63 results
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2. Distinct distances on regular varieties over finite fields.
- Author
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Hieu, Do Duy and Pham, Van Thang
- Subjects
- *
FINITE fields , *GENERALIZATION , *SET theory , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset E in a regular variety satisfies | E | ≫ q d − 1 2 + 1 k − 1 , then Δ k , F ( E ) : = { F ( x 1 + ⋯ + x k ) : x i ∈ E , 1 ≤ i ≤ k } ⊇ F q ∖ { 0 } , for some certain families of polynomials F ( x ) ∈ F q [ x 1 , … , x d ] . [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
3. Some new classes of permutation trinomials over finite fields with even characteristic.
- Author
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Gupta, Rohit and Sharma, R.K.
- Subjects
- *
FINITE fields , *PERMUTATIONS , *SET theory , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
Let F q denote the finite field of order q . In this paper, we present four new classes of permutation trinomials of the form x r h ( x 2 m − 1 ) over F 2 2 m . [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
4. The semiclassical small-h limit of loci of roots of subdominant solutions for polynomial potentials.
- Author
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Giller, Stefan
- Subjects
- *
POLYNOMIALS , *POTENTIAL theory (Mathematics) , *GRAPH theory , *MODULES (Algebra) , *MATHEMATICAL analysis , *SET theory - Abstract
In this paper, a description of the small-h limit of loci of zeros of subdominant solutions for polynomial potentials is given called as fundamental solutions in our earlier papers. The considered potentials are those which provide us with the simple turning points only. Three types of Stokes graphs (SG's) associated with the potentials are investigated - the general non-critical ones, the general critical ones but with only single internal Stokes line (SL), and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized h. In particular due to the fact that the small-h limit is semiclassical it is shown that loci of roots of subdominant solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of subdominant solutions on such lines escaping to infinity and a finite number of them on internal Stokes lines. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
5. Matching polynomials for chains of cycles
- Author
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Bian, Hong, Zhang, Fuji, Wang, Guoping, and Yu, Haizheng
- Subjects
- *
POLYNOMIALS , *ALGEBRAIC cycles , *SET theory , *MATHEMATICAL analysis , *STATISTICAL matching - Abstract
Abstract: Došlić and Måløy (2010) obtained the extremal 6-cactus chains with respect to the number of matchings and of independent sets. Motivated by the prior paper, in this paper we give recurrences for matching polynomials of ortho-chains and meta-chains, and show that they are the -cactus chains with the most matchings. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
6. Projections of planar sets in well-separated directions.
- Author
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Orponen, Tuomas
- Subjects
- *
SET theory , *ORTHOGRAPHIC projection , *POLYNOMIALS , *MATHEMATICAL constants , *MATHEMATICAL analysis - Abstract
This paper contains two new projection theorems in the plane. First, let K ⊂ B ( 0 , 1 ) ⊂ R 2 be a set with H ∞ 1 ( K ) ∼ 1 , and write π e ( K ) for the orthogonal projection of K into the line spanned by e ∈ S 1 . For 1 / 2 ≤ s < 1 , write E s : = { e : N ( π e ( K ) , δ ) ≤ δ − s } , where N ( A , r ) is the r -covering number of the set A . It is well-known – and essentially due to R. Kaufman – that N ( E s , δ ) ⪅ δ − s . Using the polynomial method, I prove that N ( E s , r ) ⪅ min { δ − s ( δ r ) 1 / 2 , r − 1 } , δ ≤ r ≤ 1 . I construct examples showing that the exponents in the bound are sharp for δ ≤ r ≤ δ s . The second theorem concerns projections of 1-Ahlfors–David regular sets. Let A ≥ 1 and 1 / 2 ≤ s < 1 be given. I prove that, for p = p ( A , s ) ∈ N large enough, the finite set of unit vectors S p : = { e 2 π i k / p : 0 ≤ k < p } has the following property. If K ⊂ B ( 0 , 1 ) is non-empty and 1-Ahlfors–David regular with regularity constant at most A , then 1 p ∑ e ∈ S p N ( π e ( K ) , δ ) ≥ δ − s for all small enough δ > 0 . In particular, dim ‾ B π e ( K ) ≥ s for some e ∈ S p . [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
7. A MAXIMUM RESONANT SET OF POLYOMINO GRAPHS.
- Author
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HEPING ZHANG and XIANGQIAN ZHOU
- Subjects
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SET theory , *POLYNOMIALS , *GRAPH theory , *MATHEMATICAL proofs , *MATHEMATICAL analysis - Abstract
A polyomino graph P is a connected finite subgraph of the infinite plane grid such that each finite face is surrounded by a regular square of side length one and each edge belongs to at least one square. A dimer covering of P corresponds to a perfect matching. Different dimer coverings can interact via an alternating cycle (or square) with respect to them. A set of disjoint squares of P is a resonant set if P has a perfect matching M so that each one of those squares is M-alternating. In this paper, we show that if K is a maximum resonant set of P, then P - K has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to the cardinality of a maximum resonant set. This confirms a conjecture of Xu et al. [26]. We also show that if K is a maximal alternating set of P, then P - K has a unique perfect matching. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
8. Some properties of classes of real self-reciprocal polynomials.
- Author
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Botta, Vanessa, Bracciali, Cleonice F., and Pereira, Junior A.
- Subjects
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SET theory , *POLYNOMIALS , *DISTRIBUTION (Probability theory) , *MONOTONIC functions , *MATHEMATICAL analysis - Abstract
The purpose of this paper is twofold. Firstly we investigate the distribution, simplicity and monotonicity of the zeros around the unit circle and real line of the real self-reciprocal polynomials R n ( λ ) ( z ) = 1 + λ ( z + z 2 + ⋯ + z n − 1 ) + z n , n ≥ 2 and λ ∈ R . Secondly, as an application of the first results we give necessary and sufficient conditions to guarantee that all zeros of the self-reciprocal polynomials S n ( λ ) ( z ) = ∑ k = 0 n s n , k ( λ ) z k , n ≥ 2 , with s n , 0 ( λ ) = s n , n ( λ ) = 1 , s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , ⌊ n / 2 ⌋ when n is odd, and s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , n / 2 − 1 , s n , n / 2 ( λ ) = ( n / 2 ) λ when n is even, lie on the unit circle, solving then an open problem given by Kim and Park in 2008. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
9. Complete $$r$$ -partite Graphs Determined by their Domination Polynomial.
- Author
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Anthony, Barbara and Picollelli, Michael
- Subjects
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DOMINATING set , *POLYNOMIALS , *SET theory , *CARDINAL numbers , *LOGICAL prediction , *UNIQUENESS (Mathematics) , *MATHEMATICAL analysis - Abstract
The domination polynomial of a graph is the polynomial whose coefficients count the number of dominating sets of each cardinality. A recent question asks which graphs are uniquely determined (up to isomorphism) by their domination polynomial. In this paper, we completely describe the complete $$r$$ -partite graphs which are; in the bipartite case, this settles in the affirmative a conjecture of Aalipour et al. (Ars Comb, ). [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
10. Smooth hyperbolicity cones are spectrahedral shadows.
- Author
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Netzer, Tim and Sanyal, Raman
- Subjects
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POLYNOMIALS , *HYPERBOLIC spaces , *SET theory , *CONES , *LINEAR systems , *MATHEMATICAL analysis - Abstract
Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone, that is, a spectrahedral shadow. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
11. A proof of Alon–Babai–Suzuki’s conjecture and multilinear polynomials.
- Author
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Hwang, Kyung-Won and Kim, Younjin
- Subjects
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POLYNOMIALS , *LOGICAL prediction , *PRIME numbers , *MATHEMATICAL proofs , *SET theory , *MATHEMATICAL analysis - Abstract
Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be disjoint subsets of { 0 , 1 , ⋯ p − 1 } , where p is a prime and F = { F 1 , F 2 , … , F m } be a family of subsets of [ n ] such that | F i | (mod p ) ∈ K for all F i ∈ F and | F i ∩ F j | (mod p ) ∈ L for i ≠ j . In 1991 Alon, Babai and Suzuki conjectured that if n ≥ s + max 1 ≤ i ≤ r k i , then | F | ≤ n s + n s − 1 + ⋯ + n s − r + 1 . In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
12. GENERALIZED q-CALKIN-WILF TREES AND c-HYPER m-EXPANSIONS OF INTEGERS.
- Author
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Mansour, Toufik and Shattuck, Mark
- Subjects
- *
MATHEMATICAL expansion , *INTEGERS , *POLYNOMIALS , *SET theory , *STATISTICS , *MATHEMATICAL analysis - Abstract
A hyperbinary expansion of a positive integer n is a partition of n into powers of 2 in which each part appears at most twice. In this paper, we consider a generalization of this concept and a certain statistic on the corresponding set of expansions of n. We then define q-generalized m-ary trees whose vertices are labeled by ratios of two consecutive terms within the sequence of distribution polynomials for the aforementioned statistic. When m = 2, we obtain a variant of a previously considered q-Calkin-Wilf tree. [ABSTRACT FROM AUTHOR]
- Published
- 2015
13. 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
14. A note on α-type polynomial sets.
- Author
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Rapeli, S.J. and Shukla, A.K.
- Subjects
- *
POLYNOMIALS , *SET theory , *GENERALIZATION , *MATHEMATICAL variables , *APPLIED mathematics , *MATHEMATICAL analysis - Abstract
In this paper, we discuss α-type polynomial sets and also generalized α -type polynomial sets of type zero in two variables. Some properties of certain polynomials have also been shown, in support of α-type zero in two variables. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
- View/download PDF
15. Nil Bohr0-sets and polynomial recurrence.
- Author
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Tu, Siming
- Subjects
- *
POLYNOMIALS , *RECURSIVE sequences (Mathematics) , *SET theory , *MATHEMATICAL proofs , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
Abstract: In Huang et al. [17] it was proved that for any Nil d Bohr0-set , there are a minimal system and a non-empty open subset of with , and for any minimal system and any open non-empty , the set is an almost Nil d Bohr0-set. The polynomial form of this problem is considered in this paper. It is shown that the latter is still true in the polynomial case, while the former is not in general. We also consider the special case when the system is a nilsystem. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
16. Nonexistence of exceptional 5-class association schemes with two Q-polynomial structures.
- Author
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Ma, Jianmin and Wang, Kaishun
- Subjects
- *
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
17. Several classes of complete permutation polynomials.
- Author
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Tu, Ziran, Zeng, Xiangyong, and Hu, Lei
- Subjects
- *
PERMUTATIONS , *POLYNOMIALS , *SET theory , *FINITE fields , *MATHEMATICAL proofs , *MATHEMATICAL analysis - Abstract
Abstract: In this paper, three classes of monomials and one class of trinomials over finite fields of even characteristic are proposed. They are proved to be complete permutation polynomials. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
18. A class of permutation binomials over finite fields.
- Author
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Hou, Xiang-dong
- Subjects
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SET theory , *PERMUTATIONS , *BINOMIAL theorem , *FINITE fields , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
Abstract: Let q be a prime power and , where . It was recently conjectured that f is a permutation polynomial of if and only if one of the following holds: (i) , ; (ii) , ; (iii) , . We confirm this conjecture in the present paper. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
19. 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
20. Some order relations induced by fuzzy subsets.
- Author
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Jeong Soon Han, Young Hee Kim, and Keum Sook So
- Subjects
- *
FUZZY sets , *SET theory , *GROUPOIDS , *ALGEBRA , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
In this paper, we introduce the notion of groupoids belongs to fuzzy subalgebras, and show that the induced poset of a groupoid belongs to fuzzy subalgebras is an ordinal sum of anti-chains, and obtain several properties of induced posets by fuzzy sets. [ABSTRACT FROM AUTHOR]
- Published
- 2013
21. Further results on a class of permutation polynomials over finite fields.
- Author
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Li, Nian, Helleseth, Tor, and Tang, Xiaohu
- Subjects
- *
SET theory , *PERMUTATIONS , *POLYNOMIALS , *FINITE fields , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
Abstract: A class of permutation polynomials with given form over finite fields is investigated in this paper, which is a further study on a recent work of Zha and Hu. Based on some particular techniques over finite fields, two results obtained by Zha and Hu are improved and new permutation polynomials are also obtained. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
22. An explicit construction of -existentially closed graphs.
- Author
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Vinh, Le Anh
- Subjects
- *
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
23. A proof of Crouzeix’s conjecture for a class of matrices
- Author
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Choi, Daeshik
- Subjects
- *
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
24. A new algorithmic scheme for computing characteristic sets
- Author
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Jin, Meng, Li, Xiaoliang, and Wang, Dongming
- Subjects
- *
SCHEMES (Algebraic geometry) , *TRIANGULARIZATION (Mathematics) , *POLYNOMIALS , *SET theory , *ALGORITHMS , *MATHEMATICAL analysis - Abstract
Abstract: Ritt–Wuʼs algorithm of characteristic sets is the most representative for triangularizing sets of multivariate polynomials. Pseudo-division is the main operation used in this algorithm. In this paper we present a new algorithmic scheme for computing generalized characteristic sets by introducing other admissible reductions than pseudo-division. A concrete subalgorithm is designed to triangularize polynomial sets using selected admissible reductions and several effective elimination strategies and to replace the algorithm of basic sets (used in Ritt–Wuʼs algorithm). The proposed algorithm has been implemented and experimental results show that it performs better than Ritt–Wuʼs algorithm in terms of computing time and simplicity of output for a number of non-trivial test examples. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
25. Half-integrality of node-capacitated multiflows and tree-shaped facility locations on trees.
- Author
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Hirai, Hiroshi
- Subjects
- *
TREE graphs , *INTEGRALS , *DUALITY theory (Mathematics) , *PROBLEM solving , *POLYNOMIALS , *MATHEMATICAL analysis , *SET theory - Abstract
In this paper, we establish a novel duality relationship between node-capacitated multiflows and tree-shaped facility locations. We prove that the maximum value of a tree-distance-weighted maximum node-capacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in a tree, and the maximum is attained by a half-integral multiflow. Utilizing this duality, we show that a half-integral optimal multiflow and an optimal location can be found in strongly polynomial time. These extend previously known results in the maximum free multiflow problems. We also show that the set of tree-distance weights is the only class having bounded fractionality in maximum node-capacitated multiflow problems. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
26. Classification of Poset-Block Spaces Admitting MacWilliams-Type Identity.
- Author
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Pinheiro, Jerry Anderson and Firer, Marcelo
- Subjects
- *
PARTIALLY ordered sets , *MATHEMATICAL analysis , *ORDERED sets , *SET theory , *MATHEMATICAL physics - Abstract
In this paper, we prove that a poset-block space admits a MacWilliams-type identity if and only if the poset is hierarchical, and at any level of the poset, all the blocks have the same dimension. When the poset-block admits the MacWilliams-type identity, we explicitly state the relation between the weight enumerators of a code and its dual. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
27. Weighted sums of consecutive values of a polynomial
- Author
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Chen, Feng-Juan and Chen, Yong-Gao
- Subjects
- *
INTEGERS , *POLYNOMIALS , *SET theory , *DIVISOR theory , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
Abstract: In this paper, we determine all sets A of integers such that, for any integral-valued polynomial which has no fixed divisor, for all integers and n, there are infinitely many integers and a choice of such that . The earlier result shows that is such a set. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
28. Oriented unicyclic graphs with the first largest skew energies
- Author
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Zhu, Jianming
- Subjects
- *
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
- View/download PDF
29. Polynomials non-negative on strips and half-strips
- Author
-
Nguyen, Ha and Powers, Victoria
- Subjects
- *
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
30. ON GROUPS WHOSE GEODESIC GROWTH IS POLYNOMIAL.
- Author
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BRIDSON, MARTIN R., BURILLO, JOSÉ, ELDER, MURRAY, ŠUNIĆ, ZORAN, and Meakin, J.
- Subjects
- *
ABELIAN groups , *POLYNOMIALS , *SET theory , *CYCLIC groups , *GEODESICS , *MATHEMATICAL analysis - Abstract
This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups). [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
31. On a Diophantine equation of Ayad and Kihel.
- Author
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El Bachraoui, Mohamed and Luca, Florian
- Subjects
- *
POLYNOMIALS , *DIOPHANTINE equations , *NUMBER theory , *SET theory , *MATHEMATICAL sequences , *NATURAL numbers , *MATHEMATICAL analysis - Abstract
Let f(n) denote the number of relatively prime sets in {1, … , n}. This is sequence A085945 in Sloane's On-Line Encyclopedia of Integer Sequences. Motivated by a paper of Ayad and Kihel [1], we show that there are at most finitely many positive integers n such that f(n) is a perfect power of exponent > 1 of some other integer. We also show that the sequence {f(n)} n≥1 is not holonomic; that is, it satisfies no recurrence relation of finite order with polynomial coefficients. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
32. On cosets in Coxeter groups.
- Author
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Hart, Sarah B. and Rowley, Peter J.
- Subjects
- *
COXETER groups , *SET theory , *PARTIALLY ordered sets , *GENERALIZATION , *MATHEMATICAL analysis , *POLYNOMIALS - Abstract
In this paper the notion of Coxeter length for a subset of a Coxeter group, as introduced in [9], is investigated for various subsets of a Coxeter group. Mostly cosets of various subgroups are examined as well as the associated idea of X-posets, which is a vast generalization of the Bruhat order. [ABSTRACT FROM AUTHOR]
- Published
- 2012
33. Some New Classes of Generalized Apostol-Euler and Apostol-Genocchi Polynomials.
- Author
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Tremblay, R., Gaboury, S., and Fugère, B.-J.
- Subjects
- *
POLYNOMIALS , *SET theory , *GENERALIZATION , *EULER method , *MATHEMATICAL formulas , *MATHEMATICAL analysis - Abstract
The main object of this paper is to introduce and investigate two new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials. In particular,we obtain a new addition formula for the new class of the generalized Apostol-Euler polynomials. We also give an extension and some analogues of the Srivastava-Pintér addition theorem obtained in the works by Srivastava and Pintér (2004) and R. Tremblay, S. Gaboury, B.-J. Fugère, and Tremblay et al. (2011). for both classes. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
34. Constructing a spanning tree with many leaves.
- Author
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Gravin, N.
- Subjects
- *
SPANNING trees , *MAXIMA & minima , *SET theory , *POLYNOMIALS , *NUMBER theory , *ALGORITHMS , *MATHEMATICAL analysis - Abstract
It was shown by Griggs and Wu that a graph of minimal degree 4 on n vertices has a spanning tree with at least $$ \frac{2}{5} $$ n leaves, which is asymptomatically the best possible bound for this class of graphs. In this paper, we present a polynomial time algorithm that finds in any graph with k vertices of degree greater than or equal to 4 and k′ vertices of degree 3 a spanning tree with $$ \left[ {\frac{2}{5} \cdot k + \frac{2}{{15}} \cdot k'} \right] $$ leaves. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
35. Exact algorithms for dominating set
- Author
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van Rooij, Johan M.M. and Bodlaender, Hans L.
- Subjects
- *
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
36. On the (non-)contractibility of the order complex of the coset poset of a classical group
- Author
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Patassini, Massimiliano
- Subjects
- *
GROUP theory , *MATHEMATICAL complexes , *SET theory , *ZETA functions , *AUTOMORPHISMS , *MATHEMATICAL proofs , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
Abstract: Let G be a classical group and suppose that G does not contains non-trivial graph automorphisms. In this paper we prove that the order complex of the coset poset of G is non-contractible. In order to prove it, we show that does not vanish, where is the Dirichlet polynomial associated to the group G. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
37. Cyclotomic polynomial coefficients with n and k in prescribed residue classes
- Author
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Fintzen, Jessica
- Subjects
- *
CYCLOTOMY , *POLYNOMIALS , *SET theory , *BINOMIAL coefficients , *DIRICHLET problem , *NUMBER theory , *MATHEMATICAL analysis - Abstract
Abstract: Let be the kth coefficient of the nth cyclotomic polynomial. Ji, Li and Moree (2009) showed that . In this paper we will determine . [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
38. More about dicriticals.
- Subjects
- *
SET theory , *ALGEBRAIC surfaces , *MATHEMATICAL functions , *DIACRITICS , *POLYNOMIALS , *MATHEMATICAL analysis - Abstract
This paper attempts to characterize the dicritical set of a rational function which generates a special pencil at a simple point of an algebraic or arithmetical surface. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
39. Polynomials over A*-Algebras.
- Author
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Rao, J. Venkateswara and Rao, P. Koteswara
- Subjects
- *
POLYNOMIALS , *MATHEMATICAL functions , *BOOLEAN algebra , *SET theory , *MATHEMATICAL analysis , *NUMERICAL analysis , *ALGEBRAIC logic - Abstract
This paper is a study on the polynomials over A*-algebras and their useful characterizations. It also contributes to the understanding of the polynomial functions over A*-algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2011
40. On Fractional Differential Operator Involving H̄-function.
- Author
-
Sharma, S. C. and Bhargava, Rachna
- Subjects
- *
MATHEMATICAL functions , *FRACTIONAL calculus , *DIFFERENTIAL operators , *POLYNOMIALS , *SET theory , *GENERALIZATION , *MATHEMATICAL analysis - Abstract
In the present paper we establish an important result involving a fractional differential operator given by Misra [5] for the product of H̄ - function, general polynomial set and two general class of polynomials. On account of the general nature of the functions and the polynomials occurring in our main results a large number of simple results follow as its special cases. For the sake of illustration, we present here two special cases involving product of general class of polynomials and the generalized Riemann Zeta function. [ABSTRACT FROM AUTHOR]
- Published
- 2011
41. On a class of equations stemming from various quadrature rules.
- Author
-
Koclȩga-Kulpa, B. and Szostok, T.
- Subjects
- *
NUMERICAL solutions to functional equations , *SET theory , *POLYNOMIALS , *MATHEMATICAL forms , *APPROXIMATION theory , *MATHEMATICAL analysis - Abstract
We deal with the functional equation motivated by quadrature rules of approximate integration. In previous results the solutions of this equation were found only in some particular cases. For example the coefficients λ were supposed to be rational or the equation in question was solved only for n=2. In the current paper we do not assume any particular form of coefficients occurring in this equation and we allow n to be any positive integer. Moreover, we obtain a solution of our equation without any regularity assumptions concerning the functions f and F. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
42. 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
- *
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
43. Characterization of graphs using domination polynomials
- Author
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Akbari, Saieed, Alikhani, Saeid, and Peng, Yee-hock
- Subjects
- *
GRAPH theory , *POLYNOMIALS , *DOMINATING set , *SET theory , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
Abstract: Let be a simple graph of order . The domination polynomial of is the polynomial , where is the number of dominating sets of of size . A root of is called a domination root of . We denote the set of distinct domination roots by . Two graphs and are said to be -equivalent, written as , if . The -equivalence class of is . A graph is said to be -unique if . In this paper, we show that if a graph has two distinct domination roots, then . Also, if is a graph with no pendant vertex and has three distinct domination roots, then . Also, we study the -equivalence classes of some certain graphs. It is shown that if , then is -unique, and if , then consists of exactly two graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
44. On a polynomial approximation problem
- Author
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Danielyan, Arthur A.
- Subjects
- *
APPROXIMATION theory , *POLYNOMIALS , *ANALYTIC functions , *SET theory , *MATHEMATICAL analysis - Abstract
Abstract: Let be a closed subset of the unit circle and let . We investigate the problem of uniform approximation of on by polynomials which are uniformly bounded on the unit disk . In a particular case when is a closed arc of , the problem was solved by L. Zalcman in 1982, who has also pointed out the possibility of considering more general approximation sets instead of an arc. The present paper gives a necessary and sufficient solution of the above problem. In fact we show that the (simple) description of given by Zalcman for the case of an arc also holds in the general case. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
45. More explicit classes of permutation polynomials of
- Author
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Yuan, Pingzhi
- Subjects
- *
PERMUTATIONS , *POLYNOMIALS , *LINEAR algebra , *NONLINEAR theories , *MATHEMATICAL analysis , *SET theory - Abstract
Abstract: Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, more linearized permutation polynomials and non-linearized permutation polynomials over are presented. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
46. 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
- *
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
47. Vertex fusion under distance constraints
- Author
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Comas, Marc and Serna, Maria
- Subjects
- *
GRAPH theory , *POLYNOMIALS , *SET theory , *MATHEMATICAL analysis , *COMBINATORICS - Abstract
Abstract: In this paper, we analyze parameter improvement under vertex fusion in a graph . This is a setting in which a new graph is obtained after identifying a subset of vertices of in a single vertex. We are interested in distance parameters, in particular diameter, radius and eccentricity of a vertex . We show that the corresponding problem is NP-Complete for the three parameters. We also find graph classes in which the problem can be solved in polynomial time. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
48. Polynomial-time algorithm for fixed points of nontrivial morphisms
- Author
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Holub, Štěpán
- Subjects
- *
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
49. Determination of the normalization level of database schemas through equivalence classes of attributes.
- Author
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Vitalie, Cotelea
- Subjects
- *
DATABASES , *POLYNOMIALS , *EQUIVALENCE classes (Set theory) , *EQUIVALENCE relations (Set theory) , *SET theory , *NORMAL forms (Mathematics) , *MATHEMATICAL analysis , *COMPUTER algorithms , *COMPUTER science - Abstract
In this paper, based on equivalence classes of attributes there are formulated necessary and sufficient conditions that constraint a database schema to be in the second, third or Boyce-Codd normal forms. These conditions offer a polynomial complexity for the testing algorithms of the normalizations level. [ABSTRACT FROM AUTHOR]
- Published
- 2009
50. Tell Me Who I Am: An Interactive Recommendation System.
- Author
-
Alon, Noga, Awerbuch, Baruch, Azar, Yossi, and Patt-Shamir, Boaz
- Subjects
- *
ALGORITHMS , *POLYNOMIALS , *APPROXIMATION theory , *VECTOR analysis , *UNIVERSAL algebra , *FUNCTIONAL analysis , *MATHEMATICAL functions , *SET theory , *MATHEMATICAL analysis - Abstract
We consider a model of recommendation systems, where each member from a given set of players has a binary preference to each element in a given set of objects: intuitively, each player either likes or dislikes each object. However, the players do not know their preferences. To find his preference of an object, a player may probe it, but each probe incurs unit cost. The goal of the players is to learn their complete preference vector (approximately) while incurring minimal cost. This is possible if many players have similar preference vectors: such a set of players with similar “taste” may split the cost of probing all objects among them, and share the results of their probes by posting them on a public billboard. The problem is that players do not know a priori whose taste is close to theirs. In this paper we present a distributed randomized peer-to-peer algorithm in which each player outputs a vector which is close to the best possible approximation of the player’s real preference vector after a polylogarithmic number of rounds. The algorithm works under adversarial preferences. Previous algorithms either made severely limiting assumptions on the structure of the preference vectors, or had polynomial overhead. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
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