General Cauchy Research Articles

On certain new CAUCHY–TYPE fracitioanl integral inequalities and OPIAL–TYPE fractional derivative inequalities

The aim of this paper is to establish several new fractional integral and derivative inequalities for non-negative and integrable functions. These inequalities related to the extension of general Cauchy type inequalities and involving Saigo, Riemann-Louville type fractional integral operators together with multiple Erdelyi-Kober operator. Furthermore the Opial-type fractional derivative inequality involving H-function is also established. The generosity of H-function could leads to several new inequalities that are of great interest of future research.

Generalized complex variable boundary integral equation for stress fields and torsional rigidity in torsion problems

Theory of complex variables is a very powerful mathematical technique for solving two-dimensional problems satisfying the Laplace equation. On the basis of the conventional Cauchy integral formula, the conventional complex variable boundary integral equation (CVBIE) can be constructed. The limitation is that the conventional CVBIE is only suitable for holomorphic (analytic) functions, however. To solve for a complex-valued harmonic-function pair without satisfying the Cauchy–Riemann equations, we propose a new boundary element method (BEM) based on the general Cauchy integral formula. The general Cauchy integral formula is derived by using the Borel–Pompeiu formula. The difference between the present CVBIE and the conventional CVBIE is that the former one has two boundary integrals instead of only one boundary integral in the latter one. When the unknown field is a holomorphic function, the present CVBIE can be reduced to the conventional CVBIE. Therefore, the conventional Cauchy integral formula can be viewed as a special case applicable to a holomorphic function. To examine the present CVBIE, we consider several torsion problems in this paper since the two shear stress fields satisfy the Laplace equation but do not satisfy the Cauchy–Riemann equations. Using the present CVBIE, we can directly solve the stress fields and the torsional rigidity simultaneously. Finally, several examples, including a circular bar containing an eccentric inclusion (with dissimilar materials) or hole, a circular bar, elliptical bar, equilateral triangular bar, rectangular bar, asteroid bar and circular bar with keyway, were demonstrated to check the validity of the present method.

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

The paper studies the existence, uniqueness, regularity and the approximation of solutions to the reaction–diffusion equation endowed with a general nonlinear regular potential and Cauchy–Neumann boundary conditions. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic equation.

Generalization of Cauchy’s characteristics method to construct smooth solutions to Hamilton-Jacobi-Bellman equations in optimal control problems with singular regimes

An approach to constructing optimal control synthesis, based on studying the allocation of characteristics to the Cauchy problem for the Hamilton-Jacobi-Bellman (HJB) equation (i.e., determining how the extended phase space is filled with these characteristics), is proposed. A method for finding a global solution to the Cauchy problem for the HJB equation by setting boundary conditions on the surface of singular characteristics corresponding to singular optimal controls is developed. Control is considered to be one-dimensional and linear within the system. In describing the method, it is assumed that this surface is unique, and that the switching of any admissible process satisfying Pontryagin’s maximum principle can occur only on it and not more than once. The corresponding sufficient conditions are obtained, and the smoothness of the cost function constructed in this way is verify. The resulting approach is demonstrated via the example of a mathematical model for the treatment of viral infections.

Polar angle tangent vectors follow Cauchy distributions under spherical symmetry

Abstract Let X = ( X 1 , … , X n ) ′ follow a spherically or elliptically symmetric distribution centered at zero, and Y i = X i + 1 / X 1 , Y = ( Y 1 , … , Y n − 1 ) ′ . It is shown that under spherical symmetry Y has a symmetric Cauchy distribution and under elliptical symmetry a general Cauchy distribution. Geometrically, Y is the tangent (or cotangent) vector of the polar angle θ 1 . The simple case of one ratio is treated in Arnold and Brockett (1992), Jones (1999, 2008). Moreover, it is shown that n − 1 cot θ 1 follows the t n − 1 distribution, so that the normal theory distributions of Student’s t and correlation coefficient r hold under spherical symmetry.

General Cauchy–Lipschitz theory for Δ-Cauchy problems with Carathéodory dynamics on time scales

The aim of this paper was to complete some aspects of the classical Cauchy–Lipschitz (or Picard–Lindelöf) theory for general nonlinear systems posed on time scales. Despite a rich literature on Cauchy–Lipschitz type results on time scales, most of the existing results are concerned with rd-continuous dynamics (and -solutions) and do not cover the framework of general Carathéodory dynamics encountered for instance in control theory with measurable controls (which are not rd-continuous in general). In this paper, our main objective was to study -Cauchy problems with general Carathéodory dynamics. We introduce the notion of absolutely continuous solution (weaker regularity than ) and then the notion of maximal solution. We state and prove a Cauchy–Lipschitz theorem, providing existence and uniqueness of the maximal solution of a given -Cauchy problem under suitable assumptions such as regressivity and local Lipschitz continuity. Three new related issues are also discussed in this paper: the boundary value is not necessarily an initial or a final condition, the solutions are constrained to take their values in a non-empty open subset and the behaviour of maximal solutions at terminal points is studied.

The solution of Maxwell’s equations in multiphysics

We consider the solution of the fully-coupled equations of electromagnetics with fluid flows and structures. The electromagnetic effects are governed by the general Maxwell’s equations, the fluid flows by the Navier–Stokes equations, and the solids and structures by the general Cauchy equations of motion. We present an effective general finite element formulation for the solution of the Maxwell’s equations and demonstrate the coupling to the equations for fluids and structures. For the solution, we can use the electric field and magnetic field intensities, or the electric and magnetic potentials, with advantages depending on the problem solved. We give various example solutions that illustrate the use of the solution procedure.

A solvable string on a Lorentzian surface

It is shown that there are nonlinear sigma models which are Darboux integrable and possess a solvable Vessiot group in addition to those whose Vessiot groups are central extensions of semi-simple Lie groups. They govern harmonic maps between Minkowski space R1,1 and certain complete, non-constant curvature 2-metrics. The solvability of the Vessiot group permits a reduction of the general Cauchy problem to quadrature. We treat the specific case of harmonic maps from Minkowski space into a non-constant curvature Lorentzian 2-metric, λ. Despite the completeness of λ we exhibit a Cauchy problem with real analytic initial data which blows up in finite time. We also derive a hyperbolic Weierstrass representation formula for all harmonic maps from R1,1 into λ.

On general fractional abstract Cauchy problem

This paper is concerned with general fractional Cauchy problems oforder $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ ininfinite-dimensional Banach spaces. A new notion, named generalfractional resolvent of order $0 < \alpha < 1$ and type $0 \leq \beta \leq1$ is developed. Some of its properties are obtained.Moreover, some sufficient conditions are presented to guarantee thatthe mild solutions and strong solutions of homogeneous and inhomogeneous general fractional Cauchy problem exist.An illustrative example is presented.

On regularization and error estimates for non-homogeneous backward Cauchy problem

In this paper we study the general non-homogeneous Backward Cauchy problemut+Au=f,0<t<T,u(T)=g,for positive selfadjoint unbounded operator A on the Hilbert space H. The problem is known to be severely ill-posed. We give extensions of the quasi-boundary methods to the non-homogeneous case. We prove several sharp results on regularizations and error-estimates. Other results, including some explicit convergence rates are proved. Finally applications to the non-homogeneous backward heat equation with Bessel operator are given.

Taylor's Expansion Revisited: A General Formula for the Remainder

We give a new approach to Taylor's remainder formula, via a generalization of Cauchy's generalized mean value theorem, which allows us to include the well-known Schölomilch, Lebesgue, Cauchy, and the Euler classic types, as particular cases.

Some generalization of Cauchy’s and the quadratic functional equations

We find the solutions \({f,g,h \colon S \to H}\) of each of the functional equations $$\sum\limits_{\lambda \in \Lambda} f(x+\lambda y)=|\Lambda| g(x)+h(y),\quad x,y\in S,$$ where (S, +) is an abelian semigroup, Λ is a finite subgroup of the automorphism group of S, (H, +) is an abelian group.

New Generalizations of Cauchy Distribution

The generalized skew-normal distribution introduced by Balakrishnan (2002) is used to obtain new generalizations of univariate Cauchy distribution with two parameters, denoted by GC m, n (a, b) with m and n non-negative integer numbers and a, b ∈ R. For cases (m, n) = (1, 2), (m, n) = (2, 1), (m, n) = (0, 3) and (m, n) = (3, 0) explicit forms of the density functions are derived and compared to previous generalizations of Cauchy and skew-Cauchy distributions.

Electro-osmotic mobility of non-Newtonian fluids

Electrokinetically driven microfluidic devices are usually used to analyze and process biofluids which can be classified as non-Newtonian fluids. Conventional electrokinetic theories resulting from Newtonian hydrodynamics then fail to describe the behaviors of these fluids. In this study, a theoretical analysis of electro-osmotic mobility of non-Newtonian fluids is reported. The general Cauchy momentum equation is simplified by incorporation of the Gouy-Chapman solution to the Poisson-Boltzmann equation and the Carreau fluid constitutive model. Then a nonlinear ordinary differential equation governing the electro-osmotic velocity of Carreau fluids is obtained and solved numerically. The effects of the Weissenberg number (Wi), the surface zeta potential (ψ¯s), the power-law exponent(n), and the transitional parameter (β) on electro-osmotic mobility are examined. It is shown that the results presented in this study for the electro-osmotic mobility of Carreau fluids are quite general so that the electro-osmotic mobility for the Newtonian fluids and the power-law fluids can be obtained as two limiting cases.

A Study of Value‐at‐Risk Based on M‐Estimators of the Conditional Heteroscedastic Models

ABSTRACTIn this paper, we investigate the performance of a class of M‐estimators for both symmetric and asymmetric conditional heteroscedastic models in the prediction of value‐at‐risk. The class of estimators includes the least absolute deviation (LAD), Huber's, Cauchy and B‐estimator, as well as the well‐known quasi maximum likelihood estimator (QMLE). We use a wide range of summary statistics to compare both the in‐sample and out‐of‐sample VaR estimates of three well‐known stock indices. Our empirical study suggests that in general Cauchy, Huber and B‐estimator have better performance in predicting one‐step‐ahead VaR than the commonly used QMLE. Copyright © 2011 John Wiley &amp; Sons, Ltd.

A new convergence proof of the Adomian decomposition method for a mixed hyperbolic elliptic system of conservation laws

In this paper, we propose a new convergence proof of the Adomian’s decomposition method (ADM), applied to the generalized nonlinear system of partial differential equations (PDE’s) based on new formula for Adomian polynomials. The decomposition scheme obtained from the ADM yields an analytical solution in the form of a rapidly convergent series for a system of conservation laws. Systems of conservation laws is presented, we obtain the stability of the approximate solution when the system changes type. We show with an explicit example that the latter property is true for general Cauchy problem satisfying convergence hypothesis. The results indicate that the ADM is effective and promising.

Quantitative approximation by fractional smooth Poisson Cauchy singular operators

In this article we study the very general fractional smooth Poisson Cauchy singular integral operators on the real line, regarding their convergence to the unit operator with fractional rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of the associated right and left Caputo fractional derivatives of the engaged function. Furthermore, we produce a fractional Voronovskaya type of result giving the fractional asymptotic expansion of the basic error of our approximation. We finish with applications. Our operators are not in general positive. We are mainly motivated by Anastassiou (submitted for publication) [1] .

Approximate homomorphisms and derivations between C∗-ternary algebras

In 1940, Ulam proposed the famous Ulam stability problem. In this paper we introduce a general Cauchy–Jensen functional equation and prove the generalized Ulam stability of C∗-ternary homomorphisms and C∗-ternary derivations in C∗-ternary algebras for the general Cauchy–Jensen equation.

Modern system of mathematics and general Cauchy theater in its theoretical foundation

PurposeThe paper aims to show using a different method that any uncountable set is a self‐contradictory non‐set.Design/methodology/approachThe paper discusses the concept.FindingsElsewhere it is shown that in the framework of ZFC, various countable infinite sets are all self‐contradicting non‐sets. In this paper, the authors will generalize the concept of Cauchy theater, and establish the concept of transfinite Cauchy theaters. After that, employing a new method, they will prove that various uncountable infinite sets, as studied in naive set theory and modern axiomatic set theory, are also self‐contradicting non‐sets.Originality/valueThe concept of general Cauchy theater is introduced.

Random data Cauchy theory for supercritical wave equations II: a global existence result

We prove that the subquartic wave equation on the three dimensional ball Θ, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\bigcap_{s<1/2} (H^s(\Theta)\times H^{s-1}(\Theta))$ . We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations, inspired by earlier works by Bourgain [2, 3] on the non linear Schrödinger equation, which allow us to obtain also precise large time dynamical informations on our solutions.