On the value distribution of <Emphasis Type=Italic>ƒ</Emphasis> + <Emphasis Type=Italic>a</Emphasis>(<Emphasis Type=Italic>ƒ</Emphasis>′)<Superscript><Emphasis Type=Italic>n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

<Emphasis Type=Italic>∂</Emphasis>-operations and <Emphasis Type=Italic>H</Emphasis><Subscript><Emphasis Type=Italic>V</Emphasis></Subscript>-fields

The hyperoperations, called theta-operations （δ）, are motivated from the usual property, which the derivative has on the derivation of a product of functions. Using any map on a set, one can define δ-operations. In this paper, we continue our study on the δ-operations on groupoids, rings, fields and vector spaces or on the corresponding hyperstructures. Using δ-operations one obtains, mainly, Hwstructures, which form the largest class of the hyperstructures. For representation theory of hyperstructures, by hypermatrices, one needs special Hv-rings or Hy-fields, so these hyperstructures can be used. Moreover, we study the relation of these δ-structures with other classes of hyperstructures, especially with the Hv-structures.     

<Emphasis Type=Italic>L</Emphasis><Superscript><Emphasis Type=Italic>p</Emphasis></Superscript>(<Emphasis Type=Italic>AP</Emphasis>) spaces (1 ≤ <Emphasis Type=Italic>p</Emphasis> ≤ ∞) and their adjoint ones

We study Besicovitch-type spaces of generalized almost periodic functions. The main result is a theorem on representation of linear continuous functionals that is similar to the classical result of F. Riesz.     

Littlewood–Paley theorem on spaces <Emphasis Type=Italic>L</Emphasis><Superscript><Emphasis Type=Italic>p</Emphasis>(<Emphasis Type=Italic>t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type=Italic>n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

An ABC inequality for Mahler’s measure

We prove an upper bound for the Mahler measure of the Wronskian of a collection of N linearly independent polynomials with complex coefficients. If the coefficients of the polynomials are algebraic numbers we obtain an inequality for the absolute Weil heights of the roots of the polynomials. This later inequality is analogous to the abc inequality for polynomials, and also has applications to Diophantine problems.     

Calibration relations for <Emphasis Type=Italic>B</Emphasis><Subscript><Emphasis Type=Italic>φ</Emphasis></Subscript>-splines of fourth order

We study necessary and sufficient conditions for the smoothness of (not necessarily polynomial) splines of fourth order. We establish the uniqueness of a space of splines of maximal smoothness and prove embeddings on refined grids. We also obtain the corresponding calibration relations. Bibliography: 12 titles.     

Duality for <Emphasis Type=Italic>ε</Emphasis>-Variational Inequality

In this paper, following the method in the proof of the composition duality principle due to Robinson and using some basic properties of the ε-subdifferential and the conjugate function of a convex function, we establish duality results for an ε-variational inequality problem. Then, we give Fenchel duality results for the ε-optimal solution of an unconstrained convex optimization problem. Moreover, we present an example to illustrate our Fenchel duality results for the ε-optimal solutions. The authors thank the referees for valuable suggestions and comments. This work was supported by Grant No. R01-2003-000-10825-0 from the Basic Research Program of KOSEF.     

On <Emphasis Type=Italic>n</Emphasis> × <Emphasis Type=Italic>m</Emphasis>-valued Łukasiewicz-Moisil algebras

n×m-valued Łukasiewicz algebras with negation were introduced and investigated in [20, 22, 23]. These algebras constitute a non trivial generalization of n-valued Łukasiewicz-Moisil algebras and in what follows, we shall call them n×m-valued Łukasiewicz-Moisil algebras (or LM _n×m -algebras). In this paper, the study of this new class of algebras is continued. More precisely, a topological duality for these algebras is described and a characterization of LM _n×m -congruences in terms of special subsets of the associated space is shown. Besides, it is determined which of these subsets correspond to principal congruences. In addition, it is proved that the variety of LM _n×m -algebras is a discriminator variety and as a consequence, certain properties of the congruences are obtained. Finally, the number of congruences of a finite LM _n×m -algebra is computed.      

Sur la rigidité des groupes de triangles (<Emphasis Type=Italic>r</Emphasis>, <Emphasis Type=Italic>p</Emphasis>, <Emphasis Type=Italic>q</Emphasis>)

We show the length spectral rigidity for the family of fuchsian triangular groups.     

The <Emphasis Type=Italic>M</Emphasis>/<Emphasis Type=Italic>M</Emphasis>/∞ queue in a random environment

We consider the M/M/∞ queueing system with arrival and service rate depending on the state of an auxiliary semi-Markov process (which can be viewed as an external environment) and find the mean number of customers in the system in steady state. In a particular case when the external environment can be only in two states we find the distribution of the number of customers in the system.      

A method for the resolution of the Jacobi equation <Emphasis Type=Italic>Y</Emphasis>″ + <Emphasis Type=Italic>RY</Emphasis> = 0 on the manifold <Emphasis Type=Italic>Sp</Emphasis>(2)/<Emphasis Type=Italic>SU</Emphasis>(2)

In this paper a method for the resolution of the differential equation of the Jacobi vector fields in the manifold V ₁ = Sp(2)/SU(2) is exposed. These results are applied to determine areas and volumes of geodesic spheres and balls. Work partially supported by DGI (Spain) and FEDER Projects MTM 2004-06015-C02-01 and MTM 2007-65852 (first author) and by Research Project PGIDIT05PXIB16601PR (second author). Authors’ addresses: A. M. Naveira, Departamento de Geometría y Topología. Facultad de Matemáticas, Avda. Andrés Estellés, N1, 46100 – Burjassot, Valencia, Spain; A. D. Tarrío Tobar, E. U. Arquitectura Técnica, Campus A Zapateira. Universidad de A Coru?a, 15192 – A Coru?a, Spain     

<Emphasis Type=Italic>ω</Emphasis>-Convergence theory of filters in <Emphasis Type=Italic>Lω</Emphasis> -spaces

In this paper, an ω-convergence theory of filters in an Lω-space is established. By means of the ω-convergence theory, some important characterizations with respective to the ω-closed sets, ω T ₂ separation and (ω ₁, ω ₂)-continuous mappings are obtained. Moreover, the mutual relationships among ω-convergence of molecular nets, ω-convergence of ideals and ω-convergence of filters are given.     

<Emphasis Type=Italic>Eω</Emphasis>-Clifford congruences and <Emphasis Type=Italic>Eω</Emphasis>-<Emphasis Type=Italic>E</Emphasis>-reflexive congruences on an inverse semigroup

This paper enlarges the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower T and lower K. Two classes of inverse semigroups, namely Eω-Clifford semigroups and Eω-E-reflexive semigroups, are studied. ((ω _T ) _K ) _T and (((ω _T ) _K ) _T ) _K are found to be the least Eω-Clifford and Eω-E-reflexive congruences on S, respectively. Characterizations of the congruences ((ω _T ) _K ) _T and (((ω _T ) _K ) _T ) _K are developed. The lattices of all Eω-Clifford and Eω-E-reflexive congruences are also studied.     

On Rump’s characterization of <Emphasis Type=Italic>P</Emphasis>-matrices

The necessary and sufficient P-matrix condition by Rump (Linear Algebra Appl 363:237–250, 2003) is simplified by showing that one of its assumptions can be deleted without affecting validity of the result.     

On the non-escaping set of <Emphasis Type=Italic>e</Emphasis><Superscript>2<Emphasis Type=Italic>πiθ</Emphasis></Superscript> sin (<Emphasis Type=Italic>z</Emphasis>)

Let 0 < θ < 1 be an irrational number of bounded type. We proved that almost every point in the Julia set of e ^2πiθ sin (z) is an escaping point. Partially supported by National Basic Research Program of China (973 Program) 2007CB814800.     

Generating <Emphasis Type=Italic>Sym</Emphasis>(<Emphasis Type=Italic>ω</Emphasis>) by Permutations Preserving Dense Orders

We show how generic permutations of ω can be presented by products of permutations preserving dense orders.     

New criteria for <Emphasis Type=Italic>λ</Emphasis>-Koszul algebras

The main purpose of this paper is to provide some new criteria for a standard graded algebra A = ⊕ _i≥0 A _i to be a λ-Koszul algebra, which was first introduced in [12] and was another class of “Koszul-type” algebras including Koszul and d-Koszul algebras as special examples.