Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

The best polynomial approximations of entire transcendental functions with generalized growth order in Banach spaces \mathcal{E}_p^{\prime}(G) and {\mathcal{E}_p}(G) , p ? 1

A theorem of the Hadamard type for entire transcendental functions f, which have a generalized ??-order of growth ?? _?? (f), has been obtained. This theorem connects the values $ \widetilde{M}\left( {f,r} \right)\;\left( {r > 1} \right) $ and the coefficients a _n (f) $ \left( {n \in {\mathbb{Z}_{+} }} \right) $ of the expansion of f in Faber series in a finite domain D whose boundary ?? belongs to the Al??per class. This result is the extension of a result obtained by M. N. Sheremeta onto a simply connected domain. The necessary and sufficient conditions for an analytic function $ f \in \mathcal{E}_p^{\prime}(G) $ or $ f \in {\mathcal{E}_p}(G)\;\left( {1 \leqslant p \leqslant \infty } \right) $ to be entire transcendental with a generalized ??-order of growth ?? _?? (f) are obtained. These conditions include the best polynomial approximations of the function f and determine the rate of their convergence to zero, as the degree of polynomials increases.     

Topological transitivity for a class of monotonic mod one transformations

Suppose that f : [0, 1] ?? [0, 2] is a continuous strictly increasing piecewise differentiable function, and define T _f x :=?f(x) (mod 1). Let ${\beta \geq \sqrt[3]{2}}$ . It is proved that T _f is topologically transitive if inf f???????? and ${f(0)\geq\frac{1}{\beta+1}}$ . Counterexamples are provided if the assumptions are not satisfied. For ${\sqrt[3]{2}\leq\beta < \sqrt{2}}$ and 0????????? 2 ? ?? it is shown that ??x?+??? (mod 1) is topologically transitive if and only if ${\alpha < \frac{1}{\beta^2+\beta}}$ or ${\alpha >2 -\beta-\frac{1}{\beta^2+\beta}}$ .     

Triangular Ces??ro summability of two dimensional Fourier series

It is proved that the maximal operator of the triangular Ces??ro means of a two-dimensional Fourier series is bounded from the periodic Hardy space $H_{p}(\mathbb{T}^{2})$ to $L_{p}(\mathbb{T}^{2})$ for all 2/(2+??)<p?Q?? and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular Ces??ro means of a function $f \in L_{1}(\mathbb{T}^{2})$ converge a.e. to?f.     

Fixed points of automorphisms preserving the length of words in free solvable groups

Let ?? be an automorphism of prime order p of the free group F _n . Suppose ?? has no fixed points and preserves the length of words. By ?? :=??? ^(m) we denote the automorphism of the free solvable group ${F_{n}/F_n^{(m)} }$ induced by ??. We show that every fixed point of ?? has the form ${cc^{\sigma} \ldots c^{\sigma^{p-1}}}$ , where ${c\in F_n^{(m-1)}/F_n^{(m)}}$ . This is a generalization of some known results, including the Macedo??ska?CSolitar Theorem [10].     

On Utiyama’s Theme Through “{\mathcal {A}} -Invariance”

The term ?? ${\mathcal {A}}$ -invariance?? refers to the invariance of our results, with respect to the ??arithmetic?? employed, viz. to an appropriate algebra sheaf ${\mathcal {A}}$ . This, combined with the categorical notion of ??adjunction??, particularized here with the homological, in nature Hom-? adjunction, affords the classical perspective of Utiyama, pertaining to the characteristic type of field interactions. Yet, all this, without any ??space-time?? support, in the classical sense of the term (: smooth manifolds), but, just based on the ??functorial?? character of ADG (acronym of ??Abstract Differential Geometry??) and the aforementioned two fundamental principles (:?? ${\mathcal {A}}$ -invariance?? and ??adjunction??).     

Isothermic Submanifolds

We propose an answer to a question raised by F. Burstall: Is there any interesting theory of isothermic submanifolds of ? ⁿ of dimension greater than two? We call an n-immersion f(x) in ? ^m isothermic _k if the normal bundle of f is flat and x is a line of curvature coordinate system such that its induced metric is of the form $\sum_{i=1}^{n} g_{ii}\,\mathrm{d} x_{i}^{2}$ with $\sum_{i=1}^{n} \epsilon_{i} g_{ii}=0$ , where ?? _i =1 for 1??i??n?k and ?? _i =?1 for n?k<i??n. A smooth map (f ₁,??,f _n ) from an open subset ${\mathcal{O}}$ of ? ⁿ to the space of m×n matrices is called an n-tuple of isothermic _k n-submanifolds in ? ^m if each f _i is an isothermic _k immersion, $(f_{i})_{x_{j}}$ is parallel to $(f_{1})_{x_{j}}$ for all 1??i,j??n, and there exists an orthonormal frame (e ₁,??,e _n ) and a GL(n)-valued map (a _ij ) such that $\mathrm{d}f_{i}= \sum_{j=1}^{n} a_{ij} e_{j}\,\mathrm {d} x_{j}$ for 1??i??n. Isothermic₁ surfaces in ?³ are the classical isothermic surfaces in ?³. Isothermic _k submanifolds in ? ^m are invariant under conformal transformations. We show that the equation for n-tuples of isothermic _k n-submanifolds in ? ^m is the $\frac{O(m+n-k,k)}{O(m)\times O(n-k,k)}$ -system, which is an integrable system. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries.     

Strengthening Kazhdan’s property (T) by Bochner methods

In this paper, we propose a property which is a natural generalization of Kazhdan??s property (T) and prove that many, but not all, groups with property (T) also have this property. Let ?? be a finitely generated group. One definition of ?? having property (T) is that ${H^{1}(\Gamma, \pi, {\mathcal{H}}) = 0}$ where the coefficient module ${{\mathcal{H}}}$ is a Hilbert space and ?? is a unitary representation of ?? on ${{\mathcal{H}}}$ . Here we allow more general coefficients and say that ?? has property ${F \otimes {H}}$ if ${H^{1}(\Gamma, \pi_{1}{\otimes}\pi_{2}, F{\otimes} {\mathcal{H}}) = 0}$ if (F, ?? ₁) is any representation with dim(F) <??? and ${({\mathcal{H}}, \pi_{2})}$ is a unitary representation. The main result of this paper is that a uniform lattice in a semisimple Lie group has property ${F \otimes {H}}$ if and only if it has property (T). The proof hinges on an extension of a Bochner-type formula due to Matsushima?CMurakami and Raghunathan. We give a new and more transparent derivation of this formula as the difference of two classical Weitzenb?ck formula??s for two different structures on the same bundle. Our Bochner-type formula is also used in our work on harmonic maps into continuum products (Fisher and Hitchman in preparation; Fisher and Hitchman in Int Math Res Not 72405:1?C19, 2006). Some further applications of property ${F\otimes {H}}$ in the context of group actions will be given in Fisher and Hitchman (in preparation).     

The fractal “Frog”

In [1?C3] some analytical properties were investigated of the Von Koch curve ?? _?? , ?? ?? $(0,\tfrac{\pi } {4}) $ . In particular, it was shown that ?? _?? is quasiconformal and not AC-removable. The natural question arises: Can one find a quasiconformal and not AC-removable curve essentially different from ?? _?? in the sense that it is not diffeomorphic to ?? _?? ? The present paper is an answer to the question. Namely, we construct a quasiconformal curve, calling the ??Frog,?? which is not AC-removable and not diffeomorphic to ?? _?? for any ?? ?? $(0,\tfrac{\pi } {4}) $ .     

Lower semicontinuity for higher order integrals below the growth exponent

We study the lower semicontinuity of functionals of the form $$ \mathcal{F}(u)=\int\limits_{\Omega}f(x, u(x), \mathcal{L}u(x))\,dx $$ with respect to the weak convergence in W ^k,p (??), where ${{\mathcal L}}$ is a linear differential operator of order k??? 1 and f is quasiconvex with respect to the operator ${{\mathcal L}}$ and satisfies 0??? f(x, s, ??) ?? c (1?+ |??| ^q ) with q ?? p?>?1.     

A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

A partial orthomorphism of ${\mathbb{Z}_{n}}$ is an injective map ${\sigma : S \rightarrow \mathbb{Z}_{n}}$ such that ${S \subseteq \mathbb{Z}_{n}}$ and ??(i)?Ci ? ??(j)? j (mod n) for distinct ${i, j \in S}$ . We say ?? has deficit d if ${|S| = n - d}$ . Let ??(n, d) be the number of partial orthomorphisms of ${\mathbb{Z}_{n}}$ of deficit d. Let ??(n, d) be the number of partial orthomorphisms ?? of ${\mathbb{Z}_n}$ of deficit d such that ??(i) ? {0, i} for all ${i \in S}$ . Then ??(n, d) =???(n, d)n ²/d ² when ${1\,\leqslant\,d < n}$ . Let R _{k, n} be the number of reduced k ×?n Latin rectangles. We show that $$R_{k, n} \equiv \chi (p, n - p)\frac{(n - p)!(n - p - 1)!^{2}}{(n - k)!}R_{k-p,\,n-p}\,\,\,\,(\rm {mod}\,p)$$ when p is a prime and ${n\,\geqslant\,k\,\geqslant\,p + 1}$ . In particular, this enables us to calculate some previously unknown congruences for R _{n, n} . We also develop techniques for computing ??(n, d) exactly. We show that for each a there exists??? _a such that, on each congruence class modulo??? _a , ??(n, n-a) is determined by a polynomial of degree 2a in n. We give these polynomials for ${1\,\leqslant\,a\,\leqslant 6}$ , and find an asymptotic formula for ??(n, n-a) as n ?? ??, for arbitrary fixed a.     

Lyapunov exponents of Green��s functions for random potentials tending to zero

We consider quenched and annealed Lyapunov exponents for the Green??s function of ????+? ??V, where the potentials ${V(x),\ x\in\mathbb {Z}^d}$ , are i.i.d.? nonnegative random variables and ?? > 0 is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like ${c\sqrt{\gamma}}$ as ?? tends to 0. Here the constant c is the same for the quenched as for the annealed exponent and is computed explicitly. This improves results obtained previously by Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.     

Inscribing nonmeasurable sets

Our main inspiration is the work in paper (Gitik and Shelah in Isr J Math 124(1):221?C242, 2001). We will prove that for a partition ${\mathcal{A}}$ of the real line into meager sets and for any sequence ${\mathcal{A}_n}$ of subsets of ${\mathcal{A}}$ one can find a sequence ${\mathcal{B}_n}$ such that ${\mathcal{B}_{n}}$ ??s are pairwise disjoint and have the same ??outer measure with respect to the ideal of meager sets??. We get?also generalization of this result to a class of ??-ideals posessing Suslin property. However, in that case we use additional set-theoretical assumption about non-existing of quasi-measurable cardinal below continuum.     

On definability in some lattices of semigroup varieties

An identity of the form x ₁?x _n ??x _1?? x _2?? ?x _n?? where ?? is a non-trivial permutation on the set {1,??,n} is called a permutation identity. If u??v is a permutation identity, then ?(u??v) [respectively r(u??v)] is the maximal length of the common prefix [suffix] of the words u and v. A variety that satisfies a permutation identity is called permutative. If $\mathcal{V}$ is a permutative variety, then $\ell=\ell(\mathcal{V})$ [respectively $r=r(\mathcal{V})$ ] is the least ? [respectively r] such that $\mathcal{V}$ satisfies a permutation identity ?? with ?(??)=? [respectively r(??)=r]. A?variety that consists of nil-semigroups is called a nil-variety. If ?? is a set of identities, then $\operatorname {var}\varSigma$ denotes the variety of semigroups defined by ??. If $\mathcal{V}$ is a variety, then $L (\mathcal{V})$ denotes the lattice of all subvarieties of $\mathcal{V}$ . For ?,r??0 and n>1 let $\mathfrak{B}_{\ell,r,n}$ denote the set that consists of n! identities of the form $$t_1\cdots t_\ell x_1x_2 \cdots x_n z_{1}\cdots z_{r}\approx t_1\cdots t_\ell x_{1\pi}x_{2\pi} \cdots x_{n\pi}z_{1}\cdots z_{r}, $$ where ?? is a permutation on the set {1,??,n}. We prove that for each permutative nil-variety $\mathcal{V}$ and each $\ell\ge\ell(\mathcal{V})$ and $r\ge r(\mathcal{V})$ there exists n>1 such that $\mathcal{V}$ is definable by a first-order formula in $L(\operatorname{var}{\mathfrak{B}}_{l,r,n})$ if ???r or $\mathcal{V}$ is definable up to duality in $L(\operatorname{var}{\mathfrak{B}}_{\ell,r,n})$ if ?=r.     

Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities

In this paper, we provide the Euler?CMaclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f??C ^??(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms where $\widehat{P}(y),P_{s}(y)$ and $\widehat{Q}(y),Q_{s}(y)$ are polynomials in y. The ?? _s and ?? _s are distinct, complex in general, and different from ?1. They also satisfy The results we obtain in this work extend the results of a recent paper [A.?Sidi, Numer. Math. 98:371?C387, 2004], which pertain to the cases in which $\widehat{P}(y)\equiv0$ and $\widehat{Q}(y)\equiv0$ . They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x??a+ and x??b?. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let $D_{\omega}=\frac{d}{d\omega}$ , h=(b?a)/n, where n is a positive integer, and define $\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)$ . Then with $\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}$ and $\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}$ , one of these results reads where ??(z) is the Riemann Zeta function and ?? _i are Stieltjes constants defined via $\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]$ , i=0,1,???.     

Pointwise Convergence of the Calderón Reproducing Formula

In this paper, we study the pointwise convergence of the Calderón reproducing formula, which is also known as an inversion formula for wavelet transforms. We show that for every $f\in L_{w}^{p}(\mathbb {R}^{d})$ with an $\mathcal{A}_{p}$ weight w, 1??p<??, the integral is convergent at every Lebesgue point of f, and therefore almost everywhere. Moreover, we prove the convergence without any assumption on the smoothness of wavelet functions.     

An extension of Mercer’s theory to L p

Let X be a topological space, either locally compact or first countable, endowed with a strictly positive measure ?? and ${\mathcal{K}:L^2(X,\nu)\to L^2(X,\nu)}$ an integral operator generated by a Mercer like kernel K. In this paper we extend Mercer??s theory for K and ${\mathcal{K}}$ under the assumption that the function ${x\in X\to K(x,x)}$ belongs to some L ^p/2(X, ??), p??? 1. In particular, we obtain series representations for K and some powers of ${\mathcal{K}}$ , with convergence in the p-mean, and show that the range of certain powers of ${\mathcal{K}}$ contains continuous functions only. These results are used to estimate the approximation numbers of a modified version of ${\mathcal{K}}$ acting on L ^p (X, ??).     

Sturmian maximizing measures for the piecewise-linear cosine family

Let T be the angle-doubling map on the circle $\mathbb{T}$ , and consider the 1-parameter family of piecewise-linear cosine functions $f_\theta :\mathbb{T} \to \mathbb{R}$ , defined by $f_\theta (x) = 1 - 4d_\mathbb{T} (x,\theta )$ . We identify the maximizing T-invariant measures for this family: for each ?? the f _?? -maximizing measure is unique and Sturmian (i.e. with support contained in some closed semi-circle). For rational p/q, we give an explicit formula for the set of functions in the family whose maximizing measure is the Sturmian measure of rotation number p/q. This allows us to analyse the variation with ?? of the maximum ergodic average for f _?? .     

A Bijective Proof of a Factorization Formula for Specialized Macdonald Polynomials

Let??? and ?? = (?? ₁, . . . , ?? _k ) be partitions such that??? is obtained from ?? by adding m parts of size r. Descouens and Morita proved algebraically that the modified Macdonald polynomials ${{\tilde{H}_\mu}(X; q, t)}$ satisfy the identity ${{\tilde{H}_\mu} = \tilde{H}_\nu \tilde{H}_{(r^m)}}$ when the parameter t is specialize to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ?? ?? _k and ${r \in \{1, 2\}}$ . This note gives a bijective proof of the formula for all r ?? ?? _k .