Topological-algebraic properties of function spaces with set-open topologies

For a Tychonoff space X, we denote by Cλ(X) the space of all real-valued continuous functions on X with set-open topology. In this paper, we study the topological-algebraic properties of Cλ(X). Our main results state that (1) Cλ(X) is a topological vector space (a topological group) iff λ is a family of C-compact sets and Cλ(X)=Cλ^′(X), where λ^′ consists of all C-compact subsets of every set of λ. In particular, if Cλ(X) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family λ; (2) a topological group Cλ(X) is ω-narrow iff λ is a family of metrizable compact subsets of X.     

An incidence algebra for t-designs with automorphisms

A t-design or generalized Steiner systemS(λ; t, k, υ) is a pair (X, B) with a υ-set X of points and a family B of k-subsets of X called blocks such that, each block has k points and any t points are contained in exactly λ blocks. An automorphism of (X, B) is a permutation of X which preserves B. In this paper the algebra of matrices, whose rows and columns are indexed by the members of P(X), that are invariant under the natural action of a group G ⩽Sym(X) is introduced. An epimorphism τ from this algebra onto the matrices whose rows and columns are indexed by the orbits of G acting on P(X) is discovered. This mapping carries the matrices of Wilson onto the matrices of Kramer and Mesner and therefore can be used to generalize the t-design inequalities of Fisher, Wilson, and Ray-Chaudhuri. A conjecture of Earl Kramer is settled and an elementary proof of a theorem of Livingstone and Wagner is presented.     

On Parameter Depending Differential Systems Under Carathéodory Conditions

For every λ in a complex domain G, consider on some interval I the initial value problem y′(λ,x) = A(λ,x)y(λ,x) + b(λ,x), y(λ,x₀) - y₀. If this problem satisfies the Carathéodory conditions for every A, then there exist locally absolutely continuous and almost everywhere differentiable solutions y(λ,· ) of the initial value problem. In general, the union N of the exceptional sets N _λ ? I where y(λ, ·) is not differentiate or does not fulfill the differential equation, is not of Lebesgue measure zero. It will be shown that N is of Lebesgue measure zero provided that A and b are holomorphic with respect to λ and their integrals with respect to x are locally bounded on G × I.     

A graph which embeds all small graphs on any large set of vertices

For certain cardinals λ and κ a colouring P:[λ]²→λ is constructed such that if X ϵ[λ]^λ and Q:[κ]²→λ, then there is a one-to-one function i:κ→X such that P(i″A)=Q(A) for every Aϵ[κ]². Additional results are obtained.     

Symmetry and bifurcation near families of solutions

Let X, Z and Λ be Banach spaces, M: X × Λ → Z a C¹-function, and assume that the equation M(x, λ) = 0 has a family of solutions for λ = 0. In this paper we consider the bifurcation of solutions from this family, for ¦λ¦ small, under the condition that both the unperturbed (λ = 0) and the perturbed (λ ≠ 0) equations have certain symmetry properties. The problem is reduced by the Liapunov-Schmidt method, and the bifurcation equations are solved by a straightforward use of the symmetry. As an application we obtain existence of certain periodic solutions for the undamped Duffing equation, a result recently obtained by Schmitt and Mazzanti using different methods.     

Generalized bi-circular projections

Recall that a projection P on a complex Banach space X is a generalized bi-circular projection if P+λ(I−P) is a (surjective) isometry for some λ such that |λ|=1 and λ≠1. It is easy to see that every hermitian projection is generalized bi-circular. A generalized bi-circular projection is said to be nontrivial if it is not hermitian. Botelho and Jamison showed that a projection P on C([0,1]) is a nontrivial generalized bi-circular projection if and only if P−(I−P) is a surjective isometry. In this article, we prove that if P is a projection such that P+λ(I−P) is a (surjective) isometry for some λ, then either P is hermitian or λ is an nth unit root of unity. We also show that for any nth unit root λ of unity, there are a complex Banach space X and a nontrivial generalized bi-circular projection P on X such that P+λ(I−P) is an isometry.     

On the <Emphasis Type="Italic">σ</Emphasis>-countable compactness of spaces of continuous functions with the set-open topology

For a Tychonoff space X, we obtain a criterion of the σ-countable compactness of the space of continuous functions C(X) with the set-open topology. In particular, for the class of extremally disconnected spaces X, we prove that the space C _λ(X) is σ-countably compact if and only if X is a pseudocompact space, the set X(P) of all P-points of the space X is dense in X, and the family λ consists of finite subsets of the set X(P).     

Direct construction methods for incomplete perfect Mendelsohn designs with block size four

Let v, k, λ, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by (v,n,k,λ)-IPMD, is a triple (X,Y,B) where X is a v-set (of points), Y is an n-subset of X, and B is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a,b) E (X × X)\(Y × Y) appears t-apart in exactly λ blocks of B and no ordered pair (a,b) E Y × Y appears in any block of B for any t, where 1 ≤ t ≤ k − 1. In this article, we introduce an effective and easy way to construct IPMDs for k = 4 and even v − n, and use it to construct some small examples for λ = 1 and 2. Obviously, these results will play an important role to completely solve the existence of (v,n,4,λ)-IPMDs. Furthermore, we also use this method to construct some small examples for HPMDs. © 1996 John Wiley & Sons, Inc.     

On matrices leaving invariant a nontrivial convex set

A real square matrix A leaves a nontrivial convex set invariant if there exists a convex set C, which is not a linear subspace, such that A(C) ? C. It is shown that this is equivalent to the statement that A has an eigenvalue λ with λ?0 or |λ|?1.     

Measurability of semimartingale characteristics with respect to the probability law

Given a càdlàg process X$X$ on a filtered measurable space, we construct a version of its semimartingale characteristics which is measurable with respect to the underlying probability law. More precisely, let _Psem${\mathfrak{P}}_{sem}$ be the set of all probability measures P$P$ under which X$X$ is a semimartingale. We construct processes (B^P,C,ν^P)$(B^P,C,{\nu }^P)$ which are jointly measurable in time, space, and the probability law P$P$, and are versions of the semimartingale characteristics of X$X$ under P$P$ for each P∈_Psem$P\in {\mathfrak{P}}_{sem}$. This result gives a general and unifying answer to measurability questions that arise in the context of quasi-sure analysis and stochastic control under the weak formulation.     

On three-designs of small order

For positive integers t?k?v and λ we define a t-design, denoted B_i[k,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (B_i:i?I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |B_i|=k for each i?I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3?k?v?32 and λ>0.Wilson has shown that there exists a constant N(t, k, v) such that designs B_t[k,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3?k?v?32. We give explicit constructions for all the designs needed.     

Scaling limits for point random fields

If X is a point random field on $\text{R}$^d then convergence in distribution of the renormalization C_λ|X_λ ? α_λ| as λ → ∞ to generalized random fields is examined, where C_λ > 0, α_λ are real numbers for λ > 0, and X_λ(f) = λ^?dX(f_λ) for $\text{f}{}_{\mathrm{\lambda }}\text{(x) = f(}\text{x}\text{λ}\text{)}$. If such a scaling limit exists then C_λ = λ^θg(λ), where g is a slowly varying function, and the scaling limit is self-similar with exponent θ. The classical case occurs when $\text{θ =}\text{d}\text{2}$ and the limit process is a Gaussian white noise. Scaling limits of subordinated Poisson (doubly stochastic) point random fields are calculated in terms of the scaling limit of the environment (driving random field). If the exponent of the scaling limit is $\text{θ =}\text{d}\text{2}$ then the limit is an independent sum of the scaling limit of the environment and a Gaussian white noise. If $\text{θ <}\text{d}\text{2}$ the scaling limit coincides with that of the environment while if $\text{θ >}\text{d}\text{2}$ the limit is Gaussian white noise. Analogous results are derived for cluster processes as well.     

On a Problem Related to Homomorphism Groups in the Theory of Abelian Groups

In this paper, for any reduced Abelian group A whose torsion-free rank is infinite, we construct a countable set A(A) of Abelian groups connected with the group A in a definite way and such that for any two different groups B and C from the set A(A) the groups B and C are isomorphic but Hom(B,X) ? Hom(C,X) for any Abelian group X. The construction of such a set of Abelian groups is closely connected with Problem 34 from L. Fuchs’ book “Infinite Abelian Groups,” Vol. 1.     

C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.     

Injective Banach spaces of typeC(T)

A Banach spaceX is aP _λ-space if wheneverX is isometrically embedded in another Banach spaceY there is a projection ofY ontoX with norm at most λ.C(T) denotes the Banach space of continuous real-valued functions on the compact Hausdorff spaceT. T satisfies the countable chain condition (CCC) if every family of disjoint non-empty open sets inT is countable.T is extremally disconnected if the closure of every open set inT is open. The main result is that ifT satisfies the CCC andC(T) is aP _λ-space, thenT is the union of an open dense extremally disconnected subset and a complementary closed setT _Asuch thatC(T_A) is aP _λ?1-space.     

Poisson thickening

Let × be a Poisson point process of intensity λ on the real line. A thickening of it is a (deterministic) measurable function f such that X∪f(X) is a Poisson point process of intensity λ′ where λ′ > λ. An equivariant thickening is a thickening which commutes with all shifts of the line. We show that a thickening exists but an equivariant thickening does not. We prove similar results for thickenings which commute only with integer shifts and in the discrete and multi-dimensional settings. This answers 3 questions of Holroyd, Lyons and Soo. We briefly consider also a much more general setup in which we ask for the existence of a deterministic coupling satisfying a relation between two probability measures. We present a conjectured sufficient condition for the existence of such couplings.     

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X?D(T) → 2 ^X * is a maximal monotone multi-valued operator and C: X?D(C) → X* is a generalized pseudomonotone quasibounded operator with L ? D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T _x + λ C _x , with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.     

Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory

Let (X_n) be a positive recurrent Harris chain on a general state space, with invariant probability measure π. We give necessary and sufficient conditions for the geometric convergence of λPⁿf towards its limit π(f), and show that when such convergence happens it is, in fact, uniform over f and in L¹(π)-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, ∝ π(dx)6Pⁿ(x,·)-π6 converges to zero geometrically fast. We also characterize the geometric ergodicity of (X_n) in terms of hitting time distributions. We show that here the so-called small sets act like individual points of a countable state space chain. We give a test function criterion for geometric ergodicity and apply it to random walks on the positive half line. We apply these results to non-singular renewal processes on [0,∞) providing a probabilistic approach to the exponencial convergence of renewal measures.     

Equivalent Gaussian measure whose R-N derivative is the exponential of a diagonal form

A simple necessary and sufficient condition, on a trace-class kernel K, is given in order to demonstrate the existence of a measurable (relative to the completed product σ-algebra) Gaussian process with covariance K. Using this result, sufficient conditions are given on the means and the covariances (relative to two equivalent (~) Gaussian measures P and P_λ) of a process X so that the Radon-Nikodým (R-N) derivative $\text{dp}{}_{\mathrm{\lambda }}\text{dP}$ is the exponential of the diagonal form in X. Analogues of the last two results in the setup of Hilbert space are also proved.