On a question of Erdös and Graham

We prove that for any $ \varepsilon > 0 $ \varepsilon > 0 there is k (e) k (\varepsilon) such that for any prime p and any integer c there exist k \leqq k(e) k \leqq k(\varepsilon) pairwise distinct integers x_i with 1 \leqq x_i \leqq p^e, i = 1, ?, k 1 \leqq x_{i} \leqq p^{\varepsilon}, i = 1, \ldots, k , and such that¶¶?_i=1^k [1/(x_i)] o c    (mod p). \sum\limits_{i=1}^k {{1}\over{x_i}} \equiv c\quad (\mathrm{mod}\, p). ¶¶ This gives a positive answer to a question of Erdös and Graham.     

A linear functional equation with a singularity at a fixed point

Summary. Local solutions of the functional equation¶¶z^k f( z) = ?_k=1ⁿG_k( z) f( s_kz ) +g( z) z{^\kappa} \phi \left( z\right) =\sum_{k=1}^nG_k\left( z\right) \phi \left( s_kz \right) +g\left( z\right) ¶with k > 0 \kappa > 0 and | s_k| \gt 1 \left| s_k\right| \gt 1 are considered. We prove that the equation is solvable if and only if a certain system of k \kappa conditions on G_k (k = 1, 2, ... , n) and g is fulfilled.     

Equality of two variable weighted means: reduction to differential equations

Summary. Let F, Y \Phi, \Psi be strictly monotonic continuous functions, F,G be positive functions on an interval I and let n ? \Bbb N \{1} n \in {\Bbb N} \setminus \{1\} . The functional equation¶¶F^-1 ([(?_i=1ⁿF(x_i)F(x_i))/(?_i=1ⁿ F(x_i)]) Y^-1 ([(?_i=1ⁿY(x_i)G(x_i))/(?_i=1ⁿ G(x_i))])  (x₁,?,x_n ? I) \Phi^{-1}\,\left({\sum\limits_{i=1}^{n}\Phi(x_{i})F(x_{i})\over\sum\limits_{i=1}^{n} F(x_{i}}\right) \Psi^{-1}\,\left({\sum\limits_{i=1}^{n}\Psi(x_{i})G(x_{i})\over\sum\limits_{i=1}^{n} G(x_{i})}\right)\,\,(x_{1},\ldots,x_{n} \in I) ¶was solved by Bajraktarevi' [3] for a fixed n 3 3 n\ge 3 . Assuming that the functions involved are twice differentiable he proved that the above functional equation holds if and only if¶¶Y(x) = [(aF(x) + b)/(cF(x) + d)],       G(x) = kF(x)(cF(x) + d) \Psi(x) = {a\Phi(x)\,+\,b\over c\Phi(x)\,+\,d},\qquad G(x) = kF(x)(c\Phi(x) + d) ¶where a,b,c,d,k are arbitrary constants with k(c²+d²)(ad-bc) 1 0 k(c^2+d^2)(ad-bc)\ne 0 . Supposing the functional equation for all n = 2,3,... n = 2,3,\dots Aczél and Daróczy [2] obtained the same result without differentiability conditions.¶The case of fixed n = 2 is, as in many similar problems, much more difficult and allows considerably more solutions. Here we assume only that the same functional equation is satisfied for n = 2 and solve it under the supposition that the functions involved are six times differentiable. Our main tool is the deduction of a sixth order differential equation for the function j = F°Y^-1 \varphi = \Phi\circ\Psi^{-1} . We get 32 new families of solutions.     

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.     

Extension theorems for functional equations with bisymmetric operations

Summary. In this paper we deal with the extension of the following functional equation¶¶ f (x) = M (f (m₁(x, y)), ..., f (m_k(x, y)))        (x, y ? K) f (x) = M \bigl(f (m_{1}(x, y)), \dots, f (m_{k}(x, y))\bigr) \qquad (x, y \in K) , (*)¶ where M is a k-variable operation on the image space Y, m₁,..., m_k are binary operations on X, K ì X K \subset X is closed under the operations m₁,..., m_k, and f : K ? Y f : K \rightarrow Y is considered as an unknown function.¶ The main result of this paper states that if the operations m₁,..., m_k, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m₁,..., m_k)-affine hull K^* of K, such that (*) holds over K^*. (The set K^* is defined as the smallest subset of X that contains K and is (m₁,..., m_k)-affine, i.e., if x ? X x \in X , and there exists y ? K^* y \in K^* such that m₁(x, y), ?, m_k(x, y) ? K^* m_{1}(x, y), \ldots, m_{k}(x, y) \in K^* then x ? K^* x \in K^* ). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.     

A note on a conjecture of Borwein and Choi

A polynomial P(X) with coefficients {ǃ} of odd degree N - 1 is cyclotomic if and only if¶¶P(X) = ±F_p1 (±X)F_p2(±X^p1) ?F_pr(±X^{p1 p2 ?p_r-1}) P(X) = \pm \Phi_{p1} (\pm X)\Phi_{p2}(\pm X^{p1}) \cdots \Phi_{p_r}(\pm X^{p1 p2 \cdots p_r-1}) ¶where N = p₁ p₂ · · · p_r and the p_i are primes, not necessarily distinct, and where F_p(X) : = (X^p - 1) / (X - 1) \Phi_{p}(X) := (X^{p} - 1) / (X - 1) is the p-th cyclotomic polynomial. This is a conjecture of Borwein and Choi [1]. We prove this conjecture for a class of polynomials of degree N - 1 = 2^r p^l - 1 N - 1 = 2^{r} p^{\ell} - 1 for any odd prime p and for integers r, l\geqq 1 r, \ell \geqq 1 .     

Transcendence of the values of certain series with Hadamard's gaps

Transcendence of the number ?_k=0^￥ a^r_k \sum_{k=0}^\infty \alpha^{r_k} , where a \alpha is an algebraic number with 0 < | a | \mid\alpha\mid > 1 and {r_k}_k\geqq0 \{r_k\}_{k\geqq0} is a sequence of positive integers such that lim_k?￥ r_k+1/r_k = d ? \mathbbN \{1} \lim_{k\to\infty}\, r_{k+1}/r_k = d \in \mathbb{N}\, \backslash \{1\} , is proved by Mahler's method. This result implies the transcendence of the number ?_k=0^￥ a^kdk \sum_{k=0}^\infty \alpha^{kd^k} .     

Generalized (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-ray systems of points and inequalities for nonoverlapping domains and open sets

We solve the extremal problem of finding the maximum of the functional

Funktionen, deren Maxima eine asymptotisch konstante Folge bilden

Es sei M₁ die Klasse der reellwertigen Funktionen f derart, daß lim_{tk? ￥} |f(t_k)| = c\lim \limits _{t_k\to \infty } |f(t_k)| = c gilt, wobei |f(t_k)||f(t_k)| die lokalen Maxima von | f | und c = c (f) eine positive Konstante ist. Diese Klasse tritt häufig im Zusammenhang mit Differentialgleichungen auf und ist kürzlich Gegenstand von mehreren Arbeiten geworden. Oft gelingt es, nur mf ? M₁mf \in M_1 mit einer passenden Funktion m ~ 1m\sim 1 zu zeigen, und es erhebt sich die Frage, ob daraus f ? M₁f \in M_1 folgt. Diese Frage wird hier für eine Unterklasse M ì M₁M \subset M_1 untersucht. Unsere Ergebnisse füllen eine Lücke in früheren Arbeiten und sind unserer Meinung nach auch für sich selbst von Interesse.     

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 standard Auslander-Reiten sequences for finite groups

Let G be a finite group and O{\cal O} a complete discrete valuation ring of characteristic zero with maximal ideal (p)(\pi ) and residue field k = O/(p)k = {\cal O}/(\pi ) of characteristic p > 0. Let S be a simple kG-module and Q_S a projective O G{\cal O} G-lattice such that Q_S / pQ_SQ_S / \pi Q_S is a projective cover of S. We show that if S is liftable and Q_S belongs to a block of O G{\cal O} G of infinite representation type, then the standard Auslander-Reiten sequence terminating in W^-1S\Omega ^{-1}S is a direct summand of the short exact sequence obtained from some Auslander-Reiten sequence of OG{\cal O}G-lattices by reducing each term mod (p)(\pi ).     

Polar factorization of maps on Riemannian manifolds

Let (M,g) be a connected compact manifold, C³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ? M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = exp_x[-?y(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ? M u : M \to M , where y: M ? \bold R \psi : M \to {\bold R} satisfies the additional property that (y^c)^c = y (\psi^c)^c = \psi with y^c(y) :=inf{c(x,y) - y(x) | x ? M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d²(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f 3 0 f \ge 0 of mass in L¹(M) onto another. Parallel results for other strictly convex cost functions c(x,y) 3 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.     

A wavelike functional equation of Pexider type

Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f₁, f₂, f₃, f₄, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f₁(x + t, y + s) + f₂(x - t, y - s) = f₃(x + s, y - t) + f₄(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f₁, f₂, f₃, and f₄ are given by f₁ = w + h, f₂ = w - h, f₃ = w + k, f₄ = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D_y,t³g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D_x,t³g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .     

Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, L ? Pic(X) L \in \textrm{Pic}(X) and H an ample divisor on X. Set M_X,H(2; L, c₂) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c₂(F) = c₂. In this paper, we show that the geometry of X and of M_X,H(2; L, c₂) are closely related. More precisely, we prove that for any ample divisor H on X and any L ? Pic(X) L \in \textrm{Pic}(X) , there exists n₀ ? \mathbbZ n_0 \in \mathbb{Z} such that for all n₀ \leqq c₂ ? \mathbbZ n_0 \leqq c_2 \in \mathbb{Z} , M_X,H(2; L, c₂) is rational if and only if X is rational.     

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.     

Inequalities of Hlawka type for matrices

A generalized Hlawka's inequality says that for any n (\geqq 2) (\geqq 2) complex numbers¶ x₁, x₂, ..., x_n,¶¶ ?_i=1ⁿ|x_i - ?_j=1ⁿx_j| \leqq ?_i=1ⁿ|x_i| + (n - 2)|?_j=1ⁿx_j|. \sum_{i=1}^n\Bigg|x_i - \sum_{j=1}^{n}x_j\Bigg| \leqq \sum_{i=1}^{n}|x_i| + (n - 2)\Bigg|\sum_{j=1}^{n}x_j\Bigg|. ¶¶ We generalize this inequality to the trace norm and the trace of an n x n matrix A as¶¶ ||A - Tr A ||₁ \leqq ||A||₁ + (n - 2)| Tr A|. ||A - {\rm Tr} A ||_1\ \leqq ||A||_1 + (n - 2)| {\rm Tr} A|. ¶¶ We consider also the related inequalities for p-norms (1 \leqq p \leqq ￥) (1 \leqq p \leqq \infty) on matrices.     

On conjugacy of some systems of functions

Summary. We investigate the bounded solutions j:[0,1]? X \varphi:[0,1]\to X of the system of functional equations¶¶j(f_k(x))=F_k(j(x)),    k=0,?,n-1,x ? [0,1] \varphi(f_k(x))=F_k(\varphi(x)),\;\;k=0,\ldots,n-1,x\in[0,1] ,(*)¶where X is a complete metric space, f₀,?,f_n-1:[0,1]?[0,1] f_0,\ldots,f_{n-1}:[0,1]\to[0,1] and F₀,...,F_n-1:X? X F_0,...,F_{n-1}:X\to X are continuous functions fulfilling the boundary conditions f₀(0) = 0, f_n-1(1) = 1, f_k+1(0) = f_k(1), F₀(a) = a,F_n-1(b) = b,F_k+1(a) = F_k(b), k = 0,?,n-2 f_{0}(0) = 0, f_{n-1}(1) = 1, f_{k+1}(0) = f_{k}(1), F_{0}(a) = a,F_{n-1}(b) = b,F_{k+1}(a) = F_{k}(b),\,k = 0,\ldots,n-2 , for some a,b ? X a,b\in X . We give assumptions on the functions f_k and F_k which imply the existence, uniqueness and continuity of bounded solutions of the system (*). In the case X = \Bbb C X= \Bbb C we consider some particular systems (*) of which the solutions determine some peculiar curves generating some fractals. If X is a closed interval we give a collection of conditions which imply respectively the existence of homeomorphic solutions, singular solutions and a.e. nondifferentiable solutions of (*).     

On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

Let M ⁿ be a Riemannian n-manifold. Denote by S(p) and [`(Ric)](p)\overline {Ric}(p) the Ricci tensor and the maximum Ricci curvature on M <sup>n</sup> at a point p ? M<sup>n</sup>p\in M^n, respectively. First we show that every isotropic submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies S ￡ ((n-1)c+ [(n<sup>2</sup>)/4] H<sup>2</sup>)gS\leq ((n-1)c+ {n^2 \over 4} H^2)g, where H<sup>2</sup> and g are the squared mean curvature function and the metric tensor on M <sup>n</sup>, respectively. The equality case of the above inequality holds identically if and only if either M <sup>n</sup> is totally geodesic submanifold or n = 2 and M <sup>n</sup> is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies [`(Ric)] = (n-1)c+ [(n²)/4] H²\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2 identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.     

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 \,.     

On groups acting freely on a tree

Suppose we are given a group G\mit\Gamma and a tree X on which G\mit\Gamma acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers a_d: = # {w ? vertX : w=gv, g ? G , d(v₀,w)=d }a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \} if d? ￥d\rightarrow \infty , where v, v_o are some fixed vertices in X.¶ In this paper we consider the case where G\mit\Gamma is a finitely generated group acting freely on a tree X. The growth function ?a_d x^d\textstyle\sum\limits a_d x^d is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where G\mit\Gamma is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.