An approximate Taylor method for Stochastic Functional Differential Equations via polynomial condition

The subject of this paper is an analytic approximate method for a class of stochastic functional differential equations with coefficients that do not necessarily satisfy the Lipschitz condition nor linear growth condition but they satisfy some polynomial conditions. Also, equations from the observed class have unique solutions with bounded moments. Approximate equations are defined on partitions of the time interval and their drift and diffusion coefficients are Taylor approximations of the coefficients of the initial equation. Taylor approximations require Fréchet derivatives since the coefficients of the initial equation are functionals. The main results of this paper are the Lp and almost sure convergence of the sequence of the approximate solutions to the exact solution of the initial equation. An example that illustrates the theoretical results and contains the proof of the existence, uniqueness and moment boundedness of the approximate solution is displayed.

On the approximations of solutions to stochastic differential equations under polynomial condition

The subject of this paper is an analytic approximate method for a class of stochastic differential equations with coefficients that do not necessarily satisfy the Lipschitz and linear growth conditions but behave like a polynomials. More precisely, equations from the observed class have unique solutions with bounded moments and their coefficients satisfy polynomial condition. Approximate equations are defined on partitions of a time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. The rate of Lp convergence increases when degrees in Taylor approximations of coefficients increase. At the end of the paper, an example is provided to support the main theoretical result.

Transiente diffusion equation: an alternative finite difference approach

This work is concerned with development of an alternative finite difference method approach to the solution of the transient diffusion equation. In this approach is assumed that the potential function has a linear variation in some time interval. Thus an integral with respect to time is carried out in the initial diffusion equation. A constant time weighting function is adopted. The time integration reduces the order of the time derivative that appears in the initial equation. Two numerical examples are presented to verify the accuracy and applicability of the proposed approaches. The results are compared with the standard corresponding analytical solutions.

Correlation in Disordered Systems

In this paper, authors consider the theoretical aspects of the dynamics of liquid metals and the issues of correlations in disordered systems. The goal is to demonstrate that in choosing variables it is necessary to bring the equation of Markov processes to an approximation in which among the selected variables there are long-range components. However, this choice should include persistent variables. In numerous studies it is shown that selection cannot be made for the initial equation of motion with dynamic variables, and memory function analysis is required. As always, it contains a high-order memory function that can quickly decrease, and finally it can be again converted by the Markov process. Ways of simplifying the equation of Markov processes are given. In this work, authors derive the connection between the Fourier transform of the autocorrelation function and the imaginary part of the dynamic susceptibility and the result of the fluctuation-dissipation theorem.

A hypersingular integro-differential equation of the Euler type

In this paper, we study an integro-differential equation on a closed curve located on the complex plane. The integrals included in the equation are understood as a finite part by Hadamard. The coefficients of the equation have a particular structure. The analytical continuation method is applied. The equation is reduced to a boundary value linear conjugation problem for analytic functions and linear Euler differential equations in the domains of the complex plane. Solutions of the Euler equations, which are unambiguous analytical functions, are sought. The conditions of solvability of the initial equation are given explicitly. The solution of the initial equation obtained under these conditions is also given explicitly. Examples are considered.

A World of Mannequins

This chapter looks into Mario Bava's aesthetic sensibility and synthesis of the stylistic and thematic thread that underlies the film Blood and Black Lace (6 donne per l'assassino). It explains how viewers of the film associated the gimmick of using mannequins with the opening pages of Gialli Mondadori, which include a list of characters and their qualification in order to give the reader a “human map” on which to rely when reading. It also discusses how the same concept of a human map is reinvented in the film Blood and Black Lace in a visual key with a surrealistic effect through the use of mannequins. The chapter analyzes the poses of mannequins that are considered artificial, enigmatic, and disturbing. It highlights how mannequins reiterate the initial equation between men and anthropomorphic objects, in which mannequins provide cautionary shadows for actors and actors are sometimes caught in the act of making the same gesture as the mannequins.

Some analytic approximations for backward stochastic differential equations

We consider an analytic iterative method to approximate the solution of the backward stochastic differential equation of general type. More precisely, we define a sequence of approximate equations and give sufficient conditions under which the approximate solutions converge with probability one and in pth moment sense, p ? 2, to the solution of the initial equation under Lipschitz condition. The Z-algorithm for this iterative method is introduced and some examples are presented to illustrate the theory.

Pencils of circles with a straight line and circle as the basic elements

Pencils of circles with are a straight line and a circle as the basic elements are investigated. Three cases of arrangement of a basic straight line and a circle are considered: when the straight line does not intersect a circle, when the straight line and a circle have one generic point, and when the straight line intersects a circle in two points. A parameter is entered and the equations of new pencils of circles are registered. By means of mathematical manipulations the obtained equations are given to the initial equation of a circle. Different values are attached to the parameter and the circles belonging to new pencils are constructed. Based on the obtained graphs it is concluded that the pencil with not intersecting basic straight line and a circle forms a hyperbolic pencil of circles, a pencil with a basic straight line and a circle having one generic point forms a parabolic pencil, and a pencil with the intersecting basic straight line and a circle forms an elliptic pencil.

PROSES PEMECAHAN MASALAH MATEMATKA SISWA BERKEMAMPUAN TINGGI BERDASARKAN LANGKAH POLYA

The purpose of this study is to determine the process of solving the problem of students who are high-ability based on Polya step. This research uses the qualitative approach. The subject in this study was a high-ability student. The results of this study indicate that the ability to solve the problem of high-ability students based on Polya steps are: (1) Understanding the problem begins by understanding the vocabulary, identifying all the facts in the form of existing information data, connecting all information from the identification result, and ending by identifying the question ; (2) A plan for splitting begins with the selection of operations and establishes the formulation of equations; (3) Implement the plan of splitting; (4) Re-checking of the settlement result obtained by substituting the result obtained to the initial equation.

CONSTRUCTIVE ASPECTS OF INTEGRATION. ISSUES OF THE MATERIAL INTEGRAL EQUATIONS

The differential equation with the continuous disturbance effect in the right part is considered and solid approaches to the initial equation solution, to the integration of an initial problem in relation to the variant of o gravitational interaction model represented by the differential equation in a normal form are proposed.