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1.
In this paper we give a necessary and sufficient condition for the oscillation of the second order linear differential equation where p is a locally integrable function and either or where We give some applications which show how these results unify and imply some classical results in oscillation theory.  相似文献   

2.
We consider the half‐linear boundary value problem where and the weight function q is assumed to change sign. We prove the existence of two sequences , of eigenvalues and derive asymptotic estimates for as .  相似文献   

3.
Let be bounded Lipschitz and relatively open. We show that the solution to the linear first order system 1 : (1) vanishes if and , (e.g. ). We prove to be a norm if with , for some p, q > 1 with 1/p + 1/q = 1 and . We give a new proof for the so called ‘in-finitesimal rigid displacement lemma’ in curvilinear coordinates: Let , satisfy for some with . Then there are and a constant skew-symmetric matrix , such that . (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

4.
We investigate the evolution problem where H is a Hilbert space, A is a self‐adjoint linear non‐negative operator on H with domain D(A), and is a continuous function. We prove that if , and , then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if for all , then this problem is well posed in H. On the contrary, if for some it happens that for all , then this problem has no solution if with β small enough. We apply these results to degenerate parabolic PDEs with non‐local non‐linearities. Copyright © 1999 John Wiley & Sons, Ltd.  相似文献   

5.
The generalized Randi?; index of a tree T is the sum over the edges of T of where is the degree of the vertex x in T. For all , we find the minimal constant such that for all trees on at least 3 vertices, , where is the number of vertices of T. For example, when . This bound is sharp up to the additive constant—for infinitely many n we give examples of trees T on n vertices with . More generally, fix and define , where is the number of leaves of T. We determine the best constant such that for all trees on at least 3 vertices, . Using these results one can determine (up to terms) the maximal Randi?; index of a tree with a specified number of vertices and leaves. Our methods also yield bounds when the maximum degree of the tree is restricted. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 270–286, 2007  相似文献   

6.
This paper deals with the Neumann problem of the pre-Maxwell partial differential equations for a vector field v defined in a region G ? R 3. We approximate its uniquely determined solution (integrability conditions assumed) uniformly on G by explicitly computable particular integrals and linear combinations of vector fields with a “fundamental” sequence of points .  相似文献   

7.
In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2sL, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.  相似文献   

8.
We consider the following mean field equations: (0.1) where M is a compact Riemann surface with volume 1, h is a positive continuous function on M, ρ is a real number, (0.2) and where Ω is a bounded smooth domain, h is a C1 positive function on Ω, and ρ ∈ ?. Based on our previous analytic work [14], we prove, among other things, that the degree‐counting formula for ( 0.1 ) is given by () for ρ ∈ (8mπ, 8(m + 1)π). © 2003 Wiley Periodicals, Inc.  相似文献   

9.
Let x? be a computed solution to a linear system Ax=b with , where is a proper subclass of matrices in . A structured backward error (SBE) of x? is defined by a measure of the minimal perturbations and such that (1) and that the SBE can be used to distinguish the structured backward stability of the computed solution x?. For simplicity, we may define a partial SBE of x? by a measure of the minimal perturbation such that (2) Can one use the partial SBE to distinguish the structured backward stability of x?? In this note we show that the partial SBE may be much larger than the SBE for certain structured linear systems such as symmetric Toeplitz systems, KKT systems, and dual Vandermonde systems. Besides, certain backward errors for linear least squares are discussed. Copyright © 2004 John Wiley & Sons, Ltd.  相似文献   

10.
An asymmetric covering is a collection of special subsets S of an n‐set such that every subset T of the n‐set is contained in at least one special S with . In this paper we compute the smallest size of any for We also investigate “continuous” and “banded” versions of the problem. The latter involves the classical covering numbers , and we determine the following new values: , , , , and . We also find the number of non‐isomorphic minimal covering designs in several cases. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 218–228, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10022  相似文献   

11.
We prove the existence of the wave operator for the system of the massive Dirac–Klein–Gordon equations in three space dimensions x∈ R 3 where the masses m, M>0. We prove that for the small final data , (?, ?)∈ H 2 + µ, 1 × H 1 + µ, 1, with and , there exists a unique global solution for system (1) with the final state conditions Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

12.
The Neumann boundary value problem for the chemotaxis system is considered in a smooth bounded domain Ω??n, n?2, with initial data and v0W1, ∞(Ω) satisfying u0?0 and v0>0 in . It is shown that if then for any such data there exists a global‐in‐time classical solution, generalizing a previous result which asserts the same for n=2 only. Furthermore, it is seen that the range of admissible χ can be enlarged upon relaxing the solution concept. More precisely, global existence of weak solutions is established whenever . Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

13.
The electric potential u in a solution of an electrolyte around a linear polyelectrolyte of the form of a cylinder satisfies We study the problem when R → ∞.  相似文献   

14.
In this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems: 1. RCA0 ? ‐Det* ? ‐Det* ? WKL0. 2. RCA0 ? ( )2‐Det* ? ACA0. 3. RCA0 ? ‐Det* ? ‐Det* ? ‐Det ? ‐Det ? ATR0. 4. For 1 < k < ω, RCA0 ? ( )k ‐Det* ? ( )k –1‐Det. 5. RCA0 ? ‐Det* ? ‐Det. Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and ( )k is the collection of formulas built from formulas by applying the difference operator k – 1 times. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

15.
For each 0 < s < 1, define where , denote respectively the s‐dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E ⊂ R with . In this paper we give a nontrivial estimation of c(s), namely, for each 0 < s < 1, where . As an application, we obtain a lower density theorem for Hausdorff measures.  相似文献   

16.
What is the minimum order of a Hadamard matrix that contains an a by b submatrix of all 1's? Newman showed that where c? denotes the smallest order greater than or equal to c for which a Hadamard matrix exists. It follows that if 4 divides both a and b, and if the Hadamard conjecture is true, then . We establish the improved bounds for min {a,b} ≥ 2. The Hadamard conjecture therefore implies that if 4 divides both 2ab and ?a/2? ?b/2?, then (a, b) = 2 · max {?a/2?b, ?b/2?a}. Our lower bound comes from a counting argument, while our upper bound follows from a sub‐multiplicative property of : Improvements in our upper bound occur when suitable conference matrices or Bush‐type Hadamard matrices exist. We conjecture that any (1,?1)‐matrix of size a by b occurs as a submatrix of some Hadamard matrix of order at most . © 2005 Wiley Periodicals, Inc. J Combin Designs  相似文献   

17.
Let S* (f be the majorant function of the partial sums of the trigonometric Fourier series of f. In this paper we consider the Orlicz space Lπ and give a generalization of Soria's result [S1]. Let π (t) be a concave function with some nice properties and . If there exists a positive constant a0 < 1 such that then we have .  相似文献   

18.
The work deals with a combinatorial problem of P. Erd?s and L. Lovász concerning simple hypergraphs. Let denote the minimum number of edges in an n‐uniform simple hypergraph with chromatic number at least . The main result of the work is a new asymptotic lower bound for . We prove that for large n and r satisfying the following inequality holds where . This bound improves previously known bounds for . The proof is based on a method of random coloring. We have also obtained results concerning colorings of h‐simple hypergraphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012  相似文献   

19.
We consider the existence of a nontrivial solution of the following equation: where g is a nondecreasing function defined on R1, satisfies g(O) = O, and some other additional conditions. Our results and methods are quite similar to those associated with recent work on the nonlinear wave equation [1]-[8]: .  相似文献   

20.
We consider the equation ℝ, where , for ℝ, (ℝ), (ℝ), (ℝ), (ℝ) := C(ℝ)). We give necessary and sufficient conditions under which, regardless of , the following statements hold simultaneously: I) For any (ℝ) Equation (0.1) has a unique solution (ℝ) where $\int ^{\infty}_{-\infty}$ ℝ. II) The operator (ℝ) → (ℝ) is compact. Here is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator.  相似文献   

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