Discontinuous solutions in the classical theory of bending of plates

We present necessary and sufficient conditions for existence of a class of analytic solutions of the equations of classical theory of bending of plates in a domain formed by removing from the infinite plane a set of points belonging to an arbitrary piecewise-smooth contour. Here we assume that all the components of a solution of a given class possess the property of continuous extendability onto almost all points of the boundary of the domain and that the boundary values are locally summable functions on the boundary.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1598–1605, December, 1990.     

On cracked plates subjected to classical bending

The classical Westergaard type stress function analysis for in-plane stress crack problems is adapted to classical bending of cracked plates for determination of mode-1 stress intensity factors.

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Zusammenfassung Die klassische Methode der Westergaard'schen Spannungsfunktionen für ebene Rißprobleme wird auf den Fall der klassischen Biegung von Platten mit Rissen unter mode-1 Belastung verallgemeinert. Spannungsintensitätsfaktoren für mode-1 Biegung werden bestimmt.

     

On Gibbs stability in the classical theory of fluid mixtures

We prove that Gibbs' criterion for stability is a necessary condition for the dynamic stability of a classical (non-linear) inviscid fluid mixture.

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Resumé On démontre que le critère de Gibbs pour la stabilité est une condition nécessaire pour la stabilité dynamique d'une mixture classique (non linéaire) des fluids inviscides.

     

On the stability of some classical operators from approximation theory

We obtain some results on Hyers–Ulam stability for some classical operators from approximation theory. For Bernstein operators we determine the Hyers–Ulam constant using a result concerning coefficient bounds of Lorentz representation for a polynomial.     

On stability in the classical linear theory of fluid mixures

Summary In this paper we study the stability(1) of linear inviscid fluid mixtures. In particular, we show that thestatical stability criterion of Gibbs is both necessary and sufficient for thedynamical stability of the mixture(2), using, as our main hypotheses, only those inequalities and symmetries which are consequences of the second law of thermodynamics(3). A Dario Graffi nel suo 70° compleanno Entrata in Redazione il 28 giugno 1975.     

On a classical theorem in the theory of Fourier integrals

In this note we give a short proof of a classical theorem in the theory of Fourier integrals.

     

On the numerical optimization of KAM estimates by classical perturbation theory

A recently proposed algorithm for the estimate of the threshold above which a certain torus disappears, which combines classical Birkhoff normalization procedure with KAM theory, is reconsidered and improved. This is done by studying the particular case of the forced pendulum Hamiltonian , and considering the golden torus. We find that there exists an optimal orderN of normalization, and the ratio between the estimate thus obtained and the experimental value turns out to be 19. A possible explanation of such result is also suggested.     

On the theory of thermoelastic bending of reinforced shells and plates

The author examines an approach to the construction of a theory of thermoelastic bending of arbitrarily reinforced shells and plates that takes into account the actual structure, the deformation characteristic, and the actual thermomechanical properties of the elements which make up the composite. The final equations are obtained with and without allowance for transverse shear strains. The problem of the thermoelastic bending of a thin, arbitrarily reinforced rectangular plate hinged at the edges, when transverse shear strains can be neglected, is considered as an example.Institute of Hydrodynamics, Siberian Division, Academy of Sciences of the USSR, Novosibirsk. Translated from Mekhanika Polimerov, No. 5, pp. 861–873, September–October, 1972.     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the problem of the ergodic constant in the theory of classical Green's functions

Moscow Institute of Electronic Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 3, pp. 459–473, September, 1990.     

Construction of the solutions of the equations of vibration of the classical theory of plates with parameters periodic on one coordinate

The system of classical equations of transverse vibrations of plates that are inhomogeneous on one coordinate and have exponential dependence of the solution on the other coordinate and time are presented in the canonical Hamiltonian form with suitably chosen canonical variables. For periodically varying parameters we use the general properties of periodic Hamiltonian systems to study the structure of the solutions of boundary-value problems for stationary vibrations of plates. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 109–113.     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspace? _d ^m of the noncommutative invariant algebra? _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(? _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition of? _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspaceĨ _d ^m of the noncommutative invariant algebraĨ _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(Ĩ _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition ofĨ _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

Addition to the classical theory of Favard

A study is made of the linear equation p_t = A(t) p +f(t) with an almost periodic matrix A(t) and an almost periodic free termf(t). It is shown that if a bounded solution exists, then an almost periodic solution in the sense of Besicovitch also exists.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 239–246, February, 1970.     

The mathematical theory of classical thermodynamics

In this paper we systematically present a new approach to classical thermodynamics, using asymptotic distribution from number theory and generalizing the Bose-Einstein distribution. The phase transition from gas to liquid, the thermodynamics of fluids, as well as the behavior of liquids under negative pressure are elucidated.     

Contributions to the theory of the classical Banach spaces

The paper contains several results on the linear topological structure of the spaces C(K), K compact metric, and L_p(0, 1), 1 ? p < ∞. The topics which are studied include: complemented subspaces, special Schauder bases, and equivalent norms in these spaces.