In 1923, Hardy and Littlewood[1] conjectured that each integer n can be written asp+m12+ m22 = n,and Linnik[2,3] proved that this conjecture is true. But if these mi with i = 1,2 are restricted to primes Pi, the corresponding result is out of reach at present. We consider the following Diophantine equation 相似文献
We establish a duality principle for arrangements of pseudolines in the projective plane, and thereby prove the conjecture of Burr, Grünbaum, and Sloane that the solution of the “orchard problem” for pseudoline arrangements and the solution t?(p) of the dual problem are equal. 相似文献
Summary Fix a curve X of genus g and L Picd(X). Let L(X) be the image of X through the complete linear system H0(X, L). Here we prove that a general projection of L(X) intoPN has maximal rank if either (a) N4, 0gN–1, dg+N, or (b) dd (g, N) for suitable d(g, N). 相似文献
For any integer s≥ 2, let μsbe the least integer so that every integer l μs is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpi′nski asked for the determination of μs. Let pibe the i-th prime and let μs= p2 + p3 + + ps+1+ cs. Recently, the authors solved this problem. In particular,we have(1) cs=-2 if and only if s = 2;(2) the set of integers s with cs= 1100 has asymptotic density one;(3) cs∈ A for all s ≥ 3, where A is an explicit set with A ■[2, 1100] and |A| = 125. In this paper, we prove that,(1) for every a ∈ A, there exists an index s with cs= a;(2) under Dickson's conjecture, for every a ∈ A,there are infinitely many s with cs= a. We also point out that recent progress on small gaps between primes can be applied to this problem. 相似文献
Let F={H1,...,Hk}(k> 1) be a family of graphs.The Turán number of the family F is the maximum number of edges in an n-vertex {H1,...,Hk)-free graph,denoted by ex(n,F) or ex(n,{H1,H2,...,Hk}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Turán number of the family cons... 相似文献
Wavelet packets provide an algorithm with many applications in signal processing together with a large class of orthonormal
bases of L2(ℝ), each one corresponding to a different splitting of L2(ℝ) into a direct sum of its closed subspaces. The definition of wavelet packets is due to the work of Coifman, Meyer, and
Wickerhauser, as a generalization of the Walsh system. A question has been posed since then: one asks if a (general) wavelet
packet system can be an orthonormal basis for L2(ℝ) whenever a certain set linked to the system, called the “exceptional set” has zero Lebesgue measure. This answer to this
question affects the quality of wavelet packet approximation. In this paper we show that the answer to this question is negative
by providing an explicit example. In the proof we make use of the “local trace function” by Dutkay and the generalized shift-invariant
system machinery developed by Ron and Shen. 相似文献
It is conjectured by Erd?s, Graham and Spencer that if 1≤a1≤a2≤?≤as are integers with , then this sum can be decomposed into n parts so that all partial sums are ≤1. This is not true for as shown by a1=?=an−2=1, . In 1997 Sandor proved that Erd?s-Graham-Spencer conjecture is true for . Recently, Chen proved that the conjecture is true for . In this paper, we prove that Erd?s-Graham-Spencer conjecture is true for . 相似文献
Call a bypergraphsimple if for any pairu, v of distinct vertices, there is at most one edge incident to bothu andv, and there are no edges incident to exactly one vertex. A conjecture of Erds, Faber and Lovász is equivalent to the statement that the edges of any simple hypergraph onn vertices can be colored with at mostn colors. We present a simple proof that the edges of a simple hypergraph onn vertices can be colored with at most [1.5n-2 colors].This research was partially supported by N.S.F. grant No. MCS-8311422. 相似文献
For a Banach space B of functions which satisfies for some m>0
$ \max ({\|F+G\|}_B,{\|F-G\|}_B)\geqq ({\|F\|}^s_B+m{\|G\|}^s_B)^{1/s},\quad \forall \,F,G\in B $
(?)
a significant improvement for lower estimates of the moduli of smoothness ωr(f,t)B is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on ?d or \(\mathbb{T}^{d}\) for which translations are isometries or on Sd?1 for which rotations are isometries. Results for C0 semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An Lp space with 1<p<∞ satisfies (?) where s=max??(p,2), and many Orlicz spaces are shown to satisfy (?) with appropriate s.
We consider the study of an eigenvalue problem obtained by linearizing about solitary wave solutions of a Boussinesq equation. Instead of using the technique of Evans functions as done by Pego and Weinstein in [R. Pego and M. Weinstein, Convective Linear Stability of Solitary Waves for Boussinesq equation. AMS, 99, 311–375] for this particular problem, we perform Fourier analysis to characterize solutions of the eigenvalue problem in terms of a multiplier operator and use the strong relationship between the eigenvalue problem for the linearized Boussinesq equation and the eigenvalue problem associated with the linearization about solitary wave solutions of a special form of the KdV equation. By using a generalization of the Rouché Theorem and the asymptotic behavior of the Fourier symbol corresponding to the eigenvalues problem for the Boussinesq equation and the Fourier symbol corresponding to the eigenvalues problem for the KdV equation, we show nonexistence of eigenvalues with respect to weighted space in a planar region containing the right-half plane. 相似文献
1. Introduction A theorem of Carleson,as generalized by Duren characterizes those positive measure μ on the unit disc U={z∈C:|z|<1} for which the H~p norm domiates the L~q(μ) norm of elements of H~p. Later on, Hasting proved an analogous results with H~p replaced by A~p, the Bergman space of fuctions f 相似文献
Given a homeomorphism ? ∈ WM1, we determine the conditions that guarantee the belonging of the inverse of ? in some Sobolev–Orlicz space WF1. We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair. 相似文献
In this paper we are concerned with the exceptional set of the sum of two squares of primesand a cube of a prime P;+p;+p;.Noting that竹三1 or 3(mod 6)is a necessary conditionfor the solvability of the equation n=P}+P;+P;(see【1]),we define E(N)=Card{n:n≤N,礼∈三and n≠P;+Pi+P;for any Pi,1≤i s 3), (1)where三={n:n三1 or 3(mod 6)). This and similar problems have been studied by a number of authors.In 1937,Davenportand Heilbronn[2J proved that if后2 3 is an odd integer then almost all posi… 相似文献
In this paper k-sets of type (a, b) with respect to hyperplanes are constructed in finite projective spaces using powers of Singer cycles. These are then used to construct further examples of sets of type (a, b) using various disjoint sets. The parameters of the associated strongly regular graphs are also calculated. The construction technique is then related to work of Foulser and Kallaher classifying rank three subgroups of AL(1, pR). It is shown that the sets of type (a, b) arising from the Foulser and Kallaher construction in the case of projective spaces are isomorphic to some of those constructed in the present paper. 相似文献
In this work we consider a nonlinear hyperbolic one-dimensional viscoelastic nonlocal problem with a nonlocal boundary condition.
We establish a blow up result for large initial data and a decay result for small enough initial data. 相似文献
A hyperplane of the symplectic dual polar space , , is said to be of subspace-type if it consists of all maximal singular subspaces of meeting a given -dimensional subspace of . We show that a hyperplane of is of subspace-type if and only if every hex of intersects it in either , a singular hyperplane of or the extension of a full subgrid of a quad. In the case is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane of is of subspace-type or arises from the spin-embedding of if and only if every hex intersects it in either , a singular hyperplane of , a hexagonal hyperplane of or the extension of a full subgrid of a quad. 相似文献