Congruence modularity implies cyclic terms for finite algebras

An n-ary operation f : A ⁿ → A is called cyclic if it is idempotent and ${f(a_1, a_2, a_3, \ldots , a_n) = f(a_2, a_3, \ldots , a_n, a_1)}$ for every ${a_1, \ldots, a_n \in A}$ . We prove that every finite algebra A in a congruence modular variety has a p-ary cyclic term operation for any prime p greater than |A|.     

Weighted approximation of entire functions of exponential type of several variables by trigonometric polynomials

Letf be an entire function (in Cⁿ) of exponential type for whichf(x)=0(?(x)) on the real subspace \(\mathbb{R}^w (\phi \geqslant 1,{\mathbf{ }}\mathop {\lim }\limits_{\left| x \right| \to \infty } \phi (x) = \infty )\) and ?δ>0?C_δ>0 $$\left| {f(z)} \right| \leqslant C_\delta \exp \left\{ {h_s (y) + S\left| z \right|} \right\},z = x + iy$$ where h, (x)=sup〈3, x〉, S being a convex set in ?ⁿ. Then for any ?, ?>0, the functionf can be approximated with any degree of accuracy in the form p→ \(\mathop {\sup }\limits_{x \in \mathbb{R}^w } \frac{{\left| {P(x)} \right|}}{{\varphi (x)}}\) by linear combinations of functions x→expi〈λx〉 with frequenciesX belonging to an ?-neighborhood of the set S.     

On the distortion required for embedding finite metric spaces into normed spaces

We investigate the minimum dimensionk such that anyn-point metric spaceM can beD-embedded into somek-dimensional normed spaceX (possibly depending onM), that is, there exists a mappingf: M→X with $$\frac{1}{D}dist_M (x,y) \leqslant \left| {f(x) - f(y)} \right| \leqslant dist_M (x,y) for any$$ Extending a technique of Arias-de-Reyna and Rodríguez-Piazza, we prove that, for any fixedD≥1,k≥c(D)n ^1/2D for somec(D)>0. For aD-embedding of alln-point metric spaces into the samek-dimensional normed spaceX we find an upper boundk≤12Dn ^1/[(D+1)/2]lnn (using thel _∞ ^k space forX), and a lower bound showing that the exponent ofn cannot be decreased at least forD?[1,7)∪[9,11), thus the exponent is in fact a jumping function of the (continuously varied) parameterD.     

Zeros ofA p functions and related classes of analytic functions

The functionf(z), analytic in the unit disc, is inA ^p if \(\int {\int {_{\left| z \right|< 1} \left| {f(z)} \right|^p dxdy< \infty } } \) . A necessary condition on the moduli of the zeros ofA ^p functions is shown to be best possible. The functionf(z) belongs toB ^p if \(\int {\int {_{\left| z \right|< 1} \log ^ + \left| {f(z)} \right|)^p } } \) . Let {z _n } be the zero set of aB ^p function. A necessary condition on |z _n | is obtained, which, in particular, implies that Σ(1?|z _n |)^1+(1/p)+g <∞ for all ε>0 (p≧1). A condition on the Taylor coefficients off is obtained, which is sufficient for inclusion off inB ^p. This in turn shows that the necessary condition on |z _n | is essentially the best possible. Another consequence is that, forq≧1,p<q, there exists aB ^p zero set which is not aB ^q zero set.     

Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

Estimates related to sumfree subsets of sets of integers

A subsetA of the positive integers ?₊ is called sumfree provided (A+A)∩A=ø. It is shown that any finite subsetB of ?₊ contains a sumfree subsetA such that |A|≥1/3(|B|+2), which is a slight improvement of earlier results of P. Erdös [Erd] and N. Alon-D. Kleitman [A-K]. The method used is harmonic analysis, refining the original approach of Erdös. In general, defines _k (B) as the maximum size of ak-sumfree subsetA ofB, i.e. (A) _k = $\underbrace {A + ... + A}_{k times}$ % MathType!End!2!1! is disjoint fromA. Elaborating the techniques permits one to prove that, for instance, $s_3 \left( B \right) > \frac{{\left| B \right|}}{4} + c\frac{{\log \left| B \right|}}{{\log \log \left| B \right|}}$ % MathType!End!2!1!as an improvement of the estimate $s_k \left( B \right) > \frac{{\left| B \right|}}{4}$ % MathType!End!2!1! resulting from Erdös’ argument. It is also shown that in an inequalitys _k (B)>δ _k |B|, valid for any finite subsetB of ?₊, necessarilyδ _k → 0 fork → ∞ (which seemed to be an unclear issue). The most interesting part of the paper are the methods we believe and the resulting harmonic analysis questions. They may be worthwhile to pursue.     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.     

Metric diophantine approximation in Julia sets of expanding rational maps

Let T : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J andf : J →R denote a Hölder continuous function satisfyingf(x)?log | T′(x vb for allx in J. Then for any pointz ₀ in J define the set D_{z 0}(f) of “well-approximable” points to be the set of points in J which lie in the Euclidean ball $B(\gamma ,{\text{ exp(}} - \sum {_{i - 0}^{\mathfrak{n} - 1} } f(T^\ell x)))$ for infinitely many pairs (y, n) satisfying T ⁿ (y)=z₀. We prove that the Hausdorff dimension of D_{z 0}(f) is the unique positive numbers(f) satisfying the equation P(T,?s(f).f)=0, where P is the pressure on the Julia set. This result is then shown to have consequences for the limsups of ergodic averages of Hölder continuous functions. We also obtain local counting results which are analogous to the orbital counting results in the theory of Kleinian groups.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

On Hölder maps of cubes

A map of metric spaces f: X → Y satisfying the inequality $$ \left| {f(x) - f(y)} \right| \leqslant C\left| {x - y} \right|^\alpha $$ for some C and α and all x, y ∈ X is called a Hölder map with exponent α. V. I. Arnold posed the following problem: Does there exist a Höldermap from the square onto the cube with exponent 2/3? The firstmain theorem of this paper gives a general method for constructing Höldermaps of compact metric spaces. This construction yields, in particular, a dimension-raising map f: I ⁿ → I ^m with Hölder exponent arbitrarily close to m/n for m > n > 1 and a map I ¹ → I ^m with Hölder exponent 1/m. The second main theorem states the nonexistence of a regular fractal map f: I ⁿ → I ^m with Hölder exponent n/m from the n-cube onto the m-cube for m < 2n.     

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyf _n :n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F?X and every real number ε>0 one can choosen∈ω such that ∥f(x)?f_n(x)∥<ε for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A?X, there exists a continuous mapf _A:X→R ^ω such that A=f _A ^?1 (A). Splitting spaces will be studied systematically.     

Equicontinuity of homeomorphisms with unbounded characteristic

The article is devoted to the study of the boundary properties of homeomorphisms f: D → D′, D,D′ ? ? ⁿ , satisfying some geometric conditions responsible for the control of the measure of distortion of families of curves in D. Under additional requirements on the boundaries ?D and ?D′ of the domains, we prove that the family of all such homeomorphisms is equicontinuous in $\bar D$ .     

Complete hypersurfaces with constant scalar curvature in spheres

To a given immersion ${i:M^n\to \mathbb S^{n+1}}$ with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|². We prove the existence of a constant C _n (R) depending on R and n so that R ≥ 1 and sup |A|² = C _n (R) imply that the hypersurface is a H(r)-torus ${\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}$ . For R > (n ? 2)/n we use rotation hypersurfaces to show that for each value C > C _n (R) there is a complete hypersurface in ${\mathbb S^{n+1}}$ with constant scalar curvature R and sup |A|² = C, answering questions raised by Q. M. Cheng.     

Total curvature and L 2 harmonic 1-forms on complete submanifolds in space forms

Let M ⁿ be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., ${\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}$ , in an (n + p)-dimensional simply connected space form N ^n+p (c) of constant curvature c, where |H| and |A|² are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n ≥ 3, c = 0 and ${\int_M|H|^n < \infty}$ ; (ii) n ≥ 5, c = ?1 and ${|H| < 1-\frac{2}{\sqrt n}}$ ; (iii) n ≥ 3, c = 1 and |H| is bounded, then the dimension of the space of L ² harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.     

Shilov boundary and p-adic injective analytic functions

Let K be an algebraically closed field complete with respect to a dense ultrametric absolute value |.|. Let D be an infraconnected affinoid subset of K and let H(D) be the Banach algebra of analytic elements on D. Let f ∈ H(D) be injective in D and let f _* be the mapping defined on the multiplicative spectrum of H(D) that identifies with the set of circular filters on D. We show that f _* is injective and maps bijectively the Shilov boundary of H(D) onto this of H(f(D)). Thanks to this property we give a new proof of the equality $\left| {f(x) - f(y)} \right| = \left| {x - y} \right|\sqrt {\left| {f'(x)f'(y)} \right|} $ .     

Semi-linear Wave Equations with Effective Damping

The authors study the Cauchy problem for the semi-linear damped wave equation $$u_{tt} - \Delta u + b\left( t \right)u_t = f\left( u \right), u\left( {0,x} \right) = u_0 \left( x \right), u_t \left( {0,x} \right) = u_1 \left( x \right)$$ in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for |f(u)| ≈ |u| ^p in the supercritical case of $p > \tfrac{2} {n}$ and $p \leqslant \tfrac{n} {{n - 2}}$ for n ≥ 3 is proved.     

Algebraic-polynomial approximation of functions satisfying a Lipschitz condition

For functionsf(x) ε KH^(α) [satisfying the Lipschitz condition of order α (0 < α < 1) with constant K on [?1, 1], the existence is proved of a sequence P_n (f; x) of algebraic polynomials of degree n = 1, 2,..., such that $$|f(x) - P_{n - 1} (f;x)| \leqslant \mathop {\sup }\limits_{f \in KH^{(\alpha )} } E_n (f)[(1 - x^2 )^{a/2} + o(1)],$$ when n → ∞, uniformly for x ε [?1, 1], where E_n(f) is the best approximation off(x) by polynomials of degree not higher than n.     

Approximation of continuous functions by trigonometric polynomials almost everywhere

We consider the problem of the rate of approximation of continuous 2π-periodic functions of class W^rH[ω]C by trigonometric polynomials of order n on sets of total measure. We prove that when r≥0,ω(δ)δ ^?1 → ∞ (δ → 0) there exists a function f ε W^rH[ω]C such thatf ε W^rH[ω]C and for any sequence {t_{n n=1} ^∞ we have almost everywhere on [0, 2π] $\begin{array}{l} \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0, \\ \overline {\mathop {\lim }\limits_{n \to \infty } } \left| {\tilde f(x) - t_n (x)} \right|n^r \omega ^{ - 1} (1/n) > C_x > 0. \\ \end{array}$