18 results
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2. Offset polygon and annulus placement problems.
- Author
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Barequet, Gill and Goryachev, Alex
- Subjects
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POLYGONS , *BOUNDARY value problems , *SET theory , *MATHEMATICAL transformations , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
Abstract: The δ-annulus of a polygon P is the closed region containing all points in the plane at distance at most δ from the boundary of P. An inner (resp., outer) δ-offset polygon is the polygon defined by the inner (resp., outer) boundary of its δ-annulus. In this paper we address three major problems of covering a given point set S by an offset version or a polygonal annulus of a polygon P. First, the Maximum Cover objective is, given a value of δ, to cover as many points from S as possible by the δ-offset (or by the δ-annulus) of P, allowing translation and rotation. Second, the Containment problem is to minimize the value of δ such that there is a rigid transformation of the δ-offset (or the δ-annulus) of P that covers all points from S. Third, in the Partial Containment problem we seek the minimum offset of P covering points. These problems arise in many applications where one needs to match a given polygonal figure (a known model) to a set of points (usually, obtained measures). We address several variants of these problems, including convex and simple polygons, as well as polygons with holes and sets of polygons, and obtain algorithms with low-degree polynomial running times in all cases. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
3. Nonexistence of exceptional 5-class association schemes with two Q-polynomial structures.
- Author
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Ma, Jianmin and Wang, Kaishun
- Subjects
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EXISTENCE theorems , *ASSOCIATION schemes (Combinatorics) , *SET theory , *POLYNOMIALS , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
Abstract: In Suzuki (1998) [7] Suzuki gave a classification of association schemes with multiple Q-polynomial structures, allowing for one exceptional case which has five classes. In this paper, we rule out the existence of this case. Hence Suzukiʼs theorem mirrors exactly the well-known counterpart for association schemes with multiple P-polynomial structures, a result due to Eiichi Bannai and Etsuko Bannai in 1980. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
4. Approximate zero polynomials of polynomial matrices and linear systems.
- Author
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Karcanias, Nicos and Halikias, George
- Subjects
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APPROXIMATION theory , *POLYNOMIALS , *MATRICES (Mathematics) , *LINEAR systems , *SET theory , *MATHEMATICAL analysis - Abstract
Abstract: This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the “approximate”, or “almost” zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the “approximate decoupling polynomials”. The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of “approximate decoupling polynomials”. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
5. An explicit construction of -existentially closed graphs.
- Author
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Vinh, Le Anh
- Subjects
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GRAPH theory , *EXISTENCE theorems , *SET theory , *POLYNOMIALS , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
Abstract: Let be positive integers. A -edge-colored graph is -e.c. or -existentially closed if for any disjoint sets of vertices with , there is a vertex not in such that all edges from this vertex to the set are colored by the -th color. In this paper, we give an explicit construction of a -e.c. graph of polynomial order. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
6. A proof of Crouzeix’s conjecture for a class of matrices
- Author
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Choi, Daeshik
- Subjects
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MATRICES (Mathematics) , *LOGICAL prediction , *POLYNOMIALS , *ALGEBRAIC field theory , *SET theory , *MATHEMATICAL analysis - Abstract
Abstract: Crouzeix’s conjecture is that for any square matrix A and any polynomial p we havewhere is the field of values of A and denotes the spectral norm. In this paper, we show that the conjecture holds for the matrices of the formwhere and . [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
7. Oriented unicyclic graphs with the first largest skew energies
- Author
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Zhu, Jianming
- Subjects
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DIRECTED graphs , *PATHS & cycles in graph theory , *SET theory , *UNDIRECTED graphs , *EIGENVALUES , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
Abstract: Let G σ be an oriented graph obtained by assigning an orientation σ to the edge set of a simple undirected graph G such that G σ becomes a directed graph. Let S(G σ ) be the skew adjacency matrix of G σ . The skew energy of G σ is defined as the sum of the absolute values of all eigenvalues of S(G σ ). In this paper, we provide a new method to compare the skew energies of two oriented graphs whose skew characteristic polynomials satisfy a given recurrence relation and determine the oriented unicyclic graphs of order n with the first largest skew energies for . [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
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8. Polynomials non-negative on strips and half-strips
- Author
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Nguyen, Ha and Powers, Victoria
- Subjects
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POLYNOMIALS , *NONNEGATIVE matrices , *SEMIALGEBRAIC sets , *SET theory , *COMPACTIFICATION (Mathematics) , *MATHEMATICAL analysis , *LINEAR algebra - Abstract
Abstract: In 2008, Marshall (2010) settled a long-standing open problem by showing that if is a polynomial that is non-negative on the strip , then there exist sums of squares such that . In this paper, we generalize Marshall’s result to various strips and half-strips in the plane. Our results give many new examples of non-compact semialgebraic sets in for which one can characterize all polynomials which are non-negative on the set. For example, we show that if is a compact subset of the real line and a specific set of generators for as a semialgebraic set, then whenever is non-negative on , there are sums of squares such that . [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
9. Exact algorithms for dominating set
- Author
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van Rooij, Johan M.M. and Bodlaender, Hans L.
- Subjects
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DOMINATING set , *COMPUTER-aided design , *ALGORITHMS , *MATHEMATICAL proofs , *COMBINATORICS , *SET theory , *MATHEMATICAL analysis , *POLYNOMIALS - Abstract
Abstract: The measure and conquer approach has proven to be a powerful tool to analyse exact algorithms for combinatorial problems like Dominating Set and Independent Set. This approach is used in this paper to obtain a faster exact algorithm for Dominating Set. We obtain this algorithm by considering a series of branch and reduce algorithms. This series is the result of an iterative process in which a mathematical analysis of an algorithm in the series with measure and conquer results in a convex or quasiconvex programming problem. The solution, by means of a computer, to this problem not only gives a bound on the running time of the algorithm, but can also give an indication on where to look for a new reduction rule, often giving a new, possibly faster algorithm. As a result, we obtain an time and polynomial space algorithm. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
10. Decomposing polynomial sets into simple sets over finite fields: The zero-dimensional case
- Author
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Li, Xiaoliang, Mou, Chenqi, and Wang, Dongming
- Subjects
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MATHEMATICAL decomposition , *POLYNOMIALS , *SET theory , *FINITE fields , *ALGORITHMS , *MATHEMATICAL analysis - Abstract
Abstract: This paper presents algorithms for decomposing any zero-dimensional polynomial set into simple sets over an arbitrary finite field, with an associated ideal or zero decomposition. As a key ingredient of these algorithms, we generalize the squarefree decomposition approach for univariate polynomials over a finite field to that over the field product determined by a simple set. As a subprocedure of the generalized squarefree decomposition approach, a method is proposed to extract the th root of any element in the field product. Experiments with a preliminary implementation show the effectiveness of our algorithms. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
11. A randomized algorithm for determining dominating sets in graphs of maximum degree five
- Author
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Khamis, Soheir M., Daoud, Sameh S., and Essa, Hanaa A.E.
- Subjects
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ALGORITHMS , *DOMINATING set , *SET theory , *GRAPH theory , *TOPOLOGICAL degree , *POLYNOMIALS , *COMPUTER science , *MATHEMATICAL analysis - Abstract
Abstract: The paper is devoted to demonstrating a randomized algorithm for determining a dominating set in a given graph having a maximum degree of five. The algorithm follows the Las Vegas technique. Furthermore, the concept of a 2-separated collection of subsets of vertices in graphs is used. The suggested algorithm is based on a condition of the upper bound of the cardinality of a local dominating set. If the condition is not satisfied, then the algorithm halts with an appropriate message. Otherwise, the algorithm determines the dominating set. The given algorithm is considered a polynomial-time approximation one. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
12. Polynomial-time algorithm for fixed points of nontrivial morphisms
- Author
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Holub, Štěpán
- Subjects
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FIXED point theory , *POLYNOMIALS , *MATHEMATICAL analysis , *ALGORITHMS , *MORPHISMS (Mathematics) , *SET theory - Abstract
Abstract: A word is a fixed point of a nontrivial morphism if and is not the identity on the alphabet of . The paper presents the first polynomial algorithm deciding whether a given finite word is such a fixed point. The algorithm also constructs the corresponding morphism, which has the smallest possible number of non-erased letters. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
13. Super-stationary set, subword problem and the complexity
- Author
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Kamae, Teturo, Rao, Hui, Tan, Bo, and Xue, Yu-Mei
- Subjects
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COMPUTATIONAL complexity , *SET theory , *MAXIMAL functions , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
Abstract: Let be a nonempty closed set with . For and , define by and We call a super-stationary set if holds for any infinite subset of . Denoting the derived set (i.e. the set of accumulating points) of and with , it is known [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] that for any nonempty closed subset of such that there exists an infinite subset of with , there exists an infinite subset such that is a super-stationary set. Moreover, if for any infinite subset of , then the maximal pattern complexity [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] is . Thus, the uniform complexity functions are realized by the super-stationary sets [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)]. We call a super-subword of if there exists with such that . Let be the set of having no super-subword . Denote where . In this paper, we prove that the class of super-stationary sets other than coincides with the class of for nonempty finite sets . Moreover, it also coincides with the class of for nonempty finite sets , where is the set of minimal covers of . Using these expressions, we can calculate the complexity of super-stationary sets and prove that the complexity function of a super-stationary set in is either or a polynomial function of for large . We also discuss the word problems related to the super-subwords. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
14. On the chromaticity of complete multipartite graphs with certain edges added
- Author
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Lau, G.C. and Peng, Y.H.
- Subjects
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GRAPH coloring , *POLYNOMIALS , *BIPARTITE graphs , *SET theory , *VERTEX operator algebras , *MATHEMATICAL analysis - Abstract
Abstract: Let be the chromatic polynomial of a graph . A graph is chromatically unique if for any graph , implies H is isomorphic to . For integers , , denote by the complete -partite graph that has partite sets of size and one partite set of size . Let be the set of graphs obtained from by adding a set of edges to the partite set of size such that is bipartite. If , denote the only graph in by . In this paper, we shall prove that for and , each graph is chromatically unique if and only if is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph is chromatically unique for and . [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
15. Branchwidth of chordal graphs
- Author
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Paul, Christophe and Telle, Jan Arne
- Subjects
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MATHEMATICAL decomposition , *GRAPH theory , *GRAPH algorithms , *SET theory , *TREE graphs , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
Abstract: This paper revisits the ‘branchwidth territories’ of Kloks, Kratochvíl and Müller [T. Kloks, J. Kratochvíl, H. Müller, New branchwidth territories, in: 16th Ann. Symp. on Theoretical Aspect of Computer Science, STACS, in: Lecture Notes in Computer Science, vol. 1563, 1999, pp. 173–183] to provide a simpler proof, and a faster algorithm for computing the branchwidth of an interval graph. We also generalize the algorithm to the class of chordal graphs, albeit at the expense of exponential running time. Compliance with the ternary constraint of the branchwidth definition is facilitated by a simple new tool called -troikas: three sets of size at most each are a -troika of set , if any two have union . We give a straightforward algorithm, computing branchwidth for an interval graph on edges, vertices and maximal cliques. We also prove a conjecture of Mazoit [F. Mazoit, A general scheme for deciding the branchwidth, Technical Report RR2004-34, LIP — École Normale Supérieure de Lyon, 2004. http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-34.pdf], by showing that branchwidth can be computed in polynomial time for a chordal graph given with a clique tree having a polynomial number of subtrees. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
16. The Sinc–Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients
- Author
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Saadatmandi, Abbas, Dehghan, Mehdi, and Azizi, Mohammad-Reza
- Subjects
- *
SET theory , *APPROXIMATION theory , *POLYNOMIALS , *NUMERICAL analysis , *NUMERICAL solutions to heat equation , *MATHEMATICAL analysis , *COEFFICIENTS (Statistics) - Abstract
Abstract: This paper deals with the numerical solution of classes of fractional convection–diffusion equations with variable coefficients. The fractional derivatives are described based on the Caputo sense. Our approach is based on the collocation techniques. The method consists of reducing the problem to the solution of linear algebraic equations by expanding the required approximate solution as the elements of shifted Legendre polynomials in time and the Sinc functions in space with unknown coefficients. The properties of Sinc functions and shifted Legendre polynomials are then utilized to evaluate the unknown coefficients. Several examples are given and the numerical results are shown to demonstrate the efficiency of the newly proposed method. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
17. Polynomial modeling for time-varying systems based on a particle swarm optimization algorithm
- Author
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Chan, Kit Yan, Dillon, Tharam S., and Kwong, C.K.
- Subjects
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PARTICLE swarm optimization , *GENETIC programming , *POLYNOMIALS , *ALGORITHMS , *ARITHMETIC , *MATHEMATICAL models , *SET theory , *MATHEMATICAL analysis - Abstract
Abstract: In this paper, an effective particle swarm optimization (PSO) is proposed for polynomial models for time varying systems. The basic operations of the proposed PSO are similar to those of the classical PSO except that elements of particles represent arithmetic operations and variables of time-varying models. The performance of the proposed PSO is evaluated by polynomial modeling based on various sets of time-invariant and time-varying data. Results of polynomial modeling in time-varying systems show that the proposed PSO outperforms commonly used modeling methods which have been developed for solving dynamic optimization problems including genetic programming (GP) and dynamic GP. An analysis of the diversity of individuals of populations in the proposed PSO and GP reveals why the proposed PSO obtains better results than those obtained by GP. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
18. Reference points and roughness
- Author
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Kedukodi, Babushri Srinivas, Kuncham, Syam Prasad, and Bhavanari, Satyanarayana
- Subjects
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FUZZY logic , *ROUGH sets , *SET theory , *APPROXIMATION theory , *DECISION making , *FUNCTIONAL analysis , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
Abstract: In this paper, we introduce the notion of a reference point and provide local approximations for a subset of the universe. The notion of a reference point naturally gives rise to a rough approximations framework, wherein several approximations are possible on the same set. Also, we present an extension to the decision theoretic rough set model by using reference points. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
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