Applications Of Functional Equations Research Articles

Cobordisms, Manifolds with Torus Action, and Functional Equations

The paper is devoted to applications of functional equations to well-known problems of compact torus actions on oriented smooth manifolds. These include the problem of Hirzebruch genera of complex cobordism classes that are determined by complex, almost complex, and stably complex structures on a fixed manifold. We consider actions with connected stabilizer subgroups. For each such action with isolated fixed points, we introduce rigidity functional equations. This is based on the localization theorem for equivariant Hirzebruch genera. We consider actions of maximal tori on homogeneous spaces of compact Lie groups and torus actions on toric and quasitoric manifolds. The arising class of equations contains both classical and new functional equations that play an important role in modern mathematical physics.

Effective elastic constants of hexagonal array of soft fibers

Analytical formulae for the effective elastic constants of 2D composites with soft unidirectional fibers arranged in the hexagonal array are obtained for arbitrary concentration of fibers, from dilute case to percolation. It is supposed that every section of composites perpendicular to fibers is the hexagonal array with circular holes or soft inclusions. First, a polynomial approximation in concentration is obtained by application of functional equations. Further, an asymptotic analysis is applied to the obtained polynomial with using of the known asymptotic formulae in percolation regime. Finally, the formula for the effective shear modulus is suggested. It can be applied to the typical matrices made from ceramics, metals and polymers, and to other materials as well, e.g., for resin and thallium.

Some integral type fixed point theorems in Non-Archimedean Menger PM-Spaces with common property (E.A) and application of functional equations in dynamic programming

In this paper, we prove some integral type common fixed point theorems for weakly compatible mappings in Non-Archimedean Menger PM-spaces employing common property (E.A). Some examples are furnished which demonstrate the validity of our results. We extend our main result to four finite families of self-mappings employing the notion of pairwise commuting. Moreover, we give an application which supports the usability of our main theorem.

Modeling incremental pavement roughness using functional network

Incremental roughness prediction is a critical component of decision making of any pavement management systems, therefore, proper estimation is of paramount importance. This paper presents the application of functional equations and networks to incremental roughness prediction of flexible pavement. In the functional networks, neuron functions are multivariate and multiargumentative. Functional equations form the basis of functional networks, therefore, established theorem in functional equations are easily applicable in the analysis. The model is developed from validated set of incremental and interactive pavement distress functions in the highway design and maintenance standard models (HDM). The models proposed and developed are intended for use in infrastructure management (pavement) applications and as a performance model for pavement design. The paper presents a computational procedure of the functional network using the serial associative functional network. The functional equation and networks approach use the domain knowledge and data for developing the roughness models.Key words: pavement roughness, functional equation, functional networks, neural networks, pavement management.

Dynamic programming approach to a Fermat type principle for heat flow

We consider nonlinear heat conduction satisfying a variational principle of Fermat type in the case of stationary heat flow. We review origins of a physical theory and transform it into a formalism consistent with irreversible thermodynamics, where the theory emerges as a consequence of the theorem of minimum entropy production. Applications of functional equations and the Hamilton–Bellman–Jacobi equation are effective when Bellman’s method of dynamic programming is applied to propagation of thermal rays. Potential functions describing minimum resistance are obtained by analytical and numerical methods. For the latter, approximation schemes are developed. Differences between propagation of thermal and optical rays are discussed and it is shown that while simplest optical rays can be described by Riemmanian geometry, it is rather Finslerian geometry that is valid for thermal rays.

Some recent applications of functional equations to the social and behavioral sciences. Further problems

Some recent applications of functional equations to the social and behavioral sciences. Further problems

An application of functional equations to the analysis of the invariance identities of classical gauge field theory

The equations of motion for a particle in a classical gauge field are derived from the invariance identities 2 and basic assumptions about the Lagrangian. They are found to be consistent with the equations of some other approaches to classical gauge-field theory, and are expressed in terms of a set of undetermined functions Eα. The functions Eα are found to satisfy a system of differential equations which has the same formal structure as a system of equations from Yang-Mills theory. 3 These results are obtained by a new method which applies techniques from the theory of functional equations to deduce the way in which the arguments of the Lagrangian must combine. The method constitutes an aid for obtaining the equations of motion when a non-gauge-invariant Lagrangian is chosen, and it is assumed that the equations of motion can be written in a gauge-invariant manner.

Functional equations in several variables: with applications to mathematics, information theory, and to the natural and social sciences

Preface Further information 1. Axiomatic motivation of vector addition 2. Cauchy's equation: Hamel basis 3. Three further Cauchy equations: an application to information theory 4. Generalizations of Cauchy's equations to several multiplace vector and matrix functions: an application to geometric objects 5. Cauchy's equations for complex functions: applications to harmonic analysis and to information measures 6. Conditional Cauchy equations: an application to geometry and a characterization of the Heaviside functions 7. Addundancy, extensions, quasi-extensions and extensions almost everywhere: applications to harmonic analysis and to rational decision making 8. D'Alembert's functional equation: an application to noneuclidean mechanics 9. Images of sets and functional equations: applications to relativity theory and to additive functions bounded on particular sets 10. Some applications of functional equations in functional analysis, in the geometry of Banach spaces and in valauation theory 11. Characterizations of inner product spaces: an application to gas dynamics 12. Some related equations and systems of equations: applications to combinatorics and Markov processes 13. Equations for trigonometric and similar functions 14. A class of equations generalizing d'Alembert and Cauchy Pexider-type equations 15. A further generalization of Pexider's equation: a uniqueness theorem: an application to mean values 16. More about conditional Cauchy equations: applications to additive number theoretical functions and to coding theory 17. Mean values, mediality and self-distributivity 18. Generalized mediality: connection to webs and nomograms 19. Further composite equations: an application to averaging theory 20. Homogeneity and some generalizations: applications to economics 21. Historical notes Notations and symbols Hints to selected 'exercises and further results' Bibliography Author index Subject index.

Some recent applications of functional equations to geometry

The results of applications of solutions of functional equations or of methods used in the theory of functional equations to the following subjects are discussed in this paper. Determination of all Cremona transformations which reduce linear transformations with triangular matrices to translations. One-parameter subsemigroups of affine transformations and their homomorphisms. Extensions of homomorphisms from sub-semigroups to groups generated by them. Determination of all collineations on subsets of general projective planes and their extensions to the entire plane.

The utility of a communication channel and applications to suboptimal information handling procedures

This paper demonstrates the applicability of the functional equations of dynamic programming to information theory problems. Yielding the same results as those obtainable by Shannon's equations, the functional equations can be modified also to consider the many restrictions enforced upon information systems by the real world. A result of the application of functional equations to systems operating under suboptimum conditions is that the information rate of a system is dependent upon the manner in which the information is used. The Kelly concept-the gain of a gambler who wagers his capital on the outcome of a communication channel — is used to determine the information rate of the channel. The mathematical analysis follows the stochastic multistage decision process technique of Bellman and Kalaba. Together with some extensions by the author, the Kelly-Bellman-Kalaba model of communication is repeated. The models are analyzed for the optimum case and examined for various suboptimum conditions. The gambler's betting policy is analogous to information usage; restrictions upon this policy affect the information rate of the system. They can require that the policy which is best under optimum conditions be replaced by other policies which, although inferior in the ideal case, are better able to compensate for the restrictions. A null zone reception system first analyzed by Bloom and others is reanalyzed to provide a concrete example of the latitude of operation allowed by the functional equation approach. Bloom's analysis assumed that the system operates under optimum conditions. His results are duplicated, and their expression indicates their alteration by suboptimum conditions. An appendix expresses the results of this paper in the form used by Bloom.