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《Indagationes Mathematicae》2022,33(2):494-516
Current work defines Schmidt representation of a bilinear operator , where and are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that if is compact, and its singular values are ordered, then has a Schmidt representation on real Hilbert spaces. We prove that the hypothesis of existence of ordered singular values is fundamental. 相似文献
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Hao Sun 《Discrete Mathematics》2019,342(3):715-722
-operators are differential operators on the polynomial ring. Mironov, Morosov and Natanzon construct the generalized Hurwitz numbers. They use the -operator to prove a formula for the generating function of the generalized Hurwitz numbers. A special example of the -operator is the cut-and-join operator. Goulden and Jackson use the cut-and-join operator to calculate the simple Hurwitz number. In this paper, we study the relation between -operator and the central elements in . Based on the relation we find, we give another proof about a differential equation of the generating function of -Hurwitz number. 相似文献
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In this paper, we prove the energy conservation for the weak solutions of the three-dimensional ideal inhomogeneous magnetohydrodynamic (MHD) equations in a bounded domain. Two types of sufficient conditions on the regularity of the weak solutions are provided to ensure the energy conservation. Due to the presence of the boundary, we need to impose the boundedness in and the continuity in for the velocity and magnetic fields near the boundary. 相似文献
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By using critical point theory, the existence of heteroclinic solutions for a second-order difference equation involving the mean curvature operator is obtained, and the values of solutions at and are more general than existing results in the literature. An example is presented to demonstrate the applicability of our main results. 相似文献
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Let M be an -dimensional closed orientable submanifold in an -dimensional space form . We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by , where T is a general symmetric, positive definite and divergence-free -tensor on M. The upper bound is given in terms of an integration involving tr T and , where tr T is the trace of the tensor T and is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator can be regarded as a natural generalization of the well-known operator which is the linearized operator of the first variation of the -th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension. 相似文献
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For a closed symplectic manifold with compatible Riemannian metric g we study the Sobolev geometry of the group of all diffeomorphisms on M which preserve the symplectic structure. We show that, for sufficiently large s, the metric admits globally defined geodesics and the corresponding exponential map is a non-linear Fredholm map of index zero. Finally, we show that the metric carries conjugate points via some simple examples. 相似文献
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《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2021,38(5):1373-1405
We establish the regularity in 2 dimension of solutions to critical elliptic systems in divergence form involving chirality operators of finite -energy. 相似文献
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We develop a normal form theory for a nonautonomous dynamics with discrete time, based on the nonuniform spectrum of the sequence of matrices . In particular, we show that any nonresonant terms of the perturbations can be eliminated through an appropriate coordinate change, with the resonances expressed in terms of the connected components of the nonuniform spectrum. The latter is defined in terms of the notion of a nonuniform exponential dichotomy with a small nonuniform part, which is ubiquitous in the context of ergodic theory. We first make a preparation of the linear part of the dynamics that is of independent interest: we show that any sequence of matrices with a bounded nonuniform spectrum can be reduced to a sequence of matrices in block form via a Lyapunov coordinate change. This allows maintaining the Lyapunov exponents as well as the nonuniform spectrum. As further developments, we describe normal form theories in two additional contexts: we consider nonuniformly hyperbolic cocycles over a diffeomorphism of a compact manifold as well as perturbations of a sequence of compact linear operators on a Banach space. The latter includes the particular case of a sequence of matrices that need not be invertible. 相似文献
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Let be the number of monochromatic copies of a fixed connected graph in a uniformly random coloring of the vertices of the graph . In this paper we give a complete characterization of the limiting distribution of , when is a converging sequence of dense graphs. When the number of colors grows to infinity, depending on whether the expected value remains bounded, either converges to a finite linear combination of independent Poisson variables or a normal distribution. On the other hand, when the number of colors is fixed, converges to a (possibly infinite) linear combination of independent centered chi-squared random variables. This generalizes the classical birthday problem, which involves understanding the asymptotics of , the number of monochromatic -cliques in a complete graph (-matching birthdays among a group of friends), to general monochromatic subgraphs in a network. 相似文献