On Clarkson's inequality,type and cotype for the Edmunds-Triebel logarithmic spaces

We show that the (p, p') Clarkson's inequality holds in the Edmunds-Triebel logarithmic spaces A_q(logA)_b,q A_{\theta}({\log}A)_{b,q} and in the Zygmund spaces L_p(logL)_b(W) L_p({\log}L)_b(\Omega) , for b ? \mathbbR b \in \mathbb{R} and for suitable 1 ￡ p ￡ 2 1 \leq p \leq 2 . As a consequence of these results we also obtain some new information about the types and the cotypes of these spaces.     

Quermassintegrals of a random polytope in a convex body

Let K be a convex body in \mathbbRⁿ \mathbb{R}^n with volume |K| = 1 |K| = 1 . We choose N 3 n+1 N \geq n+1 points x₁,?, x_N x_1,\ldots, x_N independently and uniformly from K, and write C(x₁,?, x_N) C(x_1,\ldots, x_N) for their convex hull. Let f : \mathbbR⁺ ? \mathbbR⁺ f : \mathbb{R^+} \rightarrow \mathbb{R^+} be a continuous strictly increasing function and 0 ￡ i ￡ n-1 0 \leq i \leq n-1 . Then, the quantity¶¶E (K, N, f °W_i) = ò_K ?ò_K f[W_i(C(x₁, ?, x_N))]dx_N ?dx₁ E (K, N, f \circ W_{i}) = \int\limits_{K} \ldots \int\limits_{K} f[W_{i}(C(x_1, \ldots, x_N))]dx_{N} \ldots dx_1 ¶¶is minimal if K is a ball (W_i is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1 ￡ i ￡ n-1 1 \leq i \leq n-1 , then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x₁,?, x_N) C(x_1,\ldots, x_N) .     

Takács’ Asymptotic Theorem and Its Applications: A Survey

The book of Lajos Takács Combinatorial Methods in the Theory of Stochastic Processes has been published in 1967. It discusses various problems associated with

On the reconstruction index of permutation groups: semiregular groups

Summary. The reconstruction index of all semiregular permutation groups is determined. We show that this index satisfies 3 ￡ r(G, W) ￡ 5 3 \leq \rho(G, \Omega) \leq 5 and we classify the groups in each case.     

White noise eliminates instability

Let x₁,..., x_n be points in the d-dimensional Euclidean space E^d with || x_i-x_j|| ￡ 1\| x_{i}-x_{j}\| \le 1 for all 1 \leqq i,j \leqq n1 \leqq i,j \leqq n, where || .||\| .\| denotes the Euclidean norm. We ask for the maximum M(d,n) of \mathop?_i, j=1ⁿ|| x_i-x_j|| ²\textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| ^{2} (see [4]). This paper deals with the case d = 2. We calculate M(2, n) and show that the value M(2, n) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular triangle of edge-length 1. Moreover we give an upper bound for the value \mathop?_i, j=1ⁿ|| x_i-x_j|| \textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| , where the points x₁,...,x_n are chosen under the same constraints as above.     

Pressure in Classical Statistical Mechanics and Interacting Brownian Particles in Multi-dimensions

. Let P(u) denote the pressure at the density u defined in the Gibbs statistical mechanics determined by a 2 body potential U (_qi - _qj). The function U(x) is supposed rotationally invariant and of finite range but may be unbounded about the origin. We establish a representation of P(u) by means of the law of large numbers for the virial ?_i,j q_i ·? U(q_i-q_j)\sum_{i,j} q_i \cdot {\nabla} U(q_i-q_j), whether or not there occur phase transitions. This result on P(u) is motivated by a study of the hydrodynamic behavior of a system of a large number of interacting Brownian particles moving on a d-dimensional torus (d = 1, 2, ...) in which the interaction is given by binary potential forces of potential U. Employing our representation of P(u), we also show that in the hydrodynamic limit of such a system there arises a non linear evolution equation of the form u_t = ¹/₂ DP(u)u_t = {1\over2} \Delta P(u) under a certain hypothetical postulate concerning concentration of particles.     

An Extension of Hall's Theorem

Let (A_i) _{i ? I} (A_i) _{i \in I} and (B_i) _{i ? I} (B_i) _ {i \in I} be two (possibly infinite) families of finite sets. Let cl(P) denote the closure of the set P : = { (A_i, B_i ): i ? I } P := \{ ({A_i}, {B_i} ): i \in I \} of the pairs with respect to the componentwise union and intersection operations. Then there exists an injective map è_{i ? I} A_i ? è_{i ? I} B_i {\displaystyle \bigcup _ {i \in I}} A_i \rightarrow {\displaystyle \bigcup _ {i \in I }} B_i such that f (A_i) í B_i f (A_i) \subseteq B_i for every i if, and only if, card (A) ￡ (A) \leq card (B) for every pair (A, B) ? cl (P) (A, B) \in cl (P) .     

Maximum IPP Codes of Length 3

Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let ${D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let D(a, b) = {(x₁, . . . , x_n) ? Qⁿ : x_i ? {a_i, b_i} for 1 ￡ i ￡ n}{D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}. Let C^* = è_{a, b ? C}D(a, b){C^* = {{\bigcup}_{\rm {a,\,b}\in{C}}}D({\bf a, b})}. The code C is said to have the identifiable parent property (IPP) if, for any x ? C^*{{\rm {\bf x}} \in C^*}, ?_{x ? D(a, b)}{a, b} 1 ?{{\bigcap}_{{\rm x}{\in}D({\rm a,\,b})}\{{\bf a, b}\}\neq \emptyset} . Codes with the IPP were introduced by Hollmann et al [J. Combin. Theory Ser. A 82 (1998) 21–133]. Let F(n, q) = max{|C|: C is a q-ary code of length n with the IPP}.T? and Safavi-Naini [SIAM J. Discrete Math. 17 (2004) 548–570] showed that 3q + 6 - 6 é?{q+1}ù ￡ F(3, q) ￡ 3q + 6 - é6 ?{q+1}ù{3q + 6 - 6 \lceil\sqrt{q+1}\rceil \leq F(3, q) \leq 3q + 6 - \lceil 6 \sqrt{q+1}\rceil}, and determined F (3, q) precisely when q ≤ 48 or when q can be expressed as r ² + 2r or r ² + 3r +2 for r ≥ 2. In this paper, we establish a precise formula of F(3, q) for q ≥ 24. Moreover, we construct IPP codes of size F(3, q) for q ≥ 24 and show that, for any such code C and any x ? C^*{{\rm {\bf x}} \in C^*}, one can find, in constant time, a ? C{{\rm {\bf a}} \in C} such that if x ? D (c, d){{\rm {\bf x}} \in D ({\bf c, d})} then a ? {c, d}{{\rm {\bf a}} \in \{{\rm {\bf c, d}}\}}.     

Cancellation in primely generated refinement monoids

We prove that any primely generated refinement monoid M has separative cancellation, and even strong separative cancellation provided M has no nonzero idempotents. A form of multiplicative cancellation also holds: na ￡ nb na\leq nb implies a ￡ b a\leq b for a,b ? M a,b \in M and n ? {1,2,3,?} n \in \{1,2,3,\ldots\} . In addition, M is a semilattice in the sense that, given c₁,c₂ ? M c_1,c_2 \in M , there is an element d ? M d \in M such that c₁,c₂ ￡ d c_1,c_2 \leq d and, for all a ? M, c₁,c₂ ￡ a a \in M, c_1,c_2 \leq a implies d ￡ a d \leq a . Finally, we prove that any finitely generated refinement monoid is primely generated; in fact, this holds for any refinement monoid with a set of generators satisfying the descending chain condition.     

How quadratic are the natural numbers?

Abstract. For natural numbers n we inspect all factorizations n = ab of n with a ￡ ba \le b in \Bbb N\Bbb N and denote by n=a_n b_nn=a_n b_n the most quadratic one, i.e. such that b_n - a_nb_n - a_n is minimal. Then the quotient k(n) : = a_n/b_n\kappa (n) := a_n/b_n is a measure for the quadraticity of n. The best general estimate for k(n)\kappa (n) is of course very poor: 1/n ￡ k(n) ￡ 11/n \le \kappa (n)\le 1. But a Theorem of Hall and Tenenbaum [1, p. 29], implies(logn)^-d-e ￡ k(n) ￡ (logn)^-d(\log n)^{-\delta -\varepsilon } \le \kappa (n) \le (\log n)^{-\delta } on average, with d = 1 - (1+log₂  2)/log2=0,08607 ?\delta = 1 - (1+\log _2 \,2)/\log 2=0,08607 \ldots and for every e > 0\varepsilon >0. Hence the natural numbers are fairly quadratic.¶k(n)\kappa (n) characterizes a specific optimal factorization of n. A quadraticity measure, which is more global with respect to the prime factorization of n, is k^*(n): = ?_{1 ￡ a ￡ b, ab=n} a/b\kappa ^*(n):= \textstyle\sum\limits \limits _{1\le a \le b, ab=n} a/b. We show k^*(n) ~ \frac 12\kappa ^*(n) \sim \frac {1}{2} on average, and k^*(n)=W(2^{\frac 12(1-e) log n/log ₂n})\kappa ^*(n)=\Omega (2^{\frac {1}{2}(1-\varepsilon ) {\log}\, n/{\log} _2n})for every e > 0\varepsilon>0.     

A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces

Let C be a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let T be an asymptotically nonexpansive mapping from C into itself such that the set F (T) of fixed points of T is nonempty. Let {a_n} be a sequence of real numbers with 0 ￡ a_n ￡ 10 \leq a_n \leq 1, and let x and x₀ be elements of C. In this paper, we study the convergence of the sequence {x_n} defined by¶¶x_n+1=a_n x + (1-a_n) [1/(n+1)] ?_j=0ⁿ T^j x_n   x_{n+1}=a_n x + (1-a_n) {1\over n+1} \sum\limits_{j=0}^n T^j x_n\quad for n=0,1,2,...  . n=0,1,2,\dots \,.     

Moser's second one-dimensional inequality for a class of integral operators

Suppose that $1 < p < \infty $1 < p < \infty , q=p/(p-1)q=p/(p-1), and for non-negative f ? L^p(-￥ ,￥)f\in L^p(-\infty\! ,\infty ) and any real x we let F(x)-F(0)=ò₀^xf(t) dtF(x)-F(0)=\int _0^xf(t)\ dt; suppose in addition that ò_-￥^￥ F(t)exp(-|t|) dt=0\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0. Moser's second one-dimensional inequality states that there is a constant C_pC_p, such that ò_-￥^￥ exp[a |F(x)|^q-|x|]  dx ￡ C_p\int\limits _{-\infty }^\infty \exp [a |F(x)|^q-|x|] \ dx\le C_p for each f with ||f||_p ￡ 1||f||_p\le 1 and every a ￡ 1a\le 1. Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a.     

Partitions involving D.H. Lehmer numbers

Let n ≥ 2 be a fixed integer, let q and c be two integers with q > n and (n, q) = (c, q) = 1. For every positive integer a which is coprime with q we denote by [`(a)]<sub>c</sub>{\overline{a}_{c}} the unique integer satisfying 1 ￡ [`(a)]_c ￡ q{1\leq\overline{a}_{c} \leq{q}} and a[`(a)]_c o c(mod q){a\overline{a}_{c} \equiv{c}({\rm mod}\, q)}. Put

On Lipschitz selections of affine-set valued mappings

We prove a Helly-type theorem for the family of all k-dimensional affine subsets of a Hilbert space H. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M,r) ({\cal M},\rho) into this family.¶Let F be such a mapping satisfying the following condition: for every subset M￠ ì M {\cal M'} \subset {\cal M} consisting of at most 2^k+1 points, the restriction F|_M￠ F|_{\cal M'} of F to M￠ {\cal M'} has a selection f_M￠ (i.e. f_M￠(x) ? F(x) for all x  ? M￠) f_{\cal M'}\,({\rm i.e.}\,f_{\cal M'}(x) \in F(x)\,{\rm for\,all}\,x\,\in {\cal M'}) satisfying the Lipschitz condition ||f_M￠(x) - f_M￠(y)||  ￡ r(x,y ), x,y ? M￠ \parallel f_{\cal M'}(x) - f_{\cal M'}(y)\parallel\,\le \rho(x,y ),\,x,y \in {\cal M'} . Then F has a Lipschitz selection f : M ? H f : {\cal M} \to H such that ||f(x) - f(y) ||  ￡ gr(x,y ), x,y ? M \parallel f(x) - f(y) \parallel\,\le \gamma \rho (x,y ),\,x,y \in {\cal M} where g = g(k) \gamma = \gamma(k) is a constant depending only on k. (The upper bound of the number of points in M￠ {\cal M'} , 2^k+1, is sharp.)¶The proof is based on a geometrical construction which allows us to reduce the problem to an extension property of Lipschitz mappings defined on subsets of metric trees.     

On the number of conjugacy classes of maximal subgroups in a finite soluble group

We show that for many formations \frak F\frak F, there exists an integer n = [`(m)](\frak F)n = \overline m(\frak F) such that every finite soluble group G not belonging to the class \frak F\frak F has at most n conjugacy classes of maximal subgroups belonging to the class \frak F\frak F. If \frak F\frak F is a local formation with formation function f, we bound [`(m)](\frak F)\overline m(\frak F) in terms of the [`(m)](f(p))(p ? \Bbb P )\overline m(f(p))(p \in \Bbb P ). In particular, we show that [`(m)](\frak N^k) = k+1\overline m(\frak N^k) = k+1 for every nonnegative integer k, where \frak N^k\frak N^k is the class of all finite groups of Fitting length ￡ k\le k.     

Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

We consider the Hill operator

Concentration of Points on Two and Three Dimensional Modular Hyperbolas and Applications

Let p be a large prime number, K, L, M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I ₂(M; K, L) of solutions of the congruence

p-Frames and Shift Invariant Subspaces of Lp

In this article we investigate the frame properties and closedness for the shift invariant space V_p(F) = { ?_i=1^r ?_{j ? \Zd} d_i(j) f_i (·-j):  ( d_i(j) )_{j ? \Zd} ? l^p }, \q 1 ￡ p ￡ ￥ . \displaystyle V_p(\Phi) = \left\{ \sum_{i=1}^r \sum_{j\in \Zd} d_i(j) \phi_i (\cdot-j): \ \left( d_i(j) \right)_{j\in \Zd}\in \ell^p \right\}, \q 1\le p \le \infty~. We derive necessary and sufficient conditions for an indexed family {f_i(·-j): 1 ￡ i ￡ r, j ? \Zd}\{\phi_i(\cdot-j):\ 1\le i\le r, j\in \Zd\} to constitute a pp-frame for V_p(F)V_p(\Phi), and to generate a closed shift invariant subspace of L^pL^p. A function in the L^pL^p-closure of V_p(F)V_p(\Phi) is not necessarily generated by l^p\ell^p coefficients. Hence we often hope that V_p(F)V_p(\Phi) itself is closed, i.e., a Banach space. For p 1 2p\ne 2, this issue is complicated, but we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of pp-frames, Banach frames, and the closedness of the space they generate. The relation between a function f ? V_p(F)f \in V_p(\Phi) and the coefficients of its representations is neither obvious, nor unique, in general. For the case of pp-frames, we are in the context of normed linear spaces, but we are still able to give a characterization of pp-frames in terms of the equivalence between the norm of ff and an l^p\ell^p-norm related to its representations. A Banach frame does not have a dual Banach frame in general, however, for the shift invariant spaces V_p(F)V_p(\Phi), dual Banach frames exist and can be constructed.     

A rigidity theorem for the pair ${\cal q}{\Bbb C} P^n$ (complex hyperquadric, complex projective space)

Given a compact Kähler manifold M of real dimension 2n, let P be either a compact complex hypersurface of M or a compact totally real submanifold of dimension n. Let q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space \Bbb C Pⁿ{\Bbb C} P^n of constant holomorphic sectional curvature 4l \lambda . We prove that if the Ricci and some (n-1)-Ricci curvatures of M (and, when P is complex, the mean absolute curvature of P) are bounded from below by some special constants and volume (P) / volume (M) ￡\leq volume (q\cal q)/ volume (\Bbb C Pⁿ)({\Bbb C} P^n) (resp. ￡\leq volume (\Bbb R Pⁿ)({\Bbb R} P^n) / volume (\Bbb C Pⁿ)({\Bbb C} P^n)), then there is a holomorphic isometry between M and \Bbb C Pⁿ{\Bbb C} P^n taking P isometrically onto q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n). We also classify the Kähler manifolds with boundary which are tubes of radius r around totally real and totally geodesic submanifolds of half dimension, have the holomorphic sectional and some (n-1)-Ricci curvatures bounded from below by those of the tube \Bbb R Pⁿ_r{\Bbb R} P^n_r of radius r around \Bbb R Pⁿ{\Bbb R} P^n in \Bbb C Pⁿ{\Bbb C} P^n and have the first Dirichlet eigenvalue not lower than that of \Bbb R Pⁿ_r{\Bbb R} P^n_r.     

On the singularities of residual intertwining operators

Let G be a reductive algebraic group defined over \Bbb Q {\Bbb Q} . Let P, P' be parabolic subgroups of G, defined over \Bbb Q {\Bbb Q} , and let _boxclose_boxclose, a_P') t \in W({\frak a}_{P}, {\frak a}_{P'}) . In this paper we study the intertwining operator M_P￠|P(t,l), l ? \frak a^*_{P,\Bbb C} M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}} , acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of M_P￠|P(t,l) M_{P' \vert P}(t,\lambda) is a meromorphic function of order ￡ n + 1 \le n + 1 , where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.