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《Discrete Mathematics》2020,343(3):111721
The -additive codes are subgroups of , and can be seen as a generalization of linear codes over and . A -linear Hadamard code is a binary Hadamard code which is the Gray map image of a -additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some -linear Hadamard codes of length are equivalent, once is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to , this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on , the full classification of the -linear Hadamard codes of length is established by giving the exact number of such codes. 相似文献
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Let be a finite simple graph. For , the difference of , where is the neighborhood of and is called the critical difference of . is called a critical set if equals the critical difference and is the intersection of all critical sets. is the union of all critical independent sets. An independent set is an inclusion minimal set with if no proper subset of has positive difference.A graph is called a König–Egerváry graph if the sum of its independence number and matching number equals .In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set with is at least the critical difference of the graph.We also give a new short proof of the inequality .A characterization of unicyclic non-König–Egerváry graphs is also presented and a conjecture which states that for such a graph , the critical difference equals , is proved.We also make an observation about using Edmonds–Gallai Structure Theorem as a concluding remark. 相似文献
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In this paper, we first prove that the local time associated with symmetric -stable processes is of bounded -variation for any partly based on Barlow’s estimation of the modulus of the local time of such processes. The fact that the local time is of bounded -variation for any enables us to define the integral of the local time as a Young integral for less smooth functions being of bounded -variation with . When , Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric -stable processes for . 相似文献
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《Discrete Mathematics》2019,342(1):233-249
A Weyl arrangement is the hyperplane arrangement defined by a root system. Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of their graph. In addition, the Weyl subarrangements of type can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness of the arrangements between type and type . In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type under certain assumption. 相似文献
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Vignon Oussa 《Journal of Functional Analysis》2018,274(4):1202-1254
In this work, we provide a unified method for the construction of reproducing systems arising from unitary irreducible representations of some solvable Lie groups. In contrast to other well-known techniques such as the coorbit theory, the generalized coorbit theory and other discretization schemes, we make no assumption on the integrability or square-integrability of the representations of interest. Moreover, our scheme produces explicit constructions of frames with precise frame bounds. As an illustration of the scope of our results, we highlight that a large class of representations which naturally occur in wavelet theory and time–frequency analysis is handled by our scheme. For example, the affine group, the generalized Heisenberg groups, the shearlet groups, solvable extensions of vector groups and various solvable extensions of non-commutative nilpotent Lie groups are a few examples of groups whose irreducible representations are handled by our method. The class of representations studied in this work is described as follows. Let G be a simply connected, connected, completely solvable Lie group with Lie algebra . Next, let π be an infinite-dimensional unitary irreducible representation of G obtained by inducing a character from a closed normal subgroup of G. Additionally, we assume that , is a closed subgroup of G, is a fixed Haar measure on the solvable Lie group M and there exists a linear functional such that the representation is realized as acting in . Making no assumption on the integrability of , we describe explicitly a discrete subset Γ of G and a vector such that is a tight frame for . We also construct compactly supported smooth functions s and discrete subsets such that is a frame for . 相似文献
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Houmem Belkhechine 《Discrete Mathematics》2017,340(12):2986-2994
Given a tournament , a module of is a subset of such that for and , if and only if . The trivial modules of are ,
and . The tournament is indecomposable if all its modules are trivial; otherwise it is decomposable. The decomposability index of , denoted by , is the smallest number of arcs of that must be reversed to make indecomposable. For , let be the maximum of over the tournaments with vertices. We prove that and that the lower bound is reached by the transitive tournaments. 相似文献
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We prove a sharp estimate for the k-modulus of smoothness, modelled upon a -Lebesgue space, of a function f in , where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, , and . This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings. 相似文献
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We are concerned with the following nonlinear Schrödinger equation where , . For small enough and a class of , we show the uniqueness of the positive ground state under certain assumptions on asymptotic behavior of and its first derivatives. Here our results are suitable for a kind of which has different increasing rates at different directions. 相似文献
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In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic Lévy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise asymptotic behaviors of the tail probability of subordinators when the scaling order is between 0 and 1 including 1.The asymptotic expressions are given in terms of the radial part of characteristic exponent and its derivative. In particular, when varies regularly, as the tail probability is asymptotically equal to a constant times 相似文献
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Ping Sun 《Discrete Mathematics》2018,341(4):1144-1149
This paper considers the enumeration problem of a generalization of standard Young tableau (SYT) of truncated shape. Let be the SYT of shape truncated by whose upper left cell is , where and are partitions of integers. The summation representation of the number of SYT of the truncated shape is derived. Consequently, three closed formulas for SYT of hollow shapes are obtained, including the cases of (i). , (ii). , and (iii). . Finally, an open problem is posed. 相似文献
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In this paper we study the domain of the generator of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder–Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process , for functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed if f satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes. 相似文献
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《Discrete Mathematics》2019,342(5):1351-1360
We study functions defined on the vertices of the Hamming graphs . The adjacency matrix of has distinct eigenvalues with corresponding eigenspaces for . In this work, we consider the problem of finding the minimum possible support (the number of nonzeros) of functions belonging to a direct sum for . For the case and we find the minimum cardinality of the support of such functions and obtain a characterization of functions with the minimum cardinality of the support. In the case and we also find the minimum cardinality of the support of functions, and obtain a characterization of functions with the minimum cardinality of the support for , and . In particular, we characterize eigenfunctions from the eigenspace with the minimum cardinality of the support for cases , and , . 相似文献
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John Asplund Kossi Edoh Ruth Haas Yulia Hristova Beth Novick Brett Werner 《Discrete Mathematics》2018,341(10):2938-2948
For a graph and, the shortest path reconfiguration graph of with respect to and is denoted by . The vertex set of is the set of all shortest paths between and in . Two vertices in are adjacent, if their corresponding paths in differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth 5 or greater. We include an infinite family of well structured examples, showing that the shortest path graph of a grid graph is an induced subgraph of a lattice. 相似文献
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We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in , where the intraspecies interaction and the interspecies interaction ?β are both attractive, , , and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated -critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as , where () is fixed and w is the unique positive solution of in . The semi-trivial limit behavior of ground states is tackled in the companion paper [12]. 相似文献