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In this paper, we consider the problem (Pε) : Δ2u=un+4/n-4+εu,u>0 in Ω,u=Δu=0 on ∂Ω, where Ω is a bounded and smooth domain in Rn,n>8 and ε>0. We analyze the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev inequality as ε→0 and we prove existence of solutions to (Pε) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε small, (Pε) has at least as many solutions as the Ljusternik–Schnirelman category of Ω. 相似文献
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We construct frequency-dependent rules to interpolate oscillatory functions y(x) with frequency ω of the form,
at equidistant nodes on the interval of interest where the functions f1 and f2 are smooth. Error analysis of the rules is investigated and numerical results are discussed. We provide numerical illustrations to compare the accuracy of classical Hermite polynomials and newly constructed frequency-dependent rules. 相似文献
y(x)=f1(x)cos(ωx)+f2(x)sin(ωx),
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Let F be an infinite field with characteristic not equal to two. For a graph G=(V,E) with V={1,…,n}, let S(G;F) be the set of all symmetric n×n matrices A=[ai,j] over F with ai,j≠0, i≠j if and only if ij∈E. We show that if G is the complement of a partial k -tree and m?k+2, then for all nonsingular symmetric m×m matrices K over F, there exists an m×n matrix U such that UTKU∈S(G;F). As a corollary we obtain that, if k+2?m?n and G is the complement of a partial k-tree, then for any two nonnegative integers p and q with p+q=m, there exists a matrix in S(G;R) with p positive and q negative eigenvalues. 相似文献
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Let H be a k -uniform hypergraph whose vertices are the integers 1,…,N. We say that H contains a monotone path of length n if there are x1<x2<?<xn+k−1 so that H contains all n edges of the form {xi,xi+1,…,xi+k−1}. Let Nk(q,n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n . While the study of Nk(q,n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erd?s and Szekeres, the problem of bounding Nk(q,n) for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk. 相似文献
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We examine a class of Grushin type operators Pk where k∈N0 defined in (1.1). The operators Pk are non-elliptic and degenerate on a sub-manifold of RN+?. Geometrically they arise via a submersion from a sub-Laplace operator on a nilpotent Lie group of step k+1. We explain the geometric framework and prove some analytic properties such as essential self-adjointness. The main purpose of the paper is to give an explicit expression of the fundamental solution of Pk. Our methods rely on an appropriate change of coordinates and involve the theory of Bessel and modified Bessel functions together with Weber's second exponential integral. 相似文献
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A plane partition is a p×q matrix A=(aij), where 1?i?p and 1?j?q, with non-negative integer entries, and whose rows and columns are weakly decreasing. From a geometric point of view plane partitions are equivalent to pyramids , subsets of the integer lattice Z3 which play an important role in Discrete Tomography. As a consequence, some typical problems concerning the tomography of discrete lattice sets can be rephrased and considered via plane partitions. In this paper we focus on some of them. In particular, we get a necessary and sufficient condition for additivity, a canonical procedure for checking the existence of (weakly) bad configurations, and an algorithm which constructs minimal pyramids (with respect to the number of levels) with assigned projection of a bad configurations. 相似文献
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M. Gürdal 《Expositiones Mathematicae》2009,27(2):153-160
In the present paper we consider the Volterra integration operator V on the Wiener algebra W(D) of analytic functions on the unit disc D of the complex plane C. A complex number λ is called an extended eigenvalue of V if there exists a nonzero operator A satisfying the equation AV=λVA. We prove that the set of all extended eigenvalues of V is precisely the set C?{0}, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V. The similar result for some weighted shift operator on ?p spaces is also obtained. 相似文献
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We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables x1,x2,…∈D. The setting is based on a reproducing kernel k for functions on D, a family of non-negative weights γu, where u varies over all finite subsets of N, and a probability measure ρ on D. We consider the weighted superposition K=∑uγuku of finite tensor products ku of k. Under mild assumptions we show that K is a reproducing kernel on a properly chosen domain in the sequence space DN, and that the reproducing kernel Hilbert space H(K) is the orthogonal sum of the spaces H(γuku). Integration on H(K) can be defined in two ways, via a canonical representer or with respect to the product measure ρN on DN. We relate both approaches and provide sufficient conditions for the two approaches to coincide. 相似文献
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For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow the residual norm ξk≥1 of the inner linear system at outer iteration k+1 and can be considerably weaker than the condition ξk≤ξ<1 with ξ a constant not near one commonly used in the literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditioned MINRES methods for the linear systems. Some attractive properties are derived for the residuals obtained by MINRES. Based on them and the new general theory, we make a refined analysis and establish a number of new convergence results. Let ‖rk‖ be the residual norm of approximating eigenpair at outer iteration k. Then all the available cubic and quadratic convergence results require ξk=O(‖rk‖) and ξk≤ξ with a fixed ξ not near one, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that ξk≤ξ with a constant ξ<1 not near one, ξk=1−O(‖rk‖) and ξk=1−O(‖rk‖2), respectively. The new convergence conditions are much more relaxed than ever before. The theory can be used to design practical stopping criteria to implement the method more effectively. Numerical experiments confirm our results. 相似文献
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We prove that if G is a finite simple group which is the unit group of a ring, then G is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2k−1 for some k; or (c) a projective special linear group PSLn(F2) for some n≥3. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G with no non-trivial normal 2-subgroup. 相似文献
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Let k be any field, G be a finite group acting on the rational function field k(xg:g∈G) by h⋅xg=xhg for any h,g∈G. Define k(G)=k(xg:g∈G)G. Noether’s problem asks whether k(G) is rational (= purely transcendental) over k. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether’s problem. We prove that, if G is a Frobenius group with abelian Frobenius kernel, then k(G) is retract k-rational for any field k satisfying some mild conditions. As an application, we show that, for any algebraic number field k, for any Frobenius group G with Frobenius complement isomorphic to SL2(F5), there is a Galois extension field K over k whose Galois group is isomorphic to G, i.e. the inverse Galois problem is valid for the pair (G,k). The same result is true for any non-solvable Frobenius group if k(ζ8) is a cyclic extension of k. 相似文献
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Paul-Emile Maing 《Nonlinear Analysis: Theory, Methods & Applications》2008,68(12):3913-3922
This paper is concerned with the Cauchy problem for the fast diffusion equation ut−Δum=αup1 in RN (N≥1), where m∈(0,1), p1>1 and α>0. The initial condition u0 is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value u0 so that u(t,x) blows up in finite time, and we show how to get estimates on the profile of u(t,x) for small enough values of t>0. 相似文献