Form Of Functional Equation Research Articles

The polynomial-like iterative equation for PM functions

The polynomial-like iterative equation is an important form of functional equations, in which iterates of the unknown function are linked in a linear combination. Most of known results were given for the given function to be monotone. We discuss this equation for continuous solutions in the case that the given function is a PM (piecewise monotone) function, a special class of non-monotonic functions. Using extension method, we give a general construction of solutions for the polynomial-like iterative equation.

A novel approach for solving fractional Fisher equation using differential transform method

In the present paper, an analytic solution of nonlinear fractional Fisher equation is deduced with the help of the powerful differential transform method (DTM). To illustrate the method, two examples have been prepared. The method for this equation has led to an exact solution. The reliability, simplicity and cost-effectiveness of the method are confirmed by applying this method on different forms of functional equations.

Functional Characterizations of Basic Properties of Utility Representations

Consider uncertain alternatives for which an event has two consequences (binary gambles, “gambles” for short) and over them an operation of joint receipt which need not be closed and may be non-commutative. The two structures are linked by a distributivity property called segregation and a preference order. Utility functions order nonnegative numbers to consequences and gambles. Utility representations describe how the utility of a gamble depends on the utilities of consequences and on the “weight” of the event (a number in [0,1], depending on the event). Functional characterizations give necessary and sufficient conditions, often in form of functional equations, for certain properties of representations. We first give a functional characterization of the often postulated event commutativity stating that two events can be interchanged in special composite gambles where one outcome is a consequence but the other is itself a gamble. A utility representation is separable if it is multiplicative for gambles with one consequence having 0 utility. We give three more specific characterizations of separable representations by segregation, by homogeneity and event commutativity, and by homogeneity and segregation, and show that in the last case event commutativity follows.

Moisture permeability data presented as a mathematical relationship

A detailed statistical analysis is presented of the correlation between moisture permeability and relative humidity based on experimental data obtained by the authors and a variety of other investigators using standard gravimetric measurement methods. In total, five proposed mathematical formulations are examined for five different types of material commonly used in building construction. The suitability of each formulation is investigated using non-linear regression techniques including an analysis of collinearity and relative standard error. This statistical approach allows the best functional form of equation for describing the variability of moisture permeability with relative humidity to be identified for building materials. A database covering a total of ten materials is then presented on the basis of this equation. Cet article présente une analyse statistique détaillée de la corrélation entre la perméabilité à l'humidité et l'humidité relative; elle est fondée sur des données expérimentales obtenues par les auteurs et par d'autres chercheurs qui ont utilisé des méthodes standard de mesure gravimétrique. Au total, on a examiné cinq formulations proposées pour cinq types différents de matériaux couramment utilisés pour la construction de bâtiments. Le bien-fondé de chaque formulation est étudié au moyen de techniques de régression nonlinéaires, notamment une analyse de colinéarité et d'erreur standard relative. Cette méthode permet d'établir meilleure forme fonctionnelle d'équation pour décrire la variabilité de la perméabilité à l'humidité en function de l'humidité relative pour des matériaux de construction. Les auteurs s'appuient sur cette équation pour présenter ensuite une base de données couvrant un total de dix matériaux.

Invariant imbedding and parallelism in dynamic programming for feedback control

This paper investigates related areas such as invariant imbedding, state feedback, and numerical and parallel methods in order to specify the range of control problems amenable to a dynamic programming approach. Several forms of functional equations are classified according to different applications of the invariant imbedding principle and corresponding closed-loop control structures. Computational methods to implement these algorithms are described, and a complexity analysis is made to determine their effectiveness and to explain their application domain. The design of parallel algorithms is also considered. Alternative descriptions are compared; the frame of a distributed computational method delivering tabular feedback solutions is highlighted.

Program specification and synthesis using systems of equations over data structures

We consider the analysis of specifications in the form of functional equations and synthesis of programs from such equational specifications. A systematic discussion of previously known and new methods for the solution of these analysis and synthesis problems is presented.

Star-triangle relations in the exactly solvable statistical models

A regular method for analysis of lattice spin models with a nearest neighbour interaction is proposed. Star-triangle relations in the form of functional equations are used. Parametric families of transfer matrices commuting due to star-triangle relations are constructed. The eigenvalues of transfer matrices as functions of the spectral parameter are shown to obey two functional equations. The solution of these equations for the maximal eigenvalue yields the partition function of the model. The method is applied for evaluation of the partition function of the critical Potts models, the Ising model, the Ashkin-Teller model equivalent to the eight-vertex model.

Dynamic Programming Formulation of the Redundancy Allocation Problem

Summary Dynamic programming formulation for obtaining the optimum number of redundancies at each stage in a system is explained in detail. It is also emphasized and illustrated by examples, that the ‘summation’ form of functional equations, as suggested in the paper, provide a faster solution to an otherwise computationally voluminous method.

Prior Probabilities

In decision theory, mathematical analysis shows that once the sampling distribution, loss function, and sample are specified, the only remaining basis for a choice among different admissible decisions lies in the prior probabilities. Therefore, the logical foundations of decision theory cannot be put in fully satisfactory form until the old problem of arbitrariness (sometimes called "subjectiveness") in assigning prior probabilities is resolved. The principle of maximum entropy represents one step in this direction. Its use is illustrated, and a correspondence property between maximum-entropy probabilities and frequencies is demonstrated. The consistency of this principle with the principles of conventional "direct probability" analysis is illustrated by showing that many known results may be derived by either method. However, an ambiguity remains in setting up a prior on a continuous parameter space because the results lack invariance under a change of parameters; thus a further principle is needed. It is shown that in many problems, including some of the most important in practice, this ambiguity can be removed by applying methods of group theoretical reasoning which have long been used in theoretical physics. By finding the group of transformations on the parameter space which convert the problem into an equivalent one, a basic desideratum of consistency can be stated in the form of functional equations which impose conditions on, and in some cases fully determine, an "invariant measure" on the parameter space.