NIST RANDOMNESS TESTS (IN)DEPENDENCE

In this paper we focus on three open questions regarding NIST SP 800-22 randomness test: the probability of false acceptance, the number of minimum sample size to achieve a given probability error and tests independence. We shall point out statistical testing assumptions, source of errors, sample constructions and a computational method for determining the probability of false acceptance and estimating the correlation between the statistical tests

Generating Functions for the Mean Value of a Function on a Sphere and Its Associated Ball in Rn

We define two functions which determine the properties and the representation of the mean value of a function on a ball and on its associated sphere. Using these two functions, we obtain Pizzetti's formula in Rn as well as a similar formula for the mean value of a function on the ball associated to the sphere. We also give the expressions of the remainders in these two formulas, using the surface integral on a sphere

Symmetries for Nonconservative Field Theories on Time Scale

Symmetries and their associated conserved quantities are of great importance in the study of dynamic systems. In this paper, we describe nonconservative field theories on time scales—a model that brings together, in a single theory, discrete and continuous cases. After defining Hamilton’s principle for nonconservative field theories on time scales, we obtain the associated Lagrange equations. Next, based on the Hamilton’s action invariance for nonconservative field theories on time scales under the action of some infinitesimal transformations, we establish symmetric and quasi-symmetric Noether transformations, as well as generalized quasi-symmetric Noether transformations. Once the Noether symmetry selection criteria are defined, the conserved quantities for the nonconservative field theories on time scales are identified. We conclude with two examples to illustrate the applicability of the theory

A numerical method to solve fractional Fredholm-Volterra integro-differential equations

The Goolden ratio is famous for the predictability it provides both in the microscopic world as well as in the dynamics of macroscopic structures of the universe. The extension of the Fibonacci series to the Fibonacci polynomials gives us the opportunity to use this powerful tool in the study of Fredholm-Volterra integro-differential equations. In this paper, we define a new hybrid fractional function consisting of block-pulse functions and Fibonacci polynomials (FHBPF). For this, in the Fibonacci polynomials we perform the transformation x  → xα, with α a real parameter. In the method developed in this paper, we propose that the unknown function Dαf(x) be written as a linear combination of FHBPF. We consider the fractional derivative Dα in the Caputo sense. Using theoretical considerations, we can write both the function f(x) and other involved functions of type Dβf(x) on the same basis. For this operation, we have to define an integral operator of Riemann-Liouville type associated to FHBPF, and with the help of hypergeometric functions, we can express this operator exactly. All these ingredients together with the collocation in the Newton-Cotes nodes transform the integro-differential equation into an algebraic system that we solve by applying Newton’s iterative method. We conclude the paper with some examples to demonstrate that the proposed method is simple to implement and accurate. There are situations when by simply considering α ≠ 1, we obtain an improvement in accuracy by 12 orders of magnitude

The univalence conditions for a general integral operator

AbstractIn this paper we extend a general integral operator which was introduced in the paper (Breaz, 2010) [3]. We denote this operator by Hγ1,γ2,…,γ[|η|],β,η. For this integral operator we show some conditions of univalence on the class of analytical functions

Fractional Complex Euler–Lagrange Equation: Nonconservative Systems

Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding the stationary values associated with the fractional integral of complex order. The complex Hamiltonian obtained from the Lagrangian is suitable for describing nonconservative systems. We conclude by presenting the conserved quantities attached to Noether symmetries corresponding to complex systems. We illustrate the theory with the aid of the damped oscillatory system

Enhancing the Accuracy of Solving Riccati Fractional Differential Equations

In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing x with xα, with positive α. Fractional derivatives are in the Caputo sense. With the help of incomplete beta functions, we are able to build exactly the Riemann–Liouville fractional integral operator associated with FOHBPB. This operator, together with the Newton–Cotes collocation method, allows the reduction of fractional differential equations to a system of algebraic equations, which can be solved by Newton’s iterative method. The simplicity of the method recommends it for applications in engineering and nature. The accuracy of this method is illustrated by five examples, and there are situations in which we obtain accuracy eleven orders of magnitude higher than if α=1

Distribution theory: with applications in engineering and physics

In this comprehensive monograph, the authors apply modern mathematical methods to the study of mechanical and physical phenomena or techniques in acoustics, optics, and electrostatics, where classical mathematical tools fail.They present a general method of approaching problems, pointing out different aspects and difficulties that may occur. With respect to the theory of distributions, only the results and the principle theorems are given as well as some mathematical results. The book also systematically deals with a large number of applications to problems of general Newtonian mechanics

Distribution theory : with applications in engineering and physics

xii, 381 p.; 24,5 c