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Orthogonal designs via computational algebra

JOURNAL OF COMBINATORIAL DESIGNS 2006年第5期14卷 351-362页

作者： Kotsireas, Ilias S. Koukouvinos, Christos Wilfrid Laurier Univ Dept Phys & Comp Sci Waterloo ON N2L 3C5 Canada Natl Tech Univ Athens Dept Math GR-15773 Athens Greece

We detail the Williamson array construction based on quaternions, following the description by Baumert and Hall. By analogy, we extend the construction to larger arrays using matrix representations of the algebras of octonions and sedenions. In the case of octonions, we obtain the full orthogonal design OD(8;1, 1, 1, 1, 1, 1, 1, 1) or order 8 with 8 variables. In the case of sedenions we obtain the full orthogonal design OD(16;1, 1, 7, 7) of order 16 with 4 variables and the full orthogonal design OD(16;1, 1, 2, 2, 2, 2, 2, 2, 2) of order 16 with 9 variables. We use OD(16;1, 1, 2, 2, 2, 2, 2, 2, 2) to search for inequivalent Hadamard matrices of orders 112, 144, 176 and we establish constructively three new lower bounds for the numbers of inequivalent Hadamard matrices of these three orders. (C) 2006 Wiley Periodicals, Inc.

关键词： orthogonal designs Hadamard matrices supercomputing computational algebra Grobner bases algorithms

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A computational algebra approach to the reverse engineering of gene regulatory networks

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JOURNAL OF THEORETICAL BIOLOGY 2004年第4期229卷 523-537页

作者： Laubenbacher, R Stigler, B Virginia Tech Virginia Bioinformat Inst Blacksburg VA 24061 USA

This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is time-discrete deterministic dynamical systems, with a finite set of states for each of the variables. The simplest examples of such models are Boolean networks, in which variables have only two possible states. The use of a larger number of possible states allows a finer discretization of experimental data and more than one possible mode of action for the variables, depending on threshold values. Furthermore, with a suitable choice of state set, one can employ powerful tools from computational algebra, that underlie the reverse-engineering algorithm, avoiding costly enumeration strategies. To perform well, the algorithm requires wildtype together with perturbation time courses. This makes it suitable for small to meso-scale networks rather than networks on a aenome-wide scale. An analysis of the complexity of the algorithm is performed. The algorithm is validated on a recently published Boolean network model of segment polarity development in Drosophila melanogaster. (C) 2004 Elsevier Ltd. All rights reserved.

关键词： reverse engineering gene regulatory networks discrete modeling computational algebra

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Data Identification for Improving Gene Network Inference using computational algebra

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BULLETIN OF MATHEMATICAL BIOLOGY 2014年第11期76卷 2923-2940页

作者： Dimitrova, Elena Stigler, Brandilyn Clemson Univ Dept Math Sci Clemson SC 29634 USA So Methodist Univ Dept Math Dallas TX 75275 USA

Identification of models of gene regulatory networks is sensitive to the amount of data used as input. Considering the substantial costs in conducting experiments, it is of value to have an estimate of the amount of data required to infer the network structure. To minimize wasted resources, it is also beneficial to know which data are necessary to identify the network. Knowledge of the data and knowledge of the terms in polynomial models are often required a priori in model identification. In applications, it is unlikely that the structure of a polynomial model will be known, which may force data sets to be unnecessarily large in order to identify a model. Furthermore, none of the known results provides any strategy for constructing data sets to uniquely identify a model. We provide a specialization of an existing criterion for deciding when a set of data points identifies a minimal polynomial model when its monomial terms have been specified. Then, we relax the requirement of the knowledge of the monomials and present results for model identification given only the data. Finally, we present a method for constructing data sets that identify minimal polynomial models.

关键词： Gene network inference Model identification Data selection Polynomial dynamical system computational algebra Grobner basis

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Geometric interpretations of compatibility for fundamental matrices

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JOURNAL OF SYMBOLIC COMPUTATION 2025年 131卷

作者： Connelly, Erin Rydell, Felix Univ Osnabrueck Osnabruck Germany KTH Royal Inst Technol Stockholm Sweden

In recent work, algebraic computational software was used to provide the exact algebraic conditions under which a six-tuple of fundamental matrices, corresponding to 4 images, is compatible, i.e., there exist 4 cameras such that each pair has the appropriate fundamental matrix. It has been further demonstrated that quadruplewise compatibility is sufficient when the number of cameras greater than 4. We expand on these prior results by proving equivalent geometric conditions for compatibility. We find that compatibility can be characterized via the intersections of epipolar lines in one of the images. (c) 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// ***/licenses/by/4.0/).

关键词： Computer vision computational algebra Projective geometry

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Automorphism groups of axial algebras

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JOURNAL OF algebra 2025年 661卷 657-712页

作者： Gorshkov, I. B. McInroy, J. Shumba, T. M. Mudziiri Shpectorov, S. Sobolev Inst Math Novosibirsk Russia Univ Chester Dept Math Exton PkParkgate Rd Chester CH1 4BJ England Univ Bristol Sch Math Fry BldgWoodland Rd Bristol BS8 1UG England Univ Birmingham Sch Math Birmingham B15 2TT England

Axial algebras are a class of commutative non-associative algebras which have a natural group of automorphisms, called the Miyamoto group. The motivating example is the Griess algebra which has the Monster sporadic simple group as its Miyamoto group. Previously, using an expansion algorithm, about 200 examples of axial algebras in the same class as the Griess algebra have been constructed in dimensions up to about 300. In this list, we see many reoccurring dimensions which suggests that there may be some unexpected isomorphisms. Such isomorphisms can be found when the full automorphism groups of the algebras are known. Hence, in this paper, we develop methods for computing the full automorphism groups of axial algebras and apply them to a number of examples of dimensions up to 151. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://

关键词： Axial algebra Non-associative algebra Monster Jordan algebra Automorphism Automorphism group Idempotent computational algebra

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The New is the Well-Forgotten Old-F4 Algorithm Optimization

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computational MATHEMATICS AND MATHEMATICAL PHYSICS 2025年第3期65卷 582-590页

作者： Styopkin, S. M. Yandex Moscow 119034 Russia

The Grobner basis is a fundamental concept in computational algebra. F4 is one of the fastest algorithms for computing Grobner basis. In this paper, we will discuss the process of writing effective F4. Despite the fact that this work focuses on algorithms from computational algebra, some of the results and ideas presented here may have broader applications beyond this specific subject area. In general, the theory described below can be regarded as an abstraction, as it progresses through the text. This is because the text is not actually about the F4 algorithm itself, but rather about the power of profiling, unconventional techniques, and selecting the appropriate memory model. We will provide examples of inefficient usage of the standard library, recall the fundamental principles of optimization in order to apply them as efficiently as possible to obtain the fastest F4 algorithm, using non-traditional approaches.

关键词： computational algebra F4 software optimization C plus profiling

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On an Integrated computational Environment for Numerical algebra 13th

On an Integrated Computational Environment for Numerical Alg...

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13th International Scientific Conference on Parallel computational Technologies (PCT)

作者： Il'in, Valery Novosibirsk State Univ RAS SB Inst Computat Math & Math Geophys Novosibirsk Russia

ISBN: (纸本)9783030281632;9783030281625

This paper describes the conception, general architecture, data structure, and main components of an Integrated computational Environment (ICE) for the high-performance solution of a wide class of numerical algebraic problems on heterogeneous supercomputers with distributed and hierarchical shared memory. The tasks considered include systems of linear algebraic equations (SLAEs), various eigenvalue problems, and transformations of algebraic objects with large sparse matrices. These tasks arise in various approximations of multidimensional initial boundary value problems on unstructured grids. A quite large variety of types of matrices, featuring diverse structural, spectral, and other properties are allowed;there can also be a wide diversity of algorithms for computational algebra. There are relevant issues associated with scalable parallelism through hybrid programming on heterogeneous multiprocessor systems, MPI-processes, multithread computing, and vectorization of operations, including those without formal constraints on the number of degrees of freedom and on the number of computing units. The numerical methods and technologies are implemented in the KRYLOV library, which provides the integrated subsystem of the ICE. There are various technical requirements imposed upon the software: extendibility of the set of problems and algorithms, adaptation to the evolution of supercomputer architecture, ability to reuse external products, and coordinated participation of development groups taking part in the project. The end goal of these requirements is to provide a product featuring a long life cycle, high performance, and general acceptance among end users of diverse professional backgrounds.

关键词： computational algebra Sparse matrix Iterative method High performance Krylov subspaces Scalable parallelism Heterogeneous supercomputer Domain decomposition

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computational approach to the simplicity of f₄(O_s,-) in the characteristic two case

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JOURNAL OF computational AND APPLIED MATHEMATICS 2003年第1期158卷 1-10页

作者： Bjerregaard, PA González, CM Univ Malaga Fac Sci Dept Algebra Geometry & Topol Malaga 29080 Spain Univ Malaga ETSII Dept Appl Math Malaga 29012 Spain

We present in this paper a computational approach to the study of the simplicity of the derivation Lie algebra of the quadratic Jordan algebra H-3(O-s-), denoted by f(4)(O-s,-), when the characteristic of the base field is two. We will show not only a collection of routines designed to find identities and construct principal ideals but also a philosophy of how to proceed studying the simplicity of a Lie algebra. We have first implemented the quadratic Jordan structure of H-3 (O-s,-) into the computer system Mathematica (Computing the derivation Lie algebra of the quadratic Jordon algebra H-3(O-s,-) at any characteristic, preprint, 2001) and then determined the generic expression of an element of the Lie algebra f(4) O-s, -) = Der(H-3 (Q(s), -)) (see (41)). Once the structure Of WO,,-) is Completely described, it is time to analyze the simplicity by using the strategy mentioned. If the characteristic of the base field is not two, the Lie algebra is simple, but if the characteristic is two, the Lie algebra is not simple and there exists only one proper nonzero ideal I which is 26 dimensional and simple as a Lie algebra. In order to prove this last affirmation, we have used again the set of routines to show the simplicity of the ideal and that it is isomorphic to f(4)/I, which is also a simple Lie algebra. This isomorphism is constructed from a computed Cartan decomposition of both Lie algebras. (C) 2003 Elsevier B.V. All rights reserved.

关键词： Lie algebras Jordan algebras computational algebra

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computational tools to construct semi-analytical planetary theories

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2008年第3-4期85卷 497-508页

作者： Orti, Jose A. Lopez Uso, Maria J. Martinez Castillo, Francisco J. Marco Univ Jaume 1 Dept Mat Castellon de La Plana 12071 Spain Univ Politecn Valencia Dept Matemat Aplicada Valencia Spain

The main aim of this paper is the construction of a set of computational tools in order to improve the management of large expansions used in celestial mechanics. This package involves an iterative algorithm to develop the inverse of the distance according to an appropriate set of temporal variables, a new Poisson series processor, and an iterative integration method to obtain the solution of Lagrange planetary equations. The method has been applied to compute a set of examples using several kinds of anomalies in order to test the performance of the algorithms.

关键词： celestial mechanics planetary theory computational algebra orbital mechanics perturbation theory

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computational tools to construct semi-analytical planetary theories

Computational tools to construct semi-analytical planetary t...

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6th International Conference on computational and Mathematical Methods in Science and Engineering

作者： Orti, Jose A. Lopez Uso, Maria J. Martinez Castillo, Francisco J. Marco Univ Jaume 1 Dept Mat Castellon de La Plana 12071 Spain Univ Politecn Valencia Dept Matemat Aplicada Valencia Spain

关键词： celestial mechanics planetary theory computational algebra orbital mechanics perturbation theory

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