Minimax theorems for limits of parametrized functions having at most one local minimum lying in a certain set

In this paper, we establish some minimax theorems, of purely topological nature, that, through the variational methods, can be usefully applied to nonlinear differential equations. Here is a (simplified) sample: Let X be a Hausdorff topological space, I⊆R an interval and . Assume that the function Ψ(x,⋅) is lower semicontinuous and quasi-concave in I for all x∈X, while the function Ψ(⋅,q) has compact sublevel sets and one local minimum at most for each q in a dense subset of I. Then, one has

     

Lattices generated by join of strongly closed subgraphs in d-bounded distance-regular graphs

Let Γ be a d-bounded distance-regular graph with diameter d?3. Suppose that P(x) is a set of all strongly closed subgraphs containing x and that P(x,i) is a subset of P(x) consisting of all elements of P(x) with diameter i. Let L^′(x,i) be the set generated by all joins of the elements in P(x,i). By ordering L^′(x,i) by inclusion or reverse inclusion, L^′(x,i) is denoted by or . We prove that and are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of      

Monoids with sub-log-exponential free spectra

A classic result from the 1960s states that the asymptotic growth of the free spectrum of a finite group is sub-log-exponential if and only if is nilpotent. Thus a monoid is sub-log-exponential implies , the pseudovariety of semigroups with nilpotent subgroups. Unfortunately, little more is known about the boundary between the sub-log-exponential and log-exponential monoids.The pseudovariety consists of those finite semigroups satisfying (x^ωy^ω)^ω(y^ωx^ω)^ω(x^ωy^ω)^ω≈(x^ωy^ω)^ω. Here it is shown that a monoid is sub-log-exponential implies . A quick application: a regular sub-log-exponential monoid is orthodox. It is conjectured that a finite monoid is sub-log-exponential if and only if it is , the finite monoids in having nilpotent subgroups. The forward direction of the conjecture is proved; moreover, the conjecture is proved for when is completely (0)-simple. In particular, the six-element Brandt monoid (the Perkins semigroup) is sub-log-exponential.     

Composition operators on the polydisc induced by smooth symbols

We study composition operators CΦ on the Hardy spaces Hp and weighted Bergman spaces of the polydisc Dn in Cn. When Φ is of class C² on , we show that CΦ is bounded on Hp or if and only if the Jacobian of Φ does not vanish on those points ζ on the distinguished boundary Tn such that Φ(ζ)∈Tn. Moreover, we show that if ε>0 and if , then CΦ is bounded on .     

Retractions and contractibility in hyperspaces

Given a metric continuum X, let X² denote the hyperspace of all nonempty closed subsets of X. For each positive integer k let Ck(X) stand for the hyperspace of members of X² having at most k components. Consider mappings (where B∈Cm(X)) and both defined by A?A∪B. We give necessary and sufficient conditions under which these mappings are deformation retractions (under a special convention for φB). The conditions are related to the contractibility of the corresponding hyperspaces.     

A family of invariants of rooted forests

Let A be a commutative k-algebra over a field of k and Ξ a linear operator defined on A. We define a family of A-valued invariants Ψ for finite rooted forests by a recurrent algorithm using the operator Ξ and show that the invariant Ψ distinguishes rooted forests if (and only if) it distinguishes rooted trees T, and if (and only if) it is finer than the quantity α(T)=|Aut(T)| of rooted trees T. We also consider the generating function with , where is the set of rooted trees with n vertices. We show that the generating function U(q) satisfies the equation . Consequently, we get a recurrent formula for U_n (n?1), namely, U₁=Ξ(1) and U_n=ΞS_n−1(U₁,U₂,…,U_n−1) for any n?2, where are the elementary Schur polynomials. We also show that the (strict) order polynomials and two well-known quasi-symmetric function invariants of rooted forests are in the family of invariants Ψ and derive some consequences about these well-known invariants from our general results on Ψ. Finally, we generalize the invariant Ψ to labeled planar forests and discuss its certain relations with the Hopf algebra in Foissy (Bull. Sci. Math. 126 (2002) 193) spanned by labeled planar forests.     

Generalized Haar integral

The aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) such that for each f∈A(S) and for any f∈C(Ω,R) and s∈S, , where .     

Sequences and filters of characters characterizing subgroups of compact Abelian groups

Let H be a countable subgroup of the metrizable compact Abelian group G and a (not necessarily continuous) character of H. Then there exists a sequence of (continuous) characters of G such that limn→∞χn(α)=f(α) for all α∈H and does not converge whenever α∈G?H. If one drops the countability and metrizability requirement one can obtain similar results by using filters of characters instead of sequences. Furthermore the introduced methods allow to answer questions of Dikranjan et al.     

Variations of selective separability

A space X is selectively separable if for every sequence of dense subspaces of X one can select finite Fn⊂Dn so that is dense in X. In this paper selective separability and variations of this property are considered in two special cases: Cp spaces and dense countable subspaces in κ².     

Lipschitz and piecewise-C regularity for scalar minimizers of affine simple integrals

Lipschitz, piecewise-C¹ and piecewise affine regularity is proved for AC minimizers of the “affine” integral , under general hypotheses on , , and with superlinear growth at infinity.The hypotheses assumed to obtain Lipschitz continuity of minimizers are unusual: ρ(·) and ?(·) are lsc and may be both locally unbounded (e.g., not in L_loc¹), provided their quotient ?/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.     

Countably compact hyperspaces and Frolík sums

Let H⁰(X) (H(X)) denote the set of all (nonempty) closed subsets of X endowed with the Vietoris topology. A basic problem concerning H(X) is to characterize those X for which H(X) is countably compact. We conjecture that u-compactness of X for some u∈ω^∗ (or equivalently: all powers of X are countably compact) may be such a characterization. We give some results that point into this direction.We define the property R(κ): for every family of closed subsets of X separated by pairwise disjoint open sets and any family of natural numbers, the product is countably compact, and prove that if H(X) is countably compact for a T₂-space X then X satisfies R(κ) for all κ. A space has R(1) iff all its finite powers are countably compact, so this generalizes a theorem of J. Ginsburg: if X is T₂ and H(X) is countably compact, then so is Xn for all n<ω. We also prove that, for κ<t, if the T₃ space X satisfies a weak form of R(κ), the orbit of every point in X is dense, and X contains κ pairwise disjoint open sets, then Xκ is countably compact. This generalizes the following theorem of J. Cao, T. Nogura, and A. Tomita: if X is T₃, homogeneous, and H(X) is countably compact, then so is Xω.Then we study the Frolík sum (also called “one-point countable-compactification”) of a family . We use the Frolík sum to produce countably compact spaces with additional properties (like first countability) whose hyperspaces are not countably compact. We also prove that any product _∏α<κH⁰(Xα) embeds into .     

Approximation of the Hausdorff distance by the distance of continuous surjections

The aim of this paper is to answer the following question: let (X,?) and (Y,d) be metric spaces, let A,B⊂Y be continuous images of the space X and let be a fixed continuous surjection. When is the inequality