Transfinite Extension to Q m-normality Theory

The theory of Q _m-normal families, m ∈ ?, was developed by P. Montel for the cases m = 0 (normal families) [5] and m = 1 (quasinormal families) [4] and later generalized by C.T. Chuang [2] for any m ≥ 0. In this paper, we extend the definition to an arbitrary ordinal number α as follows. Given E ? D, define the α-th derived set $E^{(\alpha)}_D$ of E with respect to D by $(E^{(\alpha-1)}_D)^{(1)}_D$ if α has an immediate predecessor and by ${\mathop \bigcap\limits_{\beta<\alpha}} E^{(\beta)}_D$ if α is a limit ordinal. Then a family ${\cal F}$ of meromorphic functions on a plane domain D is Q_α-normal if each sequence S of functions in ${\cal F}$ has a subsequence which converges locally χ-uniformaly on the domain DE, where E = E(S) ? D satisfies $E^{(\alpha)}_{D}=\emptyset$ . Inparticular, a Q ₀ -normal family is a normal family, and a Q ₁ -normal family is a quasi- normal family. We also give analogues to some basic results in Q_m-normality theory and extend Zalcman’s Lemma to Q _α -normal families where α is an infinite countable (enumerable) ordinal number.     

Strengthening Kazhdan’s property (T) by Bochner methods

In this paper, we propose a property which is a natural generalization of Kazhdan??s property (T) and prove that many, but not all, groups with property (T) also have this property. Let ?? be a finitely generated group. One definition of ?? having property (T) is that ${H^{1}(\Gamma, \pi, {\mathcal{H}}) = 0}$ where the coefficient module ${{\mathcal{H}}}$ is a Hilbert space and ?? is a unitary representation of ?? on ${{\mathcal{H}}}$ . Here we allow more general coefficients and say that ?? has property ${F \otimes {H}}$ if ${H^{1}(\Gamma, \pi_{1}{\otimes}\pi_{2}, F{\otimes} {\mathcal{H}}) = 0}$ if (F, ?? ₁) is any representation with dim(F) <??? and ${({\mathcal{H}}, \pi_{2})}$ is a unitary representation. The main result of this paper is that a uniform lattice in a semisimple Lie group has property ${F \otimes {H}}$ if and only if it has property (T). The proof hinges on an extension of a Bochner-type formula due to Matsushima?CMurakami and Raghunathan. We give a new and more transparent derivation of this formula as the difference of two classical Weitzenb?ck formula??s for two different structures on the same bundle. Our Bochner-type formula is also used in our work on harmonic maps into continuum products (Fisher and Hitchman in preparation; Fisher and Hitchman in Int Math Res Not 72405:1?C19, 2006). Some further applications of property ${F\otimes {H}}$ in the context of group actions will be given in Fisher and Hitchman (in preparation).     

Semigroups of polygons whose vertices define a centered partition of ℝ n

A partition $\mathfrak{F}$ of a Euclidean space into finite subsets has subgroup property SP if the family of the convex hulls of the leaves of $\mathfrak{F}$ constitutes a subgroup with respect to the Minkowski addition. If $\mathfrak{F}$ consists of orbits of a finite linear groups then SP is equivalent to the fact that the group is a Coxeter group. In this article, this assertion is proved only under the assumption of continuity and centrality of $\mathfrak{F}$ (this means that every leaf is inscribed in some sphere centered at zero). An example is given of a noncentered partition satisfying SP (such partitions cannot be Coxeter partitions).     

Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type

Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .     

Boundary distortion and variation of the module under an extension of a doubly connected domain

Theorem 2

Let f(z) ∈ $\mathcal{F}(\rho ,r)$ , f(z) ≠ e^iα f(z;pr), α ∈ ?, and let ?(t) be a strictly convex monotone function of t>0. Then $$\int\limits_0^{2\pi } {\Phi (|f'(e^{i\theta } )|)d\theta< } \int\limits_0^{2\pi } {\Phi (|f'(e^{i\theta } ;\rho ,r)|)d\theta } $$ . The proof of this theorem is based on the Golusin-Komatu equation. If E is a continuum in the disk U_R={z:|z|<R}, then M (R, E) denotes the conformal module of the doubly connected component of U_R/E; let $\varepsilon (m) = \{ E:\overline U _r \subset E \subset U_1 , M(1,E) = M^{ - 1} \} $ .

Problem 3

Find the maximum of M(R, E), R>1, and the minimum of cap E over all E in ε(m). This problem was posed by V. V. Kozevnikov in a lecture to the Seminar on Geometric Function Theory at the Kuban University in 1980, and by D. Gaier (see [2]). The solution of this problem is given by the following theorem.

Theorem 3

Let $E^* = \underline U _m \cup [m,s]$ . If R>1; E, E^* ∈ ε(m) and E ≠ e^iα E*, α ∈ ?, then M(R, E)<M(R, E^*), capE^*<capE. A similar statement is also proved for continua lying in the half-plane. Bibliography: 7 titles.     

A new characterization of the Poisson distribution

In this note we show that an infinitely divisible (i.d.) distribution function F is Poisson if and only if it satisfies the conditions F(+0) > 0, for any 0 < ∈ < 1 $$\int_{ - \infty }^{I - E} {\frac{{\left| x \right|}}{{1 + \left| x \right|}}} dF = 0$$ and for any 0 < β < 1 $$\int_0^\infty {e^{\alpha xln(x + 1)} } dF< \infty $$      

Existence of entire solutions for a class of quasilinear elliptic equations

The paper deals with the existence of entire solutions for a quasilinear equation ${(\mathcal E)_\lambda}$ in ${\mathbb{R}^N}$ , depending on a real parameter λ, which involves a general elliptic operator in divergence form A and two main nonlinearities. The competing nonlinear terms combine each other, being the first subcritical and the latter supercritical. We prove the existence of a critical value λ* > 0 with the property that ${(\mathcal E)_\lambda}$ admits nontrivial non-negative entire solutions if and only if λ ≥ λ*. Furthermore, when ${\lambda > \overline{\lambda} \ge \lambda^*}$ , the existence of a second independent nontrivial non-negative entire solution of ${(\mathcal{E})_\lambda}$ is proved under a further natural assumption on A.     

On the dual positive Schur property in Banach lattices

The paper contains several characterizations of Banach lattices $E$ with the dual positive Schur property (i.e., $0 \le f_n \xrightarrow {\sigma (E^*,E)} 0$ implies $\Vert f_n\Vert \rightarrow 0$ ) and various examples of spaces having this property. We also investigate relationships between the dual positive Schur property, the positive Schur property, the positive Grothendieck property and the weak Dunford–Pettis property.     

Loops and the Lagrange Property

Let $\cal F$ be a family of finite loops closed under subloops and factor loops. Then every loop in $\cal F$ has the strong Lagrange property if and only if every simple loop in $\cal F$ has the weak Lagrange property. We exhibit several such families, and indicate how the Lagrange property enters into the problem of existence of finite simple loops.     

On permutations of central quadrics which preserve an inner distance

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques.     

Saturating Sperner Families

A family ${\mathcal{F} \subseteq 2^{[n]}}$ saturates the monotone decreasing property ${\mathcal{P}}$ if ${\mathcal{F}}$ satisfies ${\mathcal{P}}$ and one cannot add any set to ${\mathcal{F}}$ such that property ${\mathcal{P}}$ is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the k-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of l-sets and (l + 1)-sets.     

On Minimum Saturated Matrices

Motivated both by the work of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let ${\mathcal{F}}$ be a family of k-row matrices. A matrix M is called ${\mathcal{F}}$ -admissible if M contains no submatrix ${F \in \mathcal{F}}$ (as a row and column permutation of F). A matrix M without repeated columns is ${\mathcal{F}}$ -saturated if M is ${\mathcal{F}}$ -admissible but the addition of any column not present in M violates this property. In this paper we consider the function sat( ${n, \mathcal{F}}$ ) which is the minimal number of columns of an ${\mathcal{F}}$ -saturated matrix with n rows. We establish the estimate sat ${(n, \mathcal{F})=O(n^{k-1})}$ for any family ${\mathcal{F}}$ of k-row matrices and also compute the sat-function for a few small forbidden matrices.     

Weak and generic bang-bang properties for continuous evolution inclusions and Baire’s method

We consider evolution inclusions, in a separable and reflexive Banach space ${\mathbb{E}}$ , of the form ${(\ast) x'(t) \in Ax(t) + F(t, x(t)), x(t_0) = c}$ and ${(**) x'(t) \in Ax(t) + {\rm ext} F(t,x(t)), x(t_0) = c}$ , where A is the infinitesimal generator of a C ₀-semigroup, F is a continuous and bounded multifunction defined on ${[t_0, t_1] \times \mathbb{E}}$ with values F(t, x) in the space of all closed convex and bounded subsets of ${\mathbb{E}}$ with nonempty interior, and ext F(t, x(t)) denotes the set of the extreme points of F(t, x(t)). For (*) and (**) we prove a weak form of the bang-bang property, namely, the closure of the set of the mild solutions of (**) contains the set of all internal solutions of (*). The proof is based on the Baire category method. This result is used to prove the following generic bang-bang property, that is, if A is the infinitesimal generator of a compact C ₀-semigroup then for most (in the sense of the Baire categories) continuous and bounded multifunctions, with closed convex and bounded values ${F(t, x) \subset \mathbb{E}}$ , the bang-bang property is actually valid, that is, the closure of the the set of the mild solutions of (**) is equal to the set of the mild solutions of (*).     

Strong approximation by Fourier series and differentiability properties of functions

ПустьΦ —N-функция Юнг а со свойствами $$\Phi (x)x^{ - 1} \downarrow 0, \exists \alpha > 1 \Phi (x)x^{ - \alpha } \uparrow (x \downarrow 0),$$ илиΦ(х)=х, {λ_k} — положи тельная, неубывающая последовательность и $$S_\Phi \{ \lambda \} = \left\{ {f:\left\| {\sum\limits_{k = 0}^\infty \Phi (\lambda _k |f - s_k |)} \right\|_\infty< \infty } \right\}.$$ В работе найдены необ ходимые и достаточны е условия для вложений $$S_\Phi \{ \lambda \} \subset W^r F(r \geqq 0),$$ , гдеF=C, L ^∞, Lip α (0<α≦1). С этой то чки зрения рассматриваются и др угие классы (например, \(W^r H^\omega ,\tilde W^r F\) ).     

On generalizing transitivity, persistence, and sensitivity

A subgroup property $\alpha $ is transitive in a group $G$ if $U \alpha V$ and $V \alpha G$ imply that $U \alpha G$ whenever $U \le V \le G$ , and $\alpha $ is persistent in $G$ if $U \alpha G$ implies that $U \alpha V$ whenever $U \le V \le G$ . Even though a subgroup property $\alpha $ may be neither transitive nor persistent, a given subgroup $U$ may have the property that each $\alpha $ -subgroup of $U$ is an $\alpha $ -subgroup of $G$ , or that each $\alpha $ -subgroup of $G$ in $U$ is an $\alpha $ -subgroup of $U$ . We call these subgroup properties $\alpha $ -transitivity and $\alpha $ -persistence, respectively. We introduce and develop the notions of $\alpha $ -transitivity and $\alpha $ -persistence, and we establish how the former property is related to $\alpha $ -sensitivity. In order to demonstrate how these concepts can be used, we apply the results to the cases in which $\alpha $ is replaced with “normal” and the “cover-avoidance property.” We also suggest ways in which the theory can be developed further.     

Approximating minimum-cost edge-covers of crossing biset-families

Part of this paper appeared in the preliminary version [16]. An ordered pair ? = (S, S ₊) of subsets of a groundset V is called a biset if S ? S+; (V S ⁺;V S) is the co-biset of ?. Two bisets \(\hat X,\hat Y\) intersect if ^X X ∩ Y ≠ \(\not 0\) and cross if both X ∩ Y \(\not 0\) and X ⁺ ∪ Y ⁺ ≠= V. The intersection and the union of two bisets \(\hat X,\hat Y\) are defined by \(\hat X \cap \hat Y = (X \cap Y,X^ + \cap Y^ + )\) and \(\hat X \cup \hat Y = (X \cup Y,X^ + \cup Y^ + )\) . A biset-family \(\mathcal{F}\) is crossing (intersecting) if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) that cross (intersect). A directed edge covers a biset ? if it goes from S to V S ⁺. We consider the problem of covering a crossing biset-family \(\mathcal{F}\) by a minimum-cost set of directed edges. While for intersecting \(\mathcal{F}\) , a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing \(\mathcal{F}\) is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently or is not known. Let us say that a biset-family \(\mathcal{F}\) is k-regular if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) with |V (X∪Y)≥k+1 that intersect. In this paper we obtain an O(log |V|)-approximation algorithm for arbitrary crossing \(\mathcal{F}\) if in addition both \(\mathcal{F}\) and the family of co-bisets of \(\mathcal{F}\) are k-regular, our ratios are: \(O\left( {\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) , and \(O\left( {\frac{{|V|}} {{|V| - k}}\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) . Using these generic algorithms, we derive for some network design problems the following approximation ratios: \(O\left( {\log k \cdot \log \tfrac{n} {{n - k}}} \right) \) for k-Connected Subgraph, and O(logk) \(\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\} \) for Subset k-Connected Subgraph when all edges with positive cost have their endnodes in the subset.     

The Duality Property of Positive R-Compact Operators

We study the duality of r-compact operator. We establish if an operator ${T:E\rightarrow F}$ is r-compact, then its adjoint ${T^{\prime}: F^{\prime }\rightarrow E^{\prime }}$ is also r-compact. We also provide some sufficient condition on the pair of Banach lattices E and F which guarantees that a regular operator ${T:E\rightarrow F}$ such that ${T^{\prime }:F^{\prime }\rightarrow E^{\prime }}$ is r-compact, must itself be r-compact.     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

On the Intersection of the Normalizers of the \boldsymbol{\cal{F}}-Residuals of Subgroups of a Finite Group

In this paper we give criteria for a finite group to belong to a formation. As applications, recent theorems of Li, Shen, Shi and Qian are generalized. Let G  be a finite group, $\cal F$ a formation and p  a prime. Let $D_{\mathcal {F}}(G)$ be the intersection of the normalizers of the $\cal F$ -residuals of all subgroups of G, and let $D_{\mathcal {F}}^{p}(G)$ be the intersection of the normalizers of $(H^{\cal F}O_{p'}(G))$ for all subgroups H of G. We then define $D_{\mathcal F}^{0}(G)=D_{\mathcal F, p}^{~0}(G)=1$ and $D_{\mathcal F}^{i+1}(G)/D_{\mathcal F}^{i}(G)=D_{\mathcal F}(G/D_{\mathcal F}^{i}(G))$ , $D_{\mathcal F, p}^{i+1}(G)/D_{\mathcal F, p}^{~i}(G)=D_{\mathcal F, p}(G/D_{\mathcal F, p}^{~i}(G))$ . Let $D_{\mathcal {F}}^{\infty}(G)$ and $D_{\mathcal {F}, p}^{~\infty}(G)$ denote the terminal member of the ascending series of $D_{\mathcal F}^{i}(G)$ and $D_{\mathcal F, p}^{~i}(G)$ respectively. In this paper we prove that under certain hypotheses, the the $\cal F$ -residual $G^{\cal F}$ is nilpotent (respectively,p-nilpotent) if and only if $G=D_{\mathcal {F}}^{\infty}(G)$ (respectively, $G=D_{\mathcal {F}, p}^{~\infty}(G)$ ). Further more, if the formation $\cal F$ is either the class of all nilpotent groups or the class of all abelian groups, then $G^{\cal F}$ is p-nilpotent if and only if and only if every cyclic subgroup of G order p and 4 (if p?=?2) is contained in $D_{\mathcal {F}, p}^{~\infty}(G)$ .     

Pairs of Projections: Geodesics,Fredholm and Compact Pairs

A pair \((P, Q)\) of orthogonal projections in a Hilbert space \( \mathcal{H} \) is called a Fredholm pair if $$\begin{aligned} QP : R(P) \rightarrow R(Q) \end{aligned}$$ is a Fredholm operator. Let \( \mathcal{F} \) be the set of all Fredholm pairs. A pair is called compact if \(P-Q\) is compact. Let \( \mathcal{C} \) be the set of all compact pairs. Clearly \( \mathcal{C} \subset \mathcal{F} \) properly. In this paper it is shown that both sets are differentiable manifolds, whose connected components are parametrized by the Fredholm index. In the process, pairs \(P, Q\) that can be joined by a geodesic (or equivalently, a minimal geodesic) of the Grassmannian of \( \mathcal{H} \) are characterized: this happens if and only if $$\begin{aligned} \dim (R(P)\cap N(Q))=\dim (R(Q)\cap N(P)). \end{aligned}$$