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In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces , for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in -topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. 相似文献
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This paper is concerned with the Cauchy problem on the Boltzmann equation without angular cutoff assumption for hard potential in the whole space. When the initial data is a small perturbation of a global Maxwellian, the global existence of solution to this problem is proved in unweighted Sobolev spaces with . But if we want to obtain the optimal temporal decay estimates, we need to add the velocity weight function, in this case the global existence and the optimal temporal decay estimate of the Boltzmann equation are all established. Meanwhile, we further gain a more accurate energy estimate, which can guarantee the validity of the assumption in Chen et al. (0000). 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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Michael Greenblatt 《Journal of Functional Analysis》2019,276(5):1510-1527
to boundedness results are proven for translation invariant averaging operators over hypersurfaces in Euclidean space. The operators can either be Radon transforms or averaging operators with multiparameter fractional integral kernel. In many cases, the amount of smoothing proven is optimal up to endpoints, and in such situations this amount of smoothing can be computed explicitly through the use of appropriate Newton polyhedra. 相似文献
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Cédric Arhancet 《Journal of Functional Analysis》2019,276(7):2279-2314
We prove that any weak* continuous semigroup of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ?-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative -spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's functional calculus of the generators of these semigroups on the associated noncommutative -spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of . 相似文献
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Let be the Möbius function and real. In this paper, we investigated the best possible estimates for the sum under the weak Generalized Riemann Hypothesis. A similar result also holds for the Liouville function . 相似文献
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We consider edge decompositions of the -dimensional hypercube into isomorphic copies of a given graph . While a number of results are known about decomposing into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if is even, and divides the number of edges of , then the path of length decomposes . Tapadia et al. proved that any path of length , where , satisfying these conditions decomposes . Here, we make progress toward resolving Erde’s conjecture by showing that cycles of certain lengths up to decompose . As a consequence, we show that can be decomposed into copies of any path of length at most dividing the number of edges of , thereby settling Erde’s conjecture up to a linear factor. 相似文献
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《Indagationes Mathematicae》2022,33(2):322-333
We apply the techniques developed by I. Panin for the proof of the equicharacteristic case of the Serre–Grothendieck conjecture for isotropic reductive groups (Panin et al., 2015; Panin, 2019) to obtain similar injectivity and -invariance theorems for non-stable -functors associated to isotropic reductive groups. Namely, let be a reductive group over a commutative ring . We say that has isotropic rank , if every non-trivial normal semisimple -subgroup of contains . We show that if has isotropic rank and is a regular domain containing a field, then , where is the corresponding non-stable -functor, also called the Whitehead group of . If is, moreover, local, then we show that is injective, where is the field of fractions of . 相似文献
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We introduce a class of -semigroups that is broader and more flexible than the class of pure -semigroups, and characterize the states of the spectral -algebra of a product system that give rise to them. 相似文献
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Hai Q. Dinh Xiaoqiang Wang Hongwei Liu Songsak Sriboonchitta 《Discrete Mathematics》2019,342(11):3062-3078
Let be an odd prime, and be a nonzero element of the finite field . The -constacyclic codes of length over are classified as the ideals of quotient ring in terms of their generator polynomials. Based on these generator polynomials, the symbol-pair distances of all such -constacyclic codes of length are obtained in this paper. As an application, all MDS symbol-pair constacyclic codes of length over are established, which produce many new MDS symbol-pair codes with good parameters. 相似文献
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Huffman (2013) [12] studied -linear codes over and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative -algebra. An -linear code over S of length n is an -submodule of . In this paper, we study -linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes of MDR codes are introduced. We generalize the ordinary and Hermitian trace products over -algebras and we prove the MacWilliams identity with respect to the generalized form. In particular, we obtain Huffman's results on the MacWilliams identity. Among other results, we give a theory to construct a class of quantum codes and the structure of -linear codes over finite commutative graded -algebras. 相似文献
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We define a ribbon category , depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for the monoidal category of representations of generated by exterior powers of the vector representation and their duals. We identify this category with a direct limit of quotients of a dual idempotented quantum group , proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category gives a unified natural setting for defining the colored link invariant (for ) and the colored HOMFLY-PT polynomial (for β generic). 相似文献