On the equation of generalized bisymmetry with outer functions injective in each argument

Summary. It is shown that provided F and G are injective in every argument, the functional equation of generalized m ×n m \times n bisymmetry (m,n 3 2) (m,n \ge 2) ,¶¶ G(F₁(x₁₁, \hdots , x_1n),\hdots , F_m(x_m1,\hdots, x_mn)) G(F_1(x_{11}, \hdots , x_{1n}),\hdots , F_m(x_{m1},\hdots, x_{mn})) ¶ = F(G₁(x₁₁,\hdots , x_m1),\hdots , G_n(x_1n,\hdots , x_mn)) = F(G_1(x_{11},\hdots , x_{m1}),\hdots , G_n(x_{1n},\hdots , x_{mn})) ¶may be reduced to ¶¶ G([`(F)]<sub>1</sub>(u<sub>11</sub>, \hdots , u<sub>1n</sub>),\hdots ,[`(F)]_m(u_m1,\hdots, u_mn)) G(\overline{F}_1(u_{11}, \hdots , u_{1n}),\hdots , \overline{F}_m(u_{m1},\hdots, u_{mn})) ¶ = F([`(G)]<sub>1</sub>(u<sub>11</sub>,\hdots , u<sub>m1</sub>),\hdots ,[`(G)]_n(u_1n,\hdots , u_mn)) = F(\overline{G}_1(u_{11},\hdots , u_{m1}),\hdots ,\overline{G}_n(u_{1n},\hdots , u_{mn})) ¶where¶¶ F_i(x_i1,\hdots , x_in) = [`(F)]<sub>i</sub> (j<sub>i1</sub>(x<sub>i1</sub>),\hdots , j<sub>in</sub>(x<sub>in</sub>)), G<sub>j</sub>(x<sub>1j</sub>, \hdots , x<sub>mj</sub>) = [`(G)]_j(j_1j (x_1j),\hdots, j_mj(x_mj)) F_i(x_{i1},\hdots , x_{in}) = \overline{F}_i (\varphi_{i1}(x_{i1}),\hdots , \varphi_{in}(x_{in})), G_j(x_{1j}, \hdots , x_{mj}) = \overline{G}_j(\varphi_{1j} (x_{1j}),\hdots, \varphi_{mj}(x_{mj})) ,¶¶j_ij < /FORMULA > are surjections and < FORMULA > \varphi_{ij} are surjections and \overline{F}_i, \overline{G}_j < /FORMULA > are injective in every argument for all < FORMULA > are injective in every argument for all 1\le i \le m,\ 1\le j\le n $. The result is also shown to hold for a wider class of functional equations.     

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.     

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.     

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.     

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.     

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 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).     

Smooth 2-homogeneous polynomials on Hilbert spaces

We determine the smooth points of the unit ball of the space of 2-homogeneous polynomials on a Hilbert space H. Working separately for the real and the complex cases we show that a smooth polynomial attains its norm. We deduce that the polynomial P is smooth if and only if there exists a unit vector x₀ in H such that P(x)=± á x,x₀ ñ ²+P₁(x₁)P(x)=\pm \left \langle x,x_{0}\right \rangle ^{2}+P_{1}(x_{1}) where x= á x,x₀ ñ x₀+x₁ x=\left \langle x,x_{0}\right \rangle x_{0}+x_{1} is the decomposition of x in H=span{ x₀} ?H₁H={\rm {span}}\{ x_{0}\} \oplus H_{1} and P₁ is a 2-homogeneous polynomial on H₁ of norm strictly less than 1.     

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 (*).     

Extremal property of some surfaces in n-dimensional Euclidean space

A surface Γ=(f ₁(X₁,..., x_m),...,f _n(x₁,..., x_m)) is said to be extremal if for almost all points of Γ the inequality $$\parallel a_1 f_1 (x_1 , \ldots ,x_m ) + \ldots + a_n f_n (x_1 , \ldots ,x_m )\parallel< H^{ - n - \varepsilon } ,$$ , where H=max(¦a _i¦) (i=1, 2, ..., n), has only a finite number of solutions in the integersa ₁, ...,a _n. In this note we prove, for a specific relationship between m and n and a functional condition on the functionsf ₁, ...,f _n, the extremality of a class of surfaces in n-dimensional Euclidean space.     

The Theorem of the k-1 Happy Divorces

Given a binary relation R between the elements of two sets X and Y and a natural number k, it is shown that there exist k injective maps f₁, f₂,...,f_k: X \hookrightarrow Y X \hookrightarrow Y with # {f₁(x), f₂(x),...,f_k(x)}=k    and    (x,f₁(x)), (x, f₂(x)),...,(x, f_k(x)) ? R \# \{f_1(x), f_2(x),...,f_k(x)\}=k \quad{\rm and}\quad (x,f_1(x)), (x, f_2(x)),...,(x, f_k(x)) \in R for all x ? X x \in X if and only if the inequality k ·# A ￡ ?_{y ? Y} min(k, #{a ? A | (a,y) ? R}) k \cdot \# A \leq \sum_{y \in Y} min(k, \#\{a \in A \mid (a,y) \in R\}) holds for every finite subset A of X, provided {y ? Y | (x,y) ? R} \{y \in Y \mid (x,y) \in R\} is finite for all x ? X x \in X .¶Clearly, as suggested by this paper's title, this implies that, in the context of the celebrated Marriage Theorem, the elements x in X can (simultaneously) marry, get divorced, and remarry again a partner from their favourite list as recorded by R, for altogether k times whenever (a) the list of favoured partners is finite for every x ? X x \in X and (b) the above inequalities all hold.¶In the course of the argument, a straightforward common generalization of Bernstein's Theorem and the Marriage Theorem will also be presented while applications regarding (i) bases in infinite dimensional vector spaces and (ii) incidence relations in finite geometry (inspired by Conway's double sum proof of the de Bruijn-Erdös Theorem) will conclude the paper.     

A functional equation arising in multi-agent statistical decision theory

Using recent results of Járai we show that the measurable solutions of the functional equationf(x ₁ y ₁,...,x _n y _n )f((1?x ₁)(1?y ₁),..., (1?x _n )(1?y _n ))=f(x ₁(1?y ₁),...,x _n (1 ? (y _n ))f(y ₁(1?x ₁),...,y _n(1 ?x _n )), wheref: (0, 1) ⁿ → (0, ∞) and 0<x _i ,y _i <1,i=1,...,n, are of the form $$f(x_1 ,...,x_n ) = c \exp \left( {\sum\limits_{i = 1}^n {a_i (x_1 - x_1^2 ))} \prod\limits_{i = 1}^n {x_i^{b_1 } ,} } \right.$$ wherec>0,a ₁,...,a _n andb ₁,..., _b are arbitrary real constants. This result enables one to characterize certain independence-preserving methods of aggregating probability distributions over four alternatives.     

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.     

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) .     

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 .     

Uncorrelatedness and correlatedness of powers of random variables

Let x₁, ?, x_n \xi_1, \ldots, \xi_n be random variables and U be a subset of the Cartesian product \mathbbZ₊ⁿ, \mathbbZ₊ \mathbb{Z}_+^n, \mathbb{Z}_+ being the set of all non-negative integers. The random variables are said to be strictly U-uncorrelated if¶¶E(x₁^j₁ ?x_n^j_n) = E(x₁^j₁) ?E(x_n^j_n) ? (j₁, ... ,j_n) ? U. \textbf {E}\big(\xi_1^{j_1} \cdots \xi_n^{j_n}\big) = \textbf {E}\big(\xi_1^{j_1}\big) \cdots \textbf {E}\big(\xi_n^{j_n}\big) \iff (j_1, \dots ,j_n) \in U. ¶It is proved that for an arbitrary subset U \subseteqq \mathbbZ₊ⁿ U \subseteqq \mathbb{Z}_+^n containing all points with 0 or 1 non-zero coordinates there exists a collection of n strictly U-uncorrelated random variables.     

Integration of the lifting formulas and the cyclic homology of the algebras of differential operators

We integrate the Lifting cocycles Y_2n+1, Y_2n+3, Y_2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Dif_n of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H^·_Lie(\frak g\frak l^fin_￥(Dif_n);\Bbb C) = ù^·(Y_2n+1, Y_2n+3, Y_2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .     

Friabilité des valeurs d’un polynôme

Let F(X) be an absolutely irreducible polynomial in \mathbbZ [X₁,..., X_n]{\mathbb{Z} [X_{1},\dots, X_{n}]}, with degree d. We prove that, for any δ < 4/3, for any sufficiently large x, there exists a positive density of integral n-tuples m = (m ₁, . . . , m _n ) in the hypercube max |m _i | ≤ x such that every prime divisor of F(m) is smaller than x ^d–δ . This result is improved when F satisfies some geometrical hypotheses.     

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 .     

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.