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This paper is concerned with the distribution of normalized zero-sets of random meromorphic functions. The normalization of the zero-set plays the same role as the counting function for a meromorphic function in Nevanlinna theory. The results generalize the theory of Shiffman and Zelditch on the distribution of the zeroes of random holomorphic sections of powers of positive Hermitian holomorphic line bundles. As in a very special case, our paper resembles a form of First Main Theorem in classical Nevanlinna Theory.

This paper studies the structure of operators on Σ 1 type Banach spaces. It solves the problem of the small compact perturbations of operators with connected spectra. Namely, it shows that every operator with a connected spectrum on separable Σ 1 type Banach spaces is a small compact perturbation of a strongly irreducible operator. Based on this result, this paper establishes the approximate Jordan forms of operators on Σ 1 type Banach spaces with Schauder bases.

We in this paper give a decomposition concerning the general matrix triplet over an arbitrary division ring $\mathcal{F}$ with the same row or column numbers. We also design a practical algorithm for the decomposition of the matrix triplet. As applications, we present necessary and sufficient conditions for the existence of the general solutions to the system of matrix equations $DXA = C_1 , EXB = C_2 , FXC = C_3$ and the matrix equation $AXD + BYE + CZF = G$ over $\mathcal{F}$ . We give the expressions of the general solutions to the system and the matrix equation when the solvability conditions are satisfied. Moreover, we present numerical examples to illustrate the results of this paper. We also mention the other applications of the equivalence canonical form, for instance, for the compression of color images.

In this note, we show that the main result in the paper ‘On an extension of the Frobenius’ theorem about \(p\)-nilpotency of a finite group’ (Monatsh Math doi:10.1007/s00605-013-0492-3, 2013) by Kong can be seen as a corollary of a theorem in the paper ‘On finite groups with prime-power order \(S\)-quasinormally embedded subgroups’ (Monatsh Math 162:329–339, 2011) by Wei and Guo.

In this paper we will first introduce the notion of affine structures on a ringed space and then obtain several properties. Affine structures on a ringed space, arising mainly from complex analytical spaces of algebraic schemes over number fields, behave like differential structures on a smooth manifold. As one does for differential manifolds, we will use pseudogroups of affine transformations to define affine atlases on a ringed space. An atlas on a space is said to be an affine structure if it is maximal. An affine structure is admissible if there is a sheaf on the underlying space such that they are coincide on all affine charts, which are in deed affine open sets of a scheme. In a rigour manner, a scheme is defined to be a ringed space with a specified affine structure if the affine structures are in action in some special cases such as analytical spaces of algebraic schemes. Particularly, by the whole of affine structures on a space, we will obtain respectively necessary and sufficient conditions that two spaces are homeomorphic and that two schemes are isomorphic, which are the two main theorems of the paper. It follows that the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively., Final version. 22 pages. to appear in Chinese Ann of Math, Series B

The theory of average case complexity studies the expected complexity of computational tasks under various specific distributions on the instances, rather than their worst case complexity. Thus, this theory deals with distributional problems, defined as pairs each consisting of a decision problem and a probability distribution over the instances. While for applications utilizing hardness, such as cryptography, one seeks an efficient algorithm that outputs random instances of some problem that are hard for any algorithm with high probability, the resulting hard distributions in these cases are typically highly artificial, and do not establish the hardness of the problem under “interesting” or “natural” distributions. This paper studies the possibility of proving generic hardness results (i.e., for a wide class of $${\mathcal{NP}}$$-complete problems), under “natural” distributions. Since it is not clear how to define a class of “natural” distributions for general $${\mathcal{NP}}$$-complete problems, one possibility is to impose some strong computational constraint on the distributions, with the intention of this constraint being to force the distributions to “look natural”. Levin, in his seminal paper on average case complexity from 1984, defined such a class of distributions, which he called P-computable distributions. He then showed that the $${\mathcal{NP}}$$-complete Tiling problem, under some P-computable distribution, is hard for the complexity class of distributional $${\mathcal{NP}}$$problems (i.e. $${\mathcal{NP}}$$with P-computable distributions). However, since then very few $${\mathcal{NP}}$$- complete problems (coupled with P-computable distributions), and in particular “natural” problems, were shown to be hard in this sense. In this paper we show that all natural $${\mathcal{NP}}$$-complete problems can be coupled with P-computable distributions such that the resulting distributional problem is hard for distributional $${\mathcal{NP}}$$.

The aim of this paper is to study the correlation between the properties of rings and the structure of their additive groups. In this paper, all multiplications on reduced algebraically compact groups and on divisible torsion-free Abelian groups are described. Absolute Jacobson radicals and absolute nil-radicals in some classes of Abelian groups are found. Using these results, we will give a description of semisimple groups and also of groups on which every ring is a nilpotent ring (a nil-ring, a radical ring) in some concerned classes of Abelian groups.

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(ℝn) is a Banach space as well as an algebra, while B(ℝn, ℝm) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σr be the set of all operators of finite rank r in B(E, F) (or B(ℝn, ℝm)). In fact, we have that 1) suppose Σr ∈ B(ℝn, ℝm), and then Σr is a smooth and path-connected submanifold of B(ℝn, ℝm) and dimΣr = (n + m)r − r2, for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σr and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σr ∈ B(E, F), and dimF = ∞, and then Σr is a smooth and path-connected submanifold of B(E, F) with the tangent space TAΣr = {B ∈ B(E, F): BN(A) ⊂ R(A)} at each A ∈ Σr for 0 ⩽ r ⩽ ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(ℝn), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.

Define two operators In and It, the inner product operator \( I_n \left( g \right)\left( x \right): = \sum\nolimits_{j \in \mathbb{Z}^s } {\left( {g,f\left( { \cdot - j} \right)} \right)f\left( {x - j} \right)} \) and the interpolation operator \( I_t \left( g \right)\left( x \right): = \sum\nolimits_{j \in \mathbb{Z}^s } {g\left( j \right)f\left( {x - j} \right)} \), where f belongs to some space and integer s ⩾ 1. We call f the generator of the operators In and It. It is well known that there are many results on operators In and It. But there remain some important problems to be further explored. For application we first need to find the available generators (that can recover polynomials as It(p) = p or In(p) = p, p ɛ Φm−1) for constructing the relative operators. In this paper, we focus on the available generator in the class of spline functions. We shall see that not all spline functions can be used to construct available generators. Fortunately, we do find a spline function in S, of degree m − 1, where m is even and S is a class of splines. But for odd m the problem is still open. Results on spline functions in this paper are new.

This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (Hs(ℝ), H−s(ℝ)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ Hs(ℝ) with s > 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L2(ℝ), whose Sobolev exponents are greater than 1/2.

This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value mk(n) equal to the minimum number of edges in an n-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains k vertices of each color. In this paper, we obtain the exact values of m2(5) and m2(4), and the upper bounds for m3(7) and m4(9).

[EN] In this paper, we study the Windy Postman Problem (WPP). This is a well-known Arc Routing Problem that contains the Mixed Chinese Postman Problem (MCPP) as a special case. We extend to arbitrary dimension some new inequalities that complete the description of the polyhedron associated with the Windy Postman Problem over graphs with up to four vertices and ten edges. We introduce two new families of facet-inducing inequalities and prove that these inequalities, along with the already known odd zigzag inequalities, are Chvátal-Gomory inequalities of rank at most 2. Moreover, a branch-and-cut algorithm that incorporates two new separation algorithms for all the previously mentioned inequalities and a new heuristic procedure to obtain upper bounds are presented. Finally, the performance of a branch-and-cut algorithm over several sets of large WPP and MCPP instances, with up to 3,000 nodes and 9,000 edges (and arcs in the MCPP case), shows that, to our knowledge, this is the best algorithm to date for the exact resolution of the WPP and the MCPP. © 2010 Springer and Mathematical Optimization Society., The authors want to thank the three referees for their careful reading of the manuscript and for their many comments and suggestions that have contributed to improve the paper content and readability. In particular, several remarks regarding the discussion of C-G and mod-k inequalities were pointed out by one of the referees. A. Corberan, I. Plana and J.M. Sanchis wish to thank the Ministerio de Educacion y Ciencia of Spain (projects MTM2006-14961-C05-02 and MTM2009-14039-C06-02) for its support.

We give a new approach to the infinitesimal structure of Lipschitz maps into L^1. As a first application, we give an alternative proof of the main theorem from an earlier paper, that the Heisenberg group does not admit a bi-Lipschitz embedding in L^1. The proof uses the metric differentiation theorem of Pauls and the cut metric decomposition to reduce the nonembedding argument to a classification of monotone subsets of the Heisenberg group., Comment: Added a missing condition to Definition 5.3. Also made a number of minor corrections, and added a reference to a paper by Lee-Raghavendra

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in \({\mathbb{Z}^n}\). The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point.

We present a new generalization of the classical trisecant lemma. Our approach is quite different from previous generalizations. Let $X$ be an equidimensional projective variety of dimension $d$. For a given $k \leq d + 1$, we are interested in the study of the variety of $k$-secants. The classical trisecant lemma just considers the case where $k = 3$ while elsewhere the case $k = d + 2$ is considered. Secants of order from $4$ to $d + 1$ provide service for our main result. In this paper, we prove that if the variety of $k$-secants ($k \leq d + 1$) satisfies the three following conditions: (i) trough every point in $X$, passes at least one $k$-secant, (ii) the variety of $k$-secant satisfies a strong connectivity property that we defined in the sequel, (iii) every $k$-secant is also a ($k+1$)-secant, then the variety $X$ can be embedded into $P^{d+1}$. The new assumption, introduced here, that we called strong connectivity is essential because a naive generalization that does not incorporate this assumption fails as we show in some example. The paper concludes with some conjectures concerning the essence of the strong connectivity assumption., Comment: arXiv admin note: text overlap with arXiv:0712.3878

A paper by Chow [3] contains (i.a.) a strong law for delayed sums, such that the length of the edge of the nth window equals n α for 0 < α < 1. In this paper we consider the kind of intermediate case when edges grow like n=L(n), where L is slowly varying at infinity, thus at a higher rate than any power less than one, but not quite at a linear rate. The typical example one should have in mind is L(n) = log n. The main focus of the present paper is on random field versions of such strong laws.

In a series of papers Mauduit and Sarkozy introduced measures of pseudorandomness and they constructed large families of sequences with strong pseudorandom properties. In later papers the structure of families of binary sequences was also studied. In these constructions fields with prime order were used. Throughout this paper the structure of a family of binary sequences based on GF(2 k ) will be studied.

For every hyperbolic toral automorphism T, the present author has defined in his previous paper some unbounded T-invariant second-order difference operators related to the so-called homoclinic group of T. These operators were considered in the space L2 with respect to the Haar measure. It is shown in the present paper that such operators give rise to transition semigroups in the space of continuous functions on the torus and generate dynamically invariant Markov processes. This leads almost immediately to a family of invariant measures for the automorphism T.Along with a short discussion, some open questions about properties of these measures are posed. Bibliography: 9 titles.

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic equation with integer coefficients; say, α = % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWaaOaaaeaacqaIYaGmaSqabaaaaa!3CE4! $$ \sqrt 2 $$ . Given any rational number 0 < x < 1 (say, x = 1/2) and any positive integer n, we count the number of elements of the sequence α, 2α, 3α, …, nα modulo 1 that fall into the subinterval [0, x]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number” nx from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ n ≤ N. Depending on α and x, we may need an extra additive correction of constant times logarithm of N; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of N. If N is large, the distribution of this renormalized counting number, as n runs in 1 ≤ n ≤ N, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as N tends to infinity. This is the main result of the paper (see Theorem 1.1). The proof is rather complicated and long; it has many interesting detours and byproducts. For example, the exact determination of the key constant factors (in the additive and multiplicative norming), which depend on α and x, requires surprisingly deep algebraic tools such as Dedeking sums, the class number of quadratic fields, and generalized class number formulas. The crucial property of a quadratic irrational is the periodicity of its continued fraction. Periodicity means self-similarity, which leads us to Markov chains: our basic probabilistic tool to prove the central limit theorem. We also use a lot of Fourier analysis. Finally, I just mention one byproduct of this research: we solve an old problem of Hardy and Littlewood on diophantine sums. The whole paper consists of an introduction and 17 sections. Part 1 contains the Introduction and Sections 1–7.

The paper deals with various centering problems for probability measures on finite-dimensional vector spaces. We show that for every such measure, there exists a vector h satisfying μδ(h)=S(μδ(h)) for each symmetry S of μ, generalizing thus Jurek’s result obtained for full measures. An explicit form of the h is given for infinitely divisible μ. The main result of the paper consists in the analysis of quasi-decomposable (operator-semistable and operator-stable) measures and finding conditions for the existence of a “universal centering” of such a measure to a strictly quasi-decomposable one.

By introducing state payoff vector to every state node on the connected graph in this paper, dynamic game is researched on finite graphs. The concept of simple strategy about games on graph defined by Berge is introduced to prove the existence theorem of absolute equilibrium about games on the connected graph with state payoff vector. The complete algorithm and an example in the three-dimensional connected mesh-like graph are given in this paper.

In this paper, we continue to discuss the properties of iterates generated by a strict pseudocontraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336–349 (2007)]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51–60 (2005)] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.

The purpose of this paper is to study the weak convergence problems of the implicity iteration process for Lipschitzian pseudocontraction semigroups in general Banach spaces. The results presented in this paper extend and improve the corresponding results of Zhou [Nonlinear Anal., 68, 2977–2983 (2008)], Chen, et al. [J. Math. Anal. Appl., 314, 701–709 (2006)], Xu and Ori [Numer. Funct. Anal. Optim, 22, 767–773 (2001)] and Osilike [J. Math. Anal. Appl., 294, 73–81 (2004)].

In this paper, we consider structured automata without output signals whose state sets are endowed with an algebraic structure of hypergraphs. The main result of the paper is a theorem where we obtain necessary and sufficient conditions for the possibility of defining on the state set of some automaton A a structure of a hypergraph H such that the automaton A will be the universal hypergraphic automaton.

In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables., 20 pages. This paper generalizes a previous version where the result was proven for a power series ring in two variables

In earlier papers Mauduit and Sarkozy have introduced and studied the measures of pseudorandomness for finite binary sequences and sequences of k symbols. Later they (with further coauthors) extended the notation of binary sequences to binary lattices. In this paper measures of pseudorandom lattices of k symbols are introduced and studied for “truly random” lattices.

Doust and Weston introduced a new method called "enhanced negative type" for calculating a non trivial lower bound p(T) on the supremal strict p-negative type of any given finite metric tree (T,d). In the context of finite metric trees any such lower bound p(T) > 1 is deemed to be non trivial. In this paper we refine the technique of enhanced negative type and show how it may be applied more generally to any finite metric space (X,d) that is known to have strict p-negative type for some non negative p. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given by Doust and Weston and, moreover, leads in to one of our main results: The supremal p-negative type of a finite metric space cannot be strict. By way of application we are then able to exhibit large classes of finite metric spaces (such as finite isometric subspaces of Hadamard manifolds) that must have strict p-negative type for some p > 1. We also show that if a metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q in [0,p). This generalizes a well known theorem of Schoenberg and leads to a complete classification of the intervals on which a metric space may have strict p-negative type. (Several of the results in this paper hold more generally for semi-metric spaces.), 15 pages - this updated version of the said paper contains additional new theory. The paper is accepted for publication in the journal POSITIVITY

Let $$f : {\mathbb{R}} \longrightarrow {\mathbb{R}}$$ denote the function given by $$f(t) = \sum \limits^{\infty}_{n=0} a_{n}t^{n}, t \in {\mathbb{R}}$$ , where $$a_n \geq 0, n \in {\mathbb{N}}$$ . Let K be a closed convex cone in a real Banach space and $$H : K \longrightarrow cc(K)$$ a linear and continuous set-valued function with nonempty, convex and compact values in K. The set-valued function $$F^t(x) := \sum \limits_{n=0}^{\infty} a_{n}t^{n}H^{n}(x), x \in K$$ is linear and continuous for all $$t \geq 0$$ and $$F^{t} \circ F^{s}(x) \subseteq \sum \limits_{n=0}^{\infty} c_{n}H^{n} (x), x \in K$$ where $$c_{n} = \sum \limits_{k=0}^{n} a_{k} a_{n-k}{{t^{k}}s^{n-k}}, t, s \geq 0$$ . The paper generalizes papers of J. Plewnia and M. Piszczek where f is an exponential and a cosine function respectively.

In this paper, we prove the generalized Hyers-Ulam stability of homomorphisms in quasi-Banach algebras associated with the following Pexiderized Jensen functional equation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGaemOzay2aaeWaaeaadaWcaaqaaiabdIha4fXafv3y % SLgzGmvETj2BSbacgaGae83kaSIaemyEaKhabaGae8Nmaidaaiab-T % caRiabdQha6bGaayjkaiaawMcaaiab-jHiTiabdEgaNnaabmaabaWa % aSaaaeaacqWG4baEcqWFsislcqWG5bqEaeaacqWFYaGmaaGae83kaS % IaemOEaOhacaGLOaGaayzkaaGae8xpa0JaemiAaGMae8hkaGIaemyE % aKNae8xkaKIae8Nla4caaa!5ABC! $$ f\left( {\frac{{x + y}} {2} + z} \right) - g\left( {\frac{{x - y}} {2} + z} \right) = h(y). $$ This is applied to investigating homomorphisms between quasi-Banach algebras. The concept of the generalized Hyers-Ulam stability originated from Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297–300 (1978).

In this paper we find the structure for the automorphism group of a split metacyclic 2-group G. It can be seen as a continuation of the paper (Curran in Arch. Math. 89 (2007), 10–23) and it makes it complete. We propose a different approach to the problem than in the paper (Curran in Arch. Math. 89 (2007), 10–23). Our intention is to show that apart from some cases of 2-groups AutG has a structure similar to that of a direct product of two groups with no common direct factor [which was considered in Bidwell, Curran, and McCaughan (Arch. Math. 86 (2006), 481–489)].

In this short note we respond to some concerns raised by Y. Censor, G. Herman, and M. Jiang about the randomized Kaczmarz method that we proposed in [5]. The Kaczmarz method is a well-known iterative algorithm for solving a linear system of equations Ax = b. For more than seven decades, this method was useful in practical applications, and it was studied in many research papers. Despite of this, little is known about the rate of convergence of this method. The classical scheme of Kaczmarz’s method sweeps through the rows of A in a cyclic manner, projecting in each substep the last iterate orthogonally onto a hyperplane associated with a row of A. One variation of Kaczmarz’s method consists of randomly choosing in each iteration the row for the projection. Our algorithm in [5] (labeled Algorithm 1 there) is based on this approach. The idea of choosing the rows randomly is certainly not new. It has been mentioned for instance by Natterer [4], and later also by Feichtinger et al. [1] and by G. Herman and L. Meyer [3]. In these papers, improvement of performance is observed in numerical experiments, but none of these papers contain any proof of the rate of convergence. We consider the main contribution of [5] that (i) it contains the first proof for a rate of convergence for the Kaczmarz’s method that is applicable to general matrices (and not just to very restricted special cases); (ii) the algorithm achieves an exponential rate of convergence; and (iii) the rate

Let T be an operator on a Hilbert space \( \mathcal{H} \). The problem of computing of the norm of T, norm of selfcommutators of T, and the numerical radius of T are discussed in many papers and a number of textbooks. In this paper we determine the relationships between these values for self inverse operators and explain how we can determine any three of these (‖T‖, ‖[T*,T]‖, ‖{T*,T}‖, and the numerical radius of T) by knowing any one of them. Also, we find the spectrum of T*T,[T*,T] and {T*,T} in the case that T is self inverse and the spectrum of T*T is an interval. Finally, by giving some examples on automorphic composition operators, we show that these results make it possible to replace lengthy computation with quick ones.

A composite dilation wavelet is a collection of functions generating an orthonormal basis for L2(ℝn) under the actions of translations from a full rank lattice and dilations by products of elements of non-commuting groups A and B. A minimally supported frequency composite dilation wavelet has generating functions whose Fourier transforms are characteristic functions of a lattice tiling set. In this paper, we study the case where A is the group of integer powers of some expanding matrix while B is a finite subgroup of the invertible n×n matrices. This paper establishes that with any finite group B together with almost any full rank lattice, one can generate a minimally supported frequency composite dilation wavelet system. The paper proceeds by demonstrating the ability to find such minimally supported frequency composite dilation wavelets with a single generator.

In this paper we pursue and deep the study of ring extensions \({R \subset S}\) such that R is a maximal non-valuation subring of S [Ben Nasr and Jarboui in Houston J Math, 2009 (in press)]. It is proved in Ben Nasr and Jarboui [Houston J Math, 2009 (in press), Theorem 3.2] that if R is integrally closed with finite Krull dimension, then R is a maximal non-valuation subring of qf (R) iff R is not local and |[R, qf (R)]| = dim(R) + 3. This result encourages us to pose the following question: Let n be a nonzero positive integer greater than 2 and let R be a finite-dimensional domain such that |[R, qf (R)]| = dim(R) + n, does there exists an overring S of R such that R is a maximal non-valuation subring of S? This paper deals mostly with this question. We solve this question in case R is integrally closed.

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0,1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q Σ)), i.e., there is a homeomorphism h:↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q Σ), where Q = [−1,1]ω, Σ = {(xn)n∈ℕ ∈ Q: sup|xn| < 1} and c0 = {(xn)n∈ℕ ∈ Σ: limn→+∞xn = 0}. Combining this statement with a result in our previous paper, we have $$ ( \downarrow USCC(X), \downarrow CC(X)) \approx \left\{ \begin{gathered} (Q,c_0 \cup (Q\backslash \Sigma )), if the set of isolanted points is dense in X, \hfill \\ (Q,c_0 ),otherwise, \hfill \\ \end{gathered} \right. $$ if X is an infinite compact metric space. We also prove that, for a metric space X, (↓USCC(X), ↓CC(X)) ≈ (Σ, c0) if and only if X is non-compact, locally compact, non-discrete and separable.

The aim of the present (mostly expository) paper is to show the relationship of a generalization of Kazhdan’s property (T) for C*-algebras introduced in our recent paper to that of B. Bekka. It is shown that our definition coincides with Bekka’s definition for group C*-algebras of locally compact groups, whereas, in general, these definitions are distinct. Criteria for a C*-algebra to possess our property (T) are given. A number of examples of C*-algebras with and without property (T) are considered. Relations to K-theory are studied.

In this paper we establish the general solution of the functional equation $$f(x + 2y) + f(x - 2y) + 4f(x) = 3[f(x + y) + f(x - y)] + f(2y) - 2f(y)$$ and investigate the Hyers–Ulam–Rassias stability of this equation in quasi-Banach spaces. The concept of Hyers–Ulam–Rassias stability originated from Th. M. Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.

In the recent paper [1] the claim is made that a probabilistic version of a common fixed point theorem of Pant holds. We provide some examples to demonstrate that this claim is false unless some additional conditions are imposed. Our note is desired to complete the interesting results in the quoted paper.

This paper is a follow-up to a previous paper where the logical characterization of n-ary synchronous relations due to Eilenbeig, Elgot, and Shepherdson was investigated in the case where the alphabet has infinitely many letters. Here we show that modifying one of the predicates leads to a completely different picture for infinite alphabets, though it does not change the expressive power for finite alphabets. Indeed, roughly speaking, being able to express the fact that two words end with the same symbol leads to an undecidable theory, already for the Σ2 fragment. Finally, we show that the existential fragment is decidable. Bibliography: 19 titles.

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If {a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly 2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in I}c_s=0. Our main theorem in this paper unifies many results in the two realms and also implies an extension of the Alon-Friedland-Kalai result on regular subgraphs.

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Abdollahi et al. (J Algebra 298(2):468–492, 2006) put forward a conjecture called AAM’s Conjecture in as follows: If M is a finite nonabelian simple group and G is a group such that ∇(G) ≅ ∇(M), then G ≅ M. Even though this conjecture is well known to hold for all simple groups with nonconnected prime graphs and the alternating group A10 [see Darafsheh (Groups with the same non-commuting graph. Discrete Appl Math (2008) doi:10.1016/j.dam.2008.06.010), Wang and Shi (Commun Algebra 36(2):523–528, 2008)], it is still unknown for all simple groups with connected prime graphs except A10. In the present paper, we prove that this conjecture is also true for the projective special linear simple group L4(9). The new method used in this paper also works well in the cases L4(4), L4(7), U4(7), etc.

Let A be a bounded linear operator defined on a separable Banach space X. Then A is said to be supercyclic if there exists a vector x ∈ X (later called supercyclic for A), such that the projective orbit \(\{\lambda A^{n} x\,:\,n \in {\mathbb{N}},\,\lambda \in {\mathbb{C}}\}\) is dense in X. On the other hand, A is said to be positive supercyclic if for each supercyclic vector x, the positive projective orbit, \(\{rA^nx\,:\, r \in {\mathbb{R}}_{+},\,n \in {\mathbb{N}}\}\) is dense in X. Sometimes supercyclicity and positive supercyclicity are equivalent. The study of this relationship was initiated in [14] by F. Leon and V. Muller. In this paper we study positive supercyclicity for operators A of the form \(A=T \oplus \alpha 1_{{\mathbb{C}}}\), with \(\alpha \in {\mathbb{C}}{\setminus}\{0\}\), defined on \(X \oplus {\mathbb{C}}\). We will see that such a problem is related with the study of regular orbits. The notion of positive directions will be central throughout the paper.

The paper is devoted to Koosis' approach to the Carleson theorem. We show that this approach also works for other related questions. The main emphasis in this paper is on the method. Bibliography: 10 titles.

Regular polynomials with quaternionic coefficients admit only isolated zeroes and spherical zeroes. In this paper we prove a factorization theorem for such polynomials. Specifically, we show that every regular polynomial can be written as a product of degree one binomials and special second degree polynomials with real coefficients. The degree one binomials are determined (but not uniquely) by the knowledge of the isolated zeroes of the original polynomial, while the second degree factors are uniquely determined by the spherical zeroes. We also show that the number of zeroes of a polynomial, counted with their multiplicity as defined in this paper, equals the degree of the polynomial. While some of these results are known in the general setting of an arbitrary division ring, our proofs are based on the theory of regular functions of a quaternionic variable, and as such they are elementary in nature and offer explicit constructions in the quaternionic setting.

In their seminal paper, Valiant, Skyum, Berkowitz and Rackoff proved that arithmetic circuits can be balanced. That is, they showed that for every arithmetic circuit Ф of size s and degree r, there exists an arithmetic circuit Ψ of size poly (r, s) and depth O (log(r) log(s)) computing the same polynomial. In the first part of this paper, we follow the proof of Valiant el al. and show that syntactically multilinear arithmetic circuits can be balanced. That is, we show that if Ф is syntactically multilinear, then so is Ψ. Recently, a super-polynomial separation between multilinear arithmetic formula and circuit size was shown. In the second part of this paper, we use the result of the first part to simplify the proof of this separation. That is, we construct a (simpler) polynomial f (x 1, ... , x n ) such that Every multilinear arithmetic formula computing f is of size n Ω(log(n)). There exists a syntactically multilinear arithmetic circuit of size poly(n) and depth O(log2(n)) computing f.

A partial order on the set of the prime knots can be defined by the existence of a surjective homomorphism between knot groups. In the previous paper, we determined the partial order in the knot table. In this paper, we prove that 31 and 41 are minimal elements. Further, we study which surjection a pair of a periodic knot and its quotient knot induces, and which surjection a degree one map can induce.

This paper is a sequel to the author's papers [1–8] devoted to lower multiplicative bounds on L 1 norms and their applications. Now we give estimates for L 1 norms of exponential sums and prove the result announced in [8].

In 1975 Philipp showed that for any increasing sequence (nk) of positive integers satisfying the Hadamard gap condition nk+1/nk > q > 1, k ≥ 1, the discrepancy DN of (nkx) mod 1 satisfies the law of the iterated logarithm $$ 1/4 \leq {\mathop {\rm lim\,sup} \limits _{N\to\infty}}\, N D_N(n_k x) (N \log \log N)^{-1/2}\leq C_q\quad \textup{a.e.}$$ Recently, Fukuyama computed the value of the lim sup for sequences of the form nk = θk, θ > 1, and in a preceding paper the author gave a Diophantine condition on (nk) for the value of the limsup to be equal to 1/2, the value obtained in the case of i.i.d. sequences. In this paper we utilize this number-theoretic connection to construct a lacunary sequence (nk) for which the lim sup in the LIL for the star-discrepancy \({D_N^*}\) is not a constant a.e. and is not equal to the lim sup in the LIL for the discrepancy DN.

Let A be a class of Abelian groups, A ∈ A, and End(A) be the additive endomorphism group of the group A. The group A is said to be defined by its endomorphism group in the class {ie208-01} if for every group B ∈ B such that End(B) ≅ End(A) the isomorphism B ≅ A holds. The paper considers the problem of definability of a periodic Abelian group A such that End-End(A) ≅ End(A). The classes of periodical Abelian groups, of divisible Abelian groups, of reduced Abelian groups, of nonreduced Abelian groups, and of all Abelian groups are investigated in this paper.

This paper is devoted to relations between the Kurosh problem and the Shirshov height theorem. The central point and main technical tool is the identity of algebraicity. The main result of this paper is the following. Let A be a finitely generated PI-algebra and Y be a finite subset of A. For any Noetherian associative and commutative ring {ie125-01}, let any factor of R ⊗ A such that all projections of elements from Y are algebraic over π(R) be a Noetherian R-module. Then A has bounded essential height over Y. If, furthermore, Y generates A as an algebra, then A has bounded height over Y in the Shirshov sense.