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1.
A coloring of a q-ary n-dimensional cube (hypercube) is called perfect if, for every n-tuple x, the collection of the colors of the neighbors of x depends only on the color of x. A Boolean-valued function is called correlation-immune of degree n?m if it takes value 1 the same number of times for each m-dimensional face of the hypercube. Let f=χS be a characteristic function of a subset S of hypercube. In the present paper we prove the inequality ρ(S)q(cor(f)+1)α(S), where cor(f) is the maximum degree of the correlation immunity of f, α(S) is the average number of neighbors in the set S for n-tuples in the complement of a set S, and ρ(S)=|S|/qn is the density of the set S. Moreover, the function f is a perfect coloring if and only if we have an equality in the formula above. Also, we find a new lower bound for the cardinality of components of a perfect coloring and a 1-perfect code in the case q>2.  相似文献   

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Given two coprime polynomials P and Q in Z[x,y] of degree bounded by d and bitsize bounded by τ, we address the problem of solving the system {P,Q}. We are interested in certified numerical approximations or, more precisely, isolating boxes of the solutions. We are also interested in computing, as intermediate symbolic objects, rational parameterizations of the solutions, and in particular Rational Univariate Representations (RURs), which can easily turn many queries on the system into queries on univariate polynomials. Such representations require the computation of a separating form for the system, that is a linear combination of the variables that takes different values when evaluated at the distinct solutions of the system.We present new algorithms for computing linear separating forms, RUR decompositions and isolating boxes of the solutions. We show that these three algorithms have worst-case bit complexity O˜B(d6+d5τ), where O˜ refers to the complexity where polylogarithmic factors are omitted and OB refers to the bit complexity. We also present probabilistic Las Vegas variants of our two first algorithms, which have expected bit complexity O˜B(d5+d4τ). A key ingredient of our proofs of complexity is an amortized analysis of the triangular decomposition algorithm via subresultants, which is of independent interest.  相似文献   

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Zemin Jin  Kun Ye 《Discrete Mathematics》2018,341(10):2846-2858
The rainbow numberrb(G,H) for the graph H in G is defined to be the minimum integer c such that any c-edge-coloring of G contains a rainbow H. As one of the most important structures in graphs, the rainbow number of matchings has drawn much attention and has been extensively studied. Jendrol et al. initiated the rainbow number of matchings in planar graphs and they obtained bounds for the rainbow number of the matching kK2 in the plane triangulations, where the gap between the lower and upper bounds is O(k3). In this paper, we show that the rainbow number of the matching kK2 in maximal outerplanar graphs of order n is n+O(k). Using this technique, we show that the rainbow number of the matching kK2 in some subfamilies of plane triangulations of order n is 2n+O(k). The gaps between our lower and upper bounds are only O(k).  相似文献   

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Suppose that (Xt)t0 is a one-dimensional Brownian motion with negative drift ?μ. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like X conditioned to hit 0, after which time it behaves as X killed at the last time X visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift +μ when it is negative, like Brownian motion with negative drift ?μ when it is positive, and is killed according to the local time spent at 0.An extension of this result holds in great generality for a Borel right process conditioned to be in some state a at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the “bang–bang” construction for a general Markov process.As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the h-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.  相似文献   

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Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances k:l, but not the semisimple cases 1:1 and 1:?1. A two-degree-of-freedom Hamiltonian system can be approximated to any order by an integrable normal form. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. For a rational frequency ratio this leads to S1-symmetric systems. The question we wish to address is about the co-dimension of such a system in 1:1 resonance with respect to left-right-equivalence, where the right action is S1-equivariant. The result is a co-dimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining non-modal parameters are the ones found in the linear unfolding of this system.  相似文献   

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We develop the asymptotic expansion theory for vector-valued sequences (FN)N1 of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence FN. Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of FN and we illustrate our results by several examples.  相似文献   

10.
This paper is concerned with the rate of convergence in the normal approximation of the sequence {Fn}, where each Fn is a functional of an infinite-dimensional Gaussian field. We develop new and powerful techniques for computing the exact rate of convergence in distribution with respect to the Kolmogorov distance. As a tool for our works, the Edgeworth expansion of general orders, with an explicitly expressed remainder, will be obtained, and this remainder term will be controlled to find upper and lower bounds of the Kolmogorov distance in the case of an arbitrary sequence {Fn}. As applications, we provide the optimal fourth moment theorem of the sequence {Fn} in the case when {Fn} is a sequence of random variables living in a fixed Wiener chaos or a finite sum of Wiener chaoses. In the former case, our results show that the conditions given in this paper seem more natural and minimal than ones appeared in the previous works.  相似文献   

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We prove several improved versions of Bohr’s inequality for the harmonic mappings of the form f=h+g¯, where h is bounded by 1 and |g(z)||h(z)|. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk D(0,r) under the mapping f is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.  相似文献   

14.
《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 A 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 B 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 A1 and type B. In this paper, we give a characterization of the freeness and supersolvability of the Weyl subarrangements of type B under certain assumption.  相似文献   

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We exploit the structure of the critical orbital sets of symmetry classes of tensors associated to sign uniform partitions and we establish new connections between symmetry classes of tensors, matchings on bipartite graphs and coding theory. In particular, we prove that the orthogonal dimension of the critical orbital sets associated to single hook partitions λ=(w,1n-w) equals the value of the coding theoretic function A(n,4,w). When w=2 we reobtain this number as the independence number of the Dynkin diagram An-1.  相似文献   

17.
In [4] we studied the group invariance of the inner product of supervectors as introduced in the framework of Clifford analysis in superspace. The fundamental group SO0 leaving invariant such an inner product turns out to be an extension of SO(m)×Sp(2n) and gives rise to the definition of the spin group in superspace through the exponential of the so-called extended superbivectors, where the spin group can be seen as a double covering of SO0 by means of the representation h(s)[x]=sxs. In the present paper, we study the invariance of the Dirac operator in superspace under the classical H and L actions of the spin group on superfunctions. In addition, we consider the Hermitian Clifford setting in superspace, where we study the group invariance of the Hermitian inner product of supervectors introduced in [3]. The group of complex supermatrices leaving this inner product invariant constitutes an extension of U(m)×U(n) and is isomorphic to the subset SO0J of SO0 of elements that commute with the complex structure J. The realization of SO0J within the spin group is studied together with the invariance under its actions of the super Hermitian Dirac system. It is interesting to note that the spin element leading to the complex structure can be expressed in terms of the n-dimensional Fourier transform.  相似文献   

18.
《Discrete Mathematics》2020,343(3):111721
The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z2s-linear Hadamard codes of length 2t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t=11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s{2,3}, the full classification of the Z2s-linear Hadamard codes of length 2t is established by giving the exact number of such codes.  相似文献   

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