The growth of Dirichlet series

We define Knopp-Kojima maximum modulus and the Knopp-Kojima maximum term of Dirichlet series on the right half plane by the method of Knopp-Kojima, and discuss the relation between them. Then we discuss the relation between the Knopp-Kojima coefficients of Dirichlet series and its Knopp-Kojima order defined by Knopp-Kojima maximum modulus. Finally, using the above results, we obtain a relation between the coefficients of the Dirichlet series and its Ritt order. This improves one of Yu Jia-Rong’s results, published in Acta Mathematica Sinica 21 (1978), 97–118. We also give two examples to show that the condition under which the main result holds can not be weakened.     

Coupling of a structural analysis and flow simulation for short-fiber-reinforced polymers: property prediction and transfer of results

The aim of this paper is to discuss the basic theories of interfaces able to transfer the results of an injection molding analyis of fiber-reinforced polymers, performed by using the commercial computer code Moldflow, to the structural analysis program ABAQUS. The elastic constants of the materials, such as Young’s modulus, shear modulus, and Poisson’s ratio, which depend on both the fiber content and the degree of fiber orientation, were calculated not by the usual method of “orientation averaging,” but with the help of linear functions fitted to experimental data. The calculation and transfer of all needed data, such as material properties, geometry, directions of anisotropy, and so on, is performed by an interface developed. The interface is suit able for midplane elements in Moldflow. It calculates and transfers to ABAQUS all data necessary for the use of shell elements. In addition, a method is described how a nonlinear orthotropic behavior can be modeled starting from the generalized Hooke’s law. It is also shown how such a model can be implemented in ABAQUS by means of a material subroutine. The results obtained according to this subroutine are compared with those based on an orthotropic, linear, elastic simulation. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 3, pp. 367-378, May-June, 2009.     

Polytetramethylene glycol-modified polycyanurate matrices reinforced with nanoclays: synthesis and thermomechanical performance

The outstanding improvement in the physical properties of cyanate esters (CEs) compared with those of competitor resins, such as epoxies, has attracted appreciable attention recently. Cyanate esters undergo thermal polycyclotrimerization to give polycyanurates (PCNs). However, like most thermo setting resins, the main draw back of CEs is brittleness. To over come this disadvan tage, CEs can be toughened by the introduction of polytetramethylene glycol (PTMG), a hydroxyl-terminated polyether. How ever, PTMG has a detrimental impact on Young’s modulus. To simultaneously enhance both the ductility and the stiffness of CE, we added PTMG and an organoclay (mont morillonite, MMT) to it. A series of PCN/PTMG/MMT nanocomposites with a constant PTMG weight ratio was pre pared, and the resulting nanophase morphology, i.e., the degree of filler dispersion and distribution in the composite and the thermomechanical properties, in terms of glass-transition behaviour, Young’s modulus, tensile strength, and elongation at break, were examined using the scanning elec tron micros copy (SEM), a dynamic mechanical analysis (DMA), and stress–strain measurements, re spectively. It was found that, at a content of MMT below 2 wt.%, MMT nanoparticles were distributed uniformly in the matrix, suggesting a lower degree of agglomeration for these materials. In the glassy state, the significant increase in the storage modulus revealed a great stiffening effect of MMT due to its high Young’s modulus. The modification with PTMG led to a 233% greater elongation at break compared with that of neat PCN. The nanocomposites exhibited an invariably higher Young’s modulus than PCN/PTMG for all the volume factors of organoclay examined, with the 2 wt.% material displaying the most pronounced in crease in the modulus, in agreement with micros copy results. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 255–268, March–April, 2009.     

Structural elastic and creep models of a UD composite in longitudinal shear

The transient creep of a UD composite with a quadratic arrangement of elastic fibers of quadratic cross section is investigated. The deformational properties of the composite are determined from the known properties of its constituents. A structural model of the UD composite is developed, whose minimal elementary cell contains four elements. The stress-strain state of the elements is assumed homogeneous. Two types of basic and resolving governing equations of transient creep are deduced, which are based on static or kinematic assumptions. In each of the cases, a formula for the longitudinal elastic shear modulus of the composite is found. The stationary solutions of creep equations allow one to obtain formulas of the steady-state creep of the composite in a form similar to Norton’s law. Numerical calculations are also performed, and a comparison of the results with data given in the literature bears witness to the efficiency of the models developed and the solutions obtained. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 437–448, July–August, 2007.     

Beurling algebra analogues of the classical theorems of Wiener and Lévy on absolutely convergent fourier series

Letf be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integersZ. If f is nowhere vanishing on Γ, then there exists a weightv onZ such that 1/f hadv-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if φ is holomorphic on a neighbourhood of the range off, then there exists a weight Χ on Z such that φ ◯f has Χ-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems,v and Χ are non-constant if and only if ω is non-constant. In general, the results fail ifv or Χ is required to be the same weight ω.     

Untypical methods of convergence acceleration

Iterated Aitken’s method is one of classical procedures which permit to accelerate series or sequences convergence. It may be a starting point of constructing better methods in some classes of series whose important parameters are known. Such untypical modifications are here proposed and investigated. They based on a common idea and refer to two kinds of series; cf. Section 2 (series with rational coefficients, hypergeometric series and many others) and Section 3 (so-called quasi-geometric series). The second kind of series is associated with a class of infinite products whose convergence may be also accelerated. Behaviour of Levin’s and Weniger’s methods depends on a parameter β. In Section 4 its role is investigated and possibility of an improvement of their initial steps is showed.     

Response of fly ash-reinforced functionally graded rubber composites subjected to mechanical loading

A novel approach to estimate the Young’s modulus of a functionally graded rubber composite (FGRC) from the damping ratio is demonstrated with the examples of unreinforced and fly ash-reinforced materials. FGRC coupons were prepared using the conventional casting technique. The occurrence of gradation in the specimens was attributed to the variable density of particles present in the fly ash, settling at different depths. The technique of free vibrations was used for experimentation. The damping response of the FGRC specimens was studied. The results obtained from the experiments showed that, with growing filler weight fraction, the Young’s modulus of the composite increased. The empirical model developed to predict the magnitude of the modulus turned out to be in good agreement with experimental data.     

An application of the Fourier method of separation of variables to constructing exactly solvable deformations of partial differential operators

As is known, in mathematical physics there are differential operators with constant coefficients whose fundamental solutions can be constructed explicitly; such operators are said to be exactly solvable. In this paper, the problem of adding lower-order terms with variable coefficients to exactly solvable operators in such a way that the new operators (deformations) admit constructing fundamental solutions in explicit form is posed. This problem is directly related to Hadamard’s problem of describing differential operators satisfying the Huygens’ principle. On the basis of the Fourier method of separation of variables and the method of gauge-equivalent operators, an effective method for finding exactly solvable deformations depending on one variable is constructed. An application of such deformations to constructing Huygens’ differential operators associated with the cone of real symmetric positive-definite matrices is suggested. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Thermoelastic homogenization of periodic composites using an eigenstrain-based micromechanical model

This paper presents a study on effective thermoelastic properties of composite materials with periodic microstructures. The overall elastic moduli and coefficients of thermal expansion of such materials are evaluated by a micromechanical model based on the Eshelby equivalent inclusion approach. The model employs Fourier series in the representation of the periodic strain and displacement fields involved in the homogenization procedures and uses the Levin's formula for determining the effective coefficients of thermal expansion. Two main objectives can be highlighted in the work. The first of them is the implementation and application of an efficient strategy for computation of the average eigenstrain vector which represents a crucial task required by the thermoelastic homogenization model. The second objective consists in a detailed investigation on the behavior of the model, considering the convergence of results and efficiency of the strategy used to obtain the approximate solution of the elastic homogenization problem. Analyses on the complexity of the eigenstrain fields in function of the inclusion volume fractions and contrasts between the elastic moduli of the constituent phases are also included in the investigation. Comparisons with results provided by other micromechanical methods and experimental data demonstrate the very good performance of the presented model.     

Application of Markov’s quadrature in orthogonal expansions

A method of using Markov’s quadrature with a fixed node is proposed to calculate the coefficients of the expansion of a function in a shifted Chebyshev series. Approximation properties of a partial sum of a series with approximate coefficients are considered. This approach can be used to construct some numerical-analytic methods for solving ordinary differential equations.     

Clifford algebra approach to pointwise convergence of Fourier series on spheres

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter’s Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini’s type pointwise convergence theorem are proved. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Equirépartition des coefficients de Fourier des fonctions L de carrés et de cubes symétriques

Dans ce travail, on étudie l’équirépartition et l’équirépartition harmonique des coefficients de Fourier des fonctions L de carrés et de cubes symétriques d’une forme primitive.     

Two-Sided Probabilistic Versions of Hardy’s Inequality

We prove two probabilistic versions of Hardy’s inequality concerning the Fourier coefficients of an integrable function on the unit circle.     

Hardy Spaces of Dirichlet Series and Their Composition Operators

 In [9], Hedenmalm, Lindqvist and Seip introduce the Hilbert space of Dirichlet series with square summable coefficients , and begin its study, with modern functional and harmonic analysis tools. The space is an analogue for Dirichlet series of the space for Fourier series. We continue their study by introducing , an analogue to the spaces . Thanks to Bohr’s vision of Dirichlet series, we identify with the Hardy space of the infinite polydisk . Next, we study a variant of the Poisson semigroup for Dirichlet series. We give a result similar to the one of Weissler ([25]) about the hypercontractivity of this semigroup on the spaces . Finally, following [8], we determine the composition operators on , and we compare some properties of such an operator and of its symbol. Received October 3, 2001; in revised form January 16, 2002 Published online July 12, 2002     

Intersection numbers of cycles on locally symmetric spaces and fourier coefficients of holomorphic modular forms in several complex variables

Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p, q) (resp. U(p, q)) of degreenq (resp. Hodge typenq, nq) to the space of classical holomorphic Siegel modular forms of weight (p +q)/2 and genusn (resp. holomorphic hermitian modular forms of weightp +q and genusn). It is important to note that the cohomology with compact supports contains the cuspidal harmonic forms by Borel [3]. We can express the Fourier coefficients of the lift of η in terms of periods of η over certain totally geodesic cycles—generalizing Shintani’s solution [21] of a conjecture of Shimura. We then choose η to be the Poincaré dual of a (finite) cycle and obtain a collection of formulas analogous to those of Hirzebruch-Zagier [8]. In our previous work we constructed the above lifting but we were unable to prove that it took values in theholomorphic forms. Moreover, we were unable to compute the indefinite Fourier coefficients of a lifted class. By Koecher’s Theorem we may now conclude that all such coefficients are zero. Partially supported by NSF Grant # MCS-82-01660. Partially supported by NSF Grant # DMS-85-01742.     

The banach algebra A and its properties

Beurling’s algebra is considered. A* arises quite naturally in problems of summability of the Fourier series at Lebesgue points, whereas Wiener’s algebra A of functions with absolutely convergent Fourier series arises when studying the norm convergence of linear means. Certainly, both algebras are used in some other areas. A* has many properties similar to those of A, but there are certain essential distinctions. A* is a regular Banach algebra, its space of maximal ideals coincides with[−π, π], and its dual space is indicated. Analogs of Herz’s and Wiener-Ditkin’s theorems hold. Quantitative parameters in an analog of the Beurling-Pollard theorem differ from those for A. Several inclusion results comparing the algebra A* with certain Banach spaces of smooth functions are given. Some special properties of the analogous space for Fourier transforms on the real axis are presented. The paper ends with a summary of some open problems.     

ON ABEL-GONTSCHAROFF-GOULD＇S POLYNOMIALS

In this paper a connective study of Gould‘s annihilation coefficients and Abel-Gontscharoff polynomials is presented. It is shown that Gould‘s annihilation coefficients and Abel-Gontscharoff polynomials are actually equivalent to each other under certain linear substitutions for the variables. Moreover, a pair of related expansion formulas involving Gontscharoff‘s remainder and a new form of it are demonstrated, and also illustrated with several examples.     

On dynamics of fluids in astrophysics

The object of this article is to show the existence of weak solutions to the Navier–Stokes–Fourier Poisson system on (in general) unbounded domains. The topic is a natural continuation of the author’s results on the existence of weak solutions to the problem on Lipschitz domains and to the Oxenius system on unbounded domains. Technique of the proof is based on the tools developed in a series of works by Feireisl (Oxford lecture series in mathematics and its applications, 26, Oxford University Press, Oxford) and others during the recent years. The weak solution’s sensitivity to a change of the domain is discussed as well.     

Explicit jump immersed interface method for virtual material design of the effective elastic moduli of composite materials

Virtual material design is the microscopic variation of materials in the computer, followed by the numerical evaluation of the effect of this variation on the material’s macroscopic properties. The goal of this procedure is an in some sense improved material. Here, we give examples regarding the dependence of the effective elastic moduli of a composite material on the geometry of the shape of an inclusion. A new approach on how to solve such interface problems avoids mesh generation and gives second order accurate results even in the vicinity of the interface. The Explicit Jump Immersed Interface Method is a finite difference method for elliptic partial differential equations that works on an equidistant Cartesian grid in spite of non-grid aligned discontinuities in equation parameters and solution. Near discontinuities, the standard finite difference approximations are modified by adding correction terms that involve jumps in the function and its derivatives. This work derives the correction terms for two dimensional linear elasticity with piecewise constant coefficients, i.e. for composite materials. It demonstrates numerically convergence and approximation properties of the method.      

On the Siegel-Eisenstein measure and its applications

An Eisenstein measure on the symplectic group over rational number field is constructed which interpolatesp-adically the Fourier expansion of Siegel-Eisenstein series. The proof is based on explicit computation of the Fourier expansions by Siegel, Shimura and Feit. As an application of this result ap-adic family of Siegel modular forms is given which interpolates Klingen-Eisenstein series of degree two using Boecherer’s integral representation for the Klingen-Eisenstein series in terms of the Siegel-Eisenstein series.