Averaging the physical properties of fibrous piezocomposites

Elastic, piezoelectric, and dielectric properties of fibrous composites with piezoelectric components are averaged for antiplane strains. Methods for determining coupled electroelastic fields in piezocomposites are proposed. Calculation results of the effective physical characteristics of some composites are given.Sumskii State University, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 116–123, January–February, 1998.     

Explicit formulas for wavelet-homogenized coefficients of elliptic operators

Wavelet-based homogenization provides a method for constructing a coarse-grid discretization of a variable–coefficient differential operator that implicitly accounts for the influence of the fine scale medium parameters on the coarse scale of the solution. The method is applied to discretizations of operators of the form in one dimension and μ(x)Δ in one and more dimensions. The resulting homogenized matrices are shown to correspond to differential operators of the same (or closely related) form. In dimension one, results are obtained for periodic two-phase and for arbitrary coefficients μ(x). For periodic two-phase coefficients, the homogenized coefficients may be computed exactly as the harmonic mean of the function μ. For non-periodic coefficients, the “mass-lumping” approximation results in an explicit formula for the homogenized coefficients. In higher dimensions, results are obtained for operators of the form μ(x)Δ, where μ(x) may or may not be periodic; explicit formulae for the homogenized coefficients are also derived. Numerical examples in 1D and 2D are also presented.     

Effect of ultrasound on the properties of a glass-reinforced plastic

It is shown that removing paraffin size from the glass strands by means of ultrasound has little effect on their strength, improves the structure of the material, and raises the mechanical strength of the GRP by 11–12%. An ultrasonic strand cleaner employing focusing piezoelectric ceramic transducers is described.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 546–548, May–June, 1969.     

A generalized knapsack problem with variable coefficients

The ordinary knapsack problem is to find the optimal combination of items to be packed in a knapsack under a single constraint on the total allowable resources, where all coefficients in the objective function and in the constraint are constant.In this paper, a generalized knapsack problem with coefficients depending on variable parameters is proposed and discussed. Developing an effective branch and bound algorithm for this problem, the concept of relaxation and the efficiency function introduced here will play important roles. Furthermore, a relation between the algorithm and the dynamic programming approach is discussed, and subsequently it will be shown that the ordinary 0–1 knapsack problem, the linear programming knapsack problem and the single constrained linear programming problem with upper-bounded variables are special cases of the interested problem. Finally, practical applications of the problem and its computational experiences will be shown.     

Extended Gauss–Markov Theorem for Nonparametric Mixed-Effects Models

The Gauss–Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss–Markov theorem to include nonparametric mixed-effects models. The extended Gauss–Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss–Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented.     

Modal analysis of piezoelectric bodies with voids. II. Finite element simulation

The paper is concerned with the eigenvalue problems for piezoelectric bodies with voids in contact with massive rigid plane punches and coved by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin–Nunziato model is used. The generalized statements for eigenvalue problem are obtained in the extended and reduced forms. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. The orthogonality relations for eigenvectors are obtained in different forms. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric solid with voids change, are established.     

Electromechanical properties of continuous fibre-reinforced piezoelectric composites

The objective of this paper is to present analytical expressions of the basic effective constants (for transducer applications) of fibrous piezoelectric composites with a periodic structure. These expressions were obtained on the basis of the asymptotic averaging method, following the same procedure used in [1] for the purely elastic case. The averaged electromechanical properties for these piezocomposites can be calculated from the above formulas. The results obtained from the theoretical predictions with the use of homogenization method show good agreement with the existing experimental results.Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, pp. 670–680, September–October, 1997.     

Some features of the frictional behavior of polymeric crystalline materials

The coefficients of friction for thin polymeric films are calculated, and it is shown that the coefficient of friction of the films is much less than that of a homogenous block. This may be due to some features of the antifriction properties of polymeric crystalline materials.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1053–1058, November–December, 1972.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

Higher Intersection Theory on Algebraic Stacks: I

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks locally of finite type over a field and establish their basic properties. This includes algebraic stacks in the sense of Deligne–Mumford as well as Artin. An intrinsic difference between our approach and earlier approaches is that the higher Chow groups of Bloch enter into our theory early on and depends heavily on his fundamental work. Our theory may be more appropriately called the (Lichtenbaum) motivic homology and cohomology of algebraic stacks. One of the main themes of these papers is that such a motivic homology does provide a reasonable intersection theory for algebraic stacks (of finite type over a field), with several key properties holding integrally and extending to stacks locally of finite type. While several important properties of our higher Chow groups, like covariance for projective representable maps (that factor as the composition of a closed immersion into the projective space associated to a locally free coherent sheaf and the obvious projection), an intersection pairing and contravariant functoriality for all smooth algebraic stacks, are shown to hold integrally, our theory works best with rational coefficients.The main results of Part I are the following. The higher Chow groups are defined in general with respect to an atlas, but are shown to be independent of the choice of the atlas for smooth stacks if one uses finite coefficients with torsion prime to the characteristics or in general for Deligne–Mumford stacks. (Using some results on motivic cohomology, we extend this integrally to all smooth algebraic stacks in Part II.) Using cohomological descent, we extend Bloch's fundamental localization sequence for quasi-projective schemes to long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks: this is one of the main results of the paper. We show that these higher Chow groups modulo torsion are covariant for all proper representable maps between stacks of finite type while being contravariant for all representable flat maps and, in Part II, that they are independent of the choice of an atlas for all stacks of finite type over the given field k. The comparison with motivic cohomology, as is worked out in Part II, enables us to provide an explicit comparison of our theory for quotient stacks associated to actions of linear algebraic groups on quasi-projective schemes with the corresponding Totaro–Edidin–Graham equivariant intersection theory. As an application of our theory we compute the higher Chow groups of Deligne–Mumford stacks and show that they are isomorphic modulo torsion to the higher Chow groups of their coarse moduli spaces. As a by-product of our theory we also produce localization sequences in (integral) higher Chow groups for all schemes locally of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.     

Harmonic Analysis on the SU(2) Dynamical Quantum Group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey–Wilson polynomials, and the Haar measure with the Askey–Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch–Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn–Elliott identity satisfied by quantum 6j-symbols.     

On the Huang Class of Variable Metric Methods

Several authors have created one-parameter families of variable metric methods for function minimization. These families contain the methods known as Davidon–Fletcher–Powell, Broyden–Fletcher–Goldfarb–Shanno, and symmetric rank one. It is shown here that the same one-parameter families of methods are obtained from the Huang update by requiring the update to be symmetric.     

A class of Noetherian overdetermined elliptic boundary-value problems

In this paper one investigates overdetermined elliptic systems with constant coefficients for which one can pose general boundary-value problems without overdetermination in the boundary conditions. One finds the algebraic condition which characterizes the indicated class of systems. It is shown that the general boundary-value problems for such systems are Noetherian in the subspace of vector-functions satisfying some purely differential consistency conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. nA. Steklova AN SSSR, Vol. 47, pp. 138–154, 1974.     

Determination of the resonant frequencies of an anisotropic piezoelectric disk

An equation for the resonant frequencies of a thin anisotropic circular disk is obtained assuming exponential particular solutions. The roots of the equation are determined numerically. The resonant frequencies of two piezoelectric crystals are computed and compared with the resonant frequencies when the piezoelectric effect is not taken into account.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 72–74, 1990.     

Inverse nonstationary problem of scattering for a hyperbolic system of n equations of the first order on a semiaxis

In connection with a joint study of n-1 problems with diverse boundary conditions we construct a scattering operator for a hyperbolic system on a semiaxis for the case of n-1 incident and one scattered wave. The possibility of the unique recovery of the coefficients of the system with respect to the scattering operator is shown.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1638–1646, December, 1991.     

A Rational Approximant for the Digamma Function

Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use Padé approximants or other rational functions constructed from sequence transformations instead. However, neither Padé approximants nor sequence transformation utilize the information which is avaliable in the case of a special function – all power series coefficients as well as the truncation errors are explicitly known – in an optimal way. Thus, alternative rational approximants, which can profit from additional information of that kind, would be desirable. It is shown that in this way a rational approximant for the digamma function can be constructed which possesses a transformation error given by an explicitly known series expansion.     

Confluence of fuchsian second-order differential equations

Ordinary linear homogeneous second-order differential equations with polynomial coefficients including one in front of the second derivative are studied. Fundamental definitions for these equations: of s-rank of the singularity (different from Poincaré rank), of s-multisymbol of the equation, and of s-homotopic transformations are proposed. The generalization of Fuchs' theorem for confluent Fuchsian equations is proved. The tree structure of types of equations is shown, and the generalized confluence theorem is proved.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 233–247, August, 1995.     

A linear filtering problem in complete correlated actions

In the last paper, the geometry of the Sz.-Nagy-Foia model for contraction operators on Hilbert spaces was used to advantage in several problems of multivariate analysis. The lifting of intertwining operators, one of the basic results from the Sz.-Nagy-Foia theory, is now recognized as the most adequate operatorial form of the deep classical results of the extrapolation theory. The labeling of the exact intertwining dilations given by [1]Acta Sci. Math. (Szeged) 40 9–32] and the recursive methods used there open a broad perspective for using the Sz.-Nagy-Foia model in multivariate filtering theory. In this paper, using the notion of correlated action (see [5 and 6] Rev. Roumaine Math. Pures Appl. 23, No. 9 1393–1423]) as a time domain, a linear filtering problem is formulated and its solution in terms of the coefficients of the analytic function which factorizes the spectral distribution of the known data and the coefficients of an analytic function which describes the cross correlations is given. In some special cases it is shown that the filter coefficients can be determined using recursive methods from the intertwining dilation theory, of the autocorrelation function of the known data and an intertwining operator, interpreted as the initial estimator given by the prior statistics.     

Field Theory Analysis of Critical Behavior of a Symmetric Binary Fluid

A method is elaborated for constructing an effective field theory Hamiltonian of the Landau–Ginzburg–Wilson type for off-lattice models of binary fluids. We show that all coefficients of the effective Hamiltonian for a symmetric binary fluid can be expressed in terms of some known characteristics of the model hard-sphere fluid, namely, compressibility and its derivatives with respect to density. Application of the effective Hamiltonian is demonstrated by an example of determining the curve of critical layering points in the mean-field approximation. This curve agrees well with numerical experiment results for symmetric binary fluids.