On Uniformly Convex and Uniformly Kadec-Klee Renomings

We give a new construction of uniformly convex norms with a power type modulus on super-reflexive spaces based on the notion of dentability index. Furthermore, we prove that if the Szlenk index of a Banach space is less than or equal to ω (first infinite ordinal) then there is an equivalent weak* lower semicontinuous positively homogeneous functional on X* satisfying the uniform Kadec-Klee Property for the weak*-topology (UKK*). Then we solve the UKK or UKK* renorming problems for Lp(X) spaces and C(K) spaces for K scattered compact space.

Some Comments on Q-Irresolute and Quasi-Irresolute Functions

The aim of this paper is to continue the study of θ-irresolute and quasi-irresolute functions as well as to give an example of a function which is θ-irresolute but neither quasi-irresolute nor an R-map and thus give an answer to a question posed by Ganster, Noiri and Reilly. We prove that RS-compactness is preserved under open, quasi-irresolute surjections.

Operators Factoring through Banach Lattices and Ideal Norms

A new, unified presentation of the ideal norms of factorization of operators through Banach lattices and related ideal norms is given.

Integral Manifolds and Perturbations of the Nonlinear Part of Systems of Autonomous Differential Equations with Impulses at Fixed Moments

* This investigation was supported by the Bulgarian Ministry of Science and Education under Grant MM-7.

Branching Processes with Immigration and Integer-valued Time Series

In this paper, we indicate how integer-valued autoregressive time series Ginar(d) of ordre d, d ≥ 1, are simple functionals of multitype branching processes with immigration. This allows the derivation of a simple criteria for the existence of a stationary distribution of the time series, thus proving and extending some results by Al-Osh and Alzaid [1], Du and Li [9] and Gauthier and Latour [11]. One can then transfer results on estimation in subcritical multitype branching processes to stationary Ginar(d) and get consistency and asymptotic normality for the corresponding estimators. The technique covers autoregressive moving average time series as well.

Deformation Lemma, Ljusternik-Schnirellmann Theory and Mountain Pass Theorem on C1-Finsler Manifolds

∗Partially supported by Grant MM409/94 Of the Ministy of Science and Education, Bulgaria. ∗∗Partially supported by Grant MM442/94 of the Ministy of Science and Education, Bulgaria.

Гиперплоскости в «Множествах и Расстояниях Соответствия»: Кластеризация

Рассматриваются общие проблемы кластеризации. Предложена концепция «множеств» и «расстояний соответствия» в построении кластеров, рассмотрены модели кластеризации, в которых «множествами соответствия» являются гиперплоскости, а «расстояниями соответствия» – различные варианты расстояний в связи с соответствующими гиперплоскостями. Развит аппарат псевдообращения по Муру – Пенроузу: приведены рекуррентные формулы возмущения для ортогональных проекторов и R-операторов, связанных с псевдообращением. Рекуррентные формулы возмущения использованы для построения алгебраического варианта Jack Knife’а. Приведена сборка важных для приложений результатов, касающихся псевдообращения.

Использование Алгоритма Ускоренного Вероятностного Моделирования в Схеме Табу-Поиска

Рассматривается метаэвристический метод комбинаторной оптимизации, основанный на использовании алгоритмов табу-поиска и ускоренного вероятностного моделирования. Излагается общая вычислительная схема предложенного метода, названного алгоритмом GS-tabu. Приведены результаты серии вычислительных экспериментов по решению известных задач коммивояжера и квадратичных задач о назначении.

Об Одной Задаче Распознавания Последовательности, Включающей Повторяющийся Вектор

* Работа поддержана грантами РФФИ 09-01-00032, 07-07-00022 и грантом АВЦП Рособразования 2.1.1/3235

Powers and Logarithms

There are applied power mappings in algebras with logarithms induced by a given linear operator D in order to study particular properties of powers of logarithms. Main results of this paper will be concerned with the case when an algebra under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for x, y ∈ dom D. Note that in the Number Theory there are well-known several formulae expressed by means of some combinations of powers of logarithmic and antilogarithmic mappings or powers of logarithms and antilogarithms (cf. for instance, the survey of Schinzel S[1].

A Generalized Convolution with a Weight Function for the Fourier Cosine and Sine Transforms

A generalized convolution with a weight function for the Fourier cosine and sine transforms is introduced. Its properties and applications to solving a system of integral equations are considered.

An Expansion Formula for Fractional Derivatives and its Application

An expansion formula for fractional derivatives given as in form of a series involving function and moments of its k-th derivative is derived. The convergence of the series is proved and an estimate of the reminder is given. The form of the fractional derivative given here is especially suitable in deriving restrictions, in a form of internal variable theory, following from the second law of thermodynamics, when applied to linear viscoelasticity of fractional derivative type.

On the Mellin Transforms of Dirac’S Delta Function, The Hausdorff Dimension Function, and The Theorem by Mellin

Mathematics Subject Classification: 44A05, 46F12, 28A78

Weighted Theorems on Fractional Integrals in the Generalized Hölder Spaces via Indices mω and Mω

Mathematics Subject Classification: 26A16, 26A33, 46E15.

Renewal Processes of Mittag-Leffler and Wright Type

2000 MSC: 26A33, 33E12, 33E20, 44A10, 44A35, 60G50, 60J05, 60K05.