Smoothness and asymptotics of global positive branches of Δu+λf(u)=0

The smallest eigenvalue of v+ f(0)v=0 gives rise to a global positive branch of u+ f(u)=0 in ² together with homogeneous Dirichlet boundary conditions (f(0)=0,f(0)0). The theory, however, guarantees only that this branch is a continuum. We present a result that in a reasonably large class of problems this branch is actually an unbounded smooth curve. In the second part we study the asymptotic behavior of this positive smooth curve whenf has exponential growth. All results apply also to branches emanating at higher eigenvalues. As shown in [7] they have a fixed nodal structure when restricted to appropriate fixed-point spaces.Dedicated to Prof. Klaus Kirchgässner on the occasion of his sixtieth birthday     

Small representations of the relation algebra ɛn+1(1, 2, 3)

Applying combinatorial methods, we prove that the symmetric relation algebra _n+1(1, 2, 3) ofn+1 atoms is finitely representable for alln 1, on at most (2+o(1))n² elements asn . We explicitly construct a representation of size 4.5n², for every n >1.Presented by B. Jonsson.     

Isomorphic embedding of ℓ p n , 1<p<2, into ℓ 1 (1+ε)n

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that ℓ _p ⁿ , (C logn)^1/q(1+1/ε)-embeds into ℓ ₁ ^(1+ε)n , where 1<p<2 and 1/p+1/q=1. Supported by ISF.     

微分方程-u"=λ2u+f(u,u')边值问题无穷多解的存在性

考虑一类边值问题-u"=λ2u+f(u,u'),u(0)=u(1)=0,其中λ＞0为参数.在满足一定条件下,得到了无穷多解的存在性,并进一步研究了有关正解,负解的存在性及变号解的零点个数.     

方程△u+|u|~(p-1)u+λu=0的径向解的一些注记

当n≥7时,已经证明对任意的λ∈(0,λ_1)以及任意整数k≥0,R~n中的单位球B(0,1)上的方程⊿u+|u|~(p-1)u+λu=0在H_0~1(B(0,1))中必有一个径向解具k个结点。本文证明当3≤n≤6时这一结果不再成立。还讨论了上述方程径向正解的唯一性。     

THE NEAR DIFFERENCE SETS OF TYPE 2 WITH PARAMETERS (9λ+6, 3λ+1, λ)

It is proved in this paper that the number of the near difference sets of type 2 with parameters ( 9λ +6, 3λ+1, λ) is finite by studying the relationship between the cyclic sequence and its normai subsequences. It is determined accordingly that there exists a ( 9λ + 6, 3λ + 1, λ) -near difference set of type 2 if and only if λ. = 1 or 2. Together with the results of [1,2,3,4] we can conclude that there are only 6 groups of parameters corresponding to the near difference sets of type 2 if k ≤ 3λ 4+2 .     

Zur Theorie der Kronecker'schen Doppelreihe Ser (ξ, η,u,v, w)

Ohne Zusammenfassung     

关于SUP(λ∈G)(Min(u∈Ω) F(U;λ)/Max(u∈Ω)F(U;λ)问题的计算

本文提出了网格—循环坐标法和网格—梯度法,用于计算新型的优化问题。它们的算法简单,实施方便。用许多例题,对不同方法和初值进行了比较,说明此二方法对本优化问题是有效的。     

The sizes of optimal (n,4,λ,3) optical orthogonal codes

Let $\Phi (n,k,{\lambda }_a,{\lambda }_c)$ denote the largest possible size among all $(n,k,{\lambda }_a,{\lambda }_c)$-OOCs. An $(n,k,{\lambda }_a,{\lambda }_c)$-OOC with $\Phi (n,k,{\lambda }_a,{\lambda }_c)$ codewords is said to be optimal. In this paper, the exact value of $\Phi (n,4,\lambda ,3)$ is determined. Equivalently, the size of an optimal $(n,4,\lambda ,3)$ optical orthogonal code is calculated.     

具临界指数椭圆方程-△u=λku+|u|2*-2u+f(x,u)非平凡多解存在性

本文利用山路引理以及P.L.Lions的集中紧性原理,给出了具临界指数2*且涉及任意特征值λk的Dirichlet问题-△u=λku+|u|2*-2u+f(x,u)一对非平凡解的存在性定理,其中次临界扰动项f(x,t)可以是关于变量t的非线性项.     

Small into-isomorphism from L∞ (A, μ) into L∞(B, υ)

In this paper we shall assert that if T is an isomorphism of L_∞(Ω₁, A, μ) into L_∞(Ω₂, B, υ) satisfying the condition ‖T‖·‖T ^?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\parallel }}$ is close to an isometry with an error less than 6ε in some conditions.     

Completeness of root vectors of a Keldysh pencil perturbed by an analytic operator-valued function S(λ) with S(∞) = 0

The multiple completeness of the root vectors of the pencil $$L(\lambda ) = I - T_0 - \lambda T_1 H - \ldots - \lambda ^{n - 1} T_{n - 1} H^{n - 1} - \lambda ^n H^n - S(\lambda ),$$ where I is the identity operator in the separable Hilbert space ,S (λ) is an operator-valued function analytic for ¦λ¦ > η with S(∞) = 0, and T_k and H are completely continuous operators, is studied. The method suggested in this note for proving the completeness does not use the factorization theorems, due to which we can remove certain restrictions on the function S(λ) connected with the application of the factorization theorems.     

General constructions of c-Bhaskar Rao designs and the (c,λ) spectrum of a c-BRD(v,k,λ)

New constructions of Bhaskar Rao designs (BRDs) and new families of c-BRDs are given using the relationship of BRDs to other combinatorial structures. For example, we use different families and their properties in various ways to obtain natural signings of BIBDs in order to construct c-BRDs. We investigate the extreme possibilities for c relative to λ, and k relative to v, and prove the non-existence of an infinite series of designs. We also give a symmetricized version of Ehlich's Hadamard construction.     

On two-sided and asymptotic estimates for the norms of embedding operators of into L q (dμ)

Explicit upper and lower estimates are given for the norms of the operators of embedding of , n ∈ ?, in L _q (dµ), 0 < q < ∞. Conditions on the measure µ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ_{1, n, β} , β > 0, of the boundary value problems (?d ²/dx ²) ⁿ u(x) = λ|x| ^β?1, x ∈ (?1, 1), u ^(k)(±1) = 0, k ∈ {0, 1, ..., n ? 1}.