An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.

Elliptic biquaternion algebra

NASA Astrophysics Data System (ADS)

Özen, Kahraman Esen; Tosun, Murat

2018-01-01

In this study, we define the elliptic biquaternions and construct the algebra of elliptic biquaternions over the elliptic number field. Also we give basic properties of elliptic biquaternions. An elliptic biquaternion is in the form A0 + A1i + A2j + A3k which is a linear combination of {1, i, j, k} where the four components A0, A1, A2 and A3 are elliptic numbers. Here, 1, i, j, k are the quaternion basis of the elliptic biquaternion algebra and satisfy the same multiplication rules which are satisfied in both real quaternion algebra and complex quaternion algebra. In addition, we discuss the terms; conjugate, inner product, semi-norm, modulus and inverse for elliptic biquaternions.

(Fuzzy) Ideals of BN-Algebras

PubMed Central

Walendziak, Andrzej

2015-01-01

The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained. PMID:26125050

The Unitality of Quantum B-algebras

NASA Astrophysics Data System (ADS)

Han, Shengwei; Xu, Xiaoting; Qin, Feng

2018-02-01

Quantum B-algebras as a generalization of quantales were introduced by Rump and Yang, which cover the majority of implicational algebras and provide a unified semantic for a wide class of substructural logics. Unital quantum B-algebras play an important role in the classification of implicational algebras. The main purpose of this paper is to construct unital quantum B-algebras from non-unital quantum B-algebras.

On Weak-BCC-Algebras

PubMed Central

Thomys, Janus; Zhang, Xiaohong

2013-01-01

We describe weak-BCC-algebras (also called BZ-algebras) in which the condition (x∗y)∗z = (x∗z)∗y is satisfied only in the case when elements x, y belong to the same branch. We also characterize ideals, nilradicals, and nilpotent elements of such algebras. PMID:24311983

Earth Algebra.

ERIC Educational Resources Information Center

Schaufele, Christopher; Zumoff, Nancy

Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through…

A Richer Understanding of Algebra

ERIC Educational Resources Information Center

Foy, Michelle

2008-01-01

Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number,…

Situating the Debate on "Geometrical Algebra" within the Framework of Premodern Algebra.

PubMed

Sialaros, Michalis; Christianidis, Jean

2016-06-01

Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.

Dynamical systems defined on infinite dimensional lie algebras of the ''current algebra'' or ''Kac-Moody'' type

NASA Astrophysics Data System (ADS)

Hermann, Robert

1982-07-01

Recent work by Morrison, Marsden, and Weinstein has drawn attention to the possibility of utilizing the cosymplectic structure of the dual of the Lie algebra of certain infinite dimensional Lie groups to study hydrodynamical and plasma systems. This paper treats certain models arising in elementary particle physics, considered by Lee, Weinberg, and Zumino; Sugawara; Bardacki, Halpern, and Frishman; Hermann; and Dolan. The lie algebras involved are associated with the ''current algebras'' of Gell-Mann. This class of Lie algebras contains certain of the algebras that are called ''Kac-Moody algebras'' in the recent mathematics and mathematical physics literature.

High-Speed, Low-Cost Workstation for Computation-Intensive Statistics. Phase 1

DTIC Science & Technology

1990-06-20

routine implementation and performance. 5 The two compiled versions given in the table were coded in an attempt to obtain an optimized compiled version...level statistics and linear algebra routines (BSAS and BLAS) that have been prototyped in this study. For each routine, both the C code ( Turbo C...OISTRIBUTION /AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Unlimited distribution 13. ABSTRACT (Maximum 200 words) High-performance and low-cost

Symmetries and Invariants of Twisted Quantum Algebras and Associated Poisson Algebras

NASA Astrophysics Data System (ADS)

Molev, A. I.; Ragoucy, E.

We construct an action of the braid group BN on the twisted quantized enveloping algebra U q'( {o}N) where the elements of BN act as automorphisms. In the classical limit q → 1, we recover the action of BN on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra U q'( {sp}2n). We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras.

I CAN Learn[R] Pre-Algebra and Algebra. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2009

2009-01-01

The I CAN Learn[R] Education System is an interactive, self-paced, mastery-based software system that includes the I CAN Learn[R] Fundamentals of Math (5th-6th grade math) curriculum, the I CAN Learn[R] Pre-Algebra curriculum, and the I CAN Learn[R] Algebra curriculum. College algebra credit is also available to students in participating schools…

Macdonald index and chiral algebra

NASA Astrophysics Data System (ADS)

Song, Jaewon

2017-08-01

For any 4d N = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type ( A 1 , A 2 n ) and ( A 1 , D 2 n+1) where the chiral algebras are given by Virasoro and \\widehat{su}(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.

Macdonald index and chiral algebra

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

Song, Jaewon

For any 4dN = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. Here, we conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type (A 1, A 2n) and (A 1, D 2n+1) where the chiral algebras are given by Virasoro andmore » $$ˆ\\atop{su}$$(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.« less

Macdonald index and chiral algebra

DOE PAGES

Song, Jaewon

2017-08-10

For any 4dN = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. Here, we conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type (A 1, A 2n) and (A 1, D 2n+1) where the chiral algebras are given by Virasoro andmore » $$ˆ\\atop{su}$$(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.« less

Generalized Galilean algebras and Newtonian gravity

NASA Astrophysics Data System (ADS)

González, N.; Rubio, G.; Salgado, P.; Salgado, S.

2016-04-01

The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.

Algebra and Algebraic Thinking in School Math: 70th YB

ERIC Educational Resources Information Center

National Council of Teachers of Mathematics, 2008

2008-01-01

Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…

An algebra of reversible computation.

PubMed

Wang, Yong

2016-01-01

We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules: basic reversible processes algebra, algebra of reversible communicating processes, recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.

Assessing Algebraic Solving Ability: A Theoretical Framework

ERIC Educational Resources Information Center

Lian, Lim Hooi; Yew, Wun Thiam

2012-01-01

Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…

Prospective Teachers' Views on the Use of Calculators with Computer Algebra System in Algebra Instruction

ERIC Educational Resources Information Center

Ozgun-Koca, S. Ash

2010-01-01

Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This…

Visual Salience of Algebraic Transformations

ERIC Educational Resources Information Center

Kirshner, David; Awtry, Thomas

2004-01-01

Information processing researchers have assumed that algebra symbol skills depend on mastery of the abstract rules presented in the curriculum (Matz, 1980; Sleeman, 1986). Thus, students' ubiquitous algebra errors have been taken as indicating the need to embed algebra in rich contextual settings (Kaput, 1995; National Council of Teachers of…

Numerical Linear Algebra.

DTIC Science & Technology

1980-09-08

February 1979 through 31 March 1980 Title of Research: NUMERICAL LINEAR ALGEBRA Principal Investigators: Gene H. Golub James H. Wilkinson Research...BEFORE COMPLETING FORM 2 OTAgSSION NO. 3. RECIPIENT’S CATALOG NUMBER ITE~ btitle) ~qEE NUMERICAL LINEAR ALGEBRA #I ~ f#7&/8 PER.ORMING ORG. REPORT NUM 27R 7

Roughness in Lattice Ordered Effect Algebras

PubMed Central

Xin, Xiao Long; Hua, Xiu Juan; Zhu, Xi

2014-01-01

Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice ordered effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a function e(a, b) in a lattice ordered effect algebra E and build a relationship between it and congruence classes. Then we study some properties about approximation of lattice ordered effect algebras. PMID:25170523

Hurwitz Algebras and the Octonion Algebra

NASA Astrophysics Data System (ADS)

Burdik, Čestmir; Catto, Sultan

2018-02-01

We explore some consequences of a theory of internal symmetries for elementary particles constructed on exceptional quantum mechanical spaces based on Jordan algebra formulation that admit exceptional groups as gauge groups.

Quiver elliptic W-algebras

NASA Astrophysics Data System (ADS)

Kimura, Taro; Pestun, Vasily

2018-06-01

We define elliptic generalization of W-algebras associated with arbitrary quiver using our construction (Kimura and Pestun in Quiver W-algebras, 2015. arXiv:1512.08533 [hep-th]) with six-dimensional gauge theory.

Pre-Algebra Lexicon.

ERIC Educational Resources Information Center

Hayden, Dunstan; Cuevas, Gilberto

The pre-algebra lexicon is a set of classroom exercises designed to teach the technical words and phrases of pre-algebra mathematics, and includes the terms most commonly found in related mathematics courses. The lexicon has three parts, each with its own introduction. The first introduces vocabulary items in three groups forming a learning…

An Arithmetic-Algebraic Work Space for the Promotion of Arithmetic and Algebraic Thinking: Triangular Numbers

ERIC Educational Resources Information Center

Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos

2016-01-01

This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…

Representing k-graphs as Matrix Algebras

NASA Astrophysics Data System (ADS)

Rosjanuardi, R.

2018-05-01

For any commutative unital ring R and finitely aligned k-graph Λ with |Λ| < ∞ without cycles, we can realise Kumjian-Pask algebra KP R (Λ) as a direct sum of of matrix algebra over some vertices v with properties ν = νΛ, i.e: ⊕ νΛ=ν M |Λv|(R). When there is only a single vertex ν ∈ Λ° such that ν = νΛ, we can realise the Kumjian-Pask algebra as the matrix algebra M |ΛV|(R). Hence the matrix algebra M |vΛ|(R) can be regarded as a representation of the k-graph Λ. In this talk we will figure out the relation between finitely aligned k-graph and matrix algebra.

Assessing Elementary Algebra with STACK

ERIC Educational Resources Information Center

Sangwin, Christopher J.

2007-01-01

This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…

Filiform Lie algebras of order 3

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

Navarro, R. M., E-mail: rnavarro@unex.es

2014-04-15

The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de lamore » variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.« less

Algebraic Systems and Pushdown Automata

NASA Astrophysics Data System (ADS)

Petre, Ion; Salomaa, Arto

We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.

Ready, Set, Algebra?

ERIC Educational Resources Information Center

Levy, Alissa Beth

2012-01-01

The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…

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

Spatial-Operator Algebra For Robotic Manipulators

NASA Technical Reports Server (NTRS)

Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.

1991-01-01

Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.

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.

On the structure of quantum L∞ algebras

NASA Astrophysics Data System (ADS)

Blumenhagen, Ralph; Fuchs, Michael; Traube, Matthias

2017-10-01

It is believed that any classical gauge symmetry gives rise to an L∞ algebra. Based on the recently realized relation between classical W algebras and L∞ algebras, we analyze how this generalizes to the quantum case. Guided by the existence of quantum W algebras, we provide a physically well motivated definition of quantum L∞ algebras describing the consistency of global symmetries in quantum field theories. In this case we are restricted to only two non-trivial graded vector spaces X 0 and X -1 containing the symmetry variations and the symmetry generators. This quantum L∞ algebra structure is explicitly exemplified for the quantum W_3 algebra. The natural quantum product between fields is the normal ordered one so that, due to contractions between quantum fields, the higher L∞ relations receive off-diagonal quantum corrections. Curiously, these are not present in the loop L∞ algebra of closed string field theory.

Highest-weight representations of Brocherd`s algebras

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

Slansky, R.

1997-01-01

General features of highest-weight representations of Borcherd`s algebras are described. to show their typical features, several representations of Borcherd`s extensions of finite-dimensional algebras are analyzed. Then the example of the extension of affine- su(2) to a Borcherd`s algebra is examined. These algebras provide a natural way to extend a Kac-Moody algebra to include the hamiltonian and number-changing operators in a generalized symmetry structure.

Teaching Structure in Algebra

ERIC Educational Resources Information Center

Merlin, Ethan M.

2013-01-01

This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…

Post-Lie algebras and factorization theorems

NASA Astrophysics Data System (ADS)

Ebrahimi-Fard, Kurusch; Mencattini, Igor; Munthe-Kaas, Hans

2017-09-01

In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions of) those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang-Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.

Discrimination in a General Algebraic Setting

PubMed Central

Fine, Benjamin; Lipschutz, Seymour; Spellman, Dennis

2015-01-01

Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras. PMID:26171421

Asymptotic aspect of derivations in Banach algebras.

PubMed

Roh, Jaiok; Chang, Ick-Soon

2017-01-01

We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.

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.

Constructing Meanings and Utilities within Algebraic Tasks

ERIC Educational Resources Information Center

Ainley, Janet; Bills, Liz; Wilson, Kirsty

2004-01-01

The Purposeful Algebraic Activity project aims to explore the potential of spreadsheets in the introduction to algebra and algebraic thinking. We discuss two sub-themes within the project: tracing the development of pupils' construction of meaning for variable from arithmetic-based activity, through use of spreadsheets, and into formal algebra,…

FRT presentation of the Onsager algebras

NASA Astrophysics Data System (ADS)

Baseilhac, Pascal; Belliard, Samuel; Crampé, Nicolas

2018-03-01

A presentation à la Faddeev-Reshetikhin-Takhtajan (FRT) of the Onsager, augmented Onsager and sl_2 -invariant Onsager algebras is given, using the framework of the nonstandard classical Yang-Baxter algebras. Associated current algebras are identified, and generating functions of mutually commuting quantities are obtained.

Particle-like structure of coaxial Lie algebras

NASA Astrophysics Data System (ADS)

Vinogradov, A. M.

2018-01-01

This paper is a natural continuation of Vinogradov [J. Math. Phys. 58, 071703 (2017)] where we proved that any Lie algebra over an algebraically closed field or over R can be assembled in a number of steps from two elementary constituents, called dyons and triadons. Here we consider the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons. We describe the maximal families of pairwise compatible base dyons and triadons called clusters, and, as a consequence, we give a complete description of the coaxial Lie algebras. The remarkable fact is that dyons and triadons in clusters are self-organised in structural groups which are surrounded by casings and linked by connectives. We discuss generalisations and applications to the theory of deformations of Lie algebras.

Constraint-Referenced Analytics of Algebra Learning

ERIC Educational Resources Information Center

Sutherland, Scot M.; White, Tobin F.

2016-01-01

The development of the constraint-referenced analytics tool for monitoring algebra learning activities presented here came from the desire to firstly, take a more quantitative look at student responses in collaborative algebra activities, and secondly, to situate those activities in a more traditional introductory algebra setting focusing on…

Teaching Strategies to Improve Algebra Learning

ERIC Educational Resources Information Center

Zbiek, Rose Mary; Larson, Matthew R.

2015-01-01

Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…

Learning Activity Package, Algebra.

ERIC Educational Resources Information Center

Evans, Diane

A set of ten teacher-prepared Learning Activity Packages (LAPs) in beginning algebra and nine in intermediate algebra, these units cover sets, properties of operations, number systems, open expressions, solution sets of equations and inequalities in one and two variables, exponents, factoring and polynomials, relations and functions, radicals,…

Algorithms for computations of Loday algebras' invariants

NASA Astrophysics Data System (ADS)

Hussain, Sharifah Kartini Said; Rakhimov, I. S.; Basri, W.

2017-04-01

The paper is devoted to applications of some computer programs to study structural determination of Loday algebras. We present how these computer programs can be applied in computations of various invariants of Loday algebras and provide several computer programs in Maple to verify Loday algebras' identities, the isomorphisms between the algebras, as a special case, to describe the automorphism groups, centroids and derivations.

Working memory, worry, and algebraic ability.

PubMed

Trezise, Kelly; Reeve, Robert A

2014-05-01

Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship. Copyright © 2013 Elsevier Inc. All rights reserved.

q-Derivatives, quantization methods and q-algebras

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

Twarock, Reidun

1998-12-15

Using the example of Borel quantization on S{sup 1}, we discuss the relation between quantization methods and q-algebras. In particular, it is shown that a q-deformation of the Witt algebra with generators labeled by Z is realized by q-difference operators. This leads to a discrete quantum mechanics. Because of Z, the discretization is equidistant. As an approach to a non-equidistant discretization of quantum mechanics one can change the Witt algebra using not the number field Z as labels but a quadratic extension of Z characterized by an irrational number {tau}. This extension is denoted as quasi-crystal Lie algebra, because thismore » is a relation to one-dimensional quasicrystals. The q-deformation of this quasicrystal Lie algebra is discussed. It is pointed out that quasicrystal Lie algebras can be considered also as a 'deformed' Witt algebra with a 'deformation' of the labeling number field. Their application to the theory is discussed.« less

Generalized conformal realizations of Kac-Moody algebras

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

Palmkvist, Jakob

2009-01-15

We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n=1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3x3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f{sub 4}, e{sub 6}, e{sub 7}, e{sub 8} for n=2. Moreover, we obtain their infinite-dimensional extensions for n{>=}3. In the casemore » of 2x2 matrices, the resulting Lie algebras are of the form so(p+n,q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q)« less

Difficulties in initial algebra learning in Indonesia

NASA Astrophysics Data System (ADS)

Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja

2014-12-01

Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.

Unifying the Algebra for All Movement

ERIC Educational Resources Information Center

Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.

2015-01-01

There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…

Who Takes College Algebra?

ERIC Educational Resources Information Center

Herriott, Scott R.; Dunbar, Steven R.

2009-01-01

The common understanding within the mathematics community is that the role of the college algebra course is to prepare students for calculus. Though exceptions are emerging, the curriculum of most college algebra courses and the content of most textbooks on the market both reflect that assumption. This article calls that assumption into question…

Algebraic special functions and SO(3,2)

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

2013-06-15

A ladder structure of operators is presented for the associated Legendre polynomials and the sphericas harmonics. In both cases these operators belong to the irreducible representation of the Lie algebra so(3,2) with quadratic Casimir equals to −5/4. As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be homomorphic to the space of linear operators acting on the L{sup 2} functions defined on (−1,1)×Z and on the sphere S{sup 2}, respectively. The presence of a ladder structure is suggested to be the general condition to obtain a Lie algebra representation defining inmore » this way the “algebraic special functions” that are proposed to be the connection between Lie algebras and square-integrable functions so that the space of linear operators on the L{sup 2} functions is homomorphic to the universal enveloping algebra. The passage to the group, by means of the exponential map, shows that the associated Legendre polynomials and the spherical harmonics support the corresponding unitary irreducible representation of the group SO(3,2). -- Highlights: •The algebraic ladder structure is constructed for the associated Legendre polynomials (ALP). •ALP and spherical harmonics support a unitary irreducible SO(3,2)-representation. •A ladder structure is the condition to get a Lie group representation defining “algebraic special functions”. •The “algebraic special functions” connect Lie algebras and L{sup 2} functions.« less

Algebra for Gifted Third Graders.

ERIC Educational Resources Information Center

Borenson, Henry

1987-01-01

Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)

Semiclassical states on Lie algebras

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

Tsobanjan, Artur, E-mail: artur.tsobanjan@gmail.com

2015-03-15

The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following themore » methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.« less

A note on derivations of Murray–von Neumann algebras

PubMed Central

Kadison, Richard V.; Liu, Zhe

2014-01-01

A Murray–von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray–von Neumann algebras. We show that the “extended derivations” of a Murray–von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray–von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer’s seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements. PMID:24469831

A note on derivations of Murray-von Neumann algebras.

PubMed

Kadison, Richard V; Liu, Zhe

2014-02-11

A Murray-von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray-von Neumann algebras. We show that the "extended derivations" of a Murray-von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray-von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer's seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements.

Astro Algebra [CD-ROM].

ERIC Educational Resources Information Center

1997

Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…

Algebra for All: The Effect of Algebra Coursework and Classroom Peer Academic Composition on Low-Achieving Students

ERIC Educational Resources Information Center

Nomi, Takako; Raudenbush, Stephen W.

2014-01-01

Algebra is often considered as a gateway for later achievement. A recent report by the Mathematics Advisory Panel (2008) underscores the importance of improving algebra learning in secondary school. Today, a growing number of states and districts require algebra for all students in ninth grade or earlier. Chicago is at the forefront of this…

Catching Up on Algebra

ERIC Educational Resources Information Center

Cavanagh, Sean

2008-01-01

A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…

A double commutant theorem for Murray–von Neumann algebras

PubMed Central

Liu, Zhe

2012-01-01

Murray–von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra of the Murray–von Neumann algebra associated with a finite von Neumann algebra is the Murray–von Neumann algebra , where is a maximal abelian self-adjoint subalgebra of and, in addition, is . We also prove that the Murray–von Neumann algebra with the center of is the center of the Murray–von Neumann algebra . Von Neumann’s celebrated double commutant theorem characterizes von Neumann algebras as those for which , where , the commutant of , is the set of bounded operators on the Hilbert space that commute with all operators in . At the end of this article, we present a double commutant theorem for Murray–von Neumann algebras. PMID:22543165

Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds

NASA Astrophysics Data System (ADS)

Liu, Chiu-Chu Melissa; Sheshmani, Artan

2017-07-01

An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.

On character amenability of Banach algebras

NASA Astrophysics Data System (ADS)

Kaniuth, E.; Lau, A. T.; Pym, J.

2008-08-01

We continue our work [E. Kaniuth, A.T. Lau, J. Pym, On [phi]-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 85-96] in the study of amenability of a Banach algebra A defined with respect to a character [phi] of A. Various necessary and sufficient conditions of a global and a pointwise nature are found for a Banach algebra to possess a [phi]-mean of norm 1. We also completely determine the size of the set of [phi]-means for a separable weakly sequentially complete Banach algebra A with no [phi]-mean in A itself. A number of illustrative examples are discussed.

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.

The BMS4 algebra at spatial infinity

NASA Astrophysics Data System (ADS)

Troessaert, Cédric

2018-04-01

We show how a global BMS4 algebra appears as part of the asymptotic symmetry algebra at spatial infinity. Using linearised theory, we then show that this global BMS4 algebra is the one introduced by Strominger as a symmetry of the S-matrix.

Geometric Algebra for Physicists

NASA Astrophysics Data System (ADS)

Doran, Chris; Lasenby, Anthony

2007-11-01

Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.

Covariant deformed oscillator algebras

NASA Technical Reports Server (NTRS)

Quesne, Christiane

1995-01-01

The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.

UCSMP Algebra. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2007

2007-01-01

"University of Chicago School Mathematics Project (UCSMP) Algebra," designed to increase students' skills in algebra, is appropriate for students in grades 7-10, depending on the students' incoming knowledge. This one-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and…

Difficulties in Initial Algebra Learning in Indonesia

ERIC Educational Resources Information Center

Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja

2014-01-01

Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…

Teacher Actions to Facilitate Early Algebraic Reasoning

ERIC Educational Resources Information Center

Hunter, Jodie

2015-01-01

In recent years there has been an increased emphasis on integrating the teaching of arithmetic and algebra in primary school classrooms. This requires teachers to develop links between arithmetic and algebra and use pedagogical actions that facilitate algebraic reasoning. Drawing on findings from a classroom-based study, this paper provides an…

Elementary maps on nest algebras

NASA Astrophysics Data System (ADS)

Li, Pengtong

2006-08-01

Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.

Teachers' Understanding of Algebraic Generalization

NASA Astrophysics Data System (ADS)

Hawthorne, Casey Wayne

Generalization has been identified as a cornerstone of algebraic thinking (e.g., Lee, 1996; Sfard, 1995) and is at the center of a rich conceptualization of K-8 algebra (Kaput, 2008; Smith, 2003). Moreover, mathematics teachers are being encouraged to use figural-pattern generalizing tasks as a basis of student-centered instruction, whereby teachers respond to and build upon the ideas that arise from students' explorations of these activities. Although more and more teachers are engaging their students in such generalizing tasks, little is known about teachers' understanding of generalization and their understanding of students' mathematical thinking in this domain. In this work, I addressed this gap, exploring the understanding of algebraic generalization of 4 exemplary 8th-grade teachers from multiple perspectives. A significant feature of this investigation is an examination of teachers' understanding of the generalization process, including the use of algebraic symbols. The research consisted of two phases. Phase I was an examination of the teachers' understandings of the underlying quantities and quantitative relationships represented by algebraic notation. In Phase II, I observed the instruction of 2 of these teachers. Using the lens of professional noticing of students' mathematical thinking, I explored the teachers' enacted knowledge of algebraic generalization, characterizing how it supported them to effectively respond to the needs and queries of their students. Results indicated that teachers predominantly see these figural patterns as enrichment activities, disconnected from course content. Furthermore, in my analysis, I identified conceptual difficulties teachers experienced when solving generalization tasks, in particular, connecting multiple symbolic representations with the quantities in the figures. Moreover, while the teachers strived to overcome the challenges of connecting different representations, they invoked both productive and unproductive

Literal algebra for satellite dynamics. [perturbation analysis

NASA Technical Reports Server (NTRS)

Gaposchkin, E. M.

1975-01-01

A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.

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

Build an Early Foundation for Algebra Success

ERIC Educational Resources Information Center

Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela

2016-01-01

Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…

Algebraic Algorithm Design and Local Search

DTIC Science & Technology

1996-12-01

method for performing algorithm design that is more purely algebraic than that of KIDS. This method is then applied to local search. Local search is a...synthesis. Our approach was to follow KIDS in spirit, but to adopt a pure algebraic formalism, supported by Kestrel’s SPECWARE environment (79), that...design was developed that is more purely algebraic than that of KIDS. This method was then applied to local search. A general theory of local search was

Metric 3-Leibniz algebras and M2-branes

NASA Astrophysics Data System (ADS)

Méndez-Escobar, Elena

2010-08-01

This thesis is concerned with superconformal Chern-Simons theories with matter in 3 dimensions. The interest in these theories is two-fold. On the one hand, it is a new family of theories in which to test the AdS/CFT correspondence and on the other, they are important to study one of the main objects of M-theory (M2-branes). All these theories have something in common: they can be written in terms of 3-Leibniz algebras. Here we study the structure theory of such algebras, paying special attention to a subclass of them that gives rise to maximal supersymmetry and that was the first to appear in this context: 3-Lie algebras. In chapter 2, we review the structure theory of metric Lie algebras and their unitary representations. In chapter 3, we study metric 3-Leibniz algebras and show, by specialising a construction originally due to Faulkner, that they are in one to one correspondence with pairs of real metric Lie algebras and unitary representations of them. We also show a third characterisation for six extreme cases of 3-Leibniz algebras as graded Lie (super)algebras. In chapter 4, we study metric 3-Lie algebras in detail. We prove a structural result and also classify those with a maximally isotropic centre, which is the requirement that ensures unitarity of the corresponding conformal field theory. Finally, in chapter 5, we study the universal structure of superpotentials in this class of superconformal Chern-Simons theories with matter in three dimensions. We provide a uniform formulation for all these theories and establish the connection between the amount of supersymmetry preserved and the gauge Lie algebra and the appropriate unitary representation to be used to write down the Lagrangian. The conditions for supersymmetry enhancement are then expressed equivalently in the language of representation theory of Lie algebras or the language of 3-Leibniz algebras.

A Balancing Act: Making Sense of Algebra

ERIC Educational Resources Information Center

Gavin, M. Katherine; Sheffield, Linda Jensen

2015-01-01

For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…

Linear {GLP}-algebras and their elementary theories

NASA Astrophysics Data System (ADS)

Pakhomov, F. N.

2016-12-01

The polymodal provability logic {GLP} was introduced by Japaridze in 1986. It is the provability logic of certain chains of provability predicates of increasing strength. Every polymodal logic corresponds to a variety of polymodal algebras. Beklemishev and Visser asked whether the elementary theory of the free {GLP}-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable [1]. For every positive integer n we solve the corresponding question for the logics {GLP}_n that are the fragments of {GLP} with n modalities. We prove that the elementary theory of the free {GLP}_n-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable for all n. We introduce the notion of a linear {GLP}_n-algebra and prove that all free {GLP}_n-algebras generated by the constants \\mathbf{0}, \\mathbf{1} are linear. We also consider the more general case of the logics {GLP}_α whose modalities are indexed by the elements of a linearly ordered set α: we define the notion of a linear algebra and prove the latter result in this case.

Identities of Finitely Generated Algebras Over AN Infinite Field

NASA Astrophysics Data System (ADS)

Kemer, A. R.

1991-02-01

It is proved that for each finitely generated associative PI-algebra U over an infinite field F, there is a finite-dimensional F-algebra C such that the ideals of identities of the algebras U and C coincide. This yields a positive solution to the local problem of Specht for algebras over an infinite field: A finitely generated free associative algebra satisfies the maximum condition for T-ideals.

Algebra? A Gate! A Barrier! A Mystery!

ERIC Educational Resources Information Center

Mathematics Educatio Dialogues, 2000

2000-01-01

This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…

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.

Spatial-Operator Algebra For Flexible-Link Manipulators

NASA Technical Reports Server (NTRS)

Jain, Abhinandan; Rodriguez, Guillermo

1994-01-01

Method of computing dynamics of multiple-flexible-link robotic manipulators based on spatial-operator algebra, which originally applied to rigid-link manipulators. Aspects of spatial-operator-algebra approach described in several previous articles in NASA Tech Briefs-most recently "Robot Control Based on Spatial-Operator Algebra" (NPO-17918). In extension of spatial-operator algebra to manipulators with flexible links, each link represented by finite-element model: mass of flexible link apportioned among smaller, lumped-mass rigid bodies, coupling of motions expressed in terms of vibrational modes. This leads to operator expression for modal-mass matrix of link.

Learning Algebra from Worked Examples

ERIC Educational Resources Information Center

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

2014-01-01

For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…

The Dixmier Map for Nilpotent Super Lie Algebras

NASA Astrophysics Data System (ADS)

Herscovich, Estanislao

2012-07-01

In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by {Prim({U}({g}))} the set of (graded) primitive ideals of the enveloping algebra {{U}({g})} of a nilpotent Lie superalgebra {{g}} and {{A}d0} the adjoint group of {{g}0}, we prove that the usual Dixmier map for nilpotent Lie algebras can be naturally extended to the context of nilpotent super Lie algebras, i.e. there exists a bijective map I : {g}0^{*}/{A}d0 rightarrow Prim({U}({g})) defined by sending the equivalence class [ λ] of a functional λ to a primitive ideal I( λ) of {{U}({g})}, and which coincides with the Dixmier map in the case of nilpotent Lie algebras. Moreover, the construction of the previous map is explicit, and more or less parallel to the one for Lie algebras, a major difference with a previous approach ( cf. [18]). One key fact in the construction is the existence of polarizations for super Lie algebras, generalizing the concept defined for Lie algebras. As a corollary of the previous description, we obtain the isomorphism {{U}({g})/I(λ) ˜eq Cliffq(k) ⊗ Ap(k)}, where {(p,q) = (dim({g}0/{g}0^{λ})/2,dim({g}1/{g}1^{λ}))}, we get a direct construction of the maximal ideals of the underlying algebra of {{U}({g})} and also some properties of the stabilizers of the primitive ideals of {{U}({g})}.

Developing "Algebraic Thinking": Two Key Ways to Establish Some Early Algebraic Ideas in Primary Classrooms

ERIC Educational Resources Information Center

Ormond, Christine

2012-01-01

Primary teachers play a key role in their students' future mathematical success in the early secondary years. While the word "algebra" may make some primary teachers feel uncomfortable or worried, the basic arithmetic ideas underlying algebra are vitally important for older primary students as they are increasingly required to use "algebraic…

Color Algebras

NASA Technical Reports Server (NTRS)

Mulligan, Jeffrey B.

2017-01-01

A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. The difficulty addressed here is the fact that, because of metamerism, we cannot know with certainty the spectrum that produced a particular color solely on the basis of sensory data. Knowledge of the spectrum is not required to compute additive mixture of colors, but is critical for subtractive (multiplicative) mixture. Therefore, we cannot predict with certainty the multiplicative interactions between colors based solely on sensory data. There are two potential applications of a color algebra: first, to aid modeling phenomena of human visual perception, such as color constancy and transparency; and, second, to provide better models of the interactions of lights and surfaces for computer graphics rendering.

The Algebra of Complex Numbers.

ERIC Educational Resources Information Center

LePage, Wilbur R.

This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…

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.

Algebra: A Challenge at the Crossroads of Policy and Practice

ERIC Educational Resources Information Center

Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.

2011-01-01

The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…

Linear Algebra and Image Processing

ERIC Educational Resources Information Center

Allali, Mohamed

2010-01-01

We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)

Quantum Observables and Effect Algebras

NASA Astrophysics Data System (ADS)

Dvurečenskij, Anatolij

2018-03-01

We study observables on monotone σ-complete effect algebras. We find conditions when a spectral resolution implies existence of the corresponding observable. We characterize sharp observables of a monotone σ-complete homogeneous effect algebra using its orthoalgebraic skeleton. In addition, we study compatibility in orthoalgebras and we show that every orthoalgebra satisfying RIP is an orthomodular poset.

The Algebra of the Arches

ERIC Educational Resources Information Center

Buerman, Margaret

2007-01-01

Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…

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)

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.

Preparing Elementary Prospective Teachers to Teach Early Algebra

ERIC Educational Resources Information Center

Hohensee, Charles

2017-01-01

Researchers have argued that integrating early algebra into elementary grades will better prepare students for algebra. However, currently little research exists to guide teacher preparation programs on how to prepare prospective elementary teachers to teach early algebra. This study examines the insights and challenges that prospective teachers…

Quantum teleportation and Birman-Murakami-Wenzl algebra

NASA Astrophysics Data System (ADS)

Zhang, Kun; Zhang, Yong

2017-02-01

In this paper, we investigate the relationship of quantum teleportation in quantum information science and the Birman-Murakami-Wenzl (BMW) algebra in low-dimensional topology. For simplicity, we focus on the two spin-1/2 representation of the BMW algebra, which is generated by both the Temperley-Lieb projector and the Yang-Baxter gate. We describe quantum teleportation using the Temperley-Lieb projector and the Yang-Baxter gate, respectively, and study teleportation-based quantum computation using the Yang-Baxter gate. On the other hand, we exploit the extended Temperley-Lieb diagrammatical approach to clearly show that the tangle relations of the BMW algebra have a natural interpretation of quantum teleportation. Inspired by this interpretation, we construct a general representation of the tangle relations of the BMW algebra and obtain interesting representations of the BMW algebra. Therefore, our research sheds a light on a link between quantum information science and low-dimensional topology.

The tensor hierarchy algebra

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

Palmkvist, Jakob, E-mail: palmkvist@ihes.fr

We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of ourmore » Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.« less

Locally Compact Quantum Groups. A von Neumann Algebra Approach

NASA Astrophysics Data System (ADS)

Van Daele, Alfons

2014-08-01

In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We start with a von Neumann algebra and a comultiplication on this von Neumann algebra. We assume that there exist faithful left and right Haar weights. Then we develop the theory within this von Neumann algebra setting. In [Math. Scand. 92 (2003), 68-92] locally compact quantum groups are also studied in the von Neumann algebraic context. This approach is independent of the original C^*-algebraic approach in the sense that the earlier results are not used. However, this paper is not really independent because for many proofs, the reader is referred to the original paper where the C^*-version is developed. In this paper, we give a completely self-contained approach. Moreover, at various points, we do things differently. We have a different treatment of the antipode. It is similar to the original treatment in [Ann. Sci. & #201;cole Norm. Sup. (4) 33 (2000), 837-934]. But together with the fact that we work in the von Neumann algebra framework, it allows us to use an idea from [Rev. Roumaine Math. Pures Appl. 21 (1976), 1411-1449] to obtain the uniqueness of the Haar weights in an early stage. We take advantage of this fact when deriving the other main results in the theory. We also give a slightly different approach to duality. Finally, we collect, in a systematic way, several important formulas. In an appendix, we indicate very briefly how the C^*-approach and the von Neumann algebra approach eventually yield the same objects. The passage from the von Neumann algebra setting to the C^*-algebra setting is more or less standard. For the other direction, we use a new method. It is based on the observation that the Haar weights on the C^*-algebra extend to weights on the double dual with central support and that all these supports are the same. Of course, we get the von Neumann algebra by cutting down the double dual with this unique

Category-theoretic models of algebraic computer systems

NASA Astrophysics Data System (ADS)

Kovalyov, S. P.

2016-01-01

A computer system is said to be algebraic if it contains nodes that implement unconventional computation paradigms based on universal algebra. A category-based approach to modeling such systems that provides a theoretical basis for mapping tasks to these systems' architecture is proposed. The construction of algebraic models of general-purpose computations involving conditional statements and overflow control is formally described by a reflector in an appropriate category of algebras. It is proved that this reflector takes the modulo ring whose operations are implemented in the conventional arithmetic processors to the Łukasiewicz logic matrix. Enrichments of the set of ring operations that form bases in the Łukasiewicz logic matrix are found.

The Growing Importance of Linear Algebra in Undergraduate Mathematics.

ERIC Educational Resources Information Center

Tucker, Alan

1993-01-01

Discusses the theoretical and practical importance of linear algebra. Presents a brief history of linear algebra and matrix theory and describes the place of linear algebra in the undergraduate curriculum. (MDH)

Classical versus Computer Algebra Methods in Elementary Geometry

ERIC Educational Resources Information Center

Pech, Pavel

2005-01-01

Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…

Algebraic reasoning and bat-and-ball problem variants: Solving isomorphic algebra first facilitates problem solving later.

PubMed

Hoover, Jerome D; Healy, Alice F

2017-12-01

The classic bat-and-ball problem is used widely to measure biased and correct reasoning in decision-making. University students overwhelmingly tend to provide the biased answer to this problem. To what extent might reasoners be led to modify their judgement, and, more specifically, is it possible to facilitate problem solution by prompting participants to consider the problem from an algebraic perspective? One hundred ninety-seven participants were recruited to investigate the effect of algebraic cueing as a debiasing strategy on variants of the bat-and-ball problem. Participants who were cued to consider the problem algebraically were significantly more likely to answer correctly relative to control participants. Most of this cueing effect was confined to a condition that required participants to solve isomorphic algebra equations corresponding to the structure of bat-and-ball question types. On a subsequent critical question with differing item and dollar amounts presented without a cue, participants were able to generalize the learned information to significantly reduce overall bias. Math anxiety was also found to be significantly related to bat-and-ball problem accuracy. These results suggest that, under specific conditions, algebraic reasoning is an effective debiasing strategy on bat-and-ball problem variants, and provide the first documented evidence for the influence of math anxiety on Cognitive Reflection Test performance.

Minimal unitary representation of 5d superconformal algebra F(4) and AdS 6/CFT 5 higher spin (super)-algebras

DOE PAGES

Fernando, Sudarshan; Günaydin, Murat

2014-11-28

We study the minimal unitary representation (minrep) of SO(5, 2), obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO(5, 2) describes a massless conformal scalar field in five dimensions and admits a unique “deformation” which describes a massless conformal spinor. Scalar and spinor minreps of SO(5, 2) are the 5d analogs of Dirac’s singletons of SO(3, 2). We then construct the minimal unitary representation of the unique 5d supercon-formal algebra F(4) with the even subalgebra SO(5, 2) ×SU(2). The minrep of F(4) describes a massless conformal supermultiplet consisting of two scalar andmore » one spinor fields. We then extend our results to the construction of higher spin AdS 6/CFT 5 (super)-algebras. The Joseph ideal of the minrep of SO(5, 2) vanishes identically as operators and hence its enveloping algebra yields the AdS 6/CFT 5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a “deformed” higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS 6/CFT 5 superalgebra as the enveloping algebra of the minimal unitary realization of F(4) obtained by the quasiconformal methods.« less

Simple nuclear C*-algebras not isomorphic to their opposites

PubMed Central

Hirshberg, Ilan

2017-01-01

We show that it is consistent with Zermelo–Fraenkel set theory with the axiom of choice (ZFC) that there is a simple nuclear nonseparable C∗-algebra, which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O2 or of the canonical anticommutation relations (CAR) algebra. PMID:28559339

Meanings Given to Algebraic Symbolism in Problem-Posing

ERIC Educational Resources Information Center

Cañadas, María C.; Molina, Marta; del Río, Aurora

2018-01-01

Some errors in the learning of algebra suggest that students might have difficulties giving meaning to algebraic symbolism. In this paper, we use problem posing to analyze the students' capacity to assign meaning to algebraic symbolism and the difficulties that students encounter in this process, depending on the characteristics of the algebraic…

The effects of an integrated Algebra 1/physical science curriculum on student achievement in Algebra 1, proportional reasoning and graphing abilities

NASA Astrophysics Data System (ADS)

Lawrence, Lettie Carol

1997-08-01

The purpose of this investigation was to determine if an integrated curriculum in algebra 1/physical science facilitates acquisition of proportional reasoning and graphing abilities better than a non-integrated, traditional, algebra 1 curriculum. Also, this study was to ascertain if the integrated algebra 1/physical science curriculum resulted in greater student achievement in algebra 1. The curriculum used in the experimental class was SAM 9 (Science and Mathematics 9), an investigation-based curriculum that was written to integrate physical science and basic algebra content. The experiment was conducted over one school year. The subjects in the study were 61 ninth grade students. The experimental group consisted of one class taught concurrently by a mathematics teacher and a physical science teacher. The control group consisted of three classes of algebra 1 students taught by one mathematics teacher and taking physical science with other teachers in the school who were not participating in the SAM 9 program. This study utilized a quasi-experimental non-randomized control group pretest-posttest design. The investigator obtained end-of-algebra 1 scores from student records. The written open-ended graphing instruments and the proportional reasoning instrument were administered to both groups as pretests and posttests. The graphing instruments were also administered as a midtest. A two sample t-test for independent means was used to determine significant differences in achievement on the end-of-course algebra 1 test. Quantitative data from the proportional reasoning and graphing instruments were analyzed using a repeated measures analysis of variance to determine differences in scores over time for the experimental and control groups. The findings indicate no significant difference between the experimental and control groups on the end-of-course algebra 1 test. Results also indicate no significant differences in proportional reasoning and graphing abilities between

Using Students' Interests as Algebraic Models

ERIC Educational Resources Information Center

Whaley, Kenneth A.

2012-01-01

Fostering algebraic thinking is an important goal for middle-grades mathematics teachers. Developing mathematical reasoning requires that teachers cultivate students' habits of mind. Teachers develop students' understanding of algebra by engaging them in tasks that involve modeling and representation. This study was designed to investigate how…

Relational Algebra and SQL: Better Together

ERIC Educational Resources Information Center

McMaster, Kirby; Sambasivam, Samuel; Hadfield, Steven; Wolthuis, Stuart

2013-01-01

In this paper, we describe how database instructors can teach Relational Algebra and Structured Query Language together through programming. Students write query programs consisting of sequences of Relational Algebra operations vs. Structured Query Language SELECT statements. The query programs can then be run interactively, allowing students to…

VASP- VARIABLE DIMENSION AUTOMATIC SYNTHESIS PROGRAM

NASA Technical Reports Server (NTRS)

White, J. S.

1994-01-01

VASP is a variable dimension Fortran version of the Automatic Synthesis Program, ASP. The program is used to implement Kalman filtering and control theory. Basically, it consists of 31 subprograms for solving most modern control problems in linear, time-variant (or time-invariant) control systems. These subprograms include operations of matrix algebra, computation of the exponential of a matrix and its convolution integral, and the solution of the matrix Riccati equation. The user calls these subprograms by means of a FORTRAN main program, and so can easily obtain solutions to most general problems of extremization of a quadratic functional of the state of the linear dynamical system. Particularly, these problems include the synthesis of the Kalman filter gains and the optimal feedback gains for minimization of a quadratic performance index. VASP, as an outgrowth of the Automatic Synthesis Program, has the following improvements: more versatile programming language; more convenient input/output format; some new subprograms which consolidate certain groups of statements that are often repeated; and variable dimensioning. The pertinent difference between the two programs is that VASP has variable dimensioning and more efficient storage. The documentation for the VASP program contains a VASP dictionary and example problems. The dictionary contains a description of each subroutine and instructions on its use. The example problems include dynamic response, optimal control gain, solution of the sampled data matrix Riccati equation, matrix decomposition, and a pseudo-inverse of a matrix. This program is written in FORTRAN IV and has been implemented on the IBM 360. The VASP program was developed in 1971.

Effectiveness of Cognitive Tutor Algebra I at Scale

ERIC Educational Resources Information Center

Pane, John F.; Griffin, Beth Ann; McCaffrey, Daniel F.; Karam, Rita

2014-01-01

This article examines the effectiveness of a technology-based algebra curriculum in a wide variety of middle schools and high schools in seven states. Participating schools were matched into similar pairs and randomly assigned to either continue with the current algebra curriculum for 2 years or to adopt Cognitive Tutor Algebra I (CTAI), which…

How Middle Grade Teachers Think about Algebraic Reasoning

ERIC Educational Resources Information Center

Glassmeyer, David; Edwards, Belinda

2016-01-01

Algebraic reasoning is an essential habit of mind for building conceptual knowledge in K-12 mathematics, yet little is known about how middle school mathematics teachers think about algebraic reasoning. In this article we describe a research project examining how algebraic reasoning was considered by grades 6, 7, or 8 mathematics teachers in a…

Yang-Baxter algebras, integrable theories and Bethe Ansatz

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

De Vega, H.J.

1990-03-10

This paper presents the Yang-Baxter algebras (YBA) in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Behe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. This approachmore » permit to define and solve rigorously massive QFT as an appropriate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underlay the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized.« less

On the homotopy equivalence of simple AI-algebras

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

Aristov, O Yu

1999-02-28

Let A and B be simple unital AI-algebras (an AI-algebra is an inductive limit of C*-algebras of the form BigOplus{sub i}{sup k}C([0,1],M{sub N{sub i}}). It is proved that two arbitrary unital homomorphisms from A into B such that the corresponding maps K{sub 0}A{yields}K{sub 0}B coincide are homotopic. Necessary and sufficient conditions on the Elliott invariant for A and B to be homotopy equivalent are indicated. Moreover, two algebras in the above class having the same K-theory but not homotopy equivalent are constructed. A theorem on the homotopy of approximately unitarily equivalent homomorphisms between AI-algebras is used in the proof, whichmore » is deduced in its turn from a generalization to the case of AI-algebras of a theorem of Manuilov stating that a unitary matrix almost commuting with a self-adjoint matrix h can be joined to 1 by a continuous path consisting of unitary matrices almost commuting with h.« less

Card Games and Algebra Tic Tacmatics on Achievement of Junior Secondary II Students in Algebraic Expressions

ERIC Educational Resources Information Center

Okpube, Nnaemeka Michael; Anugwo, M. N.

2016-01-01

This study investigated the Card Games and Algebra tic-Tacmatics on Junior Secondary II Students' Achievement in Algebraic Expressions. Three research questions and three null hypotheses guided the study. The study adopted the pre-test, post-test control group design. A total of two hundred and forty (240) Junior Secondary School II students were…

Using Homemade Algebra Tiles To Develop Algebra and Prealgebra Concepts.

ERIC Educational Resources Information Center

Leitze, Annette Ricks; Kitt, Nancy A.

2000-01-01

Describes how to use homemade tiles, sketches, and the box method to reach a broader group of students for successful algebra learning. Provides a list of concepts appropriate for such an approach. (KHR)

Algebraic Thinking through Koch Snowflake Constructions

ERIC Educational Resources Information Center

Ghosh, Jonaki B.

2016-01-01

Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…

Practicing Algebraic Skills: A Conceptual Approach

ERIC Educational Resources Information Center

Friedlander, Alex; Arcavi, Abraham

2012-01-01

Traditionally, a considerable part of teaching and learning algebra has focused on routine practice and the application of rules, procedures, and techniques. Although today's computerized environments may have decreased the need to master algebraic skills, procedural competence is still a central component in any mathematical activity. However,…

Focus on Fractions to Scaffold Algebra

ERIC Educational Resources Information Center

Ooten, Cheryl Thomas

2013-01-01

Beginning algebra is a gatekeeper course into the pipeline to higher mathematics courses required for respected professions in engineering, science, statistics, mathematics, education, and technology. Beginning algebra can also be a perfect storm if the necessary foundational skills are not within a student's grasp. What skills ensure beginning…

The Ideas of Algebra, K-12. 1988 Yearbook.

ERIC Educational Resources Information Center

Coxford, Arthur F., Ed.; Shulte, Albert P., Ed.

This volume is organized into six parts. Chapters 1-5, which make up Part 1, first discuss the forces impinging on algebra in the curriculum and suggest possible directions for change. Chapters 6-8, Part 2, concentrate on concepts and teaching possibilities available prior to the formal introduction of algebra. The notion that algebraic ideas are…

Internally connected graphs and the Kashiwara-Vergne Lie algebra

NASA Astrophysics Data System (ADS)

Felder, Matteo

2018-06-01

It is conjectured that the Kashiwara-Vergne Lie algebra \\widehat{krv}_2 is isomorphic to the direct sum of the Grothendieck-Teichmüller Lie algebra grt_1 and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of \\widehat{krv}_2 whose intersection is grt_1, thus giving a way to interpolate between these two Lie algebras.

An Inquiry-Based Linear Algebra Class

ERIC Educational Resources Information Center

Wang, Haohao; Posey, Lisa

2011-01-01

Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…

The Development of Children's Algebraic Thinking: The Impact of a Comprehensive Early Algebra Intervention in Third Grade

ERIC Educational Resources Information Center

Blanton, Maria; Stephens, Ana; Knuth, Eric; Gardiner, Angela Murphy; Isler, Isil; Kim, Jee-Seon

2015-01-01

This article reports results from a study investigating the impact of a sustained, comprehensive early algebra intervention in third grade. Participants included 106 students; 39 received the early algebra intervention, and 67 received their district's regularly planned mathematics instruction. We share and discuss students' responses to a written…

Algebra for All: California's Eighth-Grade Algebra Initiative as Constrained Curricula.

PubMed

Domina, Thurston; Penner, Andrew M; Penner, Emily K; Conley, Annemarie

2014-08-01

Across the United States, secondary school curricula are intensifying as a growing proportion of students enroll in high-level academic math courses. In many districts, this intensification process occurs as early as eighth grade, where schools are effectively constraining their mathematics curricula by restricting course offerings and placing more students into Algebra I. This paper provides a quantitative single-case research study of policy-driven curricular intensification in one California school district. (1a) What effect did 8th eighth grade curricular intensification have on mathematics course enrollment patterns in Towering Pines Unified schools? (2b) How did the distribution of prior achievement in Towering Pines math classrooms change as the district constrained the curriculum by universalizing 8th eighth grade Algebra? (3c) Did 8th eighth grade curricular intensification improve students' mathematics achievement? Towering Pines is an immigrant enclave in the inner-ring suburbs of a major metropolitan area. The district's 10 middle schools together enroll approximately 4,000 eighth graders each year. The districts' students are ethnically diverse and largely economically disadvantaged. The study draws upon administrative data describing 8th eighth graders in the district in the 2004-20-05 through 2007-20-08 school years. During the study period, Towering Pines dramatically intensified middle school students' math curricula: In the 2004-20-05 school year 32% of the district's 8th eighth graders enrolled in Algebra or a higher- level mathematics course; by the 2007-20-08 school year that proportion had increased to 84%. We use an interrupted time-series design, comparing students' 8th eighth grade math course enrollments, 10th grade math course enrollments, and 10th grade math test scores across the four cohorts, controlling for demographics and prior achievement. We find that students' odds of taking higher level mathematics courses increased as this

Chinese Algebra: Using Historical Problems to Think about Current Curricula

ERIC Educational Resources Information Center

Tillema, Erik

2005-01-01

The Chinese used the idea of generating equivalent expressions for solving problems where the problems from a historical Chinese text are studied to understand the ways in which the ideas can lead into algebraic calculations and help students to learn algebra. The texts unify algebraic problem solving through complex algebraic thought and afford…

Quantum walled Brauer algebra: commuting families, Baxterization, and representations

NASA Astrophysics Data System (ADS)

Semikhatov, A. M.; Tipunin, I. Yu

2017-02-01

For the quantum walled Brauer algebra, we construct its Specht modules and (for generic parameters of the algebra) seminormal modules. The latter construction yields the spectrum of a commuting family of Jucys-Murphy elements. We also propose a Baxterization prescription; it involves representing the quantum walled Brauer algebra in terms of morphisms in a braided monoidal category and introducing parameters into these morphisms, which allows constructing a ‘universal transfer matrix’ that generates commuting elements of the algebra.

Some Applications of Algebraic System Solving

ERIC Educational Resources Information Center

Roanes-Lozano, Eugenio

2011-01-01

Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact…

Gender differences in algebraic thinking ability to solve mathematics problems

NASA Astrophysics Data System (ADS)

Kusumaningsih, W.; Darhim; Herman, T.; Turmudi

2018-05-01

This study aimed to conduct a gender study on students' algebraic thinking ability in solving a mathematics problem, polyhedron concept, for grade VIII. This research used a qualitative method. The data was collected using: test and interview methods. The subjects in this study were eight male and female students with different level of abilities. It was found that the algebraic thinking skills of male students reached high group of five categories. They were superior in terms of reasoning and quick understanding in solving problems. Algebraic thinking ability of high-achieving group of female students also met five categories of algebraic thinking indicators. They were more diligent, tenacious and thorough in solving problems. Algebraic thinking ability of male students in medium category only satisfied three categories of algebraic thinking indicators. They were sufficient in terms of reasoning and understanding in solving problems. Algebraic thinking ability group of female students in medium group also satisfied three categories of algebraic thinking indicators. They were fairly diligent, tenacious and meticulous on working on the problems.

An Algebraic Formulation of Level One Wess-Zumino Models

NASA Astrophysics Data System (ADS)

Böckenhauer, Jens

The highest weight modules of the chiral algebra of orthogonal WZW models at level one possess a realization in fermionic representation spaces; the Kac-Moody and Virasoro generators are represented as unbounded limits of even CAR algebras. It is shown that the representation theory of the underlying even CAR algebras reproduces precisely the sectors of the chiral algebra. This fact allows to develop a theory of local von Neumann algebras on the punctured circle, fitting nicely in the Doplicher-Haag-Roberts framework. The relevant localized endomorphisms which generate the charged sectors are explicitly constructed by means of Bogoliubov transformations. Using CAR theory, the fusion rules in terms of sector equivalence classes are proven.

On Maximal Subalgebras and the Hypercentre of Lie Algebras.

ERIC Educational Resources Information Center

Honda, Masanobu

1997-01-01

Derives two sufficient conditions for a finitely generated Lie algebra to have the nilpotent hypercenter. Presents a relatively large class of generalized soluble Lie algebras. Proves that if a finitely generated Lie algebra has a nilpotent maximal subalgebra, the Fitting radical is nilpotent. (DDR)

The hopf algebra of vector fields on complex quantum groups

NASA Astrophysics Data System (ADS)

Drabant, Bernhard; Jurčo, Branislav; Schlieker, Michael; Weich, Wolfgang; Zumino, Bruno

1992-10-01

We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types A n , B n , C n , and D n by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.

Spontaneous Meta-Arithmetic as a First Step toward School Algebra

ERIC Educational Resources Information Center

Caspi, Shai; Sfard, Anna

2012-01-01

Taking as the point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following five pairs of 7th grade students as they progress in algebraic discourse during 24 months, from their informal algebraic talk to the formal algebraic discourse, as taught in school. Our analysis follows changes that…

Just Say Yes to Early Algebra!

ERIC Educational Resources Information Center

Stephens, Ana; Blanton, Maria; Knuth, Eric; Isler, Isil; Gardiner, Angela Murphy

2015-01-01

Mathematics educators have argued for some time that elementary school students are capable of engaging in algebraic thinking and should be provided with rich opportunities to do so. Recent initiatives like the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) have taken up this call by reiterating the place of early algebra in…

Abstract Numeric Relations and the Visual Structure of Algebra

ERIC Educational Resources Information Center

Landy, David; Brookes, David; Smout, Ryan

2014-01-01

Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition,…

A Relational Algebra Query Language for Programming Relational Databases

ERIC Educational Resources Information Center

McMaster, Kirby; Sambasivam, Samuel; Anderson, Nicole

2011-01-01

In this paper, we describe a Relational Algebra Query Language (RAQL) and Relational Algebra Query (RAQ) software product we have developed that allows database instructors to teach relational algebra through programming. Instead of defining query operations using mathematical notation (the approach commonly taken in database textbooks), students…

Error-Detecting Identification Codes for Algebra Students.

ERIC Educational Resources Information Center

Sutherland, David C.

1990-01-01

Discusses common error-detecting identification codes using linear algebra terminology to provide an interesting application of algebra. Presents examples from the International Standard Book Number, the Universal Product Code, bank identification numbers, and the ZIP code bar code. (YP)

Using the Internet To Investigate Algebra.

ERIC Educational Resources Information Center

Sherwood, Walter

The lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approach algebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…

THE RADICAL OF A JORDAN ALGEBRA

PubMed Central

McCrimmon, Kevin

1969-01-01

In this paper we define a Jacobson radical for Jordan algebras analogous to that for associative algebras and show that it enjoys many of the properties of the associative radical. We then relate the corresponding notion of “semisimplicity” to the previously defined notion of “nondegeneracy” (Jacobson, N., these Proceedings, 55, 243-251 (1966)). PMID:16591736

Deriving the Regression Line with Algebra

ERIC Educational Resources Information Center

Quintanilla, John A.

2017-01-01

Exploration with spreadsheets and reliance on previous skills can lead students to determine the line of best fit. To perform linear regression on a set of data, students in Algebra 2 (or, in principle, Algebra 1) do not have to settle for using the mysterious "black box" of their graphing calculators (or other classroom technologies).…

Lattices of Varieties of Algebras

NASA Astrophysics Data System (ADS)

Volkov, M. V.

1980-02-01

Let A be an associative and commutative ring with 1, S a subsemigroup of the multiplicative semigroup of A, not containing divisors of zero, and \\mathfrak{X} some variety of A-algebras. A study is made of the homomorphism from the lattice L(\\mathfrak{X}) of all subvarieties of \\mathfrak{X} into the lattice of all varieties of S^{-1} A-algebras, which is induced in a certain natural sense by the functor S^{-1}. Under one weak restriction on \\mathfrak{X} a description is given of the kernel of this homomorphism, and this makes it possible to establish a good interrelation between the properties of the lattice L(\\mathfrak{X}) and the lattice of varieties of S^{-1} A-algebras. These results are applied to prove that a number of varieties of associative and Lie rings have the Specht property.Bibliography: 18 titles.

Assessing Mathematics Automatically Using Computer Algebra and the Internet

ERIC Educational Resources Information Center

Sangwin, Chris

2004-01-01

This paper reports some recent developments in mathematical computer-aided assessment which employs computer algebra to evaluate students' work using the Internet. Technical and educational issues raised by this use of computer algebra are addressed. Working examples from core calculus and algebra which have been used with first year university…

Solving the Unknown with Algebra: Poster/Teaching Guide for Pre-Algebra Students. Expect the Unexpected with Math[R

ERIC Educational Resources Information Center

Actuarial Foundation, 2013

2013-01-01

"Solving the Unknown with Algebra" is a new math program aligned with the National Council of Teachers of Mathematics (NCTM) standards and designed to help students practice pre-algebra skills including using formulas, solving for unknowns, and manipulating equations. Developed by The Actuarial Foundation with Scholastic, this program provides…

Form in Algebra: Reflecting, with Peacock, on Upper Secondary School Teaching.

ERIC Educational Resources Information Center

Menghini, Marta

1994-01-01

Discusses algebra teaching by looking back into the history of algebra and the work of George Peacock, who considered algebra from two points of view: symbolic and instrumental. Claims that, to be meaningful, algebra must be linked to real-world problems. (18 references) (MKR)

School Algebra Reform: Meeting the Grade?

ERIC Educational Resources Information Center

Telese, James A.

This paper reports on a case study that was conducted at five high schools from a large, urban school district located in South Texas. The purpose of the study was to gain an understanding of Algebra 1 teaching strategies. The research questions were: (1) What is the predominant mode of instruction for Algebra 1? and (2) What is the level of…

On special Lie algebras having a faithful module with Krull dimension

NASA Astrophysics Data System (ADS)

Pikhtilkova, O. A.; Pikhtilkov, S. A.

2017-02-01

For special Lie algebras we prove an analogue of Markov's theorem on {PI}-algebras having a faithful module with Krull dimension: the solubility of the prime radical. We give an example of a semiprime Lie algebra that has a faithful module with Krull dimension but cannot be represented as a subdirect product of finitely many prime Lie algebras. We prove a criterion for a semiprime Lie algebra to be representable as such a subdirect product.

Connections between Kac-Moody algebras and M-theory

NASA Astrophysics Data System (ADS)

Cook, Paul P.

2007-11-01

We investigate some of the motivations and consequences of the conjecture that the Kac-Moody algebra E11 is the symmetry algebra of M-theory, and we develop methods to aid the further investigation of this idea. The definitions required to work with abstract root systems of Lie algebras are given in review leading up to the definition of a Kac-Moody algebra. The motivations for the E11 conjecture are presented and the nonlinear realisation of gravity relevant to the conjecture is described. We give a beginner's guide to producing the algebras of E11, relevant to M-theory, and K27, relevant to the bosonic string theory, along with their l1 representations are constructed. Reference tables of low level roots are produced for both the adjoint and l1 representations of these algebras. In addition a particular group element, having a generic form for all G+++ algebras, is shown to encode all the half-BPS brane solutions of the maximally oxidised supergravities. Special analysis is given to the role of space-time signature in the context of this group element and subsequent to this analysis spacelike brane solutions are derived from the same solution generating group element. Finally the appearance of U-duality charge multiplets from E11 is reviewed. General formulae for finding the content of arbitrary brane charge multiplets are given and the content of the particle and string multiplets in dimensions 4,5,6,7 and 8 is shown to be contained in the l1 representation of E11.

Earth Algebra: Real-Life Mathematics in Navajoland.

ERIC Educational Resources Information Center

Schaufele, Christopher; Srivastava, Ravindra

1995-01-01

An algebra class at Navajo Community College (Shiprock, New Mexico) uses traditional algebra topics to study real-life situations, focuses on environmental issues, encourages collaborative learning, uses modern technology, and promotes development of critical thinking and decision-making skills. Students follow principles of Dine educational…

Hom Gel'fand-Dorfman bialgebras and Hom-Lie conformal algebras

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

Yuan, Lamei, E-mail: lmyuan@hit.edu.cn

2014-04-15

The aim of this paper is to introduce the notions of Hom Gel'fand-Dorfman bialgebra and Hom-Lie conformal algebra. In this paper, we give four constructions of Hom Gel'fand-Dorfman bialgebras. Also, we provide a general construction of Hom-Lie conformal algebras from Hom-Lie algebras. Finally, we prove that a Hom Gel'fand-Dorfman bialgebra is equivalent to a Hom-Lie conformal algebra of degree 2.

Assessment of polytechnic students' understanding of basic algebra

NASA Astrophysics Data System (ADS)

Mokmin, Nur Azlina Mohamed; Masood, Mona

2015-12-01

It is important for engineering students to excel in algebra. Previous studies show that the algebraic fraction is a subtopic of algebra that was found to be the most challenging for engineering students. This study is done with 191 first semester engineering students who have enrolled in engineering programs in Malaysian polytechnic. The respondents are divided into Group 1 (Distinction) and Group 2 (Credit) based on their Mathematics SPM result. A computer application is developed for this study to assess student information and understanding of the algebraic fraction topic. The result is analyzed using SPSS and Microsoft Excel. The test results show that there are significant differences between Group 1 and Group 2 and that most of the students scored below the minimum requirement.

A description of pseudo-bosons in terms of nilpotent Lie algebras

NASA Astrophysics Data System (ADS)

Bagarello, Fabio; Russo, Francesco G.

2018-02-01

We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we do not find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed into the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behavior of pseudo-bosonic operators in many quantum models.

Computing Gröbner Bases within Linear Algebra

NASA Astrophysics Data System (ADS)

Suzuki, Akira

In this paper, we present an alternative algorithm to compute Gröbner bases, which is based on computations on sparse linear algebra. Both of S-polynomial computations and monomial reductions are computed in linear algebra simultaneously in this algorithm. So it can be implemented to any computational system which can handle linear algebra. For a given ideal in a polynomial ring, it calculates a Gröbner basis along with the corresponding term order appropriately.

Quantum walks, deformed relativity and Hopf algebra symmetries.

PubMed

Bisio, Alessandro; D'Ariano, Giacomo Mauro; Perinotti, Paolo

2016-05-28

We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014Phys. Rev. A90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras-the usual Poincaré and theκ-Poincaré algebras. © 2016 The Author(s).

Algebra 1r, Mathematics (Experimental): 5215.13.

ERIC Educational Resources Information Center

Strachan, Florence

This third of six guidebooks on minimum course content for first-year algebra includes work with laws of exponents; multiplication, division, and factoring of polynomials; and fundamental operations with rational algebraic expressions. Course goals are stated, performance objectives listed, a course outline provided, testbook references specified…

Bootstrapping non-commutative gauge theories from L∞ algebras

NASA Astrophysics Data System (ADS)

Blumenhagen, Ralph; Brunner, Ilka; Kupriyanov, Vladislav; Lüst, Dieter

2018-05-01

Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying L∞ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. We recall that such field theories arise in the context of branes in WZW models and briefly comment on its appearance for integrable deformations of AdS5 sigma models. For the SU(2) WZW model, we show that the earlier proposed matrix valued gauge theory on the fuzzy 2-sphere can be bootstrapped via an L∞ algebra. We then apply this approach to the construction of non-commutative Chern-Simons and Yang-Mills theories on flat and curved backgrounds with non-constant NC-structure. More concretely, up to the second order, we demonstrate how derivative and curvature corrections to the equations of motion can be bootstrapped in an algebraic way from the L∞ algebra. The appearance of a non-trivial A∞ algebra is discussed, as well.

Cubic map algebra functions for spatio-temporal analysis

USGS Publications Warehouse

Mennis, J.; Viger, R.; Tomlin, C.D.

2005-01-01

We propose an extension of map algebra to three dimensions for spatio-temporal data handling. This approach yields a new class of map algebra functions that we call "cube functions." Whereas conventional map algebra functions operate on data layers representing two-dimensional space, cube functions operate on data cubes representing two-dimensional space over a third-dimensional period of time. We describe the prototype implementation of a spatio-temporal data structure and selected cube function versions of conventional local, focal, and zonal map algebra functions. The utility of cube functions is demonstrated through a case study analyzing the spatio-temporal variability of remotely sensed, southeastern U.S. vegetation character over various land covers and during different El Nin??o/Southern Oscillation (ENSO) phases. Like conventional map algebra, the application of cube functions may demand significant data preprocessing when integrating diverse data sets, and are subject to limitations related to data storage and algorithm performance. Solutions to these issues include extending data compression and computing strategies for calculations on very large data volumes to spatio-temporal data handling.

A non-commutative *-algebra of Borel functions

NASA Astrophysics Data System (ADS)

Hart, Robert

To the pair (E, sigma), where E is a countable Borel equivalence relation on a standard Borel space ( X, A ) and sigma a normalized Borel T -valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra B*r (E, sigma), contained in the bounded linear operators on ℓ2(E). Associated to B*r (E, sigma) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L( Bo (X)) isomorphic to the bounded Borel functions on X. Then L( Bo (X)) and its normalizer (the set of the unitaries u ∈ B*r (E, sigma) such that u* fu ∈ L( Bo (X)), f ∈ L( Bo (X))) countably generates the Borel *-algebra B*r (E, sigma). In this thesis, we study B*r (E, sigma) and in particular prove that: i) If E is smooth, then B*r (E, sigma) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then B*r (E, sigma) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist Gamma over E and its associated sequentially closed Borel *-algebra B*r (Gamma). iv) Let a Borel Cartan pair ( B,B0 ) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra B0 , where B is countably B0 -generated. Generalizing Feldman-Moore's result, we prove that any pair ( B,B0 ) can be realized uniquely as a pair ( B*r (E, sigma), L( Bo (X))). Moreover, we show that the pair ( B*r (E), L( Bo (X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras B*r (E1) and B*r (E2) are isomorphic.

The algebra of supertraces for 2+1 super de Sitter gravity

NASA Technical Reports Server (NTRS)

Urrutia, L. F.; Waelbroeck, H.; Zertuche, F.

1993-01-01

The algebra of the observables for 2+1 super de Sitter gravity, for one genus of the spatial surface is calculated. The algebra turns out to be an infinite Lie algebra subject to non-linear constraints. The constraints are solved explicitly in terms of five independent complex supertraces. These variables are the true degrees of freedom of the system and their quantized algebra generates a new structure which is referred to as a 'central extension' of the quantum algebra SU(2)q.

Solving Absolute Value Equations Algebraically and Geometrically

ERIC Educational Resources Information Center

Shiyuan, Wei

2005-01-01

The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.

The Structural Algebra Option: A Discussion Paper.

ERIC Educational Resources Information Center

Kirshner, David

The goal of this paper is to renew interest in the structural option to algebra instruction. Concern for the usual secondary school algebra curriculum related to simplifying expressions, solving equations, and rationalizing numerators and denominators is viewed from three pedagogical approaches: (1) structural approach, (2) empirical approach, and…

Playing Your Cards Right: Integers for Algebra

ERIC Educational Resources Information Center

Tillema, Erik; Gatza, Andrew; Ulrich, Catherine

2017-01-01

The number and algebra strand of the "Australian Curriculum: Mathematics" (2015) advocates for holding together the study of number and algebra across years K-8--a position that mathematics educators have endorsed in many countries. This recommendation along with the report "Shape of the Australian Curriculum: Mathematics"…

The general symmetry algebra structure of the underdetermined equation ux=(vxx)2

NASA Astrophysics Data System (ADS)

Kersten, Paul H. M.

1991-08-01

In a recent paper, Anderson, Kamran, and Olver [``Interior, exterior, and generalized symmetries,'' preprint (1990)] obtained the first- and second-order generalized symmetry algebra for the system ux=(vxx)2, leading to the noncompact real form of the exceptional Lie algebra G2. Here, the structure of the general higher-order symmetry algebra is obtained. Moreover, the Lie algebra G2 is obtained as ordinary symmetry algebra of the associated first-order system. The general symmetry algebra for ux=f(u,v,vx,...,) is established also.

Tensor Algebra Library for NVidia Graphics Processing Units

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

Liakh, Dmitry

This is a general purpose math library implementing basic tensor algebra operations on NVidia GPU accelerators. This software is a tensor algebra library that can perform basic tensor algebra operations, including tensor contractions, tensor products, tensor additions, etc., on NVidia GPU accelerators, asynchronously with respect to the CPU host. It supports a simultaneous use of multiple NVidia GPUs. Each asynchronous API function returns a handle which can later be used for querying the completion of the corresponding tensor algebra operation on a specific GPU. The tensors participating in a particular tensor operation are assumed to be stored in local RAMmore » of a node or GPU RAM. The main research area where this library can be utilized is the quantum many-body theory (e.g., in electronic structure theory).« less

Homomorphisms in C*-ternary algebras and JB*-triples

NASA Astrophysics Data System (ADS)

Park, Choonkil; Rassias, Themistocles M.

2008-01-01

In this paper, we investigate homomorphisms between C*-ternary algebras and derivations on C*-ternary algebras, and homomorphisms between JB*-triples and derivations on JB*-triples, associated with the following Apollonius type additive functional equation

Geometric interpretation of vertex operator algebras.

PubMed Central

Huang, Y Z

1991-01-01

In this paper, Vafa's approach to the formulation of conformal field theories is combined with the formal calculus developed in Frenkel, Lepowsky, and Meurman's work on the vertex operator construction of the Monster to give a geometric definition of vertex operator algebras. The main result announced is the equivalence between this definition and the algebraic one in the sense that the categories determined by these definitions are isomorphic. PMID:11607240

A Geometric Construction of Cyclic Cocycles on Twisted Convolution Algebras

NASA Astrophysics Data System (ADS)

Angel, Eitan

2010-09-01

In this thesis we give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. In his seminal book, Connes constructs a map from the equivariant cohomology of a manifold carrying the action of a discrete group into the periodic cyclic cohomology of the associated convolution algebra. Furthermore, for proper étale groupoids, J.-L. Tu and P. Xu provide a map between the periodic cyclic cohomology of a gerbe twisted convolution algebra and twisted cohomology groups. Our focus will be the convolution algebra with a product defined by a gerbe over a discrete translation groupoid. When the action is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial notions related to ideas of J. Dupont to construct a simplicial form representing the Dixmier-Douady class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial Dixmier-Douady form to the mixed bicomplex of certain matrix algebras. Finally, we define a morphism from this complex to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.

Students’ Algebraic Reasonsing In Solving Mathematical Problems With Adversity Quotient

NASA Astrophysics Data System (ADS)

Aryani, F.; Amin, S. M.; Sulaiman, R.

2018-01-01

Algebraic reasoning is a process in which students generalize mathematical ideas from a set of particular instances and express them in increasingly formal and age-appropriate ways. Using problem solving approach to develop algebraic reasoning of mathematics may enhace the long-term learning trajectory of the majority students. The purpose of this research was to describe the algebraic reasoning of quitter, camper, and climber junior high school students in solving mathematical problems. This research used qualitative descriptive method. Subjects were determined by purposive sampling. The technique of collecting data was done by task-based interviews.The results showed that the algebraic reasoning of three students in the process of pattern seeking by identifying the things that are known and asked in a similar way. But three students found the elements of pattern recognition in different ways or method. So, they are generalize the problem of pattern formation with different ways. The study of algebraic reasoning and problem solving can be a learning paradigm in the improve students’ knowledge and skills in algebra work. The goal is to help students’ improve academic competence, develop algebraic reasoning in problem solving.

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.

1991-01-01

A recently developed spatial operator algebra for manipulator modeling, control, and trajectory design is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and for control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics.

Towards classical spectrum generating algebras for f-deformations

NASA Astrophysics Data System (ADS)

Kullock, Ricardo; Latini, Danilo

2016-01-01

In this paper we revise the classical analog of f-oscillators, a generalization of q-oscillators given in Man'ko et al. (1997) [8], in the framework of classical spectrum generating algebras (SGA) introduced in Kuru and Negro (2008) [9]. We write down the deformed Poisson algebra characterizing the entire family of non-linear oscillators and construct its general solution algebraically. The latter, covering the full range of f-deformations, shows an energy dependence both in the amplitude and the frequency of the motion.

On Non-Abelian Extensions of 3-Lie Algebras

NASA Astrophysics Data System (ADS)

Song, Li-Na; Makhlouf, Abdenacer; Tang, Rong

2018-04-01

In this paper, we study non-abelian extensions of 3-Lie algebras through Maurer-Cartan elements. We show that there is a one-to-one correspondence between isomorphism classes of non-abelian extensions of 3-Lie algebras and equivalence classes of Maurer-Cartan elements in a DGLA. The structure of the Leibniz algebra on the space of fundamental objects is also analyzed. Supported by National Natural Science Foundation of China under Grant No. 11471139 and National Natural Science Foundation of Jilin Province under Grant No. 20170101050JC

The Xs and Whys of Algebra: Key Ideas and Common Misconceptions

ERIC Educational Resources Information Center

Collins, Anne; Dacey, Linda

2011-01-01

In many ways, algebra can be as challenging for teachers as it is for students. With so much emphasis placed on procedural knowledge and the manipulations of variables and symbols, it can be easy to lose sight of the key ideas that underlie algebraic thinking and the relevance algebra has to the real world. In the The Xs and Whys of Algebra: Key…

Symbolic Notations and Students' Achievements in Algebra

ERIC Educational Resources Information Center

Peter, Ebiendele E.; Olaoye, Adetunji A.

2013-01-01

This study focuses on symbolic notations and its impact on students' achievement in Algebra. The main reason for this study rests on the observation from personal and professional experiences on students' increasing hatred for Algebra. One hundred and fifty (150) Senior Secondary School Students (SSS) from Ojo Local Education District, Ojo, Lagos,…

Designing Spreadsheet-Based Tasks for Purposeful Algebra

ERIC Educational Resources Information Center

Ainley, Janet; Bills, Liz; Wilson, Kirsty

2005-01-01

We describe the design of a sequence of spreadsheet-based pedagogic tasks for the introduction of algebra in the early years of secondary schooling within the Purposeful Algebraic Activity project. This design combines two relatively novel features to bring a different perspective to research in the use of spreadsheets for the learning and…

Eighth Grade Algebra Course Placement and Student Motivation for Mathematics

PubMed Central

Simzar, Rahila M.; Domina, Thurston; Tran, Cathy

2016-01-01

This study uses student panel data to examine the association between Algebra placement and student motivation for mathematics. Changes in achievement goals, expectancy, and task value for students in eighth grade Algebra are compared with those of peers placed in lower-level mathematics courses (N = 3,306). In our sample, students placed in Algebra reported an increase in performance-avoidance goals as well as decreases in academic self-efficacy and task value. These relations were attenuated for students who had high mathematics achievement prior to Algebra placement. Whereas all students reported an overall decline in performance-approach goals over the course of eighth grade, previously high-achieving students reported an increase in these goals. Lastly, previously high-achieving students reported an increase in mastery goals. These findings suggest that while previously high-achieving students may benefit motivationally from eighth grade Algebra placement, placing previously average- and low-performing students in Algebra can potentially undermine their motivation for mathematics. PMID:26942210

Eighth Grade Algebra Course Placement and Student Motivation for Mathematics.

PubMed

Simzar, Rahila M; Domina, Thurston; Tran, Cathy

2016-01-01

This study uses student panel data to examine the association between Algebra placement and student motivation for mathematics. Changes in achievement goals, expectancy, and task value for students in eighth grade Algebra are compared with those of peers placed in lower-level mathematics courses (N = 3,306). In our sample, students placed in Algebra reported an increase in performance-avoidance goals as well as decreases in academic self-efficacy and task value. These relations were attenuated for students who had high mathematics achievement prior to Algebra placement. Whereas all students reported an overall decline in performance-approach goals over the course of eighth grade, previously high-achieving students reported an increase in these goals. Lastly, previously high-achieving students reported an increase in mastery goals. These findings suggest that while previously high-achieving students may benefit motivationally from eighth grade Algebra placement, placing previously average- and low-performing students in Algebra can potentially undermine their motivation for mathematics.

College Algebra I.

ERIC Educational Resources Information Center

Benjamin, Carl; And Others

Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra I. Topics covered include: sets; vocabulary; linear equations; inequalities; real numbers; operations; factoring; fractions; formulas; ratio, proportion, and…

libdrdc: software standards library

NASA Astrophysics Data System (ADS)

Erickson, David; Peng, Tie

2008-04-01

This paper presents the libdrdc software standards library including internal nomenclature, definitions, units of measure, coordinate reference frames, and representations for use in autonomous systems research. This library is a configurable, portable C-function wrapped C++ / Object Oriented C library developed to be independent of software middleware, system architecture, processor, or operating system. It is designed to use the automatically-tuned linear algebra suite (ATLAS) and Basic Linear Algebra Suite (BLAS) and port to firmware and software. The library goal is to unify data collection and representation for various microcontrollers and Central Processing Unit (CPU) cores and to provide a common Application Binary Interface (ABI) for research projects at all scales. The library supports multi-platform development and currently works on Windows, Unix, GNU/Linux, and Real-Time Executive for Multiprocessor Systems (RTEMS). This library is made available under LGPL version 2.1 license.

An algebra of discrete event processes

NASA Technical Reports Server (NTRS)

Heymann, Michael; Meyer, George

1991-01-01

This report deals with an algebraic framework for modeling and control of discrete event processes. The report consists of two parts. The first part is introductory, and consists of a tutorial survey of the theory of concurrency in the spirit of Hoare's CSP, and an examination of the suitability of such an algebraic framework for dealing with various aspects of discrete event control. To this end a new concurrency operator is introduced and it is shown how the resulting framework can be applied. It is further shown that a suitable theory that deals with the new concurrency operator must be developed. In the second part of the report the formal algebra of discrete event control is developed. At the present time the second part of the report is still an incomplete and occasionally tentative working paper.

Application of polynomial su(1, 1) algebra to Pöschl-Teller potentials

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

Zhang, Hong-Biao, E-mail: zhanghb017@nenu.edu.cn; Lu, Lu

2013-12-15

Two novel polynomial su(1, 1) algebras for the physical systems with the first and second Pöschl-Teller (PT) potentials are constructed, and their specific representations are presented. Meanwhile, these polynomial su(1, 1) algebras are used as an algebraic technique to solve eigenvalues and eigenfunctions of the Hamiltonians associated with the first and second PT potentials. The algebraic approach explores an appropriate new pair of raising and lowing operators K-circumflex{sub ±} of polynomial su(1, 1) algebra as a pair of shift operators of our Hamiltonians. In addition, two usual su(1, 1) algebras associated with the first and second PT potentials are derivedmore » naturally from the polynomial su(1, 1) algebras built by us.« less

On squares of representations of compact Lie algebras

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

Zeier, Robert, E-mail: robert.zeier@ch.tum.de; Zimborás, Zoltán, E-mail: zimboras@gmail.com

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the summore » of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.« less

Classical affine W-algebras associated to Lie superalgebras

NASA Astrophysics Data System (ADS)

Suh, Uhi Rinn

2016-02-01

In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras), which can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum affine W-superalgebras. Also, we show that a classical finite W-superalgebra can be obtained by a Zhu algebra of a classical affine W-superalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical W-superalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical W-algebra associated to spo(2|3) and its principal nilpotent. In the last part of this paper, we introduce a generalization of classical affine W-superalgebras called classical affine fractional W-superalgebras. We show these have Poisson vertex algebra structures and find generators of a fractional W-superalgebra associated to a minimal nilpotent.

Parabolas: Connection between Algebraic and Geometrical Representations

ERIC Educational Resources Information Center

Shriki, Atara

2011-01-01

A parabola is an interesting curve. What makes it interesting at the secondary school level is the fact that this curve is presented in both its contexts: algebraic and geometric. Being one of Apollonius' conic sections, the parabola is basically a geometric entity. It is, however, typically known for its algebraic characteristics, in particular…

Calif. Laws Shift Gears on Algebra, Textbooks

ERIC Educational Resources Information Center

Robelen, Erik W.

2012-01-01

New laws in California have set the state on a course for some potentially significant changes to the curriculum, including a measure that revisits the matter of teaching Algebra 1 in 8th grade and another that revamps the state's textbook-adoption process and hands districts greater leeway in choosing instructional materials. The algebra-related…

Gauss Elimination: Workhorse of Linear Algebra.

DTIC Science & Technology

1995-08-05

linear algebra computation for solving systems, computing determinants and determining the rank of matrix. All of these are discussed in varying contexts. These include different arithmetic or algebraic setting such as integer arithmetic or polynomial rings as well as conventional real (floating-point) arithmetic. These have effects on both accuracy and complexity analyses of the algorithm. These, too, are covered here. The impact of modern parallel computer architecture on GE is also

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Kreutz, Kenneth; Jain, Abhinandan

1989-01-01

A recently developed spatial operator algebra, useful for modeling, control, and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanizaton of specific algorithms is greatly simplified. The analytical formulation of the operator algebra, as well as its implementation in the Ada programming language are discussed.

Algebra for All: California’s Eighth-Grade Algebra Initiative as Constrained Curricula

PubMed Central

Domina, Thurston; Penner, Andrew M.; Penner, Emily K.; Conley, Annemarie

2015-01-01

Background/Context Across the United States, secondary school curricula are intensifying as a growing proportion of students enroll in high-level academic math courses. In many districts, this intensification process occurs as early as eighth grade, where schools are effectively constraining their mathematics curricula by restricting course offerings and placing more students into Algebra I. This paper provides a quantitative single-case research study of policy-driven curricular intensification in one California school district. Research Questions (1a) What effect did 8th eighth grade curricular intensification have on mathematics course enrollment patterns in Towering Pines Unified schools? (2b) How did the distribution of prior achievement in Towering Pines math classrooms change as the district constrained the curriculum by universalizing 8th eighth grade Algebra? (3c) Did 8th eighth grade curricular intensification improve students’ mathematics achievement? Setting Towering Pines is an immigrant enclave in the inner-ring suburbs of a major metropolitan area. The district’s 10 middle schools together enroll approximately 4,000 eighth graders each year. The districts’ students are ethnically diverse and largely economically disadvantaged. The study draws upon administrative data describing 8th eighth graders in the district in the 2004–20-05 through 2007–20-08 school years. Intervention/Program/Practice During the study period, Towering Pines dramatically intensified middle school students’ math curricula: In the 2004–20-05 school year 32% of the district’s 8th eighth graders enrolled in Algebra or a higher- level mathematics course; by the 2007–20-08 school year that proportion had increased to 84%. Research Design We use an interrupted time-series design, comparing students’ 8th eighth grade math course enrollments, 10th grade math course enrollments, and 10th grade math test scores across the four cohorts, controlling for demographics and

Eighth Grade Algebra Placement Policies: Promoting Equity, Achievement, and Access

ERIC Educational Resources Information Center

Wambsgans, Cynthia

2014-01-01

This study was an investigation of a standardized 8th grade Algebra I placement policy across multiple educational districts. Researchers have documented benefits of students' 8th grade Algebra I education, while others have detailed the consequences of algebra enrollment without necessary prerequisite skills. The purpose of this study was to…

The operator algebra approach to quantum groups

PubMed Central

Kustermans, Johan; Vaes, Stefaan

2000-01-01

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory. PMID:10639116

Using Linguistics in the Teaching of Developmental and Remedial Algebra.

ERIC Educational Resources Information Center

Lesnak, Richard J.

Basic algebra at Robert Morris College (RMC) in Pittsburgh, Pennsylvania, is a remedial course for students with virtually no algebra background, and for students whose previous experiences with algebra have created math blocks and math anxiety. A study was conducted in an effort to measure quantitatively the benefits of using linguistic methods…

Classical Affine W-Algebras and the Associated Integrable Hamiltonian Hierarchies for Classical Lie Algebras

NASA Astrophysics Data System (ADS)

De Sole, Alberto; Kac, Victor G.; Valeri, Daniele

2018-06-01

We prove that any classical affine W-algebra W (g, f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of generalized Adler type operators and of generalized quasideterminants, which we develop in the paper. Moreover, we show that under certain conditions, the product of two generalized Adler type operators is a Lax type operator. We use this fact to construct a large number of integrable Hamiltonian systems, recovering, as a special case, all KdV type hierarchies constructed by Drinfeld and Sokolov.

Surface defects and chiral algebras

NASA Astrophysics Data System (ADS)

Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng

2017-05-01

We investigate superconformal surface defects in four-dimensional N=2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfeld-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. In each case we find perfect agreement with the predicted characters.

Algebraic Semantics for Narrative

ERIC Educational Resources Information Center

Kahn, E.

1974-01-01

This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)

Numerical linear algebra in data mining

NASA Astrophysics Data System (ADS)

Eldén, Lars

Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.

Capitalizing on Basic Brain Processes in Developmental Algebra--Part 2

ERIC Educational Resources Information Center

Laughbaum, Edward D.

2011-01-01

Basic brain function is not a mystery. Given that neuroscientists understand its basic functioning processes, one wonders what their research suggests to teachers of developmental algebra. What if we knew how to teach so as to improve understanding of the algebra taught to developmental algebra students? What if we knew how the brain processes…

Capitalizing on Basic Brain Processes in Developmental Algebra--Part One

ERIC Educational Resources Information Center

Laughbaum, Edward D.

2011-01-01

Basic brain function is not a mystery. Given that neuroscientists understand the brain's basic functioning processes, one wonders what their research suggests to teachers of developmental algebra. What if we knew how to teach so as to improve understanding of the algebra taught to developmental algebra students? What if we knew how the brain…

The quantum holonomy-diffeomorphism algebra and quantum gravity

NASA Astrophysics Data System (ADS)

Aastrup, Johannes; Grimstrup, Jesper Møller

2016-03-01

We introduce the quantum holonomy-diffeomorphism ∗-algebra, which is generated by holonomy-diffeomorphisms on a three-dimensional manifold and translations on a space of SU(2)-connections. We show that this algebra encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Furthermore, we show that semiclassical states exist on the holonomy-diffeomorphism part of the algebra but that these states cannot be extended to the full algebra. Via a Dirac-type operator we derive a certain class of unbounded operators that act in the GNS construction of the semiclassical states. These unbounded operators are the type of operators, which we have previously shown to entail the spatial three-dimensional Dirac operator and Dirac-Hamiltonian in a semiclassical limit. Finally, we show that the structure of the Hamilton constraint emerges from a Yang-Mills-type operator over the space of SU(2)-connections.

Classical affine W-algebras associated to Lie superalgebras

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

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

2016-02-15

In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras), which can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum affine W-superalgebras. Also, we show that a classical finite W-superalgebra can be obtained by a Zhu algebra of a classical affine W-superalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical W-superalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical W-algebra associated to spo(2|3) and its principal nilpotent. In the last part of this paper, we introduce a generalizationmore » of classical affine W-superalgebras called classical affine fractional W-superalgebras. We show these have Poisson vertex algebra structures and find generators of a fractional W-superalgebra associated to a minimal nilpotent.« less

From No to Yes: The Impact of an Intervention on The Persistence of Algebraic Misconceptions among Secondary School Algebra Students

ERIC Educational Resources Information Center

Zielinski, Susan F.

2017-01-01

Many students enter high school with persistent algebraic misconceptions that limit their success in mathematics and, by extension, limit potential educational attainment and future earnings. The purpose of this study was to assess the effectiveness of a warm conceptual change based intervention on remediating algebraic misconceptions held by…

SD-CAS: Spin Dynamics by Computer Algebra System.

PubMed

Filip, Xenia; Filip, Claudiu

2010-11-01

A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples. Copyright © 2010 Elsevier Inc. All rights reserved.

Using Student Work to Develop Teachers' Knowledge of Algebra

ERIC Educational Resources Information Center

Herbel-Eisenmann, Beth A.; Phillips, Elizabeth Difanis

2005-01-01

This article describes a set of learning activities that use algebraic problems and written student work to help preservice and in-service teachers understand students' algebraic thinking. (Contains 4 figures.)

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.

Constitutive relations in optics in terms of geometric algebra

NASA Astrophysics Data System (ADS)

Dargys, A.

2015-11-01

To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p-form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.

Commutative Algebras of Toeplitz Operators in Action

NASA Astrophysics Data System (ADS)

Vasilevski, Nikolai

2011-09-01

We will discuss a quite unexpected phenomenon in the theory of Toeplitz operators on the Bergman space: the existence of a reach family of commutative C*-algebras generated by Toeplitz operators with non-trivial symbols. As it tuns out the smoothness properties of symbols do not play any role in the commutativity, the symbols can be merely measurable. Everything is governed here by the geometry of the underlying manifold, the hyperbolic geometry of the unit disk. We mention as well that the complete characterization of these commutative C*-algebras of Toeplitz operators requires the Berezin quantization procedure. These commutative algebras come with a powerful research tool, the spectral type representation for the operators under study, which permit us to answer to many important questions in the area.

On the intersection of irreducible components of the space of finite-dimensional Lie algebras

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

Gorbatsevich, Vladimir V

2012-07-31

The irreducible components of the space of n-dimensional Lie algebras are investigated. The properties of Lie algebras belonging to the intersection of all the irreducible components of this kind are studied (these Lie algebras are said to be basic or founding Lie algebras). It is proved that all Lie algebras of this kind are nilpotent and each of these Lie algebras has an Abelian ideal of codimension one. Specific examples of founding Lie algebras of arbitrary dimension are described and, to describe the Lie algebras in general, we state a conjecture. The concept of spectrum of a Lie algebra ismore » considered and some of the most elementary properties of the spectrum are studied. Bibliography: 6 titles.« less

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.

Abstract Algebra for Teachers: An Evaluative Case Study

ERIC Educational Resources Information Center

Hoffman, Andrew Joseph

2017-01-01

This manuscript describes the study of an abstract algebra course for preservice secondary mathematics teachers (PSMTs). Often, courses in abstract algebra have not been viewed as productive, beneficial learning experiences for future teachers, both by researchers and PSMTs themselves. This despite calls for increased content knowledge for…

Frobenius manifolds and Frobenius algebra-valued integrable systems

NASA Astrophysics Data System (ADS)

Strachan, Ian A. B.; Zuo, Dafeng

2017-06-01

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper, a new theory of Frobenius algebra-valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann (Int Math Res Not 19:929-952, 1996), Kontsevich and Manin (Inv Math 124: 313-339, 1996). By specializing this construction, using a fixed Frobenius algebra A, one can arrive at such a theory. More generally, one can apply the same idea to construct an A-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an A-valued modified Camassa-Holm equation is constructed.

The Great Debate: Should All 8th Graders Take Algebra?

ERIC Educational Resources Information Center

McKibben, Sarah

2009-01-01

While 8th grade algebra was once reserved as a course for the gifted, today, more U.S. 8th graders take algebra than any other math course. This article discusses a report from the Brookings Institution which chronicles the history of the 8th-grade algebra surge and its impact on today's low-performing students. The report indicates that many of…

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.

Surface defects and chiral algebras

DOE PAGES

Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng

2017-05-26

Here, we investigate superconformal surface defects in four-dimensional N = 2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfield-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. We find perfect agreement with the predicted characters, in eachmore » case.« less

Surface defects and chiral algebras

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

Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng

Here, we investigate superconformal surface defects in four-dimensional N = 2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfield-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. We find perfect agreement with the predicted characters, in eachmore » case.« less

On Orders of Observables on Effect Algebras

NASA Astrophysics Data System (ADS)

Dvurečenskij, Anatolij

2017-12-01

On the set of bounded observables on an effect algebra, the Olson order defined by spectral resolutions and the standard order defined by a system of σ-additive states are introduced. We show that sharp bounded observables form a Dedekind σ-complete sublattice of a Dedekind complete lattice under the Olson order. In addition, we compare both orders, and we illustrate them on different effect algebras.

Students’ Algebraic Thinking Process in Context of Point and Line Properties

NASA Astrophysics Data System (ADS)

Nurrahmi, H.; Suryadi, D.; Fatimah, S.

2017-09-01

Learning of schools algebra is limited to symbols and operating procedures, so students are able to work on problems that only require the ability to operate symbols but unable to generalize a pattern as one of part of algebraic thinking. The purpose of this study is to create a didactic design that facilitates students to do algebraic thinking process through the generalization of patterns, especially in the context of the property of point and line. This study used qualitative method and includes Didactical Design Research (DDR). The result is students are able to make factual, contextual, and symbolic generalization. This happen because the generalization arises based on facts on local terms, then the generalization produced an algebraic formula that was described in the context and perspective of each student. After that, the formula uses the algebraic letter symbol from the symbol t hat uses the students’ language. It can be concluded that the design has facilitated students to do algebraic thinking process through the generalization of patterns, especially in the context of property of the point and line. The impact of this study is this design can use as one of material teaching alternative in learning of school algebra.

A set for relational reasoning: Facilitation of algebraic modeling by a fraction task.

PubMed

DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J

2016-12-01

Recent work has identified correlations between early mastery of fractions and later math achievement, especially in algebra. However, causal connections between aspects of reasoning with fractions and improved algebra performance have yet to be established. The current study investigated whether relational reasoning with fractions facilitates subsequent algebraic reasoning using both pre-algebra students and adult college students. Participants were first given either a relational reasoning fractions task or a fraction algebra procedures control task. Then, all participants solved word problems and constructed algebraic equations in either multiplication or division format. The word problems and the equation construction tasks involved simple multiplicative comparison statements such as "There are 4 times as many students as teachers in a classroom." Performance on the algebraic equation construction task was enhanced for participants who had previously completed the relational fractions task compared with those who completed the fraction algebra procedures task. This finding suggests that relational reasoning with fractions can establish a relational set that promotes students' tendency to model relations using algebraic expressions. Copyright © 2016 Elsevier Inc. All rights reserved.

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

Quantum deformations of conformal algebras with mass-like deformation parameters

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

Frydryszak, Andrzej; Lukierski, Jerzy; Mozrzymas, Marek

1998-12-15

We recall the mathematical apparatus necessary for the quantum deformation of Lie algebras, namely the notions of coboundary Lie algebras, classical r-matrices, classical Yang-Baxter equations (CYBE), Froebenius algebras and parabolic subalgebras. Then we construct the quantum deformation of D=1, D=2 and D=3 conformal algebras, showing that this quantization introduce fundamental mass parameters. Finally we consider with more details the quantization of D=4 conformal algebra. We build three classes of sl(4,C) classical r-matrices, satisfying CYBE and depending respectively on 8, 10 and 12 generators of parabolic subalgebras. We show that only the 8-dimensional r-matrices allow to impose the D=4 conformal o(4,2){approx_equal}su(2,2)more » reality conditions. Weyl reflections and Dynkin diagram automorphisms for o(4,2) define the class of admissible bases for given classical r-matrices.« less

A Proposed Algebra Assessment for Use in a Problem-Analysis Framework

ERIC Educational Resources Information Center

Walick, Christopher M.; Burns, Matthew K.

2017-01-01

Algebra is critical to high school graduation and college success, but student achievement in algebra frequently falls significantly below expected proficiency levels. While existing research emphasizes the importance of quality algebra instruction, there is little research about how to conduct problem analysis for struggling secondary students.…

Associative Algebraic Approach to Logarithmic CFT in the Bulk: The Continuum Limit of the {gl(1|1)} Periodic Spin Chain, Howe Duality and the Interchiral Algebra

NASA Astrophysics Data System (ADS)

Gainutdinov, A. M.; Read, N.; Saleur, H.

2016-01-01

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed {gl(1|1)} spin-chain and its continuum limit—the {c=-2} symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245-288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289-329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL N , goes over in the continuum limit to a bigger algebra than {V}, the product of the left and right Virasoro algebras. This algebra, {S}—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field {S(z,bar{z})≡ S_{αβ} ψ^α(z)bar{ψ}^β(bar{z})}, with a symmetric form {S_{αβ}} and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL N in the {gl(1|1)} spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of {sp_{N-2}}. The semi-simple part of JTL N is represented by {U sp_{N-2}}, providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL N image represented in the spin-chain. On the continuum side, simple modules over {S} are identified with "fundamental" representations of {sp_∞}.

Chiral algebras in Landau-Ginzburg models

NASA Astrophysics Data System (ADS)

Dedushenko, Mykola

2018-03-01

Chiral algebras in the cohomology of the {\\overline{Q}}+ supercharge of two-dimensional N=(0,2) theories on flat spacetime are discussed. Using the supercurrent multiplet, we show that the answer is renormalization group invariant for theories with an R-symmetry. For N=(0,2) Landau-Ginzburg models, the chiral algebra is determined by the operator equations of motion, which preserve their classical form, and quantum renormalization of composite operators. We study these theories and then specialize to the N=(2,2) models and consider some examples.

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.

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.

Current algebra, statistical mechanics and quantum models

NASA Astrophysics Data System (ADS)

Vilela Mendes, R.

2017-11-01

Results obtained in the past for free boson systems at zero and nonzero temperatures are revisited to clarify the physical meaning of current algebra reducible functionals which are associated to systems with density fluctuations, leading to observable effects on phase transitions. To use current algebra as a tool for the formulation of quantum statistical mechanics amounts to the construction of unitary representations of diffeomorphism groups. Two mathematical equivalent procedures exist for this purpose. One searches for quasi-invariant measures on configuration spaces, the other for a cyclic vector in Hilbert space. Here, one argues that the second approach is closer to the physical intuition when modelling complex systems. An example of application of the current algebra methodology to the pairing phenomenon in two-dimensional fermion systems is discussed.

Line defect Schur indices, Verlinde algebras and U(1) r fixed points

NASA Astrophysics Data System (ADS)

Neitzke, Andrew; Yan, Fei

2017-11-01

Given an N=2 superconformal field theory, we reconsider the Schur index ℐ L ( q) in the presence of a half line defect L. Recently Cordova-Gaiotto-Shao found that ℐ L ( q) admits an expansion in terms of characters of the chiral algebra A introduced by Beem et al., with simple coefficients υ L, β ( q). We report a puzzling new feature of this expansion: the q → 1 limit of the coefficients υ L, β ( q) is linearly related to the vacuum expectation values 〈 L〉 in U(1) r -invariant vacua of the theory compactified on S 1. This relation can be expressed algebraically as a commutative diagram involving three algebras: the algebra generated by line defects, the algebra of functions on U(1) r -invariant vacua, and a Verlindelike algebra associated to A . Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type ( A 1, A 2), ( A 1, A 4), ( A 1, A 6), ( A 1, D 3) and ( A 1, D 5). In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.

Developing Meaning for Algebraic Procedures: An Exploration of the Connections Undergraduate Students Make between Algebraic Rational Expressions and Basic Number Properties

ERIC Educational Resources Information Center

Yantz, Jennifer

2013-01-01

The attainment and retention of later algebra skills in high school has been identified as a factor significantly impacting the postsecondary success of students majoring in STEM fields. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. The present study…

Algebraic Concepts: What's Really New in New Curricula?

ERIC Educational Resources Information Center

Star, Jon R.; Herbel-Eisenmann, Beth A.; Smith, John P., III

2000-01-01

Examines 8th grade units from the Connected Mathematics Project (CMP). Identifies differences in older and newer conceptions, fundamental objects of study, typical problems, and typical solution methods in algebra. Also discusses where the issue of what is new in algebra is relevant to many other innovative middle school curricula. (KHR)

Ada Linear-Algebra Program

NASA Technical Reports Server (NTRS)

Klumpp, A. R.; Lawson, C. L.

1988-01-01

Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.

Tensor models, Kronecker coefficients and permutation centralizer algebras

NASA Astrophysics Data System (ADS)

Geloun, Joseph Ben; Ramgoolam, Sanjaye

2017-11-01

We show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin decompositions into matrix blocks are given in terms of Clebsch-Gordan coefficients of symmetric groups. The matrix basis for the algebras also gives an orthogonal basis for the tensor observables which diagonalizes the Gaussian two-point functions. The centres of the algebras are associated with correlators which are expressible in terms of Kronecker coefficients (Clebsch-Gordan multiplicities of symmetric groups). The color-exchange symmetry present in the Gaussian model, as well as a large class of interacting models, is used to refine the description of the permutation centralizer algebras. This discussion is extended to a general number of colors d: it is used to prove the integrality of an infinite family of number sequences related to color-symmetrizations of colored graphs, and expressible in terms of symmetric group representation theory data. Generalizing a connection between matrix models and Belyi maps, correlators in Gaussian tensor models are interpreted in terms of covers of singular 2-complexes. There is an intriguing difference, between matrix and higher rank tensor models, in the computational complexity of superficially comparable correlators of observables parametrized by Young diagrams.

SATA II - Stochastic Algebraic Topology and Applications

DTIC Science & Technology

2017-01-30

AFRL-AFOSR-UK-TR-2017-0018 SATA II - Stochastic Algebraic Topology and Applications 150032 Robert Adler TECHNION ISRAEL INSTITUTE OF TECHNOLOGY Final...REPORT TYPE Final 3. DATES COVERED (From - To) 15 Dec 2014 to 14 Dec 2016 4. TITLE AND SUBTITLE SATA II - Stochastic Algebraic Topology and Applications ...has recently been submitted to AFOSR. 15. SUBJECT TERMS Network Theory, Sensor Technology, Mathematical Modeling, EOARD 16. SECURITY CLASSIFICATION OF

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.

Concreteness Fading of Algebraic Instruction: Effects on Learning

ERIC Educational Resources Information Center

Ottmar, Erin; Landy, David

2017-01-01

Learning algebra is difficult for many students in part because of an emphasis on the memorization of abstract rules. Algebraic reasoners across expertise levels often rely on perceptual-motor strategies to make these rules meaningful and memorable. However, in many cases, rules are provided as patterns to be memorized verbally, with little overt…

Complex quantum enveloping algebras as twisted tensor products

NASA Astrophysics Data System (ADS)

Chryssomalakos, Chryssomalis; Engeldinger, Ralf A.; Jurčo, Branislav; Schlieker, Michael; Zumino, Bruno

1994-12-01

We introduce a *-structure on the quantum double and its dual in order to make contact with various approaches to the enveloping algebras of complex quantum groups. Furthermore, we introduce a canonical basis in the quantum double, its universal R-matrices and give its relation to subgroups in the dual Hopf algebra.

Early Integration of Tutorial Support in Beginning Algebra

ERIC Educational Resources Information Center

Copus, Colleen; McKinney, Betsy

2016-01-01

Anecdotal observations reveal that most students with strong arithmetic skills will succeed in the Beginning Algebra course even if they have no previous experience with algebra. In trying to quantify this with an initial teacher-created survey of arithmetic skills, it was observed, for three consecutive semesters, that students who scored in the…

Priority in Process Algebras

NASA Technical Reports Server (NTRS)

Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.

1999-01-01

This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.

Quantization and Superselection Sectors I:. Transformation Group C*-ALGEBRAS

NASA Astrophysics Data System (ADS)

Landsman, N. P.

Quantization is defined as the act of assigning an appropriate C*-algebra { A} to a given configuration space Q, along with a prescription mapping self-adjoint elements of { A} into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here { A} is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of { A}. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of { A}.

On explicit algebraic stress models for complex turbulent flows

NASA Technical Reports Server (NTRS)

Gatski, T. B.; Speziale, C. G.

1992-01-01

Explicit algebraic stress models that are valid for three-dimensional turbulent flows in noninertial frames are systematically derived from a hierarchy of second-order closure models. This represents a generalization of the model derived by Pope who based his analysis on the Launder, Reece, and Rodi model restricted to two-dimensional turbulent flows in an inertial frame. The relationship between the new models and traditional algebraic stress models -- as well as anistropic eddy visosity models -- is theoretically established. The need for regularization is demonstrated in an effort to explain why traditional algebraic stress models have failed in complex flows. It is also shown that these explicit algebraic stress models can shed new light on what second-order closure models predict for the equilibrium states of homogeneous turbulent flows and can serve as a useful alternative in practical computations.

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Kreutz, K.; Milman, M.

1988-01-01

A powerful new spatial operator algebra for modeling, control, and trajectory design of manipulators is discussed along with its implementation in the Ada programming language. Applications of this algebra to robotics include an operator representation of the manipulator Jacobian matrix; the robot dynamical equations formulated in terms of the spatial algebra, showing the complete equivalence between the recursive Newton-Euler formulations to robot dynamics; the operator factorization and inversion of the manipulator mass matrix which immediately results in O(N) recursive forward dynamics algorithms; the joint accelerations of a manipulator due to a tip contact force; the recursive computation of the equivalent mass matrix as seen at the tip of a manipulator; and recursive forward dynamics of a closed chain system. Finally, additional applications and current research involving the use of the spatial operator algebra are discussed in general terms.

Quantum superintegrable system with a novel chain structure of quadratic algebras

NASA Astrophysics Data System (ADS)

Liao, Yidong; Marquette, Ian; Zhang, Yao-Zhong

2018-06-01

We analyse the n-dimensional superintegrable Kepler–Coulomb system with non-central terms. We find a novel underlying chain structure of quadratic algebras formed by the integrals of motion. We identify the elements for each sub-structure and obtain the algebra relations satisfied by them and the corresponding Casimir operators. These quadratic sub-algebras are realized in terms of a chain of deformed oscillators with factorized structure functions. We construct the finite-dimensional unitary representations of the deformed oscillators, and give an algebraic derivation of the energy spectrum of the superintegrable system.

Equivalent D = 3 supergravity amplitudes from double copies of three-algebra and two-algebra gauge theories.

PubMed

Huang, Yu-tin; Johansson, Henrik

2013-04-26

We show that three-dimensional supergravity amplitudes can be obtained as double copies of either three-algebra super-Chern-Simons matter theory or two-algebra super-Yang-Mills theory when either theory is organized to display the color-kinematics duality. We prove that only helicity-conserving four-dimensional gravity amplitudes have nonvanishing descendants when reduced to three dimensions, implying the vanishing of odd-multiplicity S-matrix elements, in agreement with Chern-Simons matter theory. We explicitly verify the double-copy correspondence at four and six points for N = 12,10,8 supergravity theories and discuss its validity for all multiplicity.

Generalized derivation extensions of 3-Lie algebras and corresponding Nambu-Poisson structures

NASA Astrophysics Data System (ADS)

Song, Lina; Jiang, Jun

2018-01-01

In this paper, we introduce the notion of a generalized derivation on a 3-Lie algebra. We construct a new 3-Lie algebra using a generalized derivation and call it the generalized derivation extension. We show that the corresponding Leibniz algebra on the space of fundamental objects is the double of a matched pair of Leibniz algebras. We also determine the corresponding Nambu-Poisson structures under some conditions.

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.

Deformed twistors and higher spin conformal (super-)algebras in four dimensions

DOE PAGES

Govil, Karan; Gunaydin, Murat

2015-03-05

Massless conformal scalar field in d = 4 corresponds to the minimal unitary representation (minrep) of the conformal group SU(2, 2) which admits a one-parameter family of deformations that describe massless fields of arbitrary helicity. The minrep and its deformations were obtained by quantization of the nonlinear realization of SU(2, 2) as a quasiconformal group in arXiv:0908.3624. We show that the generators of SU(2,2) for these unitary irreducible representations can be written as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group and apply them to define and study higher spin algebras and superalgebras in AdS 5.more » The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS 5 is simply the enveloping algebra of SU(2, 2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS 5. Furthermore, the enveloping algebras of the deformations of the minrep define a one parameter family of HS algebras in AdS 5 for which certain 4d covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras SU(2, 2|N) and we find a one parameter family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a family of (supersymmetric) HS theories in AdS 5 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 4d. We also discuss the corresponding picture in HS algebras in AdS 4 where the corresponding 3d conformal group Sp(4,R) admits only two massless representations (minreps), namely the scalar and spinor singletons.« less

Algebraic Thinking through Origami.

ERIC Educational Resources Information Center

Higginson, William; Colgan, Lynda

2001-01-01

Describes the use of paper folding to create a rich environment for discussing algebraic concepts. Explores the effect that changing the dimensions of two-dimensional objects has on the volume of related three-dimensional objects. (Contains 13 references.) (YDS)

A Process Algebra Approach to Quantum Electrodynamics

NASA Astrophysics Data System (ADS)

Sulis, William

2017-12-01

The process algebra program is directed towards developing a realist model of quantum mechanics free of paradoxes, divergences and conceptual confusions. From this perspective, fundamental phenomena are viewed as emerging from primitive informational elements generated by processes. The process algebra has been shown to successfully reproduce scalar non-relativistic quantum mechanics (NRQM) without the usual paradoxes and dualities. NRQM appears as an effective theory which emerges under specific asymptotic limits. Space-time, scalar particle wave functions and the Born rule are all emergent in this framework. In this paper, the process algebra model is reviewed, extended to the relativistic setting, and then applied to the problem of electrodynamics. A semiclassical version is presented in which a Minkowski-like space-time emerges as well as a vector potential that is discrete and photon-like at small scales and near-continuous and wave-like at large scales. QED is viewed as an effective theory at small scales while Maxwell theory becomes an effective theory at large scales. The process algebra version of quantum electrodynamics is intuitive and realist, free from divergences and eliminates the distinction between particle, field and wave. Computations are carried out using the configuration space process covering map, although the connection to second quantization has not been fully explored.

Analyzing Algebraic Thinking Using "Guess My Number" Problems

ERIC Educational Resources Information Center

Patton, Barba; De Los Santos, Estella

2012-01-01

The purpose of this study was to assess student knowledge of numeric, visual and algebraic representations. A definite gap between arithmetic and algebra has been documented in the research. The researchers' goal was to identify a link between the two. Using four "Guess My Number" problems, seventh and tenth grade students were asked to write…

Digital Tools for Algebra Education: Criteria and Evaluation

ERIC Educational Resources Information Center

Bokhove, Christian; Drijvers, Paul

2010-01-01

In the Netherlands, as in many other countries, the algebraic expertise of students graduating from secondary education is an issue. The use of digital tools for algebra education is expected to change epistemologies, activity structures and student achievement. Therefore, a study was set up to investigate in what way the use of ICT in upper…

Introduction to Matrix Algebra, Student's Text, Unit 23.

ERIC Educational Resources Information Center

Allen, Frank B.; And Others

Unit 23 in the SMSG secondary school mathematics series is a student text covering the following topics in matrix algebra: matrix operations, the algebra of 2 X 2 matrices, matrices and linear systems, representation of column matrices as geometric vectors, and transformations of the plane. Listed in the appendix are four research exercises in…

Emphasizing Language and Visualization in Teaching Linear Algebra

ERIC Educational Resources Information Center

Hannah, John; Stewart, Sepideh; Thomas, Mike

2013-01-01

Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his…

Numerical methods on some structured matrix algebra problems

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

Jessup, E.R.

1996-06-01

This proposal concerned the design, analysis, and implementation of serial and parallel algorithms for certain structured matrix algebra problems. It emphasized large order problems and so focused on methods that can be implemented efficiently on distributed-memory MIMD multiprocessors. Such machines supply the computing power and extensive memory demanded by the large order problems. We proposed to examine three classes of matrix algebra problems: the symmetric and nonsymmetric eigenvalue problems (especially the tridiagonal cases) and the solution of linear systems with specially structured coefficient matrices. As all of these are of practical interest, a major goal of this work was tomore » translate our research in linear algebra into useful tools for use by the computational scientists interested in these and related applications. Thus, in addition to software specific to the linear algebra problems, we proposed to produce a programming paradigm and library to aid in the design and implementation of programs for distributed-memory MIMD computers. We now report on our progress on each of the problems and on the programming tools.« less

Modular operads and the quantum open-closed homotopy algebra

NASA Astrophysics Data System (ADS)

Doubek, Martin; Jurčo, Branislav; Münster, Korbinian

2015-12-01

We verify that certain algebras appearing in string field theory are algebras over Feynman transform of modular operads which we describe explicitly. Equivalent description in terms of solutions of generalized BV master equations are explained from the operadic point of view.

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.

Exploring Teacher Noticing of Student Algebraic Thinking in a Video Club

ERIC Educational Resources Information Center

Walkoe, Janet

2015-01-01

Learning algebra is critical for students in the USA today, yet many students in the USA struggle in algebra classes. Researchers claim that one reason for these difficulties is that algebra classes often focus on symbol manipulation and procedures above, and many times at the expense of, a more conceptual understanding of the content. Teaching…

Symmetry algebra of a generalized anisotropic harmonic oscillator

NASA Technical Reports Server (NTRS)

Castanos, O.; Lopez-Pena, R.

1993-01-01

It is shown that the symmetry Lie algebra of a quantum system with accidental degeneracy can be obtained by means of the Noether's theorem. The procedure is illustrated by considering a generalized anisotropic two dimensional harmonic oscillator, which can have an infinite set of states with the same energy characterized by an u(1,1) Lie algebra.

Questions Arise about Algebra 2 for All Students

ERIC Educational Resources Information Center

Robelen, Erik W.

2013-01-01

Should all students take Algebra 2? Florida seemed to say "no" this spring with the passage of a law striking it from graduation requirements. Texas said much the same in legislation Republican Gov. Rick Perry signed this week that also backs away from Algebra 2 for all. Those steps come as the Common Core State Standards for math set…

Schroedinger operators with the q-ladder symmetry algebras

NASA Technical Reports Server (NTRS)

Skorik, Sergei; Spiridonov, Vyacheslav

1994-01-01

A class of the one-dimensional Schroedinger operators L with the symmetry algebra LB(+/-) = q(+/-2)B(+/-)L, (B(+),B(-)) = P(sub N)(L), is described. Here B(+/-) are the 'q-ladder' operators and P(sub N)(L) is a polynomial of the order N. Peculiarities of the coherent states of this algebra are briefly discussed.

An Uncommon Approach to a Common Algebraic Error

ERIC Educational Resources Information Center

Rossi, Paul S.

2008-01-01

The basic rules of elementary algebra can often appear beyond the grasp of many students. Even though most subjects, including calculus, prove to be more difficult, it is the simple rules of algebra that continue to be the "thorn in the side" of many mathematics students. In this paper we present a result intended to help students achieve a…

The Impact of New State Accountability Standards on Algebra I Students

ERIC Educational Resources Information Center

Heath, Kyle G.

2013-01-01

The purpose of this quasi-experimental quantitative study was to determine if a new Algebra I curriculum resulted in improved student performance on the state Algebra I exam. The treatment group consisted of 383 9th grade Algebra I students who received the college-ready standards-based (CRSB) curricula. The control group consisted of 338 9th…

Graph C ∗-algebras and Z2-quotients of quantum spheres

NASA Astrophysics Data System (ADS)

Hajac, Piotr M.; Matthes, Rainer; Szymański, Wojciech

2003-06-01

We consider two Z2-actions on the Podleś generic quantum spheres. They yield, as noncommutative quotient spaces, the Klimek-Lesmewski q-disc and the quantum real projective space, respectively. The C ∗-algebas of all these quantum spaces are described as graph C ∗-algebras. The K-groups of the thus presented C ∗-algebras are then easily determined from the general theory of graph C ∗-algebas. For the quantum real projective space, we also recall the classification of the classes of irreducible ∗-representations of its algebra and give a linear basis for this algebra.

Asymptotic identity in min-plus algebra: a report on CPNS.

PubMed

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions.

The Power of Algebra.

ERIC Educational Resources Information Center

Boiteau, Denise; Stansfield, David

This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative…

Algebra for Enterprise Ontology: towards analysis and synthesis of enterprise models

NASA Astrophysics Data System (ADS)

Suga, Tetsuya; Iijima, Junichi

2018-03-01

Enterprise modeling methodologies have made enterprises more likely to be the object of systems engineering rather than craftsmanship. However, the current state of research in enterprise modeling methodologies lacks investigations of the mathematical background embedded in these methodologies. Abstract algebra, a broad subfield of mathematics, and the study of algebraic structures may provide interesting implications in both theory and practice. Therefore, this research gives an empirical challenge to establish an algebraic structure for one aspect model proposed in Design & Engineering Methodology for Organizations (DEMO), which is a major enterprise modeling methodology in the spotlight as a modeling principle to capture the skeleton of enterprises for developing enterprise information systems. The results show that the aspect model behaves well in the sense of algebraic operations and indeed constructs a Boolean algebra. This article also discusses comparisons with other modeling languages and suggests future work.

Maximizing algebraic connectivity in air transportation networks

NASA Astrophysics Data System (ADS)

Wei, Peng

In air transportation networks the robustness of a network regarding node and link failures is a key factor for its design. An experiment based on the real air transportation network is performed to show that the algebraic connectivity is a good measure for network robustness. Three optimization problems of algebraic connectivity maximization are then formulated in order to find the most robust network design under different constraints. The algebraic connectivity maximization problem with flight routes addition or deletion is first formulated. Three methods to optimize and analyze the network algebraic connectivity are proposed. The Modified Greedy Perturbation Algorithm (MGP) provides a sub-optimal solution in a fast iterative manner. The Weighted Tabu Search (WTS) is designed to offer a near optimal solution with longer running time. The relaxed semi-definite programming (SDP) is used to set a performance upper bound and three rounding techniques are discussed to find the feasible solution. The simulation results present the trade-off among the three methods. The case study on two air transportation networks of Virgin America and Southwest Airlines show that the developed methods can be applied in real world large scale networks. The algebraic connectivity maximization problem is extended by adding the leg number constraint, which considers the traveler's tolerance for the total connecting stops. The Binary Semi-Definite Programming (BSDP) with cutting plane method provides the optimal solution. The tabu search and 2-opt search heuristics can find the optimal solution in small scale networks and the near optimal solution in large scale networks. The third algebraic connectivity maximization problem with operating cost constraint is formulated. When the total operating cost budget is given, the number of the edges to be added is not fixed. Each edge weight needs to be calculated instead of being pre-determined. It is illustrated that the edge addition and the

Contractions and deformations of quasiclassical Lie algebras preserving a nondegenerate quadratic Casimir operator

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

Campoamor-Stursberg, R., E-mail: rutwig@mat.ucm.e

2008-05-15

By means of contractions of Lie algebras, we obtain new classes of indecomposable quasiclassical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from noncompact real simple algebras with nonsimple complexification, where we impose that a nondegenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem and obtain sufficient conditions on integrable cocycles of quasiclassical Lie algebras in order to preserve nondegenerate quadratic Casimir operators by the associated linear deformations.

Systems with outer constraints. Gupta-Bleuler electromagnetism as an algebraic field theory

NASA Astrophysics Data System (ADS)

Grundling, Hendrik

1988-03-01

Since there are some important systems which have constraints not contained in their field algebras, we develop here in a C*-context the algebraic structures of these. The constraints are defined as a group G acting as outer automorphisms on the field algebra ℱ, α: G ↦ Aut ℱ, α G ⊄ Inn ℱ, and we find that the selection of G-invariant states on ℱ is the same as the selection of states ω on M( G M(Gmathop × limits_α F) ℱ) by ω( U g)=1∨ g∈ G, where U g ∈ M ( G M(Gmathop × limits_α F) ℱ)/ℱ are the canonical elements implementing α g . These states are taken as the physical states, and this specifies the resulting algebraic structure of the physics in M( G M(Gmathop × limits_α F) ℱ), and in particular the maximal constraint free physical algebra ℛ. A nontriviality condition is given for ℛ to exist, and we extend the notion of a crossed product to deal with a situation where G is not locally compact. This is necessary to deal with the field theoretical aspect of the constraints. Next the C*-algebra of the CCR is employed to define the abstract algebraic structure of Gupta-Bleuler electromagnetism in the present framework. The indefinite inner product representation structure is obtained, and this puts Gupta-Bleuler electromagnetism on a rigorous footing. Finally, as a bonus, we find that the algebraic structures just set up, provide a blueprint for constructive quadratic algebraic field theory.

A tensor Banach algebra approach to abstract kinetic equations

NASA Astrophysics Data System (ADS)

Greenberg, W.; van der Mee, C. V. M.

The study deals with a concrete algebraic construction providing the existence theory for abstract kinetic equation boundary-value problems, when the collision operator A is an accretive finite-rank perturbation of the identity operator in a Hilbert space H. An algebraic generalization of the Bochner-Phillips theorem is utilized to study solvability of the abstract boundary-value problem without any regulatory condition. A Banach algebra in which the convolution kernel acts is obtained explicitly, and this result is used to prove a perturbation theorem for bisemigroups, which then plays a vital role in solving the initial equations.

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.

Algebraic approach to small-world network models

NASA Astrophysics Data System (ADS)

Rudolph-Lilith, Michelle; Muller, Lyle E.

2014-01-01

We introduce an analytic model for directed Watts-Strogatz small-world graphs and deduce an algebraic expression of its defining adjacency matrix. The latter is then used to calculate the small-world digraph's asymmetry index and clustering coefficient in an analytically exact fashion, valid nonasymptotically for all graph sizes. The proposed approach is general and can be applied to all algebraically well-defined graph-theoretical measures, thus allowing for an analytical investigation of finite-size small-world graphs.

Birman—Wenzl—Murakami Algebra and Topological Basis

NASA Astrophysics Data System (ADS)

Zhou, Cheng-Cheng; Xue, Kang; Wang, Gang-Cheng; Sun, Chun-Fang; Du, Gui-Jiao

2012-02-01

In this paper, we use entangled states to construct 9 × 9-matrix representations of Temperley—Lieb algebra (TLA), then a family of 9 × 9-matrix representations of Birman—Wenzl—Murakami algebra (BWMA) have been presented. Based on which, three topological basis states have been found. And we apply topological basis states to recast nine-dimensional BWMA into its three-dimensional counterpart. Finally, we find the topological basis states are spin singlet states in special case.

Hilbert space structure in quantum gravity: an algebraic perspective

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

Giddings, Steven B.

If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime. Here, this viewpoint is supported by difficulties of such quantization, and by the apparent lack of a fundamental role for locality. In finite or discrete quantum systems, important structure is provided by tensor factorizations of the Hilbert space. However, even in local quantum field theory properties of the generic type III von Neumann algebras and of long range gauge fields indicate that factorization of themore » Hilbert space is problematic. Instead it is better to focus on the structure of the algebra of observables, and in particular on its subalgebras corresponding to regions. This paper suggests that study of analogous algebraic structure in gravity gives an important perspective on the nature of the quantum theory. Significant departures from the subalgebra structure of local quantum field theory are found, working in the correspondence limit of long-distances/low-energies. Particularly, there are obstacles to identifying commuting algebras of localized operators. In addition to suggesting important properties of the algebraic structure, this and related observations pose challenges to proposals of a fundamental role for entanglement.« less

Hilbert space structure in quantum gravity: an algebraic perspective

DOE PAGES

Giddings, Steven B.

2015-12-16

If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime. Here, this viewpoint is supported by difficulties of such quantization, and by the apparent lack of a fundamental role for locality. In finite or discrete quantum systems, important structure is provided by tensor factorizations of the Hilbert space. However, even in local quantum field theory properties of the generic type III von Neumann algebras and of long range gauge fields indicate that factorization of themore » Hilbert space is problematic. Instead it is better to focus on the structure of the algebra of observables, and in particular on its subalgebras corresponding to regions. This paper suggests that study of analogous algebraic structure in gravity gives an important perspective on the nature of the quantum theory. Significant departures from the subalgebra structure of local quantum field theory are found, working in the correspondence limit of long-distances/low-energies. Particularly, there are obstacles to identifying commuting algebras of localized operators. In addition to suggesting important properties of the algebraic structure, this and related observations pose challenges to proposals of a fundamental role for entanglement.« less

Directed Abelian algebras and their application to stochastic models.

PubMed

Alcaraz, F C; Rittenberg, V

2008-10-01

With each directed acyclic graph (this includes some D-dimensional lattices) one can associate some Abelian algebras that we call directed Abelian algebras (DAAs). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA, one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground-state wave functions (stationary state probability distributions) is an easy algebraic exercise. If one considers D-dimensional lattices and chooses Hamiltonians linear in the generators, in finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent z=D. One possible application of the DAA is to sandpile models. In the paper we present this application, considering one- and two-dimensional lattices. In the one-dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent sigma_(tau)=32 ). We study the local density of particles inside large avalanches, showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found sigma_(tau)=1.780+/-0.005 .

General Algebraic Modeling System Tutorial | High-Performance Computing |

Science.gov Websites

power generation from two different fuels. The goal is to minimize the cost for one of the fuels while Here's a basic tutorial for modeling optimization problems with the General Algebraic Modeling System (GAMS). Overview The GAMS (General Algebraic Modeling System) package is essentially a compiler for a

Emphasizing language and visualization in teaching linear algebra

NASA Astrophysics Data System (ADS)

Hannah, John; Stewart, Sepideh; Thomas, Mike

2013-06-01

Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his approach in both lectures and tutorials, and how he employed visualization and an emphasis on language to encourage conceptual thinking. We use Tall's framework of three worlds of mathematical thinking to reflect on the effect of these activities in students' learning. An analysis of students' attitudes to the course and their test and examination results help to answer questions about the value of such an approach, suggesting ways forward in teaching linear algebra.

Solving Our Algebra Problem: Getting All Students through Algebra I to Improve Graduation Rates

ERIC Educational Resources Information Center

Schachter, Ron

2013-01-01

graduation as well as admission to most colleges. But taking algebra also can turn into a pathway for failure, from which some students never recover. In 2010, a national U.S. Department of Education study…

Implementing the Curriculum and Evaluation Standards: First-Year Algebra.

ERIC Educational Resources Information Center

Kysh, Judith

1991-01-01

Described is an alternative first year algebra program developed to bridge the gap between the NCTM's Curriculum and Evaluation Standards and institutional demands of schools. Increased attention is given to graphing as a context for algebra, calculator use, solving "memorable problems," and incorporating geometry concepts, while…

Statistical Aspects of Coherent States of the Higgs Algebra

NASA Astrophysics Data System (ADS)

Shreecharan, T.; Kumar, M. Naveen

2018-04-01

We construct and study various aspects of coherent states of a polynomial angular momentum algebra. The coherent states are constructed using a new unitary representation of the nonlinear algebra. The new representation involves a parameter γ that shifts the eigenvalues of the diagonal operator J 0.

Aspects of QCD current algebra on a null plane

NASA Astrophysics Data System (ADS)

Beane, S. R.; Hobbs, T. J.

2016-09-01

Consequences of QCD current algebra formulated on a light-like hyperplane are derived for the forward scattering of vector and axial-vector currents on an arbitrary hadronic target. It is shown that current algebra gives rise to a special class of sum rules that are direct consequences of the independent chiral symmetry that exists at every point on the two-dimensional transverse plane orthogonal to the lightlike direction. These sum rules are obtained by exploiting the closed, infinite-dimensional algebra satisfied by the transverse moments of null-plane axial-vector and vector charge distributions. In the special case of a nucleon target, this procedure leads to the Adler-Weisberger, Gerasimov-Drell-Hearn, Cabibbo-Radicati and Fubini-Furlan-Rossetti sum rules. Matching to the dispersion-theoretic language which is usually invoked in deriving these sum rules, the moment sum rules are shown to be equivalent to algebraic constraints on forward S-matrix elements in the Regge limit.

The noncommutative Poisson bracket and the deformation of the family algebras

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

Wei, Zhaoting, E-mail: zhaotwei@indiana.edu

The family algebras are introduced by Kirillov in 2000. In this paper, we study the noncommutative Poisson bracket P on the classical family algebra C{sub τ}(g). We show that P controls the first-order 1-parameter formal deformation from C{sub τ}(g) to Q{sub τ}(g) where the latter is the quantum family algebra. Moreover, we will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary, and therefore, the deformation is infinitesimally trivial. In the last part of this paper, we discuss the relation between Mackey’s analogue and the quantization problem of the family algebras.

Critical role of the proton-dependent oligopeptide transporter (POT) in the cellular uptake of the peptidyl nucleoside antibiotic, blasticidin S.

PubMed

Kitamura, Kenji; Kinsui, Eldaa Zefany Banami; Abe, Fumiyoshi

2017-02-01

Blasticidin S (BlaS) interferes in the cell growth of both eukaryotes and prokaryotes. Its mode of action as a protein synthesis inhibitor has been investigated extensively. However, the mechanism of BlaS transport into the target cells is not understood well. Here, we show that Ptr2, a member of the proton-dependent oligopeptide transporter (POT) family, is responsible for the uptake of BlaS in yeasts Schizosaccharomyces pombe and Saccharomyces cerevisiae. Notably, some mutants of Ptr2 that are dysfunctional in dipeptide uptake were still competent to transport BlaS. Mouse-derived oligopeptide transporter PepT1 conferred BlaS sensitivity in the S. cerevisiae ptr2∆ mutant. Furthermore, bacterial POT family proteins also potentiated the BlaS sensitivity of E. coli. The role of the POT family oligopeptide transporters in the uptake of BlaS is conserved across species from bacteria to mammals. Copyright © 2016 Elsevier B.V. All rights reserved.

Enhanced asymptotic symmetry algebra of (2 +1 ) -dimensional flat space

NASA Astrophysics Data System (ADS)

Detournay, Stéphane; Riegler, Max

2017-02-01

In this paper we present a new set of asymptotic boundary conditions for Einstein gravity in (2 +1 ) -dimensions with a vanishing cosmological constant that are a generalization of the Barnich-Compère boundary conditions [G. Barnich and G. Compere, Classical Quantum Gravity 24, F15 (2007), 10.1088/0264-9381/24/5/F01]. These new boundary conditions lead to an asymptotic symmetry algebra that is generated by a bms3 algebra and two affine u ^(1 ) current algebras. We then apply these boundary conditions to topologically massive gravity (TMG) and determine how the presence of the gravitational Chern-Simons term affects the central extensions of the asymptotic symmetry algebra. We furthermore determine the thermal entropy of solutions obeying our new boundary conditions for both Einstein gravity and TMG.

Asymptotic Identity in Min-Plus Algebra: A Report on CPNS

PubMed Central

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions. PMID:21822446

Vague Congruences and Quotient Lattice Implication Algebras

PubMed Central

Qin, Xiaoyan; Xu, Yang

2014-01-01

The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given. PMID:25133207

Soft translations and soft extensions of BCI/BCK-algebras.

PubMed

Sultana, Nazra; Rani, Nazia; Ali, Muhammad Irfan; Hussain, Azhar

2014-01-01

The concept of soft translations of soft subalgebras and soft ideals over BCI/BCK-algebras is introduced and some related properties are studied. Notions of Soft extensions of soft subalgebras and soft ideals over BCI/BCK-algebras are also initiated. Relationships between soft translations and soft extensions are explored.

Teaching Linear Algebra: Must the Fog Always Roll In?

ERIC Educational Resources Information Center

Carlson, David

1993-01-01

Proposes methods to teach the more difficult concepts of linear algebra. Examines features of the Linear Algebra Curriculum Study Group Core Syllabus, and presents problems from the core syllabus that utilize the mathematical process skills of making conjectures, proving the results, and communicating the results to colleagues. Presents five…

Promoting Quantitative Literacy in an Online College Algebra Course

ERIC Educational Resources Information Center

Tunstall, Luke; Bossé, Michael J.

2016-01-01

College algebra (a university freshman level algebra course) fulfills the quantitative literacy requirement of many college's general education programs and is a terminal course for most who take it. An online problem-based learning environment provides a unique means of engaging students in quantitative discussions and research. This article…

A Structural Model of Algebra Achievement: Computational Fluency and Spatial Visualisation as Mediators of the Effect of Working Memory on Algebra Achievement

ERIC Educational Resources Information Center

Tolar, Tammy Daun; Lederberg, Amy R.; Fletcher, Jack M.

2009-01-01

The goal of this study was to develop and evaluate a structural model of the relations among cognitive abilities and arithmetic skills and college students' algebra achievement. The model of algebra achievement was compared to a model of performance on the Scholastic Assessment in Mathematics (SAT-M) to determine whether the pattern of relations…

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

PubMed

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

2016-01-01

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

Soft Translations and Soft Extensions of BCI/BCK-Algebras

PubMed Central

Sultana, Nazra; Rani, Nazia; Ali, Muhammad Irfan

2014-01-01

The concept of soft translations of soft subalgebras and soft ideals over BCI/BCK-algebras is introduced and some related properties are studied. Notions of Soft extensions of soft subalgebras and soft ideals over BCI/BCK-algebras are also initiated. Relationships between soft translations and soft extensions are explored. PMID:25298968

The Algebra Initiative Colloquium. Volume 2: Working Group Papers.

ERIC Educational Resources Information Center

Lacampagne, Carole B., Ed.; And Others

This volume presents recommendations from four working groups at a conference on reform in algebra held in Leesburg, Virginia, December 9-12, 1993. Working Group 1: Creating an Appropriate Algebra Experience for All Grades K-12 Students produced the following papers: (1) "Report" (A. H. Schoenfeld); (2) "Five Questions About Algebra…

Comparing the Effectiveness of Collaborative Instructional Practices in Algebra

ERIC Educational Resources Information Center

Triaga, Russell D.

2014-01-01

The use of multiple forms of collaborative instruction to teach integrated algebra makes it difficult for teachers to determine which collaborative form is best suited for the curriculum. An inconsistent approach to integrated algebra instruction at the study school needed to be addressed for the benefit of teacher effectiveness and student…

Boolean Operations with Prism Algebraic Patches

PubMed Central

Bajaj, Chandrajit; Paoluzzi, Alberto; Portuesi, Simone; Lei, Na; Zhao, Wenqi

2009-01-01

In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C1-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation. PMID:21516262

Comparison of the Effectiveness of a Traditional Intermediate Algebra Course With That of a Less Rigorous Intermediate Algebra Course in Preparing Students for Success in a Subsequent Mathematics Course

ERIC Educational Resources Information Center

Sworder, Steven C.

2007-01-01

An experimental two-track intermediate algebra course was offered at Saddleback College, Mission Viejo, CA, between the Fall, 2002 and Fall, 2005 semesters. One track was modeled after the existing traditional California community college intermediate algebra course and the other track was a less rigorous intermediate algebra course in which the…

Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*

DOE PAGES

Bank, R.; Falgout, R. D.; Jones, T.; ...

2015-10-29

In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods inmore » Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.« less

The Algebra Artist

ERIC Educational Resources Information Center

Beigie, Darin

2014-01-01

Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…

An Algebraic Approach to the Eigenstates of the Calogero Model

NASA Astrophysics Data System (ADS)

Ujino, Hideaki

2002-11-01

An algebraic treatment of the eigenstates of the (AN-1-) Calogero model is presented, which provides an algebraic construction of the nonsymmetric orthogonal eigenvectors, symmetrization, antisymmetrization and calculation of square norms in a unified way.

Some Applications Of Semigroups And Computer Algebra In Discrete Structures

NASA Astrophysics Data System (ADS)

Bijev, G.

2009-11-01

An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. A map (p) is defined, which is similar to an orthogonal projection in linear vector spaces. Some other important maps with properties similar to those of the generalized inverses (pseudoinverses) of linear transformations and matrices corresponding to them are also defined and investigated. Let Ax = b be an equation with matrix A and vectors x and b Boolean. Stochastic experiments for solving the equation, which involves the maps defined and use computer algebra methods, have been made. As a result, the Hamming distance between vectors Ax = p(b) and b is equal or close to the least possible. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra and research. Some examples for computations with binary relations using Maple are given.

Algebraic Generalization Strategies Used by Kuwaiti Pre-Service Teachers

ERIC Educational Resources Information Center

Alajmi, Amal Hussain

2016-01-01

This study reports on the algebraic generalization strategies used by elementary and middle/high school pre-service mathematics teachers in Kuwait. They were presented with 9 tasks that involved linear, exponential, and quadratic situations. The results showed that these pre-service teachers had difficulty in generalizing algebraic rules in all 3…

Resources for Teaching Linear Algebra. MAA Notes Volume 42.

ERIC Educational Resources Information Center

Carlson, David, Ed.; And Others

This book takes the position that the teaching of elementary linear algebra can be made more effective by emphasizing applications, exposition, and pedagogy. It includes the recommendations of the Linear Algebra Curriculum Study Group with their core syllabus for the first course, and the thoughts of mathematics faculty who have taught linear…

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.

College Algebra Students' Attitudes toward Mathematics in Their Careers

ERIC Educational Resources Information Center

Champion, Joe; Parker, Frieda; Mendoza-Spencer, Bernadette; Wheeler, Ann

2011-01-01

The purpose of this study was to identify the degree to which college algebra students' value mathematical skills in their prospective careers. A survey was administered to N = 144 students in 6 college algebra classes at a mid-sized doctoral granting university. Students in half the classes completed a data analysis project, and half of the…

Quantum walks, deformed relativity and Hopf algebra symmetries

PubMed Central

2016-01-01

We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014 Phys. Rev. A 90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras–the usual Poincaré and the κ-Poincaré algebras. PMID:27091171

Pawlak Algebra and Approximate Structure on Fuzzy Lattice

PubMed Central

Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai

2014-01-01

The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties. PMID:25152922

Pawlak algebra and approximate structure on fuzzy lattice.

PubMed

Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai

2014-01-01

The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties.

Thinking Visually about Algebra

ERIC Educational Resources Information Center

Baroudi, Ziad

2015-01-01

Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…

Development of abstract mathematical reasoning: the case of algebra.

PubMed

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

2014-01-01

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

Cartooning in Algebra and Calculus

ERIC Educational Resources Information Center

Moseley, L. Jeneva

2014-01-01

This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.

Designing Virtual Worlds for Use in Mathematics Education: The Example of Experiential Algebra.

ERIC Educational Resources Information Center

Winn, William; Bricken, William

1992-01-01

Discussion of the use of virtual reality (VR) to help students learn highlights the use of VR with elementary algebra. Learning theory is examined, including knowledge construction; knowledge representation is discussed, including the symbol systems of algebra; and spatial algebra is described and illustrated. (34 references) (LRW)

On Correspondence of BRST-BFV, Dirac, and Refined Algebraic Quantizations of Constrained Systems

NASA Astrophysics Data System (ADS)

Shvedov, O. Yu.

2002-11-01

The correspondence between BRST-BFV, Dirac, and refined algebraic (group averaging, projection operator) approaches to quantizing constrained systems is analyzed. For the closed-algebra case, it is shown that the component of the BFV wave function corresponding to maximal (minimal) value of number of ghosts and antighosts in the Schrodinger representation may be viewed as a wave function in the refined algebraic (Dirac) quantization approach. The Giulini-Marolf group averaging formula for the inner product in the refined algebraic quantization approach is obtained from the Batalin-Marnelius prescription for the BRST-BFV inner product, which should be generally modified due to topological problems. The considered prescription for the correspondence of states is observed to be applicable to the open-algebra case. The refined algebraic quantization approach is generalized then to the case of nontrivial structure functions. A simple example is discussed. The correspondence of observables for different quantization methods is also investigated.

A natural history of mathematics: George Peacock and the making of English algebra.

PubMed

Lambert, Kevin

2013-06-01

In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic would suggest arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's "principle of equivalent forms," which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra.

Selecting reusable components using algebraic specifications

NASA Technical Reports Server (NTRS)

Eichmann, David A.

1992-01-01

A significant hurdle confronts the software reuser attempting to select candidate components from a software repository - discriminating between those components without resorting to inspection of the implementation(s). We outline a mixed classification/axiomatic approach to this problem based upon our lattice-based faceted classification technique and Guttag and Horning's algebraic specification techniques. This approach selects candidates by natural language-derived classification, by their interfaces, using signatures, and by their behavior, using axioms. We briefly outline our problem domain and related work. Lattice-based faceted classifications are described; the reader is referred to surveys of the extensive literature for algebraic specification techniques. Behavioral support for reuse queries is presented, followed by the conclusions.

Conceptualizing Routines of Practice That Support Algebraic Reasoning in Elementary Schools: A Constructivist Grounded Theory

ERIC Educational Resources Information Center

Store, Jessie Chitsanzo

2012-01-01

There is ample literature documenting that, for many decades, high school students view algebra as difficult and do not demonstrate understanding of algebraic concepts. Algebraic reasoning in elementary school aims at meaningfully introducing algebra to elementary school students in preparation for higher-level mathematics. While there is research…

Multifractal vector fields and stochastic Clifford algebra.

PubMed

Schertzer, Daniel; Tchiguirinskaia, Ioulia

2015-12-01

In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.

Operator algebra as an application of logarithmic representation of infinitesimal generators

NASA Astrophysics Data System (ADS)

Iwata, Yoritaka

2018-02-01

The operator algebra is introduced based on the framework of logarithmic representation of infinitesimal generators. In conclusion a set of generally-unbounded infinitesimal generators is characterized as a module over the Banach algebra.

A Meta-Analysis of Algebra Interventions for Learners with Disabilities and Struggling Learners

ERIC Educational Resources Information Center

Hughes, Elizabeth M.; Witzel, Bradley S.; Riccomini, Paul J.; Fries, Karen M.; Kanyongo, Gibbs Y.

2014-01-01

The need for global competence in mathematics is apparent. Algebra is considered a gateway course to prepare students for the demands of a competitive global market. Many students demonstrate low performance in algebra; this is especially true for students with disabilities. Effective algebra instruction is essential to increase algebra…

Geometry of quantum state manifolds generated by the Lie algebra operators

NASA Astrophysics Data System (ADS)

Kuzmak, A. R.

2018-03-01

The Fubini-Study metric of quantum state manifold generated by the operators which satisfy the Heisenberg Lie algebra is calculated. The similar problem is studied for the manifold generated by the so(3) Lie algebra operators. Using these results, we calculate the Fubini-Study metrics of state manifolds generated by the position and momentum operators. Also the metrics of quantum state manifolds generated by some spin systems are obtained. Finally, we generalize this problem for operators of an arbitrary Lie algebra.

Cognitive Tutor[R] Algebra I. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2009

2009-01-01

The "Cognitive Tutor[R] Algebra I" curriculum, published by Carnegie Learning, is an approach that combines algebra textbooks with interactive software. The software is developed around an artificial intelligence model that identifies strengths and weaknesses in each individual student's mastery of mathematical concepts. It then customizes prompts…

Measuring the Readability of Elementary Algebra Using the Cloze Technique.

ERIC Educational Resources Information Center

Kulm, Gerald

The relationship to readability of ten variables characterizing structural properties of mathematical prose was investigated in elementary algebra textbooks. Readability was measured by algebra student's responses to two forms of cloze tests. Linear and currilinear correlations were calculated between each structural variable and the cloze test.…

Pay-Offs from Expanding Summer Credit Recovery in Algebra

ERIC Educational Resources Information Center

Allensworth, Elaine; Nomi, Takako; Heppen, Jessica

2013-01-01

The consequences of failing core academic courses during the first year are dire. In Chicago, over a quarter of students fail at least one semester of algebra in their ninth grade year, and only 13% of students who fail both semesters of Algebra I in ninth grade graduate in 4 years. Offering credit recovery options is one strategy to deal with…

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.

Super-BMS3 algebras from {N}=2 flat supergravities

NASA Astrophysics Data System (ADS)

Lodato, Ivano; Merbis, Wout

2016-11-01

We consider two possible flat space limits of three dimensional {N}=(1, 1) AdS supergravity. They differ by how the supercharges are scaled with the AdS radius ℓ: the first limit (democratic) leads to the usual super-Poincaré theory, while a novel `twisted' theory of supergravity stems from the second (despotic) limit. We then propose boundary conditions such that the asymptotic symmetry algebras at null infinity correspond to supersymmetric extensions of the BMS algebras previously derived in connection to non- and ultra-relativistic limits of the {N}=(1, 1) Virasoro algebra in two dimensions. Finally, we study the supersymmetric energy bounds and find the explicit form of the asymptotic and global Killing spinors of supersymmetric solutions in both flat space supergravity theories.

Relativistic Causality and Quasi-Orthomodular Algebras

NASA Astrophysics Data System (ADS)

Nobili, Renato

2006-05-01

The concept of fractionability or decomposability in parts of a physical system has its mathematical counterpart in the lattice--theoretic concept of orthomodularity. Systems with a finite number of degrees of freedom can be decomposed in different ways, corresponding to different groupings of the degrees of freedom. The orthomodular structure of these simple systems is trivially manifest. The problem then arises as to whether the same property is shared by physical systems with an infinite number of degrees of freedom, in particular by the quantum relativistic ones. The latter case was approached several years ago by Haag and Schroer (1962; Haag, 1992) who started from noting that the causally complete sets of Minkowski spacetime form an orthomodular lattice and posed the question of whether the subalgebras of local observables, with topological supports on such subsets, form themselves a corresponding orthomodular lattice. Were it so, the way would be paved to interpreting spacetime as an intrinsic property of a local quantum field algebra. Surprisingly enough, however, the hoped property does not hold for local algebras of free fields with superselection rules. The possibility seems to be instead open if the local currents that govern the superselection rules are driven by gauge fields. Thus, in the framework of local quantum physics, the request for algebraic orthomodularity seems to imply physical interactions! Despite its charm, however, such a request appears plagued by ambiguities and criticities that make of it an ill--posed problem. The proposers themselves, indeed, concluded that the orthomodular correspondence hypothesis is too strong for having a chance of being practicable. Thus, neither the idea was taken seriously by the proposers nor further investigated by others up to a reasonable degree of clarification. This paper is an attempt to re--formulate and well--pose the problem. It will be shown that the idea is viable provided that the algebra of

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.

Image-algebraic design of multispectral target recognition algorithms

NASA Astrophysics Data System (ADS)

Schmalz, Mark S.; Ritter, Gerhard X.

1994-06-01

In this paper, we discuss methods for multispectral ATR (Automated Target Recognition) of small targets that are sensed under suboptimal conditions, such as haze, smoke, and low light levels. In particular, we discuss our ongoing development of algorithms and software that effect intelligent object recognition by selecting ATR filter parameters according to ambient conditions. Our algorithms are expressed in terms of IA (image algebra), a concise, rigorous notation that unifies linear and nonlinear mathematics in the image processing domain. IA has been implemented on a variety of parallel computers, with preprocessors available for the Ada and FORTRAN languages. An image algebra C++ class library has recently been made available. Thus, our algorithms are both feasible implementationally and portable to numerous machines. Analyses emphasize the aspects of image algebra that aid the design of multispectral vision algorithms, such as parameterized templates that facilitate the flexible specification of ATR filters.

Spherical Hecke algebra in the Nekrasov-Shatashvili limit

NASA Astrophysics Data System (ADS)

Bourgine, Jean-Emile

2015-01-01

The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal variations of the roots. These operators obey the commutation relations defining the SHc algebra at first order in the equivariant parameter ɛ 2. Furthermore, their action on the bifundamental contributions reproduces the Kanno-Matsuo-Zhang transformation. We also discuss the connections with the Mayer cluster expansion approach that leads to TBA-like equations.

Learning to Apply Algebra in the Community for Adults with Intellectual Developmental Disabilities

ERIC Educational Resources Information Center

Rodriguez, Anthony M.

2016-01-01

Students with intellectual and developmental disabilities (IDD) are routinely excluded from algebra and other high-level mathematics courses. High school students with IDD take courses in arithmetic and life skills rather than having an opportunity to learn algebra. Yet algebra skills can support the learning of money and budgeting skills. This…

Advanced Algebra and Calculus. High School Mathematics Curricula. Instructor's Guide.

ERIC Educational Resources Information Center

Natour, Denise M.

This manual is an instructor's guide for the utilization of the "CCA High School Mathematics Curricula: Advanced Algebra and Calculus" courseware developed by the Computer-based Education Research Laboratory (CERL). The curriculum comprises 34 algebra lessons within 12 units and 15 calculus lessons that are computer-based and require…

A Learning Progressions Approach to Early Algebra Research and Practice

ERIC Educational Resources Information Center

Fonger, Nicole L.; Stephens, Ana; Blanton, Maria; Knuth, Eric

2015-01-01

We detail a learning progressions approach to early algebra research and how existing work around learning progressions and trajectories in mathematics and science education has informed our development of a four-component theoretical framework consisting of: a curricular progression of learning goals across big algebraic ideas; an instructional…

Weak Lie symmetry and extended Lie algebra

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

Goenner, Hubert

2013-04-15

The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found ('extended Lie algebras') which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).

Investigating Students' Modes of Thinking in Linear Algebra: The Case of Linear Independence

ERIC Educational Resources Information Center

Çelik, Derya

2015-01-01

Linear algebra is one of the most challenging topics to learn and teach in many countries. To facilitate the teaching and learning of linear algebra, priority should be given to epistemologically analyze the concepts that the undergraduate students have difficulty in conceptualizing and to define their ways of reasoning in linear algebra. After…

Image Algebra Matlab language version 2.3 for image processing and compression research

NASA Astrophysics Data System (ADS)

Schmalz, Mark S.; Ritter, Gerhard X.; Hayden, Eric

2010-08-01

Image algebra is a rigorous, concise notation that unifies linear and nonlinear mathematics in the image domain. Image algebra was developed under DARPA and US Air Force sponsorship at University of Florida for over 15 years beginning in 1984. Image algebra has been implemented in a variety of programming languages designed specifically to support the development of image processing and computer vision algorithms and software. The University of Florida has been associated with development of the languages FORTRAN, Ada, Lisp, and C++. The latter implementation involved a class library, iac++, that supported image algebra programming in C++. Since image processing and computer vision are generally performed with operands that are array-based, the Matlab™ programming language is ideal for implementing the common subset of image algebra. Objects include sets and set operations, images and operations on images, as well as templates and image-template convolution operations. This implementation, called Image Algebra Matlab (IAM), has been found to be useful for research in data, image, and video compression, as described herein. Due to the widespread acceptance of the Matlab programming language in the computing community, IAM offers exciting possibilities for supporting a large group of users. The control over an object's computational resources provided to the algorithm designer by Matlab means that IAM programs can employ versatile representations for the operands and operations of the algebra, which are supported by the underlying libraries written in Matlab. In a previous publication, we showed how the functionality of IAC++ could be carried forth into a Matlab implementation, and provided practical details of a prototype implementation called IAM Version 1. In this paper, we further elaborate the purpose and structure of image algebra, then present a maturing implementation of Image Algebra Matlab called IAM Version 2.3, which extends the previous implementation

Algebraic methods in system theory

NASA Technical Reports Server (NTRS)

Brockett, R. W.; Willems, J. C.; Willsky, A. S.

1975-01-01

Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.

Implementation of Algebra I in Eighth Grade: An "Ex-Post Facto" Study on Student Achievement

ERIC Educational Resources Information Center

Realdine, Dorothy S.

2010-01-01

Only recently have school districts across the nation begun to offer Algebra I to all eighth grade students. Currently, most eighth grade Algebra I curriculum does not have a national consistent focus of topics or level of rigor. A key issue of implementing Algebra I in eighth grade is defining national Algebra I concepts and skills that students…

Development of abstract mathematical reasoning: the case of algebra

PubMed Central

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

2014-01-01

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

Exact Baker-Campbell-Hausdorff formula for the contact Heisenberg algebra

NASA Astrophysics Data System (ADS)

Bravetti, Alessandro; Garcia-Chung, Angel; Tapias, Diego

2017-03-01

In this work we introduce the contact Heisenberg algebra which is the restriction of the Jacobi algebra on contact manifolds to the linear and constant functions. We give the exact expression of its corresponding Baker-Campbell-Hausdorff formula. We argue that this result is relevant to the quantization of contact systems.

Using CRA to Teach Algebra to Students with Math Difficulties in Inclusive Settings

ERIC Educational Resources Information Center

Witzel, Bradley S.

2005-01-01

The importance of algebra instruction has increased in the United States in the past few years. Thus, in most states, middle school students are required to take Algebra 1. Middle school students with math difficulties in inclusion algebra settings may require a different instructional approach. The purpose of this research was to compare student…

A Electro-Optical Image Algebra Processing System for Automatic Target Recognition

NASA Astrophysics Data System (ADS)

Coffield, Patrick Cyrus

The proposed electro-optical image algebra processing system is designed specifically for image processing and other related computations. The design is a hybridization of an optical correlator and a massively paralleled, single instruction multiple data processor. The architecture of the design consists of three tightly coupled components: a spatial configuration processor (the optical analog portion), a weighting processor (digital), and an accumulation processor (digital). The systolic flow of data and image processing operations are directed by a control buffer and pipelined to each of the three processing components. The image processing operations are defined in terms of basic operations of an image algebra developed by the University of Florida. The algebra is capable of describing all common image-to-image transformations. The merit of this architectural design is how it implements the natural decomposition of algebraic functions into spatially distributed, point use operations. The effect of this particular decomposition allows convolution type operations to be computed strictly as a function of the number of elements in the template (mask, filter, etc.) instead of the number of picture elements in the image. Thus, a substantial increase in throughput is realized. The implementation of the proposed design may be accomplished in many ways. While a hybrid electro-optical implementation is of primary interest, the benefits and design issues of an all digital implementation are also discussed. The potential utility of this architectural design lies in its ability to control a large variety of the arithmetic and logic operations of the image algebra's generalized matrix product. The generalized matrix product is the most powerful fundamental operation in the algebra, thus allowing a wide range of applications. No other known device or design has made this claim of processing speed and general implementation of a heterogeneous image algebra.

An algebraic interpretation of PSP composition.

PubMed

Vaucher, G

1998-01-01

The introduction of time in artificial neurons is a delicate problem on which many groups are working. Our approach combines some properties of biological models and the algebraic properties of McCulloch and Pitts artificial neuron (AN) (McCulloch and Pitts, 1943) to produce a new model which links both characteristics. In this extended artificial neuron, postsynaptic potentials (PSPs) are considered as numerical elements, having two degrees of freedom, on which the neuron computes operations. Modelled in this manner, a group of neurons can be seen as a computer with an asynchronous architecture. To formalize the functioning of this computer, we propose an algebra of impulses. This approach might also be interesting in the modelling of the passive electrical properties in some biological neurons.

A Cohomological Perspective on Algebraic Quantum Field Theory

NASA Astrophysics Data System (ADS)

Hawkins, Eli

2018-05-01

Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model. This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited. To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory.

Generic, Type-Safe and Object Oriented Computer Algebra Software

NASA Astrophysics Data System (ADS)

Kredel, Heinz; Jolly, Raphael

Advances in computer science, in particular object oriented programming, and software engineering have had little practical impact on computer algebra systems in the last 30 years. The software design of existing systems is still dominated by ad-hoc memory management, weakly typed algorithm libraries and proprietary domain specific interactive expression interpreters. We discuss a modular approach to computer algebra software: usage of state-of-the-art memory management and run-time systems (e.g. JVM) usage of strongly typed, generic, object oriented programming languages (e.g. Java) and usage of general purpose, dynamic interactive expression interpreters (e.g. Python) To illustrate the workability of this approach, we have implemented and studied computer algebra systems in Java and Scala. In this paper we report on the current state of this work by presenting new examples.

On the Primitive Ideal spaces of the C(*) -algebras of graphs

NASA Astrophysics Data System (ADS)

Bates, Teresa

2005-11-01

We characterise the topological spaces which arise as the primitive ideal spaces of the Cuntz-Krieger algebras of graphs satisfying condition (K): directed graphs in which every vertex lying on a loop lies on at least two loops. We deduce that the spaces which arise as Prim;C(*(E)) are precisely the spaces which arise as the primitive ideal spaces of AF-algebras. Finally, we construct a graph wt{E} from E such that C(*(wt{E})) is an AF-algebra and Prim;C(*(E)) and Prim;C(*(wt{E})) are homeomorphic.

Capitalizing on Basic Brain Processes in Developmental Algebra--Part 3

ERIC Educational Resources Information Center

Laughbaum, Edward D.

2011-01-01

In Part Three, the author reviews the basic ideas presented in Parts One and Two while arguing why the traditional equation-solving developmental algebra curricula is not a good choice for implementing neural response strategies presented in the first two parts. He continues by showing that the developmental algebra student audience is simply…

Celestial mechanics with geometric algebra

NASA Technical Reports Server (NTRS)

Hestenes, D.

1983-01-01

Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.

Lisa's Lemonade Stand: Exploring Algebraic Ideas.

ERIC Educational Resources Information Center

Billings, Esther M. H.; Lakatos, Tracy

2003-01-01

Presents an activity, "Lisa's Lemonade Stand," that actively engages students in algebraic thinking as they analyze change by investigating relationships between variables and gain experience describing and representing these relationships graphically. (YDS)

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.

Quantum Torus Algebras and B(C)-Type Toda Systems

NASA Astrophysics Data System (ADS)

Wang, Na; Li, Chuanzhong

2017-12-01

In this paper, we construct a new even constrained B(C)-type Toda hierarchy and derive its B(C)-type Block-type additional symmetry. Also we generalize the B(C)-type Toda hierarchy to the N-component B(C)-type Toda hierarchy which is proved to have symmetries of a coupled \\bigotimes ^NQT_+ algebra ( N-fold direct product of the positive half of the quantum torus algebra QT).

Fractional quiver W-algebras

NASA Astrophysics Data System (ADS)

Kimura, Taro; Pestun, Vasily

2018-04-01

We introduce quiver gauge theory associated with the non-simply laced type fractional quiver and define fractional quiver W-algebras by using construction of Kimura and Pestun (Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1072-1; Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1073-0) with representation of fractional quivers.

Introduction to Algebra Curriculum Guide, Grade 8, 1987. Bulletin 1802.

ERIC Educational Resources Information Center

Louisiana State Dept. of Education, Baton Rouge. Div. of Academic Programs.

Because of the high incidence of failure in algebra I among ninth-grade students, the Louisiana State Board of Elementary and Secondary Education requested the development of this guide with the intention of providing a good pre-algebra foundation. The purposes of the guide are to recognize standards that involve the application of mathematical…

Linear Algebra Revisited: An Attempt to Understand Students' Conceptual Difficulties

ERIC Educational Resources Information Center

Britton, Sandra; Henderson, Jenny

2009-01-01

This article looks at some of the conceptual difficulties that students have in a linear algebra course. An overview of previous research in this area is given, and the various theories that have been espoused regarding the reasons that students find linear algebra so difficult are discussed. Student responses to two questions testing the ability…

The Aftermath of Accelerating Algebra: Evidence from District Policy Initiatives

ERIC Educational Resources Information Center

Clotfelter, Charles T.; Ladd, Helen F.; Vigdor, Jacob L.

2014-01-01

In 2008, the California State Board of Education voted to require all students to enroll in algebra by 8th grade. This policy initiative, yet to be actually implemented, represents the culmination of a decades-long movement toward offering algebra instruction before the traditional high school years. Nationally, the proportion of 8th grade…

Properties of coupled-cluster equations originating in excitation sub-algebras

NASA Astrophysics Data System (ADS)

Kowalski, Karol

2018-03-01

In this paper, we discuss properties of single-reference coupled cluster (CC) equations associated with the existence of sub-algebras of excitations that allow one to represent CC equations in a hybrid fashion where the cluster amplitudes associated with these sub-algebras can be obtained by solving the corresponding eigenvalue problem. For closed-shell formulations analyzed in this paper, the hybrid representation of CC equations provides a natural way for extending active-space and seniority number concepts to provide an accurate description of electron correlation effects. Moreover, a new representation can be utilized to re-define iterative algorithms used to solve CC equations, especially for tough cases defined by the presence of strong static and dynamical correlation effects. We will also explore invariance properties associated with excitation sub-algebras to define a new class of CC approximations referred to in this paper as the sub-algebra-flow-based CC methods. We illustrate the performance of these methods on the example of ground- and excited-state calculations for commonly used small benchmark systems.

Quantized Algebras of Functions on Homogeneous Spaces with Poisson Stabilizers

NASA Astrophysics Data System (ADS)

Neshveyev, Sergey; Tuset, Lars

2012-05-01

Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0 < q < 1. We study a quantization C( G q / K q ) of the algebra of continuous functions on G/ K. Using results of Soibelman and Dijkhuizen-Stokman we classify the irreducible representations of C( G q / K q ) and obtain a composition series for C( G q / K q ). We describe closures of the symplectic leaves of G/ K refining the well-known description in the case of flag manifolds in terms of the Bruhat order. We then show that the same rules describe the topology on the spectrum of C( G q / K q ). Next we show that the family of C*-algebras C( G q / K q ), 0 < q ≤ 1, has a canonical structure of a continuous field of C*-algebras and provides a strict deformation quantization of the Poisson algebra {{C}[G/K]} . Finally, extending a result of Nagy, we show that C( G q / K q ) is canonically KK-equivalent to C( G/ K).

Evaluate fundamental approaches to longwall dust control. Phase III report

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

Babbitt, C.; Bartlett, P.; Kelly, J.

1984-03-31

The overall objective of the contract is to evaluate the effectiveness of available dust control technology for double-drum shearer longwall sections in a coordinated, systematic program at a few longwall test sections and to make the results available to the entire coal mining industry. This program is investigating nine different dust control techniques. These nine subprograms encompass a broad range of dust control measures ranging from administrative controls to new hardware. They span not only presently employed methods but also those recently adopted in the United States and those proposed for the future. This report documents the Phase III effortmore » on each of the subprograms. For clarity, the report is divided in sections by subprogram as follows: Section 2, Subprogram A - passive barriers/spray air movers for dust control; Section 3, Subprogram B - practical aspects of deep cutting; Section 4, Subprogram C - stage loader dust control; Section 5, Subprogram D - longwall automation technology; Section 6, Subprogram E - longwall application of ventilation curtains; Section 7, Subprogram F - reversed drum rotation; Section 8, Subprogram G - reduction of shield generated dust; Section 9, Subprogram H - air canopies for longwalls; and Section 10, Subprogram I - mining practices. 43 figures, 11 tables.« less

Shapes and stability of algebraic nuclear models

NASA Technical Reports Server (NTRS)

Lopez-Moreno, Enrique; Castanos, Octavio

1995-01-01

A generalization of the procedure to study shapes and stability of algebraic nuclear models introduced by Gilmore is presented. One calculates the expectation value of the Hamiltonian with respect to the coherent states of the algebraic structure of the system. Then equilibrium configurations of the resulting energy surface, which depends in general on state variables and a set of parameters, are classified through the Catastrophe theory. For one- and two-body interactions in the Hamiltonian of the interacting Boson model-1, the critical points are organized through the Cusp catastrophe. As an example, we apply this Separatrix to describe the energy surfaces associated to the Rutenium and Samarium isotopes.

Pre-Algebra Groups. Concepts & Applications.

ERIC Educational Resources Information Center

Montgomery County Public Schools, Rockville, MD.

Discussion material and exercises related to pre-algebra groups are provided in this five chapter manual. Chapter 1 (mappings) focuses on restricted domains, order of operations (parentheses and exponents), rules of assignment, and computer extensions. Chapter 2 considers finite number systems, including binary operations, clock arithmetic,…

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

Recurrence approach and higher order polynomial algebras for superintegrable monopole systems

NASA Astrophysics Data System (ADS)

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

2018-05-01

We revisit the MIC-harmonic oscillator in flat space with monopole interaction and derive the polynomial algebra satisfied by the integrals of motion and its energy spectrum using the ad hoc recurrence approach. We introduce a superintegrable monopole system in a generalized Taub-Newman-Unti-Tamburino (NUT) space. The Schrödinger equation of this model is solved in spherical coordinates in the framework of Stäckel transformation. It is shown that wave functions of the quantum system can be expressed in terms of the product of Laguerre and Jacobi polynomials. We construct ladder and shift operators based on the corresponding wave functions and obtain the recurrence formulas. By applying these recurrence relations, we construct higher order algebraically independent integrals of motion. We show that the integrals form a polynomial algebra. We construct the structure functions of the polynomial algebra and obtain the degenerate energy spectra of the model.

A comparison of three algebraic stress closures for combustor flow calculations

NASA Technical Reports Server (NTRS)

Nikjooy, M.; So, R. M. C.; Hwang, B. C.

1985-01-01

A comparison is made of the performance of two locally nonequilibrium and one equilibrium algebraic stress closures in calculating combustor flows. Effects of four different pressure-strain models on these closure models are also analyzed. The results show that the pressure-strain models have a much greater influence on the calculated mean velocity and turbulence field than the algebraic stress closures, and that the best mean strain model for the pressure-strain terms is that proposed by Launder, Reece and Rodi (1975). However, the equilibrium algebraic stress closure with the Rotta return-to-isotropy model (1951) for the pressure-strain terms gives as good a correlation with measurements as when the Launder et al. mean strain model is included in the pressure-strain model. Finally, comparison of the calculations with the standard k-epsilon closure results show that the algebraic stress closures are better suited for simple turbulent flow calculations.

Curriculum Reform in Irish Secondary Schools--A Focus on Algebra

ERIC Educational Resources Information Center

Prendergast, Mark; Treacy, Paraic

2018-01-01

Algebra has long been identified as an area of difficulty in the teaching and learning of mathematics. Evidence of this difficulty can be found in Irish secondary-level classrooms. Chief Examiner Reports have consistently identified algebra as an area of student weakness in State examinations. In light of poor student performance, and as part of a…

Constraints and Superspin for SuperPoincare Algebras in Diverse Dimensions

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

Pasqua, Andrea; Zumino, Bruno

2004-04-27

We generalize to arbitrary dimension the construction of a covariant and supersymmetric constraint for the massless superPoincare algebra, which was given for the eleven-dimensional case in a previous work. We also contrast it with a similar construction appropriate to the massive case. Finally we show that the constraint uniquely fixes the representation of the algebra.

Intertextuality and Sense Production in the Learning of Algebraic Methods

ERIC Educational Resources Information Center

Rojano, Teresa; Filloy, Eugenio; Puig, Luis

2014-01-01

In studies carried out in the 1980s the algebraic symbols and expressions are revealed through prealgebraic readers as non-independent texts, as texts that relate to other texts that in some cases belong to the reader's native language or to the arithmetic sign system. Such outcomes suggest that the act of reading algebraic texts submerges…

Student and Instructor Perceptions of a Flipped College Algebra Classroom

ERIC Educational Resources Information Center

Jaster, Robert W.

2017-01-01

Each year about half a million students fail to make planned academic progress due to college algebra, hence the need for researchers to find ways of improving the quality of instruction in the course. Recent research suggests that flipping college algebra to allow time for active learning in the classroom may improve student performance. Also,…

Paper 3: Content and Rigor of Algebra Credit Recovery Courses

ERIC Educational Resources Information Center

Walters, Kirk; Stachel, Suzanne

2014-01-01

This paper describes the content, organization and rigor of the f2f and online summer algebra courses that were delivered in summers 2011 and 2012. Examining the content of both types of courses is important because research suggests that algebra courses with certain features may be better than others in promoting success for struggling students.…

Stable homotopical algebra and [Gamma]-spaces

NASA Astrophysics Data System (ADS)

Schwede, Stefan

1999-03-01

In this paper we advertise the category of [Gamma]-spaces as a convenient framework for doing ‘algebra’ over ‘rings’ in stable homotopy theory. [Gamma]-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. ([minus sign]1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for [Gamma]-spaces. The study of ‘rings, modules and algebras’ based on [Gamma]-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg-MacLane spectra counterparts.

Generalized Quantum Field Theory Based on a Nonlinear Deformed Heisenberg Algebra

NASA Astrophysics Data System (ADS)

Ribeiro-Silva, C. I.; Oliveira-Neto, N. M.

We consider a quantum field theory based on a nonlinear Heisenberg algebra which describes phenomenologically a composite particle. Perturbative computation, considering the λϕ4 interaction was done and we also performed some comparison with a quantum field theory based on the q-oscillator algebra.

Multi-loop Integrand Reduction with Computational Algebraic Geometry

NASA Astrophysics Data System (ADS)

Badger, Simon; Frellesvig, Hjalte; Zhang, Yang

2014-06-01

We discuss recent progress in multi-loop integrand reduction methods. Motivated by the possibility of an automated construction of multi-loop amplitudes via generalized unitarity cuts we describe a procedure to obtain a general parameterisation of any multi-loop integrand in a renormalizable gauge theory. The method relies on computational algebraic geometry techniques such as Gröbner bases and primary decomposition of ideals. We present some results for two and three loop amplitudes obtained with the help of the MACAULAY2 computer algebra system and the Mathematica package BASISDET.

Short Round Sub-Linear Zero-Knowledge Argument for Linear Algebraic Relations

NASA Astrophysics Data System (ADS)

Seo, Jae Hong

Zero-knowledge arguments allows one party to prove that a statement is true, without leaking any other information than the truth of the statement. In many applications such as verifiable shuffle (as a practical application) and circuit satisfiability (as a theoretical application), zero-knowledge arguments for mathematical statements related to linear algebra are essentially used. Groth proposed (at CRYPTO 2009) an elegant methodology for zero-knowledge arguments for linear algebraic relations over finite fields. He obtained zero-knowledge arguments of the sub-linear size for linear algebra using reductions from linear algebraic relations to equations of the form z = x *' y, where x, y ∈ Fnp are committed vectors, z ∈ Fp is a committed element, and *' : Fnp × Fnp → Fp is a bilinear map. These reductions impose additional rounds on zero-knowledge arguments of the sub-linear size. The round complexity of interactive zero-knowledge arguments is an important measure along with communication and computational complexities. We focus on minimizing the round complexity of sub-linear zero-knowledge arguments for linear algebra. To reduce round complexity, we propose a general transformation from a t-round zero-knowledge argument, satisfying mild conditions, to a (t - 2)-round zero-knowledge argument; this transformation is of independent interest.

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

ERIC Educational Resources Information Center

Wasserman, Nicholas H.

2014-01-01

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

The Visual Syntax of Algebra.

ERIC Educational Resources Information Center

Kirshner, David

1989-01-01

A structured system of visual features is seen to parallel the propositional hierarchy of operations usually associated with the parsing of algebraic expressions. Women more than men were found to depend on these visual cues. Possible causes and consequences are discussed. Subjects were secondary and college students. (Author/DC)

Principals + Algebra (- Fear) = Instructional Leadership

ERIC Educational Resources Information Center

Carver, Cynthia L.

2010-01-01

Recent state legislation in Michigan mandates that all graduating seniors successfully pass algebra I and II. Numerous initiatives have been enacted to help mathematics teachers meet this challenge, yet school principals have had little preparation for the necessary curricular and instructional changes. To address this unmet need, university-based…

Those Do What? Connecting Eigenvectors and Eigenvalues to the Rest of Linear Algebra: Using Visual Enhancements to Help Students Connect Eigenvectors to the Rest of Linear Algebra

ERIC Educational Resources Information Center

Nyman, Melvin A.; Lapp, Douglas A.; St. John, Dennis; Berry, John S.

2010-01-01

This paper discusses student difficulties in grasping concepts from Linear Algebra--in particular, the connection of eigenvalues and eigenvectors to other important topics in linear algebra. Based on our prior observations from student interviews, we propose technology-enhanced instructional approaches that might positively impact student…

The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra.

ERIC Educational Resources Information Center

Carlson, David; And Others

1993-01-01

Presents five recommendations of the Linear Algebra Curriculum Study Group: (1) The syllabus must respond to the client disciplines; (2) The first course should be matrix oriented; (3) Faculty should consider the needs and interests of students; (4) Faculty should use technology; and (5) At least one follow-up course should be required. Provides a…

University of Chicago School Mathematics Project (UCSMP) Algebra. WWC Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2009

2009-01-01

University of Chicago School Mathematics Project (UCSMP) Algebra is a one-year course covering three primary topics: (1) linear and quadratic expressions, sentences, and functions; (2) exponential expressions and functions; and (3) linear systems. Topics from geometry, probability, and statistics are integrated with the appropriate algebra.…

Positive basis for surface skein algebras

PubMed Central

Thurston, Dylan Paul

2014-01-01

We show that the twisted SL2 skein algebra of a surface has a natural basis (the bracelets basis) that is positive, in the sense that the structure constants for multiplication are positive integers. PMID:24982193

Relationships between Classroom Schedule Types and Performance on the Algebra I Criterion-Referenced Test

ERIC Educational Resources Information Center

Murray, Gregory V.; Moyer-Packenham, Patricia S.

2014-01-01

One option for length of individual mathematics class periods is the schedule type selected for Algebra I classes. This study examined the relationship between student achievement, as indicated by Algebra I Criterion-Referenced Test scores, and the schedule type for Algebra I classes. Data obtained from the Utah State Office of Education included…

The impact of fraction magnitude knowledge on algebra performance and learning.

PubMed

Booth, Julie L; Newton, Kristie J; Twiss-Garrity, Laura K

2014-02-01

Knowledge of fractions is thought to be crucial for success with algebra, but empirical evidence supporting this conjecture is just beginning to emerge. In the current study, Algebra 1 students completed magnitude estimation tasks on three scales (0-1 [fractions], 0-1,000,000, and 0-62,571) just before beginning their unit on equation solving. Results indicated that fraction magnitude knowledge, and not whole number knowledge, was especially related to students' pretest knowledge of equation solving and encoding of equation features. Pretest fraction knowledge was also predictive of students' improvement in equation solving and equation encoding skills. Students' placement of unit fractions (e.g., those with a numerator of 1) was not especially useful for predicting algebra performance and learning in this population. Placement of non-unit fractions was more predictive, suggesting that proportional reasoning skills might be an important link between fraction knowledge and learning algebra. Copyright © 2013 Elsevier Inc. All rights reserved.

Thought beyond language: neural dissociation of algebra and natural language.

PubMed

Monti, Martin M; Parsons, Lawrence M; Osherson, Daniel N

2012-08-01

A central question in cognitive science is whether natural language provides combinatorial operations that are essential to diverse domains of thought. In the study reported here, we addressed this issue by examining the role of linguistic mechanisms in forging the hierarchical structures of algebra. In a 3-T functional MRI experiment, we showed that processing of the syntax-like operations of algebra does not rely on the neural mechanisms of natural language. Our findings indicate that processing the syntax of language elicits the known substrate of linguistic competence, whereas algebraic operations recruit bilateral parietal brain regions previously implicated in the representation of magnitude. This double dissociation argues against the view that language provides the structure of thought across all cognitive domains.

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.

Algebra. A Guidebook for Teaching.

ERIC Educational Resources Information Center

Goodman, Terry A.; And Others

This book provides a general framework for organizing the instructional program in algebra. For each topic, a general approach for instruction, together with specific strategies, activities, and resources that can be used daily, are provided. Also included are worksheet pages that can be used with students. Most activities provide for student…

Algebra, Home Mortgages, and Recessions

ERIC Educational Resources Information Center

Mariner, Jean A. Miller; Miller, Richard A.

2009-01-01

The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…

Elementary Algebra Connections to Precalculus

ERIC Educational Resources Information Center

Lopez-Boada, Roberto; Daire, Sandra Arguelles

2013-01-01

This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…

Improving Algebra Preparation: Implications from Research on Student Misconceptions and Difficulties

ERIC Educational Resources Information Center

Welder, Rachael M.

2012-01-01

Through historical and contemporary research, educators have identified widespread misconceptions and difficulties faced by students in learning algebra. Many of these universal issues stem from content addressed long before students take their first algebra course. Yet elementary and middle school teachers may not understand how the subtleties of…

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.

Algebraic and geometric structures of analytic partial differential equations

NASA Astrophysics Data System (ADS)

Kaptsov, O. V.

2016-11-01

We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.

Intuitionistic fuzzy n-fold KU-ideal of KU-algebra

NASA Astrophysics Data System (ADS)

Mostafa, Samy M.; Kareem, Fatema F.

2018-05-01

In this paper, we apply the notion of intuitionistic fuzzy n-fold KU-ideal of KU-algebra. Some types of ideals such as intuitionistic fuzzy KU-ideal, intuitionistic fuzzy closed ideal and intuitionistic fuzzy n-fold KU-ideal are studied. Also, the relations between intuitionistic fuzzy n-fold KU-ideal and intuitionistic fuzzy KU-ideal are discussed. Furthermore, a few results of intuitionistic fuzzy n-fold KU-ideals of a KU-algebra under homomorphism are discussed.

Open-Closed Homotopy Algebras and Strong Homotopy Leibniz Pairs Through Koszul Operad Theory

NASA Astrophysics Data System (ADS)

Hoefel, Eduardo; Livernet, Muriel

2012-08-01

Open-closed homotopy algebras (OCHA) and strong homotopy Leibniz pairs (SHLP) were introduced by Kajiura and Stasheff in 2004. In an appendix to their paper, Markl observed that an SHLP is equivalent to an algebra over the minimal model of a certain operad, without showing that the operad is Koszul. In the present paper, we show that both OCHA and SHLP are algebras over the minimal model of the zeroth homology of two versions of the Swiss-cheese operad and prove that these two operads are Koszul. As an application, we show that the OCHA operad is non-formal as a 2-colored operad but is formal as an algebra in the category of 2-collections.

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.

Learning to Apply Algebra in the Community for Adults With Intellectual Developmental Disabilities.

PubMed

Rodriguez, Anthony M

2016-02-01

Students with intellectual and developmental disabilities (IDD) are routinely excluded from algebra and other high-level mathematics courses. High school students with IDD take courses in arithmetic and life skills rather than having an opportunity to learn algebra. Yet algebra skills can support the learning of money and budgeting skills. This study explores the feasibility of algebra instruction for adults with IDD through an experimental curriculum. Ten individuals with IDD participated in a 6-week course framing mathematics concepts within the context of everyday challenges in handling money. The article explores classroom techniques, discusses student strategies, and proposes possible avenues for future research analyzing mathematics instructional design strategies for individuals with IDD.

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

Allada, Veerendra, Benjegerdes, Troy; Bode, Brett

Commodity clusters augmented with application accelerators are evolving as competitive high performance computing systems. The Graphical Processing Unit (GPU) with a very high arithmetic density and performance per price ratio is a good platform for the scientific application acceleration. In addition to the interconnect bottlenecks among the cluster compute nodes, the cost of memory copies between the host and the GPU device have to be carefully amortized to improve the overall efficiency of the application. Scientific applications also rely on efficient implementation of the BAsic Linear Algebra Subroutines (BLAS), among which the General Matrix Multiply (GEMM) is considered as themore » workhorse subroutine. In this paper, they study the performance of the memory copies and GEMM subroutines that are critical to port the computational chemistry algorithms to the GPU clusters. To that end, a benchmark based on the NetPIPE framework is developed to evaluate the latency and bandwidth of the memory copies between the host and the GPU device. The performance of the single and double precision GEMM subroutines from the NVIDIA CUBLAS 2.0 library are studied. The results have been compared with that of the BLAS routines from the Intel Math Kernel Library (MKL) to understand the computational trade-offs. The test bed is a Intel Xeon cluster equipped with NVIDIA Tesla GPUs.« less

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.

The role of difficulty and gender in numbers, algebra, geometry and mathematics achievement

NASA Astrophysics Data System (ADS)

Rabab'h, Belal Sadiq Hamed; Veloo, Arsaythamby; Perumal, Selvan

2015-05-01

This study aims to identify the role of difficulty and gender in numbers, algebra, geometry and mathematics achievement among secondary schools students in Jordan. The respondent of the study were 337 students from eight public secondary school in Alkoura district by using stratified random sampling. The study comprised of 179 (53%) males and 158 (47%) females students. The mathematics test comprises of 30 items which has eight items for numbers, 14 items for algebra and eight items for geometry. Based on difficulties among male and female students, the findings showed that item 4 (fractions - 0.34) was most difficult for male students and item 6 (square roots - 0.39) for females in numbers. For the algebra, item 11 (inequality - 0.23) was most difficult for male students and item 6 (algebraic expressions - 0.35) for female students. In geometry, item 3 (reflection - 0.34) was most difficult for male students and item 8 (volume - 0.33) for female students. Based on gender differences, female students showed higher achievement in numbers and algebra compare to male students. On the other hand, there was no differences between male and female students achievement in geometry test. This study suggest that teachers need to give more attention on numbers and algebra when teaching mathematics.

Effective Lagrangians and Current Algebra in Three Dimensions

NASA Astrophysics Data System (ADS)

Ferretti, Gabriele

In this thesis we study three dimensional field theories that arise as effective Lagrangians of quantum chromodynamics in Minkowski space with signature (2,1) (QCD3). In the first chapter, we explain the method of effective Langrangians and the relevance of current algebra techniques to field theory. We also provide the physical motivations for the study of QCD3 as a toy model for confinement and as a theory of quantum antiferromagnets (QAF). In chapter two, we derive the relevant effective Lagrangian by studying the low energy behavior of QCD3, paying particular attention to how the global symmetries are realized at the quantum level. In chapter three, we show how baryons arise as topological solitons of the effective Lagrangian and also show that their statistics depends on the number of colors as predicted by the quark model. We calculate mass splitting and magnetic moments of the soliton and find logarithmic corrections to the naive quark model predictions. In chapter four, we drive the current algebra of the theory. We find that the current algebra is a co -homologically non-trivial generalization of Kac-Moody algebras to three dimensions. This fact may provide a new, non -perturbative way to quantize the theory. In chapter five, we discuss the renormalizability of the model in the large-N expansion. We prove the validity of the non-renormalization theorem and compute the critical exponents in a specific limiting case, the CP^ {N-1} model with a Chern-Simons term. Finally, chapter six contains some brief concluding remarks.

Numerical algebraic geometry: a new perspective on gauge and string theories

NASA Astrophysics Data System (ADS)

Mehta, Dhagash; He, Yang-Hui; Hauensteine, Jonathan D.

2012-07-01

There is a rich interplay between algebraic geometry and string and gauge theories which has been recently aided immensely by advances in computational algebra. However, symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these shortcomings. The so-called `embarrassing parallelizability' allows us to solve many problems and extract physical information which elude symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.

A Computer Algebra Approach to Solving Chemical Equilibria in General Chemistry

ERIC Educational Resources Information Center

Kalainoff, Melinda; Lachance, Russ; Riegner, Dawn; Biaglow, Andrew

2012-01-01

In this article, we report on a semester-long study of the incorporation into our general chemistry course, of advanced algebraic and computer algebra techniques for solving chemical equilibrium problems. The method presented here is an alternative to the commonly used concentration table method for describing chemical equilibria in general…

Conformal field algebras with quantum symmetry from the theory of superselection sectors

NASA Astrophysics Data System (ADS)

Mack, Gerhard; Schomerus, Volker

1990-11-01

According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central charge c=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid group B ∞ which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.

Exploring Algebraic Misconceptions with Technology

ERIC Educational Resources Information Center

Sakow, Matthew; Karaman, Ruveyda

2015-01-01

Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…

Experts Question California's Algebra Edict

ERIC Educational Resources Information Center

Cavanagh, Sean

2008-01-01

Business leaders from important sectors of the American economy have been urging schools to set higher standards in math and science--and California officials, in mandating that 8th graders be tested in introductory algebra, have responded with one of the highest such standards in the land. Still, many California educators and school…

Monitoring Student Learning in Algebra

ERIC Educational Resources Information Center

Accardo, Amy L.; Kuder, S. Jay

2017-01-01

Mr. Perez and Mrs. Peterson co-teach a ninth-grade algebra class. Perez and Peterson's class includes four students with individualized education programs (IEPs). In response to legislation, such as the No Child Left Behind (NCLB) Act (2001) and the Individuals with Disabilities Education Improvement Act (2006), an increasing number of students…

Algebraic Procedures and Creative Thinking

ERIC Educational Resources Information Center

Tabach, Michal; Friedlander, Alex

2017-01-01

Simplifying symbolic expressions is usually perceived in middle school algebra as an algorithmic activity, achieved by performing sequences of short drill-and-practice tasks, which have little to do with conceptual learning or with creative mathematical thinking. The aim of this study is to explore possible ways by which ninth-grade students can…

Adventures in Flipping College Algebra

ERIC Educational Resources Information Center

Van Sickle, Jenna

2015-01-01

This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…

Correlates of gender and achievement in introductory algebra based physics

NASA Astrophysics Data System (ADS)

Smith, Rachel Clara

The field of physics is heavily male dominated in America. Thus, half of the population of our country is underrepresented and underserved. The identification of factors that contribute to gender disparity in physics is necessary for educators to address the individual needs of students, and, in particular, the separate and specific needs of female students. In an effort to determine if any correlations could be established or strengthened between sex, gender identity, social network, algebra skill, scientific reasoning ability, and/or student attitude, a study was performed on a group of 82 students in an introductory algebra based physics course. The subjects each filled out a survey at the beginning of the semester of their first semester of algebra based physics. They filled out another survey at the end of that same semester. These surveys included physics content pretests and posttests, as well as questions about the students' habits, attitudes, and social networks. Correlates of posttest score were identified, in order of significance, as pretest score, emphasis on conceptual learning, preference for male friends, number of siblings (negatively correlated), motivation in physics, algebra score, and parents' combined education level. Number of siblings was also found to negatively correlate with, in order of significance, gender identity, preference for male friends, emphasis on conceptual learning, and motivation in physics. Preference for male friends was found to correlate with, in order of significance, emphasis on conceptual learning, gender identity, and algebra score. Also, gender identity was found to correlate with emphasis on conceptual learning, the strongest predictor of posttest score other than pretest score.

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.

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.

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.

The smooth entropy formalism for von Neumann algebras

NASA Astrophysics Data System (ADS)

Berta, Mario; Furrer, Fabian; Scholz, Volkher B.

2016-01-01

We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy, we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.

The smooth entropy formalism for von Neumann algebras

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

Berta, Mario, E-mail: berta@caltech.edu; Furrer, Fabian, E-mail: furrer@eve.phys.s.u-tokyo.ac.jp; Scholz, Volkher B., E-mail: scholz@phys.ethz.ch

2016-01-15

We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional min- and max-entropy, we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-dimensional case. We generalize the entropic uncertainty relation with quantum side information of Tomamichel and Renner and discuss applications to quantum cryptography. In particular, we prove the possibility to perform privacy amplification and classical data compression with quantum side information modeled by a von Neumann algebra.

Strategies for Solving Fraction Tasks and Their Link to Algebraic Thinking

ERIC Educational Resources Information Center

Pearn, Catherine; Stephens, Max

2015-01-01

Many researchers argue that a deep understanding of fractions is important for a successful transition to algebra. Teaching, especially in the middle years, needs to focus specifically on those areas of fraction knowledge and operations that support subsequent solution processes for algebraic equations. This paper focuses on the results of Year 6…

Examining Heterogeneity in the Effect of Taking Algebra in Eighth Grade

ERIC Educational Resources Information Center

Rickles, Jordan H.

2013-01-01

Increased access to algebra was a focal point of the National Mathematics Advisory Panel's 2008 report on improving mathematics learning in the United States. Past research found positive effects for early access to algebra, but the focus on average effects may mask important variation across student subgroups. The author addresses whether these…

Recursive boson system in the Cuntz algebra O{sub {infinity}}

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

Kawamura, Katsunori

2007-09-15

Bosons and fermions are often written by elements of other algebras. Abe (private communication) gave a realization of bosons by formal infinite sums of the canonical generators of the Cuntz algebra O{sub {infinity}}. We show that such formal infinite sum always makes sense on a certain dense subspace of any permutative representation of O{sub {infinity}}. In this meaning, we can regard as if the algebra B of bosons was a unital *-subalgebra of O{sub {infinity}} on a given permutative representation. According to this relation, we compute branching laws arising from restrictions of representations of O{sub {infinity}} on B. For example,more » it is shown that the Fock representation of B is given as the restriction of the standard representation of O{sub {infinity}} on B.« less

Continuous family of finite-dimensional representations of a solvable Lie algebra arising from singularities

PubMed Central

Yau, Stephen S.-T.

1983-01-01

A natural mapping from the set of complex analytic isolated hypersurface singularities to the set of finite dimensional Lie algebras is first defined. It is proven that the image under this natural mapping is contained in the set of solvable Lie algebras. This approach gives rise to a continuous inequivalent family of finite dimensional representations of a solvable Lie algebra. PMID:16593401

Reading Bombelli's x-purgated Algebra.

ERIC Educational Resources Information Center

Arcavi, Abraham; Bruckheimer, Maxim

1991-01-01

Presents the algorithm to approximate square roots as reproduced from the 1579 edition of an algebra book by Rafael Bombelli. The sequence of activities illustrates that the process of understanding an original source of mathematics, first at the algorithmic level and then with respect to its mathematical validity in modern terms, can be an…

Algebra from Chips and Chopsticks

ERIC Educational Resources Information Center

Yun, Jeong Oak; Flores, Alfinio

2012-01-01

Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…

Dimension independence in exterior algebra.

PubMed Central

Hawrylycz, M

1995-01-01

The identities between homogeneous expressions in rank 1 vectors and rank n - 1 covectors in a Grassmann-Cayley algebra of rank n, in which one set occurs multilinearly, are shown to represent a set of dimension-independent identities. The theorem yields an infinite set of nontrivial geometric identities from a given identity. PMID:11607520

Inequalities, Assessment and Computer Algebra

ERIC Educational Resources Information Center

Sangwin, Christopher J.

2015-01-01

The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary…

Entanglement classification with algebraic geometry

NASA Astrophysics Data System (ADS)

Sanz, M.; Braak, D.; Solano, E.; Egusquiza, I. L.

2017-05-01

We approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry, its natural language. We show that the class of symmetric separable states has the structure of a Veronese variety and that its k-secant varieties are SLOCC invariants. Thus SLOCC classes gather naturally into families. This classification presents useful properties such as a linear growth of the number of families with the number of particles, and nesting, i.e. upward consistency of the classification. We attach physical meaning to this classification through the required interaction length of parent Hamiltonians. We show that the states W N and GHZ N are in the same secant family and that, effectively, the former can be obtained in a limit from the latter. This limit is understood in terms of tangents, leading to a refinement of the previous families. We compute explicitly the classification of symmetric states with N≤slant4 qubits in terms of both secant families and its refinement using tangents. This paves the way to further use of projective varieties in algebraic geometry to solve open problems in entanglement theory.

Implications of Eighth Grade Algebra I on High School Mathematics Achievement

ERIC Educational Resources Information Center

Bayard, Robert

2012-01-01

As of 2008, approximately 40% of eighth grade students in the United States enroll in Algebra I (National Council of Teachers of Mathematics, 2008). Although research has shown that students have more opportunities to take advanced mathematics courses by taking eighth grade Algebra I, in the United States, approximately only one-third to one-half…

Syntax and Meaning as Sensuous, Visual, Historical Forms of Algebraic Thinking

ERIC Educational Resources Information Center

Radford, Luis; Puig, Luis

2007-01-01

Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and…

Designing Cognitively Diagnostic Assessment for Algebraic Content Knowledge and Thinking Skills

ERIC Educational Resources Information Center

Zhang, Zhidong

2018-01-01

This study explored a diagnostic assessment method that emphasized the cognitive process of algebra learning. The study utilized a design and a theory-driven model to examine the content knowledge. Using the theory driven model, the thinking skills of algebra learning was also examined. A Bayesian network model was applied to represent the theory…

Individual Differences in Algebraic Cognition: Relation to the Approximate Number and Sematic Memory Systems

PubMed Central

Geary, David C.; Hoard, Mary K.; Nugent, Lara; Rouder, Jeffrey N.

2015-01-01

The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 (92 girls) 9th graders, controlling parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation, but not schema memory. Frequency of fact-retrieval errors was related to schema memory but not coordinate plane or expression evaluation accuracy. The results suggest the ANS may contribute to or is influenced by spatial-numerical and numerical only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest different brain and cognitive systems are engaged during the learning of different components of algebraic competence, controlling demographic and domain general abilities. PMID:26255604

A Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies

NASA Astrophysics Data System (ADS)

Lu, Wei

2017-09-01

We propose a Clifford algebra approach to chiral symmetry breaking and fermion mass hierarchies in the context of composite Higgs bosons. Standard model fermions are represented by algebraic spinors of six-dimensional binary Clifford algebra, while ternary Clifford algebra-related flavor projection operators control allowable flavor-mixing interactions. There are three composite electroweak Higgs bosons resulted from top quark, tau neutrino, and tau lepton condensations. Each of the three condensations gives rise to masses of four different fermions. The fermion mass hierarchies within these three groups are determined by four-fermion condensations, which break two global chiral symmetries. The four-fermion condensations induce axion-like pseudo-Nambu-Goldstone bosons and can be dark matter candidates. In addition to the 125 GeV Higgs boson observed at the Large Hadron Collider, we anticipate detection of tau neutrino composite Higgs boson via the charm quark decay channel.

Sixth SIAM conference on applied linear algebra: Final program and abstracts. Final technical report

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

NONE

1997-12-31

Linear algebra plays a central role in mathematics and applications. The analysis and solution of problems from an amazingly wide variety of disciplines depend on the theory and computational techniques of linear algebra. In turn, the diversity of disciplines depending on linear algebra also serves to focus and shape its development. Some problems have special properties (numerical, structural) that can be exploited. Some are simply so large that conventional approaches are impractical. New computer architectures motivate new algorithms, and fresh ways to look at old ones. The pervasive nature of linear algebra in analyzing and solving problems means that peoplemore » from a wide spectrum--universities, industrial and government laboratories, financial institutions, and many others--share an interest in current developments in linear algebra. This conference aims to bring them together for their mutual benefit. Abstracts of papers presented are included.« less

Constructive Learning in Undergraduate Linear Algebra

ERIC Educational Resources Information Center

Chandler, Farrah Jackson; Taylor, Dewey T.

2008-01-01

In this article we describe a project that we used in our undergraduate linear algebra courses to help our students successfully master fundamental concepts and definitions and generate interest in the course. We describe our philosophy and discuss the projects overall success.

Using Technology to Balance Algebraic Explorations

ERIC Educational Resources Information Center

Kurz, Terri L.

2013-01-01

In 2000, the "National Council of Teachers of Mathematics" recommended that Algebra Standards, "instructional programs from prekindergarten through grade 12 should enable all students to use mathematical models to represent and understand quantitative relationships." In this article, the authors suggest the "Balance"…

Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

NASA Astrophysics Data System (ADS)

Hoque, Md. Fazlul; Marquette, Ian; Post, Sarah; Zhang, Yao-Zhong

2018-04-01

We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schrödinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.

Attributions for Success and Failure in Algebra of Samoan Community College Students: A Profile Analysis.

ERIC Educational Resources Information Center

Powers, Stephen; And Others

Sex differences in attributions for success and failure in algebra of Samoan community college students were examined and compared with attributions of a large group of mainland U.S. students. study included the Mathematics Attribution Scale: Algebra Version (MAS), which assessed students' attributions of achievement in algebra to their effort,…

Reinventing Fractions and Division as They Are Used in Algebra: The Power of Preformal Productions

ERIC Educational Resources Information Center

Peck, Frederick; Matassa, Michael

2016-01-01

In this paper, we explore algebra students' mathematical realities around fractions and division, and the ways in which students reinvented mathematical productions involving fractions and division. We find that algebra students' initial realities do not include the fraction-as-quotient sub-construct. This can be problematic because in algebra,…

Application of symbolic and algebraic manipulation software in solving applied mechanics problems

NASA Technical Reports Server (NTRS)

Tsai, Wen-Lang; Kikuchi, Noboru

1993-01-01

As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.

A systematic investigation of the link between rational number processing and algebra ability.

PubMed

Hurst, Michelle; Cordes, Sara

2018-02-01

Recent research suggests that fraction understanding is predictive of algebra ability; however, the relative contributions of various aspects of rational number knowledge are unclear. Furthermore, whether this relationship is notation-dependent or rather relies upon a general understanding of rational numbers (independent of notation) is an open question. In this study, college students completed a rational number magnitude task, procedural arithmetic tasks in fraction and decimal notation, and an algebra assessment. Using these tasks, we measured three different aspects of rational number ability in both fraction and decimal notation: (1) acuity of underlying magnitude representations, (2) fluency with which symbols are mapped to the underlying magnitudes, and (3) fluency with arithmetic procedures. Analyses reveal that when looking at the measures of magnitude understanding, the relationship between adults' rational number magnitude performance and algebra ability is dependent upon notation. However, once performance on arithmetic measures is included in the relationship, individual measures of magnitude understanding are no longer unique predictors of algebra performance. Furthermore, when including all measures simultaneously, results revealed that arithmetic fluency in both fraction and decimal notation each uniquely predicted algebra ability. Findings are the first to demonstrate a relationship between rational number understanding and algebra ability in adults while providing a clearer picture of the nature of this relationship. © 2017 The British Psychological Society.

Linear algebraic methods applied to intensity modulated radiation therapy.

PubMed

Crooks, S M; Xing, L

2001-10-01

Methods of linear algebra are applied to the choice of beam weights for intensity modulated radiation therapy (IMRT). It is shown that the physical interpretation of the beam weights, target homogeneity and ratios of deposited energy can be given in terms of matrix equations and quadratic forms. The methodology of fitting using linear algebra as applied to IMRT is examined. Results are compared with IMRT plans that had been prepared using a commercially available IMRT treatment planning system and previously delivered to cancer patients.

Applications: Using Algebra in an Accounting Practice.

ERIC Educational Resources Information Center

Eisner, Gail A.

1994-01-01

Presents examples of algebra from the field of accounting including proportional ownership of stock, separation of a loan payment into principal and interest portions, depreciation methods, and salary withholdings computations. (MKR)

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.

A New Age for Algebra

ERIC Educational Resources Information Center

Oishi, Lindsay

2011-01-01

"Solve for x." While many people first encountered this enigmatic instruction in high school, the last 20 years have seen a strong push to get students to take algebra in eighth grade or even before. Today, concerns about the economy highlight a familiar worry: American eighth-graders trailed their peers in five Asian countries on the…

Weaving Geometry and Algebra Together

ERIC Educational Resources Information Center

Cetner, Michelle

2015-01-01

When thinking about student reasoning and sense making, teachers must consider the nature of tasks given to students along with how to plan to use the tasks in the classroom. Students should be presented with tasks in a way that encourages them to draw connections between algebraic and geometric concepts. This article focuses on the idea that it…

Algebra for All. Research Brief

ERIC Educational Resources Information Center

Bleyaert, Barbara

2009-01-01

The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless…