检索结果-内蒙古大学图书馆

Full Rank Representation of Real algebraic Sets and applications 1

19th International Workshop on computer algebra in Scientific Computing (CASC)

作者： Chen, Changbo Wu, Wenyuan Feng, Yong Chinese Acad Sci Chongqing Inst Green & Intelligent Technol Chongqing Key Lab Automated Reasoning & Cognit Chongqing Peoples R China Univ Chinese Acad Sci Beijing Peoples R China

ISBN: (数字)9783319663203

ISBN: (纸本)9783319663203;9783319663197

We introduce the notion of the full rank representation of a real algebraic set, which represents it as the projection of a union of real algebraic manifolds lik(F) of Rm, m > n, such that the rank of the Jacobian matrix of each at any point of VIR(F.) is the same as the number of polynomials in F. By introducing an auxiliary variable, we show that a squarefree regular chain T can be transformed to a new regular chain C having various nice properties, such as the Jacobian matrix of C attains full rank at any point of VR(C). Based on a symbolic triangular decomposition approach and a numerical critical point technique, we present a hybrid algorithm to compute a full rank representation. As an application, we show that such a representation allows to better visualize plane and space curves with singularities. Effectiveness of this approach is also demonstrated by computing witness points of polynomial systems having rank -deficient Jacobian matrices.

关键词： Jacobian matrices

来源：评论

学校读者我要写书评

暂无评论

The conditioning of block kronecker -ifications of matrix polynomials

arXiv

引用

arXiv 2018年

作者： Pérez, Javier Department of Mathematical Sciences University of Montana United States

A strong -ification of a matrix polynomial P (λ) = P Aiλi of degree d is a matrix polynomial L(λ) of degree having the same finite and infinite elementary divisors, and the same number of left and right minimal indices as P (λ). Strong -ifications can be used to transform the polynomial eigenvalue problem associated with P (λ) into an equivalent polynomial eigenvalue problem associated with a larger matrix polynomial L(λ) of lower degree. Of most interest in applications is = 1, for which L(λ) receives the name of strong linearization. However, there exist some situations, e.g., the preservation of algebraic structures, in which it is more convenient to replace strong linearizations by other low degree matrix polynomials. In this work, we investigate the eigenvalue conditioning of -ifications from a family of matrix polynomials recently identified and studied by Dopico, Pérez and Van Dooren, the so-called block Kronecker companion forms. We compare the conditioning of these -ifications with that of the matrix polynomial P (λ), and show that they are about as well conditioned as the polynomial itself, provided we scale P (λ) so that max{kAik2} = 1, and the quantity min{kA0k2, kAdk2} is not too small. Moreover, under the scaling assumption max{kAik2} = 1, we show that any block Kronecker companion form, regardless of its degree or block structure, is about as well-conditioned as the well-known Frobenius companion forms. Our theory is illustrated by numerical examples.65F15, 65F30, 65F35 Copyright © 2018, The Authors. All rights reserved.

关键词： Matrix algebra

来源：评论

学校读者我要写书评

暂无评论

A nearly optimal algorithm to decompose binary forms

arXiv

引用

arXiv 2018年

作者： Bender, Matías R. Faugère, Jean-Charles Perret, Ludovic Tsigaridas, Elias Sorbonne Université CNRS INRIA Laboratoire d'Informatique de Paris 6 LIP6 Équipe POLSYS 4 place Jussieu ParisF-75005 France

Symmetric tensor decomposition is an important problem with applications in several areas, for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous polynomials, that is to write a homogeneous polynomial in n variables of degree D as a sum of D-th powers of linear forms, using the minimal number of summands. This minimal number is called the rank of the polynomial/tensor. We focus on decomposing binary forms, a problem that corresponds to the decomposition of symmetric tensors of dimension 2 and order D, that is, symmetric tensors of order D over the vector space K2. Under this formulation, the problem finds its roots in invariant theory where the decompositions are related to canonical forms. We introduce a superfast algorithm that exploits results from structured linear algebra. It achieves a softly linear arithmetic complexity bound. To the best of our knowledge, the previously known algorithms have at least quadratic complexity bounds. Our algorithm computes a symbolic decomposition in O(M(D)log(D)) arithmetic operations, where M(D) is the complexity of multiplying two polynomials of degree D. It is deterministic when the decomposition is unique. When the decomposition is not unique, it is randomized. We also present a Monte Carlo variant as well as a modification to make it a Las Vegas one. From the symbolic decomposition, we approximate the terms of the decomposition with an error of 2−Ε, in O(Dlog2(D)(log2(D) + log(Ε))) arithmetic operations. We use results from Kaltofen and Yagati (1989) to bound the size of the representation of the coefficients involved in the decomposition and we bound the algebraic degree of the problem by min(rank,D−rank+1). We show that this bound can be tight. When the input polynomial has integer coefficients, our algorithm performs, up to poly-logarithmic factors, OeB(D`+ D4 + D3τ) bit operations, where τ is the maximum bitsize of the coefficients an

关键词： polynomials

来源：评论

学校读者我要写书评

暂无评论

The minrank of random graphs over arbitrary fields

arXiv

引用

arXiv 2018年

作者： Alon, Noga Balla, Igor Gishboliner, Lior Mond, Adva Mousset, Frank Department of Mathematics Princeton University PrincetonNJ United States Schools of Mathematics and Computer Science Tel Aviv University Tel Aviv Israel Department of Mathematics ETH Zurich8092 Switzerland School of Mathematical Sciences Tel Aviv University Tel Aviv69978 Israel

The minrank of a graph G on the set of vertices [n] over a field F is the minimum possible rank of a matrix M ∈ -n×nwith nonzero diagonal entries such that Mi,j= 0 whenever i and j are distinct nonadjacent vertices of G. This notion, over the real field, arises in the study of the Lovász theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph G(n, p) over any finite or infinite field, showing that for every field F = F(n) and every p = p(n) satisfying n-1≤ p ≤ 1-n-0.99, the minrank of G = G(n, p) over - is (Equation Presented) with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev, and Weinstein, who proved the above result for finite fields of size at most nO(1), with tools from linear algebra, including an estimate of Rónyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of polynomials. Copyright © 2018, The Authors. All rights reserved.

关键词： Graph theory

来源：评论

学校读者我要写书评

暂无评论

Sum-of-exponentials modeling and common dynamics estimation using Tensorlab

引用

IFAC-PapersOnLine 2017年第1期50卷 14150-14155页

作者： Markovsky, I. Debals, O. Lathauwer, L. De Building K Pleinlaan 2 BrusselsB-1050 Belgium ESAT/STADIUS and iMinds Medical IT Katholieke Universiteit Leuven Kasteelpark Arenberg 10 Leuven3001 Belgium Katholieke Universiteit Leuven Campus Kortrijk Group Science Engineering and Technology E. Sabbelaan 53 Kortrijk8500 Belgium

Fitting a signal to a sum-of-exponentials model is a basic problem in signal processing. It can be posed and solved as a Hankel structured low-rank matrix approximation problem. Subsequently, local optimization, subspace, and convex relaxation methods can be used for the numerical solution. In this paper, we show another approach, based on the recently developed concept of structured data fusion. Structured data fusion problems are solved in the Tensorlab toolbox by local optimization methods. The approach allows fitting of signals with missing samples and adding constraints on the model, such as fixed exponents and common dynamics in multi-channel estimation problems. These problems are non-trivial to solve by other existing methods. Tensorlab is publicly available and the results presented are reproducible. © 2017

关键词： Matrix algebra Approximation theory Data fusion Identification (control systems) Numerical methods Relaxation processes Signal processing Hankel matrix Local optimization methods Local optimizations Low rank approximations Low rank matrix approximations Structured data Sum of exponentials model tensorlab

来源：评论

学校读者我要写书评

暂无评论

Maximum Gap of (Inverse) Cyclotomic polynomials

Maximum Gap of (Inverse) Cyclotomic Polynomials

引用

作者： Ambrosino, Mary Elizabeth North Carolina State University

学位级别：Ph.D.

The cyclotomic polynomial фn is the monic polynomial whose zeroes are the n-th primitive roots of unity and the inverse cyclotomic polynomial &PSgr;n is the monic polynomial whose zeroes are the n-th non-primitive roots of unity. They have numerous applications in number theory, abstract algebra, and cryptography. Thus it is beneficial to further our understanding of their properties. In this dissertation, we present results on the size of their maximum gap, that is, the largest difference between consecutive exponents in the polynomials, denoted g(фn) and g(&PSgr; n). In this paper, we assume that n is odd, square-free. A summary of results is as follows: We present lower bounds for g(фn) and g(&PSgr;n): 1. We prove five lower bounds: α±, β±, γ ±, δ–, ϵ± 2. We observe that they are very often equal to g(ф n) and g(&PSgr;n) 3. We analyze their time complexity compared to direct computation of g(фn) and g(&PSgr; n) We discuss an exact expression for g(ф n): 1. We conjecture that, for n = mp where m is a product of odd primes and p is an odd prime, g(фmp) = ϕ(m) if and only if p > m 2. We present an algorithm which we use to check the conjecture for infinitely many values of mp 3. We prove the conjecture when m = p1p2 and p = p3, where p3 ≡ p1p2 +1, p2 ≡ p1 ±1 We discuss an exact expression for g(&PSgr;n): 1. We prove that g(&PSgr;n) = δ–. under a certain condition 2. We show the condition "almost always" holds in a certain sense

关键词：

来源：评论

学校读者我要写书评

暂无评论

New Mathematical Tools in Reach Control theory. 1

New Mathematical Tools in Reach Control Theory. 1

引用

作者： Ornik, Melkior.0 University of Toronto (Canada).bElectrical and Computer Engineering.0

学位级别：Ph.D.

Reach control theory is an approach to satisfying complex control objectives on a constrained state space. It relies on triangulating the state space into simplices, and devising a separate controller on each simplex to satisfy the control specifications. A fundamental element of reach control theory is the Reach Control Problem (RCP). The goal of the RCP is to drive system trajectories of an affine control system on a simplex to leave this simplex through a predetermined facet. This thesis discusses a number of issues pertaining to reach control theory and the RCP. Its central part is a discussion of the solvability of the RCP. We identify strong necessary conditions for the solvability of the RCP by affine feedback and continuous state feedback, and provide elegant characterizations of these conditions using methods from linear algebra and algebraic topology. Additionally, using the theory of positive systems, Z-matrices, and graph theory, we obtain several new interpretations of the currently known set of necessary and sufficient conditions for the solvability of the RCP by affine feedback. The thesis also provides a rigorous foundation for the notion of exiting a simplex through a facet, and discusses uniqueness and existence of trajectories in reach control with discontinuous feedback. Finally, building on previous results, the thesis includes novel applications of reach control theory to parallel parking and adaptive cruise control. These applications serve to motivate new directions of theoretical research in reach control.

关键词：

来源：评论

学校读者我要写书评

暂无评论

Amari Functors and Dynamics in Gauge Structures 3rd

Amari Functors and Dynamics in Gauge Structures

引用

3rd International SEE Conference on Geometric Science of Information (GSI)

作者： Boyom, Michel Nguiffo Zeglaoui, Ahmed Univ Montpellier Inst Montpellierain Alexander Grothendieck IMAG CC051Pl Eugne Bataillon F-34095 Montpellier France USTHB Fac Math Lab Algebra & Number Theory Bab Ezzouar 16111 Algeria

ISBN: (纸本)9783319684451;9783319684444

We deal with finite dimensional differentiable manifolds. All items are concerned with are differentiable as well. The class of differentiability is C-infinity. A metric structure in a vector bundle E is a constant rank symmetric bilinear vector bundle homomorphism of E x E in the trivial bundle line bundle. We address the question whether a given gauge structure in E is metric. That is the main concerns. We use generalized Amari functors of the information geometry for introducing two index functions defined in the moduli space of gauge structures in E. Beside we introduce a differential equation whose analysis allows to link the new index functions just mentioned with the main concerns. We sketch applications in the differential geometry theory of statistics.

关键词： Gauge structure Metric structure Amari functor Index functions Metric dynamic

来源：评论

学校读者我要写书评

暂无评论

Extremal functions of forbidden multidimensional matrices

引用

DISCRETE MATHEMATICS 2017年第12期340卷 2769-2781页

作者： Geneson, Jesse T. Tian, Peter M. MIT Dept Math Cambridge MA 02139 USA Harvard Univ Dept Math Cambridge MA 02138 USA

Pattern avoidance is a central topic in graph theory and combinatorics. Pattern avoidance in matrices has applications in computer science and engineering, such as robot motion planning and VLSI circuit design. A d-dimensional zero-one matrix A avoids another d-dimensional zero one matrix P if no submatrix of A can be transformed to P by changing some ones to zeros. A fundamental problem is to study the maximum number of nonzero entries in a d-dimensional nx . . . xn matrix that avoids P. This maximum number, denoted by f(n, P, d), is called the extremal function. We advance the extremal theory of matrices in two directions. The methods that we use come from combinatorics, probability, and analysis. Firstly, we obtain non-trivial lower and upper bounds on f(n, P, d) when n is large for every d-dimensional block permutation matrix P. We establish the tight bound circle minus(n(d-1)) on f(n, P, d) for every d-dimensional tuple permutation matrix P. This tight bound has the lowest possible order that an extremal function of a nontrivial matrix can ever achieve. Secondly, we show that the limit inferior of the sequence {f(n,p,d)/n(d-1)} has a lower bound 2(Omega(K1/2)) for a family of kx . . . xk permutation matrices P. We also improve the upper bound on the limit superior from 2(O(k log k)) to 2(O(k)) for all kx . . . xk permutation matrices and show that the new upper bound also holds for tuple permutation matrices. (C) 2017 Elsevier B.V. All rights reserved.

关键词： GRAPH theory VECTOR algebra TRANSMISSION line matrix methods MATRICES DIRECTED graphs

来源：评论

学校读者我要写书评

暂无评论

Using wolfram mathematica in spectral theory 3

Using wolfram mathematica in spectral theory

引用

3rd International Conference on Numerical and Symbolic Computation: Developments and applications, SYMCOMP 2017

作者： Conceicao, Ana C. Pereira, Jose C. Center for Functional Analysis Linear Structures and Applications Faculdade de Ciências e Tecnologia Universidade Do Algarve Faro8005-139 Portugal

ISBN: (纸本)9789899941038

The development of the spectral theory is motivated by the need to solve problems emerging from several fields in Mathematics and Physics. Some progress has been achieved for some classes of singular integral operators (SIOs) whose properties allow the use of particular strategies in the study of the spectral problem but there is still no general and explicit method for obtaining the spectrum of any given SIO. Furthermore, the existing algorithms allow, in general, to study the spectrum of some kind of SIOs but they are not designed to be implemented on a computer. The main goal of this work is to present our spectral algorithms [ASpec-Hankel], [ASpecPaired-Scalar], and [ASpecPaired-Matrix] which use the symbolic and numeric computation capabilities of the computer algebra system Mathematica to explore the spectra of some classes of SIOs, defined on the unit circle. These analytical algorithms allow us to check, for each considered SIO, if a complex number (chosen arbitrarily) belongs to its spectrum. © ECCOMAS, Portugal.

关键词： Computation theory

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：