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.     

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.     

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 \,.     

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.     

Decay estimates for "anisotropic" viscous Hamilton-Jacobi equations in $ \mathbb{R}^{N} $

The large time behaviour of the L^q L^q -norm of nonnegative solutions to the "anisotropic" viscous Hamilton-Jacobi equation¶¶ u_t - Du + ?_i=1^m |u_xi|^p_i = 0      in   \mathbbR₊×\mathbbR^N,u_t - \Delta u + \sum_{i=1}^m \vert u_{x_i}\vert^{p_i} = 0 \;\;\mbox{ in }\; {\mathbb{R}}_+\times{\mathbb{R}}^N,¶¶is studied for q=1 q=1 and q=￥ q=\infty , where m ? {1,?,N} m\in\{1,\ldots,N\} and p_i ? [1,+￥) p_i\in [1,+\infty) for i ? {1,?,m} i\in\{1,\ldots,m\} . The limit of the L¹ L^1 -norm is identified, and temporal decay estimates for the L^￥ L^\infty -norm are obtained, according to the values of the p_i p_i 's. The main tool in our approach is the derivation of L^￥ L^\infty -decay estimates for ?(u^a ), a ? (0,1] \nabla\left(u^\alpha \right), \alpha\in (0,1] , by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.     

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.     

Helly-type theorems for appropriate colorings of visibility sets

Summary. For positive integers q and n, think of P as the vertex set of a (qn + r)-gon, 0 ￡ r ￡ q - 1 0 \leq r \leq q - 1 . For 1 ￡ i ￡ qn + r 1 \leq i \leq qn + r , define V(i) to be a set of q consecutive points of P, starting at p(i), and let S be a subset of {V(i) : 1 ￡ i ￡ qn + r } \lbrace V(i) : 1 \leq i \leq qn + r \rbrace . A q-coloring of P = P(q) such that each member of S contains all q colors is called appropriate for S, and when 1 ￡ j ￡ q 1 \leq j \leq q , the definition may be extended to suitable subsets P(j) of P. If for every 1 ￡ j ￡ q 1 \leq j \leq q and every corresponding P(j), P(j) has a j-coloring appropriate for S, then we say P = P(q) has all colorings appropriate for S. With this terminology, the following Helly-type result is established: Set P = P(q) has all colorings appropriate for S if and only if for every (2n + 1)-member subset T of S, P has all colorings appropriate for T. The number 2n + 1 is best possible for every r 3 1 r \geq 1 . Intermediate results for q-colorings are obtained as well.     

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.     

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.     

Mass equidistribution of Hilbert modular eigenforms

Let \mathbbF\mathbb{F} be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on \operatornameGL₂/\mathbbF\operatorname{GL}_{2}/\mathbb{F} of weight (k₁,?,k_{[\mathbbF:\mathbbQ]})(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as max(k₁,?,k_{[\mathbbF:\mathbbQ]}) ? ￥\max(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}) \rightarrow \infty.     

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.     

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.     

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\,) .     

Metrizability of the space of {\mathbb{R}} -places of a real function field

For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x₁,?, x_n){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.     

Homogenization for strongly anisotropic nonlinear elliptic equations

In this paper we present homogenization results for elliptic degenerate differential equations describing strongly anisotropic media. More precisely, we study the limit as e? 0 \epsilon \to 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:¶¶ . \cases {{ -div(\alpha(\frac{x}{\epsilon}}, \nabla u) A(\frac{x}{\epsilon}) \nabla u) = f(x) \in L^{\infty}(\Omega) \atop u = 0 su \eth\Omega\ } ¶¶where, p > 1,     a: \Bbb Rⁿ ×\Bbb Rⁿ ? \Bbb R,     a(y,x) ? áA(y)x,x?^p/2-1, A ? M^{n ×n}(\Bbb R) p>1, \quad \alpha : \Bbb R^n \times \Bbb R^n \to \Bbb R, \quad \alpha(y,\xi) \approx \langle A(y)\xi,\xi \rangle ^{p/2-1}, A \in M^{n \times n}(\Bbb R) , A being a measurable periodic matrix such that A^t(x) = A(x) 3 0A^t(x) = A(x) \ge 0 almost everywhere.¶¶The anisotropy of the medium is described by the following structure hypothesis on the matrix A:¶¶l^2/p(x) |x|² ￡ áA(x)x,x? ￡ L ^2/p(x) |x|², \lambda^{2/p}(x) |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq \Lambda ^{2/p}(x) |\xi|^2, ¶¶where the weight functions l \lambda and L \Lambda (satisfying suitable summability assumptions) can vanish or blow up, and can also be "moderately" different. The convergence to the homogenized problem is obtained by a classical compensated compactness argument, that had to be extended to two-weight Sobolev spaces.     

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.     

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.     

On the selfsimilar solutions of a diffusion convection equation

This paper is concerned with the equation¶¶ div(| ?u| ^p-2?u)+e| ?U| ^q+bx?U+aU=0, for  x ? \mathbbR^N div(| \nabla u| ^{p-2}\nabla u)+\varepsilon \left| \nabla U\right| ^q+\beta x\nabla U+\alpha U=0,{\rm \ for}\;x\in \mathbb{R}^N ¶¶ where $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 and a,b, m \alpha ,\beta, \mu are positive parameters. We study the existence, uniqueness of radial solutions u(r). Also, qualitative behavior of u(r) are presented.     

Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

For x = (x ₁, x ₂, …, x _n ) ∈ (0, 1 ] ⁿ and r ∈ { 1, 2, … , n}, a symmetric function F _n (x, r) is defined by the relation