Statistical laws of fracture and statistical estimation of the static strength of structural elements made from polymeric composites

The basis of the statistical estimation of the strength of structural elements made from fibrous composites is considered; the statistical characteristics of the strength of these materials are described and the limit states are formulated. Special attention is given to the resistance to debonding. The conditions of fracture of this class of materials in plane stress are subjected to a statistical analysis.Presented at the 2nd All-Union Conference on Polymer Mechanics, Riga, November 10–12, 1971.Aviation Engineering Institute, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 466–482, May–June, 1972.     

Equivalence of Many-Photon Green's Functions in the Duffin–Kemmer–Petiau and Klein–Gordon–Fock Statistical Quantum Field Theories

Using the functional integral formalism for the statistical generating functional in the statistical (finite temperature) quantum field theory, we prove the equivalence of many-photon Green's functions in the Duffin–Kemmer–Petiau and Klein–Gordon–Fock statistical quantum field theories. As an illustration, we calculate the one-loop polarization operators in both theories and demonstrate their coincidence.     

Concerning the glass fiber strength distribution function

The statistical strength distribution functions of glass monofilaments of various composition are considered. On the basis of the experimental data it is shown that the fiber strength distributions can be described by a three-parameter function of the Weibull type.All-Union Scientific-Research Institute of Glass-Reinforced Plastics and Glass Fibers, Moscow Regions. Translated from Mekhanika Polimerov, No. 1, pp. 131–136, January–February, 1970.     

A new probabilistic fuzzy model: Fuzzification–Maximization (FM) approach

Over the past few decades, fuzzy logic systems have been used for nonlinear modeling and approximation in many fields ranging from engineering to science. In this paper, a new fuzzy model is developed from the probabilistic and statistical point of view. The proposed model decomposes the input–output characteristics into noise-free part and probabilistic noise part and identifies them simultaneously. The noise-free model recovers the nominal input–output characteristics of the target system and the noise model gives approximation to the probabilistic nature of the added noise. To identify the two submodels simultaneously, we propose the Fuzzification–Maximization (FM). Finally, some simulations are conducted and the effectiveness of the proposed method is demonstrated through the comparison with the previous methods.     

Equivalence of Many-Gluon Green’s Functions in the Duffin-Kemmer-Petieu and Klein-Gordon-Fock Statistical Quantum Field Theories

We prove the equivalence of many-gluon Green’s functions in the Duffin-Kemmer-Petieu and Klein-Gordon-Fock statistical quantum field theories. The proof is based on the functional integral formulation for the statistical generating functional in a finite-temperature quantum field theory. As an illustration, we calculate one-loop polarization operators in both theories and show that their expressions indeed coincide.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 368–374, June, 2005.     

Creep of a nonlinear viscoelastic beam bent by a random moment

The moments of the stress distribution are calculated for a beam made of unidirectional glass-reinforced plastic bent by a random moment. The method previously proposed in [1] is used to solve the statistical boundary value problem.V. A. Steklov Mathematical Institute, Sverdlovsk Branch, Academy of Sciences of the USSR, Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 731–734, July–August, 1968.     

Statistical boundary value problems of polymer mechanics

Research on statistical boundary value problems with applications in the mechanics of reinforced polymers is briefly reviewed. The common features of the models and mathematical methods used in connection with polycrystalline metals and materials of the glass-reinforced plastic type are noted.Sverdlovsk Branch of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR. Translated from Mekhanika Polimerov, No. 1, pp. 63–69, January–February, 1969.     

Vertex Algebras and Combinatorial Identities

In the 1980's, J. Lepowsky and R. Wilson gave a Lie-theoretic interpretation of Rogers–Ramanujan identities in terms of level 3 representations of affine Lie algebra sl(2,C)~. When applied to other representations and affine Lie algebras, Lepowsky and Wilson's approach yielded a series of other combinatorial identities of the Rogers–Ramanujan type. At about the same time, R. Baxter rediscovered Rogers–Ramanujan identities within the context of statistical mechanics. The work of R. Baxter initiated another line of research which yielded numerous combinatorial and analytic generalizations of Rogers–Ramanujan identities. In this note, we describe some ideas and results related to Lepowsky and Wilson's approach and indicate the connections with some results in combinatorics and statistical physics.     

Statistical analysis of some characteristics of cylindrical glass-fiber-reinforced plastic materials

The results are given of a statistical treatment of experimental data on several characteristics of glass-fiber-reinforced plastic cylindrical shells, prepared from glass fabric TS8/3-250 and epoxide binder IF-ÉD-6 (TU-26-59) at the "Électroizolit" factory by winding an impregnated fabric cloth on a mandrel. The experim ental data on the mechanical properties of the glass-fiber-reinforced plastic material were obtained in tests on representative samples cut from the margin of the cylindrical shells. The results of the statistical analysis can be used to evaluate the mechanical reliability of structures made of glass-fiber-reinforced plastic material of the above type.Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 906–910, September–October, 1973.     

Probability, geometry, and irreversibility in quantum mechanics

The main conclusion of this paper is that the Bell–Wigner–Accardi theory of quantum probabilities in spin systems may be placed within the general operator trigonometry developed independently by this author about 30 years ago. The use of the Grammian from the operator trigonometry simplifies and clarifies the analysis of Wigner. A general triangle inequality from the operator trigonometry clarifies and generalizes the analysis of Accardi. The statistical meaning of the complex numbers in quantum mechanics is seen to be that of the natural geometry of the operator trigonometry. A new connection of the operator trigonometry to CP symmetry violation is established.     

A statistical approach to the solution of ill-posed problems in geophysics

The article surveys the main results of the statistical approach to the solution of ill-posed problems of mathematical physics, in application to specific ill-posed inverse problems in geophysics.Invited paper presented at the International Seminar on Mathematical Foundations of the Interpretation of Geophysical Fields, Moscow, May–June 1972.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 79, pp. 67–81, 1978.     

Girsanov's theorem and ergodic properties of statistical solutions of nonlinear parabolic equations

Theorems on the existence and uniqueness of the statistical solutions and for the existence of stationary solutions for nonlinear stochastic equations are proved. Markov families of statistical solutions are constructed. The ergodic properties of monotone systems are investigated. A Girsanov type theorem on the absolute continuity of the statistical solution of a nonlinear stochastic parabolic equation with respect to the statistical solution of the monotone equation is proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 88–117, 1987.     

Estimation of the stability of the strength of glass-reinforced plastics in conical shells

On the basis of a statistical analysis of the test data it is shown that there is a difference between the mechanical characteristics (in tension and compression) of laminated glass-reinforced plastics obtained under laboratory and industrial conditions by impregnation under pressure in a closed mold. The stability of the strength properties of the glass-reinforced plastic in various conical shells, produced in large batches, is considered. Certain experimental relations between the material properties and the total number of shells produced are also established.Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 102–108, January–February, 1970.     

Method of the anisotropy potential in the linear mechanics of polymers

The potential of the displacements due to the anisotropy of the physical properties of a material (the anisotropy potential) is used in order to reduce the solution of the linear statistical boundary problem of elasticity theory and that of the linear creep of homogeneous anisotropic bodies to two simpler problems: one involving an anisotropic medium, and the other involving an equation analogous to the Poisson equation.S. M. Kirov Ural Polytechnic Institute, Sverdlovsk. Translated from Mekhanika Polimerov, No. 1, pp. 24–31, January–February, 1972.     

Propagation of ultrasonics in bones

Conclusions The possibility has been established of determining the propagation velocity of elastic waves in bones by an ultrasonic technique in order to study the healing of fracture. An experimental apparatus of new design is described, which has been developed and made for research purposes and diagnostic measurements. The apparatus enables a numerical evaluation of the propagation velocity of ultrasonics in a bone to be automatically portrayed on the counter display, and the complete wave pattern or its details to be watched on the oscilloscope screen. The trends and principles of the exploratory measurements and practical diagnostics are discussed and, after statistical processing, certain results from our determinations are also quoted.A paper presented to the First All-Union Conference on Engineering and Medical Biomechanics, Riga, October 28–31, 1975.Scientific-Research Institute of Traumatology, Brno, Czechoslavakian SSR. Translated from Mekhanika Polimerov, No. 6, 1084–1087, November–December, 1976.     

On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space

The concept of statistical convergence was introduced by Fast [H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244] which was later on studied by many authors. In [J.A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993) 43–51], Fridy and Orhan introduced the idea of lacunary statistical convergence. Quite recently, the concept of statistical convergence of double sequences has been studied in intuitionistic fuzzy normed space by Mursaleen and Mohiuddine [M. Mursaleen, S.A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals (2008), doi:10.1016/j.chaos.2008.09.018]. In this paper, we study lacunary statistical convergence in intuitionistic fuzzy normed space. We also introduce here a new concept, that is, statistical completeness and show that IFNS is statistically complete but not complete.     

Admissible polynomial estimators for quadratic polynomials of normal parameters

The problem of the statistical estimation of quadratic polynomials of the parameters of the normal law is considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 234–247, 1990.     

Statistical strength of glass-reinforced plastics

The failure of glass-reinforced plastics is treated as an aggregate of successive failures of the individual fibers. A method of estimating the statistical strength from the distribution curve is proposed.Mekhanika Polimerov, Vol. 3, No. 3, pp. 498–502, 1967     

Statistical estimate of a static strength criterion for AG-4s two-way glass-reinforced plastic

The statistical characteristics of the static strength of AG-4s entering into the equations of the strength criteria for anisotropic materials are determined. The agreement between a series of existing criteria and the test data is estimated. A theoretical estimate of the mathematical expectation and variance of the function for the criterion best adapted to the material in question is obtained on the basis of the method of linearization of a function of several random arguments. The theoretical and experimental data are compared.Moscow Aviation Technological Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1117–1120, November–December, 1973.