scholarly journals homalg – A META-PACKAGE FOR HOMOLOGICAL ALGEBRA

2008 ◽  
Vol 07 (03) ◽  
pp. 299-317 ◽  
Author(s):  
MOHAMED BARAKAT ◽  
DANIEL ROBERTZ
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Linear Equations ◽  
Homological Algebra ◽  
Mathematical Object ◽  
Specific Knowledge ◽  
Central Notion ◽  
Computer Algebra Software ◽  
Finitely Presented

The central notion of this work is that of a functor between categories of finitely presented modules over so-called computable rings, i.e. rings R where one can algorithmically solve inhomogeneous linear equations with coefficients in R. The paper describes a way allowing one to realize such functors, e.g. Hom R, ⊗R, [Formula: see text], [Formula: see text], as a mathematical object in a computer algebra system. Once this is achieved, one can compose and derive functors and even iterate this process without the need of any specific knowledge of these functors. These ideas are realized in the ring independent package homalg. It is designed to extend any computer algebra software implementing the arithmetics of a computable ring R, as soon as the latter contains algorithms to solve inhomogeneous linear equations with coefficients in R. Beside explaining how this suffices, the paper describes the nature of the extensions provided by homalg.

scholarly journals Closing the category of finitely presented functors under images made constructive

2020 ◽  
Vol 2 ◽  
pp. 4
Author(s):  
Sebastian Posur
Keyword(s):  
Data Structure ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Basic Problem ◽  
Explicit Construction ◽  
Graded Modules ◽  
Additive Category ◽  
Abelian Case ◽  
Coherent Ring ◽  
Finitely Presented

For an additive category P we provide an explicit construction of a category Q(P) whose objects can be thought of as formally representing im(γ)im(ρ)∩im(γ) for given morphisms γ:A→B and ρ:C→B in P, even though P does not need to admit quotients or images. We show how it is possible to calculate effectively within Q(P), provided that a basic problem related to syzygies can be handled algorithmically. We prove an equivalence of Q(P) with the smallest subcategory of the category of contravariant functors from P to the category of abelian groups Ab which contains all finitely presented functors and is closed under the operation of taking images. Moreover, we characterize the abelian case: Q(P) is abelian if and only if it is equivalent to fp(Pop,Ab), the category of all finitely presented functors, which in turn, by a theorem of Freyd, is abelian if and only if P has weak kernels.The category Q(P) is a categorical abstraction of the data structure for finitely presented R-modules employed by the computer algebra system Macaulay2, where R is a ring. By our generalization to arbitrary additive categories, we show how this data structure can also be used for modeling finitely presented graded modules, finitely presented functors, and some not necessarily finitely presented modules over a non-coherent ring.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

The Maple computer algebra system: A review

1995 ◽  
Vol 10 (3) ◽  
pp. 329-337 ◽  
Author(s):  
John Hutton ◽  
James Hutton
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
System A

scholarly journals OPTIMALISASI MOTIVASI DAN PRESTASI BELAJAR MENGGUNAKAN MOODLE BERBANTUAN COMPUTER ALGEBRA SYSTEM (CAS)

2020 ◽  
Vol 9 (1) ◽  
pp. 53
Author(s):  
Kamhar Ngado ◽  
Rosnawati Rosnawati ◽  
Heri Retnawati ◽  
Sri Andayani
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System

scholarly journals Generation of equations for shell models of turbulence in the Maple system

2021 ◽  
Vol 254 ◽  
pp. 02006
Author(s):  
Liubov Feshchenko ◽  
Gleb Vodinchar
Keyword(s):  
Computer Algebra ◽  
Power Law ◽  
Computer Algebra System ◽  
Nonlinear Interactions ◽  
Systems Of Equations ◽  
Shell Models ◽  
Quadratic Invariants ◽  
Automated Compilation ◽  
Shell Models Of Turbulence

The paper describes a technology for the automated compilation of equations for shell models of turbulence in the computer algebra system Maple. A general form of equations for the coefficients of nonlinear interactions is given, which will ensure that the required combination of quadratic invariants and power-law solutions is fulfilled in the model. Described the codes for the Maple system allowing to generate and solve systems of equations for the coefficients. The proposed technology allows you to quickly and accurately generate classes of shell models with the desired properties.

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