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 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 .     

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.     

On the existence of certain measures appearing in distance geometry

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit _X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x₁,x₂,...,x_n,y₁,y₂,...,y_n in X, there exists an x in X such that¶¶ ?_i=1ⁿ f(x_i,x)=?_i=1ⁿ f(y_i,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).     

Another logarithmic functional equation

Summary. Let f : ]0,￥[? \Bbb R f :\,]0,\infty[\to \Bbb R be a real valued function on the set of positive reals. The functional equations¶¶f(x + y) - f(x) - f(y) = f(x^-1 + y^-1) f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1}) ¶and¶f(xy) = f(x) + f(y) f(xy) = f(x) + f(y) ¶are equivalent to each other.     

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.     

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.     

Functional equations on semigroups

Summary. We determine the general solution g:S? F g:S\to F of the d'Alembert equation¶¶g(x+y)+g(x+sy)=2g(x)g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)g(y)\qquad (x,y\in S) ,¶the general solution g:S? G g:S\to G of the Jensen equation¶¶g(x+y)+g(x+sy)=2g(x)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)\qquad (x,y\in S) ,¶and the general solution g:S? H g:S\to H of the quadratic equation¶¶g(x+y)+g(x+sy)=2g(x)+2g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)+2g(y)\qquad (x,y\in S) ,¶ where S is a commutative semigroup, F is a quadratically closed commutative field of characteristic different from 2, G is a 2-cancellative abelian group, H is an abelian group uniquely divisible by 2, and s \sigma is an endomorphism of S with s(s(x)) = x \sigma(\sigma(x)) = x .     

Regularity of functional equations on manifolds

Summary. In this paper the regularity properties of the functional equation¶¶ f (t) = h(t, y, f (g₁(t, y)), ... , f (g_n(t,y))) f (t) = h(t, y, f (g_{1}(t, y)), ... , f (g_{n}(t,y))) ¶ on a \Cal C^￥ {\Cal C}^\infty manifold for the unknown function f are treated. Under general conditions it is proved that solutions which are measurable or have the Baire property are in \Cal C^￥ {\Cal C}^\infty .     

Solvability of systems of partial differential equations for functions defined on nonconvex sets

We consider systems of partial differential equations with constant coefficients of the form ( R(D_x, D_y)f = 0, P(D_x)f = g), f,g ? C^￥(W),\big ( R(D_x, D_y)f = 0, P(D_x)f = {g}\big ), f,g \in {C}^{\infty}(\Omega),, where R (and P) are operators in (n + 1) variables (and in n variables, respectively), g satisfies the compatibility condition R(D_x, D_y)g = 0  and  W ì \Bbb Rⁿ⁺¹R(D_x, D_y){g} = 0 \ {\rm and} \ \Omega \subset {\Bbb R}^{n+1} is open. Let R be elliptic. We show that the solvability of such systems for certain nonconvex sets W\Omega implies that any localization at ￥\infty of the principle part P_m of P is hyperbolic. In contrast to this result such systems can always be solved on convex open sets W\Omega by the fundamental principle of Ehrenpreis-Palamodov.     

Bounded solutions of the Golab—Schinzel equation

Summary. Let \Bbb K {\Bbb K} be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \Bbb K {\Bbb K} n be a positive integer, and f : X ? \Bbb K f : X \to {\Bbb K} be a solution of the functional equation¶¶f(x + f(x)ⁿ y) = f(x) f(y) f(x + f(x)^n y) = f(x) f(y) .¶We prove that, if there is a real positive a such that the set { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} contains a subset of second category and with the Baire property, then f is continuous or { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} for every x ? X x \in X . As a consequence of this we obtain the following fact: Every Baire measurable solution f : X ? \Bbb K f : X \to {\Bbb K} of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set A ì X A \subset X with f(X \A) = { 0 }) f(X \backslash A) = \{ 0 \}) .     

Reducibility of polynomials a0(x) + a1 (x)y + a2 (x) y2 modulo p

Let f=a₀(x)+a₁(x)y+a₂(x)y² ? \Bbb Z[x,y]f=a_0(x)+a_1(x)y+a_2(x)y^2\in {\Bbb Z}[x,y] be an absolutely irreducible polynomial of degree m in x. We show that the reduction f mod p will also be absolutely irreducible if p 3 c_m·H(f)^e_mp\ge c_m\cdot H(f)^{e_m} where H (f) is the height of f and e₁ = 4,e₂ = 6, e₃ = 6 [2/3]{2}\over{3} and e_m = 2 m for m S 4. We also show that the exponents e_m are best possible for m 1 3m\ne 3 if a plausible number theoretic conjecture is true.     

Odd order groups with the rewriting property Q3

Let n be an integer greater than 1, and let G be a group. A subset {x₁, x₂, ..., x_n} of n elements of G is said to be rewritable if there are distinct permutations p \pi and s \sigma of {1, 2, ..., n} such that¶¶x_p(1)x_p(2) ?x_p(n) = x_s(1)x_s(2) ?x_s(n). x_{\pi(1)}x_{\pi(2)} \ldots x_{\pi(n)} = x_{\sigma(1)}x_{\sigma(2)} \ldots x_{\sigma(n)}. ¶¶A group is said to have the rewriting property Q_n if every subset of n elements of the group is rewritable. In this paper we prove that a finite group of odd order has the property Q₃ if and only if its derived subgroup has order not exceeding 5.     

Solutions of linear iterative functional equations behaving like the forcing term in the equation

Summary. We are dealing with those continuous solutions j \varphi of the functional equation¶¶j°f=g·j+h \varphi\circ f=g\cdot \varphi+h ¶that are asymptotically comparable at the origin (the fixed point of f) with the function h. Connections with a linear iterative functional inequality of second order are also mentioned.     

A characterization of infinite 3-abelian groups

In this note we prove that every infinite group G is 3-abelian (i.e. (ab)³ = a³b³ for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ? Xx\in X and y ? Yy\in Y such that (xy)³ = x³y³.     

A Quantitative Version of the Non-Abelian Idempotent Theorem

Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L ²(G) to L ²(G) by g ? f *g{g \mapsto f \ast g} where

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) .     

Solutions of functional equations having bounded variation

Summary. We prove that a solution f of the functional equation¶¶f(t)=h(t,y,f(g₁(t,y)),...,f(g_n(t,y))) f(t)=h(t,y,f(g_1(t,y)),\dots,f(g_n(t,y))) ¶ having locally bounded variation is a C^￥ {\cal C}^\infty -function.     

The character values of the irreducible constituents of a transitive permutation representation

In this article we determine the irreducible ordinary characters c_r \chi_r of a finite group G occurring in a transitive permutation representation (1_M )^G of a given subgroup M of G, and their multiplicities m_r = ((1_M)^G, c_r) 1 0 m_r = ((1_{M})^G, \chi_r) \neq 0 by means of a new explicit formula calculating the coefficients a_rk of the central idempotents e_r = ?_k=1^d a_rk D_k e_r = \sum\limits_{k=1}^{d} a_{rk} D_k in the intersection algebra B \cal B of (1_M )^G generated by the intersection matrices D_k corresponding to the double coset decomposition G = è_k=1^d Mx_k M G = \bigcup\limits_{k=1}^{d} Mx_{k} M .¶Furthermore, an explicit formula is given for the calculation of the character values c_r(x) \chi_{r}(x) of each element x ? G x \in G . Using this character formula we obtain a new practical algorithm for the calculation of a substantial part of the character table of G.     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.