Methods Of Mathematical Physics Research Articles

Variational Methods for Solving Nonlinear Boundary Problems of Statics of Hyper-Elastic Membranes

A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics [1]–[6]. In the present paper, using the variational method for solving nonlinear boundary problems of statics of hyper-elastic membranes under the regular hydrostatic load, we investigate peculiarities of deformation of a circular membrane whose mechanical characteristics are described by the Bidermann-type elastic potential. We develop an algorithm for solving a singular perturbation of nonlinear problem for the case of membrane loaded by heavy liquid. This algorithm enables us to obtain approximate solutions both in the presence of boundary layer and without it. The class of admissible functions, on which the variational method is realized, is chosen with account of the structure of formal asymptotic expansion of solutions of the corresponding linearized equations that have singularities in a small parameter at higher derivatives and in the independent variable. We give examples of calculations that illustrate possibilities of the method suggested for solving the problem under consideration.

A theoretical procedure for using multiple response time-temperature integrators for the design and evaluation of thermal processes

Time-temperature integrators (TTI) offer a sound alternative to in situ and physical-mathematical methods for thermal process design and evaluation. In the present work, the use of multi-component TTI systems, in cases where a single-component TTI with a particular z value is not available, was investigated. A procedure in using double or triple response TTI systems to evaluate the impact of a thermal process on a particular target heat labile substance was proposed and evaluated, based on a large number of product time-temperature profiles. Parameters to be considered in using a multi-component TTI included the z values of the TTI, in relation to the target z value, as well as the reference temperature used for the calculations. Guidelines for proper selection of the parameters involved are given. Although, generally, triple response TTIs performed better compared with double response TTIs, the results of double response TTIs were also satisfactory, and for practical applications, the use of a double response TTI might be sufficient.

Mathematical methods for physics and engineering: a comprehensive guide

Mathematical methods for physics and engineering: a comprehensive guide

An examination of underlying physical principles. The interaction of power-line electromagnetic fields with the human body.

A detailed examination of the scientific foundations contained in a number of recent articles has revealed some serious errors. The careful application of advanced mathematical-physical methods to study the penetration of externally applied electric fields into the organs and cells of the human body has obtained new and significant results [8, 9, 32]. Specifically, Eqs. (11) and (12) establish a definite relationship between an incident low-frequency electric field and the current, current density, and electric field induced in the body when the arms are in contact with it and the incident field is parallel to its length. Corresponding formulas with the arms raised are available [10] as are formulas for the current density and electric field in the organs of the body [9]. It is these currents and electric fields that must be used by biomedical research workers to determine their effect on living cells. The proposed interpretation of epidemiological data [31] indicates a risk of leukemia in children when the incident 50-60-Hz electric field generated by a high-voltage transmission line is Einc > or = 61 V/m. (23). The corresponding maximum electric field at approximately 10 m from a 440-kV transmission line is Einc approximately 2100 V/m [8, 9]. The electric fields induced in the different organs of the body by these electric fields are in the range 25.2 < or = E1z < or = 119 microV/m for Einc2z = 61 V/m and 0.87 < or = E1z < or = 4.1 mV/m for Einc2z = 2100 V/m [9]. These are in the range of applied fields in in vitro experiments, which showed significant effects on cells [4], and should serve as guidelines for such experiments.

Solutions of axially symmetric and two-dimensional dynamic problems of thermoelasticity for cylinders in terms of stresses

We present the mathematical statement of a dynamic problem of thermoelasticity in cylindrical coordinates in terms of the components of the stress tensor. We write the systems of basic equations for axially symmetric and two-dimensional problems satisfying the consistency conditions for strains. By identical transformations, these systems are reduced to hierarchies of wave equations, which can be solved by using standard methods of mathematical physics. In the absence of the influence of temperature and bulk forces, the indicated hierarchies turn into the well-known equations.

ガリレオ・デカルト・ホイヘンス

Galileo and Descartes made an important contribution to the making of modern mechanics in the Scientific Revo-lution. However, there are crucial differences between their views on the nature of gravity and on the concept of iner-tial motion. They indicate the difference between their methods of science.Huygens supported Descartes' mechanical natural phi-losophy and based his theory of light on the corpuscular theory. But he criticized Decartes' excessive confidence in his own philosophy. Huygens gave higher priority to the solution of individual problems than to the construction of a whole system. He developed the problem-oriented method of Galileo's mathematical physics.

Estimate of error of an approximated solution by the method of moments of an operator equation

For an equationAu = f whereA is a closed densely defined operator in a Hilbert spaceH, f eH, we estimate the deviation of its approximated solution obtained by the moment method from the exact solution. All presented theorems are of direct and inverse character. The paper refers to direct methods of mathematical physics, the development of which was promoted by Yu. D. Sokolov, the well-known Ukrainian mathematician and mechanic, a great humanitarian and righteous man. We dedicate this paper to his blessed memory.

Modelling of phenomena within catalyst particles

There are strong economic forces that demand rational methods for the development of new catalysts. A basis for the solution of this problem are appropriate models of the phenomena within catalyst supports, augmented with a wide variety of experimental methods. This paper deals with models of the porous structures of catalyst supports, multi-component diffusion, chemisorption, reaction, desorption and deactivation. Mathematical methods of quantum chemistry and statistical physics will be briefly described. The powerfulness of modern modelling approaches will especially be demonstrated with zeolites.

THEORETICAL CONSIDERATIONS ON DESIGN of MULTICOMPONENT TIME TEMPERATURE INTEGRATORS IN EVALUATION of THERMAL PROCESSES

Recently, the use of Time Temperature Integrators (TTIs) as an alternative means of process evaluation, besides in situ process monitoring and the physical-mathematical method, has received a lot of attention. the stringent requirement to which an adequate functioning TTI should respond, i.e., the temperature dependence of reaction rate constants for the monitored target quality attribute and the TTI shall be identical, is hard to realize in practice. Therefore, multicomponent TTIs have been suggested to evaluate the integrated impact of time and temperature on a target quality attribute. These monitoring devices consist of a number of temperature sensitive units, each with its own response of reaction rates to temperature. the question how from the combined reading of these individual systems, the impact of the heating process on a target quality parameter can be predicted has not yet been answered. In this theoretical study, we critically consider the restrictions of two interpolation methods in determining the impact on a target quality attribute from the reading of three other temperature sensitive systems. Possibilities of the ‘Equivalent Point Method’and a polynomial expression as interpolation method are tested. Success of these transformations depends on kinetic characteristics of the components in the multicomponent TTI system and the actual product temperature history. Since only limited information is available on the product temperature

Development of direct methods in mathematical physics in the works of M. P. Kravchuk

Fundamental results in the works of M. P. Kravchuk devoted to the creation and development of the method of moments for differential and integral equations are discussed.

Mathematical methods of non-Archimedean physics

CONTENTS Introduction § 1. Generalized functions and Gaussian integrals on finite-dimensional non-Archimedean spaces § 2. Non-Archimedean Hilbert spaces § 3. Spaces of functions square-integrable with respect to Gauss and Lebesgue distributions on non-Archimedean spaces § 4. The Schrödinger and Bargmann-Fock representations in non-Archimedean quantum mechanics § 5. Non-Archimedean statistical quantum mechanics § 6. Existence and uniqueness theorems for the solution of linear partial differential equations on non-Archimedean spaces § 7. The solubility of Schrödinger's, Heisenberg's and Liouville's equations in non-Archimedean mechanics § 8. Trotter's formula in non-Archimedean commutative Banach algebras, and the chronological representation of the solution of Schrödinger's equation with a potential § 9. Generalized functions and Gaussian integrals on infinite-dimensional non-Archimedean spaces § 10. Existence and uniqueness theorems for the solution of linear partial differential equations on infinite-dimensional non-Archimedean spaces § 11. The path integral for a non-Archimedean scalar bosonic field § 12. Non-Archimedean bosonic strings in a photonic-like gauge § 13. Non-Archimedean bosonic string fields in a photonic-like gauge § 14. Singular functions of a non-Archimedean scalar bosonic field References

Short-circuit heating of insulated conductors

The author presents a calculation method, developed from measurements and mathematical-physical methods, for investigating the short-circuit heating of insulated conductors. The proposed approach permits the determination of the temperature at the conductors or of the admissible currents. Starting from common positions of adiabatic and nonadiabatic heating situations, models based on transmission line theory are developed that permit a general view and estimation of the different processes involved. A measuring device has been developed which evaluates temperature from the true resistance and the temperature coefficient of the conductor material. Results were converted for short-circuit calculation via the proposed models and confirmed the usefulness of the calculation method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Probabilistic Methods in Mathematical Physics.

Probabilistic Methods in Mathematical Physics.

Analysis of controlled rotations of an elastic rod around an arbitrary axis

Controlled plane rotations around an arbitrary axis under the action of concentrated (transverse) forces and the moments of forces /2, 3/ are investigated for a homogeneous rectilinear rod within the framework of the linear theory of small strains /1/. To be specific, we consider the case when the moments are applied relative to the ends of the rod and the axis of rotation while forces orthogonal to the rod are also concentrated at the ends. A complete solution of the Cauchy problem (in time) is constructed by analytic methods of mathematical physics /4/ and the approach in /5/ for a system described by the boundary value problem, and a foundation is given /6, 7/ and controllability is established /8/. Problems of applied interest are examined in the cases of realizations of the “kinematic” and “dynamic” controls of rod rotations and estimates of the errors due to its elasticity are also examined. The results can be utilized in investigations of problems of precision control of mechanical objects possessing substantial compliance of the structure.

Stability criteria and classification of singularities for equivariant lagrangian submanifolds

One of the useful methods of mathematical physics is the one arising from symplectic geometry and associating the singularities of Lagrangian submanifolds with the optical caustics, phase transitions, bifurcation patterns, obstacle geometry etc. In this paper we derive the stability criteria for singularities of equivariant Lagrangian submanifolds with a compact Lie group action determined by a system with symmetry. The recognition problem and classification list for stable ( Z 2) q -equivariant singularities is proved. We find that the classified stable local models occur as possible realizations for the equilibrium states in the breaking of symmetry and structural phase transitions. Additionally, the connection between two technically different (by generating functions, by Morse families) infinitesimal G-stability conditions for equivariant Lagrangian submanifols is studied and an alternative approach is proposed.

Uniqueness of limit cycles of generalised Lienard systems and predator-prey systems

Another interesting example of how mathematical methods in physics can be applied to the area of ecology is presented. By using a theorem which first appeared in a Russian paper in 1958 for a generalised Lienard system, a uniqueness theorem of limit cycles of a predator-prey system, which includes Lotka-Volterra, Gause and other systems as special cases, is obtained. Several examples show that the theorem is very useful in dealing with the uniqueness problem of limit cycles for certain ecological systems.

THEORY OF RELATIVISTIC EXTENDED OBJECTS

Physical ideas and mathematical methods of molecular and nuclear physics are extended into the relativistic domain. The resulting models are noncanonical modifications of restricted modes of the string. They describe collective motions of relativistic extended objects in terms of variables which are derived from the symmetry group (for the center-of-mass motion) and from the spectrum generating group (for the intrinsic motion). The simplest relativistic model of this kind is used to calculate spectrum and radiative transitions of hadrons.

Book Review: Nonstandard methods in stochastic analysis and mathematical physics

Book Review: Nonstandard methods in stochastic analysis and mathematical physics

The Exponential Accuracy of Fourier and Chebyshev Differencing Methods

It is shown that when differencing analytic functions using the pseudospectral Fourier or Chebyshev methods, the error committed decays to zero at an exponential rate.

Design values of Inconel 617 for high temperature reactors operating at temperatures above 800°C

For extrapolation of the time-dependent stress values the following two methods are introduced: • -extrapolation with constitutive equations • -extrapolation with time-temperature parameters. The first method is a system of coupled equations with creep and damaging parts (a physical-mathematical method), the second is an extrapolation method which combines three well-known time-temperature parameters (a statistical method). On the basis of the latter method time-dependent stress intensity values for a high temperature reactor material are derived and as an example the time safety against rupture is shown for a heat exchanger tube at a temperature of 950°C.