Solvability of Elliptic Boundary Value Problems without Standard Landesman-Lazer Condition

Abstract In this paper we prove a very general result concerning solvability of the resonant problem: Δu + λ_κ u + g(x, u) = h (x); u = 0, x ∈∂Ω, which immediately gives three generalized Landesman-Lazer conditions. The most interesting application of the general result is concerned with the problem when λ_κ = λ₁, in which case we prove solvability results for it under conditions which are not the standard Landesman-Lazer condition or only partly enjoy it. Furthermore, we propose a new sign condition and give a comprehensive extension of a main result of Figueiredo and Ni.     

A characterization of cardinalsκ such that 2λ = 2κ wheneverκ ≦λ < 2κ

We characterize cardinalsκ such that 2^λ = 2^κ wheneverκ ≦λ < 2^κ using ideals in small algebras of sets satisfying certain completeness and saturation conditions. Research of this author was supported by an NSF Grant.     

<Emphasis Type="Italic">Q</Emphasis>-measures on <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript>λ

 We give a characterization of strongly compact cardinals in terms of Q _κ λ. We also prove that weakly normal Q-measures on Q _κ λ are ⊂κ-normal. Received: 29 September 2000 / Revised version: March 2002 Published online: 5 November 2002 This project is supported by the New Zealand Marsden Fund. The author wishes to thank the referee for numerous comments and suggestions which have been incorporated into this version of the paper.     

On Closed Weingarten Surfaces

We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces of genus zero where F is a quadratic polynomial or F(λ) = cλ²ⁿ⁺¹. This generalizes results by H. Hopf on the case where F is linear and the case of ellipsoids of revolution where F(λ) = cλ³.     

On Closed Weingarten Surfaces

We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces of genus zero where F is a quadratic polynomial or F(λ) = cλ²ⁿ⁺¹. This generalizes results by H. Hopf on the case where F is linear and the case of ellipsoids of revolution where F(λ) = cλ³.     

Remarks on enlargements of generalized topologies

The concept of enlargement of a generalized topology was introduced by á. Császár [4]. He also introduced the concepts of (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity on enlargements. In this paper, we characterize the (κ,λ)-continuity and introduce the concept of strong (κ,λ)-continuity on enlargements. In particular, we study characterizations for the strong (κ,λ)-continuity and the relationships among (μ,ν)-continuity, (κ,λ)-continuity, strong (κ,λ)-continuity and (κ _μ ,λ _ν)-continuity.     

Antichains in partially ordered sets of singular cofinality

In their paper from 1981, Milner and Sauer conjectured that for any poset , if , then P must contain an antichain of size κ. We prove that for λ > cf(λ) = κ, if there exists a cardinal μ < λ such that cov(λ, μ, κ, 2) = λ, then any poset of cofinality λ contains λ ^κ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.      

On universal locally finite groups

We deal with the question of existence of a universal object in the category of universal locally finite groups; the answer is negative for many uncountable cardinalities; for example, for 2^ℵ ₀, and assuming G.C.H. for every cardinal whose confinality is >ℵ₀. However, if λ>κ when κ is strongly compact and of λ=ℵ₀, then there exists a universal locally finite group of cardinality λ. The idea is to use the failure of the amalgamation property in a strong sense. We shall also prove the failure of the amalgamation property for universal locally finite groups by transferring the kind of failure of the amalgamation property from LF into ULF. We would like to thank Simon Thomas for reading carefully a preliminary version of this paper, proving Lemma 20 and making valuable remarks. Also we thank the United States—Israel Binational Science Foundation for partially supporting this work.     

Extending the Thorin class

We prove that the Thorin class $ {T_\kappa }\left( {{\mathbb{R}_{+} }} \right),\kappa > 0 $ {T_\kappa }\left( {{\mathbb{R}_{+} }} \right),\kappa > 0 , is the minimal class of probability distributions on \mathbbR₊ {\mathbb{R}_{+} } that is closed under convolutions and weak limits and contains the Tweedie distributions Tw_(κ+1)/κ (μ, λ), μ > 0, λ > 0.     

Homogenization of eigenvalue problems in perforated domains

In this paper, we treat some eigenvalue problems in periodically perforated domains and study the asymptotic behaviour of the eigenvalues and the eigenvectors when the number of holes in the domain increases to infinity Using the method of asymptotic expansion, we give explicit formula for the homogenized coefficients and expansion for eigenvalues and eigenvectors. If we denote by ε the size of each hole in the domain, then we obtain the following aysmptotic expansion for the eigenvalues: Dirichlet: λ_ε = ε⁻² λ + λ₀ +O (ε), Stekloff: λ_ε = ελ₁ +O (ε²), Neumann: λ_ε = λ₀ + ελ₁ +O (ε²). Using the method of energy, we prove a theorem of convergence in each case considered here. We briefly study correctors in the case of Neumann eigenvalue problem.     

Exponential decay rate of survival probability in a disastrous random environment

Summary. We study the exponential decay rate of the survival probability up to time t>0 of a random walker moving in Zopf; ^d in a temporally and spatially fluctuating random environment. When the random walker has a speed parameter κ>0, we investigate the influence of κ on the exponential decay rate λ(d,κ). In particular we prove that for any fixed d≥1, λ(d,κ) behaves like as logκ as κ↘0. Received: 21 May 1996 / In revised form: 2 February 1997     

On the weight-spectrum of a compact space

The weight-spectrumSp(w, X) of a spaceX is the set of weights of all infinite closed subspaces ofX. We prove that ifκ>ω is regular andX is compactT ₂ withω(X)≥κ then some λ withκ≤λ≤2^<κ is inSp(ω, X). Under CH this implies that the weight spectrum of a compact space can not omitω ₁, and thus solves problem 22 of [M]. Also, it is consistent with 2^ω=c being anything it can be that every countable closed setT of cardinals less thanc withω ∈ T satisfiesSp(w, X)=T for some separable compact LOTSX. This shows the independence from ZFC of a conjecture made in [AT]. Research supported by OTKA grant no. 1908.     

An infinite combinatorial statement with a poset parameter

We introduce an extension, indexed by a partially ordered set P and cardinal numbers κ,λ, denoted by (κ,<λ)⇝P, of the classical relation (κ,n,λ)→ρ in infinite combinatorics. By definition, (κ,n,λ)→ρ holds if every map F: [κ] ⁿ →[κ]^<λ has a ρ-element free set. For example, Kuratowski’s Free Set Theorem states that (κ,n,λ)→n+1 holds iff κ ≥ λ ⁺ⁿ , where λ ⁺ⁿ denotes the n-th cardinal successor of an infinite cardinal λ. By using the (κ,<λ)⇝P framework, we present a self-contained proof of the first author’s result that (λ ⁺ⁿ ,n,λ)→n+2, for each infinite cardinal λ and each positive integer n, which solves a problem stated in the 1985 monograph of Erdős, Hajnal, Máté, and Rado. Furthermore, by using an order-dimension estimate established in 1971 by Hajnal and Spencer, we prove the relation $(\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1} {2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor } $(\lambda ^{ + (n - 1)} ,r,\lambda ) \to 2^{\left\lfloor {\tfrac{1} {2}(1 - 2^{ - r} )^{ - n/r} } \right\rfloor } , for every infinite cardinal λ and all positive integers n and r with 2≤r<n. For example, (ℵ₂₁₀,4,ℵ₀)→32,768. Other order-dimension estimates yield relations such as (ℵ₁₀₉,4,ℵ₀) → 257 (using an estimate by Füredi and Kahn) and (ℵ₇,4,ℵ₀)→10 (using an exact estimate by Dushnik).     

Further cardinal arithmetic

We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S _{≤ℵ 0}(κ), ⊆) for κ real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(λ,λ,θ⁺,2) in more cases even when cf(λ)=ℵ₀ (Section 1), deal with bounds of pp(λ) for λ limit of inaccessible (Section 4) and give proofs to various claims I was sure I had already written but did not find (Section 6). Done mainly 1–4/1991. I thank Alice Leonhardt for typing and retyping so beautifully and accurately. Partially supported by the Basic Research Fund, Israel Academy of Sciences. Publication number 430.     

Separating ultrafilters on uncountable cardinals

A uniform ultrafilterU on κ is said to be λ-separating if distinct elements of the ultrapower never projectU to the same uniform ultrafilterV on λ. It is shown that, in the presence of CH, an ω-separating ultrafilterU on κ>ω is non-(ω, ω₁)-regular and, in fact, if κ < ℵ_ω thenU is λ-separating for all λ. Several large cardinal consequences of the existence of such an ultrafilterU are derived.     

Splitting ρκλ into maximally many stationary sets

Letκ >ω be a regular cardinal and λ >κ a cardinal. Solovay’s classical result for κ[So] led Menas [Me] to conjecture that a stationary subset ofP _κλ would split into λ^<κ stationary set of size κ⁺ (see[BT]), the conjecture implies that the size is (κ⁺)^<κ as well. Part of this work was done during the author’s stay at Boston University as one of the Japanese Overseas Research Fellows. He gratefully acknowledge Professor Akihiro Kanamori’s hospitality. He also wishes to thank members of the set theory seminar at Waseda University for their interest at the early stage.     

Universal graphs without instances of CH: Revisited

We give a complete proof of the consistency of the existence of a universal graph of powerλ, whereκ =κ ^<κ <λ = cfλ < 2 ^κ are arbitrary. The author would like to thank the NSF for partially supporting this research, Alice Leonhardt for the beautiful typing, and M. Kojman for proofreading. Publication No. 175A. The proof in the second section of [9] is flawed.     

Super Connectivity of Line Graphs and Digraphs

The h-super connectivity κh and the h-super edge-connectivity λh are more refined network reliability indices than the conneetivity and the edge-connectivity. This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edge-connected, then κ1（L） = 2λ1 （D）, and that for a connected graph G and its line graph L, if one of κ1 （L） and λ（G） exists, then κ1（L） = λ2（G）. This paper determines that κ1（B（d, n） is equal to 4d- 8 for n = 2 and d ≥ 4, and to 4d-4 for n ≥ 3 and d ≥ 3, and that κ1（K（d, n）） is equal to 4d- 4 for d 〉 2 and n ≥ 2 except K（2, 2）. It then follows that B（d,n） and K（d, n） are both super connected for any d ≥ 2 and n ≥ 1.     

Existence results for semilinear elliptic variational inequalities with changing sign nonlinearities

In this paper we present some existence results for a class of semilinear elliptic variational inequalities, depending on a real parameter λ, with changing sign nonlinearities. The fundamental tool to prove the existence result is a penalization method combined with the Mountain Pass Theorem and the Linking Theorem, respectively in the case λ < λ ₁ and λ ≥ λ ₁, where λ₁ is the first eigenvalue of the uniformly elliptic operator A involved in the variational inequality.     

More on countably compact,locally countable spaces

Following [5], aT ₃ spaceX is called good (splendid) if it is countably compact, locally countable (andω-fair).G(κ) (resp.S(κ)) denotes the statement that a good (resp. splendid) spaceX with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2 ^ω is big+S(κ) holds unlessω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>_ω+₁+┐G(ω_ω+₁); (iii) the “Chang conjecture” (ω_ω+₁),→(ω ₁,ω) implies ┐S(κ) for allκ≧k≧ω_ω; (iv) ifP addsω ₁ dominating reals toV iteratively then, in , we haveG(λ^ω) for allλ. Research supported by Hungarian National Foundation for Scientific Research grant no. 1805.