Abstract Cauchy problems three approaches
Melnikova, Irina V
2001-01-01
Although the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems-involved in a numerous mathematical models in physics, engineering, and finance- can be approached in a variety of ways. Historically, there have been three major strategies for dealing with such problems: semigroup, abstract distribution, and regularization methods. Semigroup and distribution methods restore well-posedness, in a modern weak sense. Regularization methods provide approximate solutions to ill-posed problems. Although these approaches were extensively developed over the last decades by many researchers, nowhere could one find a comprehensive treatment of all three approaches.Abstract Cauchy Problems: Three Approaches provides an innovative, self-contained account of these methods and, furthermore, demonstrates and studies some of the profound connections between them. The authors discuss the application of different methods not only to the Cauchy problem that is not well-posed in the classical sense, b...
Beginnings of the Cauchy problem
Choquet-Bruhat, Yvonne
2014-01-01
Brief account of results on the Cauchy problem for the Einstein equations starting with early the works of Darmois and Lichnerowicz and going up to the proofs of the existence and uniqueness of solutions global in space and local in time, in Sobolev spaces, for the general equations either in vacuum or with classical sources.
The Cauchy problem and Hadamard's example
Oleg Yaremko
2013-05-01
Full Text Available Integral representation for harmonic function in the ring. We prove the existence and uniqueness of solutions of the Cauchy problem for the Laplace equation in the circle. Integral representation for the solution of the Cauchy problem was found.
CAUCHY PROBLEM OF ONE TYPE OF ATMOSPHERE EVOLUTION EQUATIONS
HE Juan-xiong; HE You-hua
2006-01-01
One type of evolution atmosphere equations was discussed. It is found that according to the stratification theory, (i) the inertial force has no influence on the criterion of the well-posed Cauchy problem; (ii) the compressibility plays no role on the well-posed condition of the Cauchy problem of the viscid atmosphere equations, but changes the well-posed condition of the viscid atmosphere equations; (iii) this type of atmosphere evolution equations is ill-posed on the hyperplane t = 0 in spite of its compressibility and viscosity; (iv) the Cauchy problem of compressible viscosity atmosphere with still initial motion is ill-posed.
The Cauchy problem of f(R) gravity
Lanahan-Tremblay, Nicolas; Faraoni, Valerio
2007-01-01
The initial value problem of metric and Palatini f(R)gravity is studied by using the dynamical equivalence between these theories and Brans-Dicke gravity. The Cauchy problem is well-formulated for metric f(R)gravity in the presence of matter and well-posed in vacuo. For Palatini f(R)gravity, instead, the Cauchy problem is not well-formulated.
Giovangigli, V.; Graille, B. [Ecole Polytechnique, CMAP, 91 - Palaiseau (France)
2005-01-01
We investigate a system of partial differential equations modeling dissipative plasmas. Transport fluxes are anisotropic linear combinations of gradients and also include zeroth order contributions due to electromagnetic forces. There are also source terms depending on the solution gradient. By using entropic variables, we first recast the system in a partially symmetric form and next in the form of a quasilinear partially symmetric hyperbolic-parabolic system. Using a result of Vol'Pert and Hudjaev, we prove local existence and uniqueness of a bounded smooth solution to the Cauchy problem. (authors)
On the Cauchy Problem Describing an Electron-Phonon Interaction
Jo(a)o-Paulo DIAS; Mário FIGUEIRA; Filipe OLIVEIRA
2011-01-01
In this paper, a model is derived to describe a quartic anharmonic interatomic interaction with an external potential involving a pair electron-phonon. The authors study the corresponding Cauchy Problem in the semilinear and quasilinear cases.
Form methods for autonomous and non-autonomous Cauchy problems
Thomaschewski, Sonja
2003-01-01
Die vorliegende Dissertation stellt einige Resultate vor, die aus Untersuchungen von Formmethoden hervorgingen. Diese spielen eine wichtige Rolle für Evolutionsgleichungen, wenn sich diese als abstraktes Cauchy-Problem schreiben lassen, bei dem der lineare Operator zu einer Sesquilinearform assoziiert ist. Dies ist der Fall für die meisten partiellen Differentialgleichungen. Falls diese Form dicht definiert, stetig und elliptisch ist, so ist das zugehörige abstrakte Cauchy-Problem wohlgestell...
Lucio R. Berrone
2005-01-01
Full Text Available The notion of invariance under transformations (changes of coordinates of the Cauchy mean-value expression is introduced and then used in furnishing a suitable two-variable version of a result by L. Losonczi on equality of many-variable Cauchy means. An assessment of the methods used by Losonczi and Matkowski is made and an alternative way is proposed to solve the problem of representation of two-variable Cauchy means.
Cauchy problem for Laplace equation: An observer based approach
Majeed, Muhammad Usman
2013-10-01
A method to solve Cauchy Problem for Laplace equation using state observers is proposed. It is known that this problem is ill-posed. The domain under consideration is simple lipschitz in 2 with a hole. The idea is to recover the solution over whole domain from the observations on outer boundary. Proposed approach adapts one of the space variables as a time variable. The observer developed to solve Cauchy problem for the Laplace\\'s equation is compuationally robust and accurate. © 2013 IEEE.
Existence and Regularity for Boundary Cauchy Problems with Infinite Delay
Jung-Chan Chang
2014-01-01
Full Text Available The aim of this work is to investigate a class of boundary Cauchy problems with infinite delay. We give some sufficient conditions ensuring the uniqueness, existence, and regularity of solutions. For illustration, we apply the result to an age dependent population equation, which covers some special cases considered in some recent papers.
Existence and Regularity for Boundary Cauchy Problems with Infinite Delay
Jung-Chan Chang
2014-01-01
The aim of this work is to investigate a class of boundary Cauchy problems with infinite delay. We give some sufficient conditions ensuring the uniqueness, existence, and regularity of solutions. For illustration, we apply the result to an age dependent population equation, which covers some special cases considered in some recent papers.
Applications of elliptic Carleman inequalities to Cauchy and inverse problems
Choulli, Mourad
2016-01-01
This book presents a unified approach to studying the stability of both elliptic Cauchy problems and selected inverse problems. Based on elementary Carleman inequalities, it establishes three-ball inequalities, which are the key to deriving logarithmic stability estimates for elliptic Cauchy problems and are also useful in proving stability estimates for certain elliptic inverse problems. The book presents three inverse problems, the first of which consists in determining the surface impedance of an obstacle from the far field pattern. The second problem investigates the detection of corrosion by electric measurement, while the third concerns the determination of an attenuation coefficient from internal data, which is motivated by a problem encountered in biomedical imaging.
The stability for the Cauchy problem for elliptic equations
Alessandrini, Giovanni; Rosset, Edi; Vessella, Sergio
2009-01-01
We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.
The Cauchy problem for higher order abstract differential equations
Xiao, Ti-Jun
1998-01-01
This monograph is the first systematic exposition of the theory of the Cauchy problem for higher order abstract linear differential equations, which covers all the main aspects of the developed theory. The main results are complete with detailed proofs and established recently, containing the corresponding theorems for first and incomplete second order cases and therefore for operator semigroups and cosine functions. They will find applications in many fields. The special power of treating the higher order problems directly is demonstrated, as well as that of the vector-valued Laplace transforms in dealing with operator differential equations and operator families. The reader is expected to have a knowledge of complex and functional analysis.
Chen Guowang; Xue Hongxia
2008-01-01
In this article, the existence, uniqueness and regularities of the global gener-alized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equation utt -uxx-auxxtt+bux4 - duxxt= f(u)xx are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.
Oscillatory solutions of the Cauchy problem for linear differential equations
Gro Hovhannisyan
2015-06-01
Full Text Available We consider the Cauchy problem for second and third order linear differential equations with constant complex coefficients. We describe necessary and sufficient conditions on the data for the existence of oscillatory solutions. It is known that in the case of real coefficients the oscillatory behavior of solutions does not depend on initial values, but we show that this is no longer true in the complex case: hence in practice it is possible to control oscillatory behavior by varying the initial conditions. Our Proofs are based on asymptotic analysis of the zeros of solutions, represented as linear combinations of exponential functions.
On the Cauchy problem for the damped Boussinesq equation
Varlamov, Vladimir
1996-01-01
A classic solution to the Cauchy problem for the damped Boussinesq equation $u_{tt}-2Bu_{txx}=-\\alpha u_{xxxx}+u_{xx}-\\beta(u^2)_{xx}$, $x\\in\\Bbb R^1$, $t>0$, $\\alpha, B=\\text{const}>0$, $\\beta=\\text{const}\\in\\Bbb R^1$, with small initial data is constructed by means of the application of both the spectral and perturbation theories. Large time asymptotics of this solution are obtained. Its main term accounts for two solitons traveling in opposite directions. Each of th...
Cauchy problems of pseudo-parabolic equations with inhomogeneous terms
Li, Zhongping; Du, Wanjuan
2015-12-01
This paper deals with Cauchy problems of pseudo-parabolic equations with inhomogeneous terms. The aim of the paper is to study the influence of the inhomogeneous term on the asymptotic behavior of solutions. We at first determine the critical Fujita exponent and then give the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity. Furthermore, the precise estimate of life span for the blow-up solution is obtained. Our results show that the asymptotic behavior of solutions is seriously affected by the inhomogeneous term.
Cauchy problem in spacetimes with closed timelike curves
The laws of physics might permit the existence, in the real Universe, of closed timelike curves (CTC's). Macroscopic CTC's might be a semiclassical consequence of Planck-scale, quantum gravitational, Lorentzian foam, if such foam exists. If CTC's are permitted, then the semiclassical laws of physics (the laws with gravity classical and other fields quantized or classical) should be augmented by a principle of self-consistency, which states that a local solution to the equations of physics can occur in the real Universe only if it can be extended to be part of a global solution, one which is well defined throughout the (nonsingular regions of) classical spacetime. The consequences of this principle are explored for the Cauchy problem of the evolution of a classical, massless scalar field Φ (satisfying □Φ=0) in several model spacetimes with CTC's
A NOTE ON "THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS"
Hongxia GUO; Guowang CHEN
2013-01-01
In this article,we prove that the Cauchy problem for a N-dimensional system of nonlinear wave equations utt-a△utt =△f(u,v),x ∈ RN,t ＞ 0,vtt-a△vtt =△g(u,v),x ∈ RN,t ＞ 0 admits a unique global generalized solution in C3([0,∞); Wm,p(RN) ∩ L∞(RN) f∩ L2(RN))(m ≥ 0 is an integer,1 ≤ p ≤ ∞) and a unique global classical solution in C3([0,∞); Wm,p ∩ L∞ ∩ L2) (m ＞ 2 + N/p),the sufficient conditions of the blow up of the solution in finite time are given,and also two examples are given.
A Generalized Cauchy Distribution Framework for Problems Requiring Robust Behavior
Carrillo RafaelE
2010-01-01
Full Text Available Statistical modeling is at the heart of many engineering problems. The importance of statistical modeling emanates not only from the desire to accurately characterize stochastic events, but also from the fact that distributions are the central models utilized to derive sample processing theories and methods. The generalized Cauchy distribution (GCD family has a closed-form pdf expression across the whole family as well as algebraic tails, which makes it suitable for modeling many real-life impulsive processes. This paper develops a GCD theory-based approach that allows challenging problems to be formulated in a robust fashion. Notably, the proposed framework subsumes generalized Gaussian distribution (GGD family-based developments, thereby guaranteeing performance improvements over traditional GCD-based problem formulation techniques. This robust framework can be adapted to a variety of applications in signal processing. As examples, we formulate four practical applications under this framework: (1 filtering for power line communications, (2 estimation in sensor networks with noisy channels, (3 reconstruction methods for compressed sensing, and (4 fuzzy clustering.
Origins and development of the Cauchy problem in general relativity
Ringström, Hans
2015-06-01
The seminal work of Yvonne Choquet-Bruhat published in 1952 demonstrates that it is possible to formulate Einstein's equations as an initial value problem. The purpose of this article is to describe the background to and impact of this achievement, as well as the result itself. In some respects, the idea of viewing the field equations of general relativity as a system of evolution equations goes back to Einstein himself; in an argument justifying that gravitational waves propagate at the speed of light, Einstein used a special choice of coordinates to derive a system of wave equations for the linear perturbations on a Minkowski background. Over the following decades, Hilbert, de Donder, Lanczos, Darmois and many others worked to put Einstein's ideas on a more solid footing. In fact, the issue of local uniqueness (giving a rigorous justification for the statement that the speed of propagation of the gravitational field is bounded by that of light) was already settled in the 1930s by the work of Stellmacher. However, the first person to demonstrate both local existence and uniqueness in a setting in which the notion of finite speed of propagation makes sense was Yvonne Choquet-Bruhat. In this sense, her work lays the foundation for the formulation of Einstein's equations as an initial value problem. Following a description of the results of Choquet-Bruhat, we discuss the development of three research topics that have their origin in her work. The first one is local existence. One reason for addressing it is that it is at the heart of the original paper. Moreover, it is still an active and important research field, connected to the problem of characterizing the asymptotic behaviour of solutions that blow up in finite time. As a second topic, we turn to the questions of global uniqueness and strong cosmic censorship. These questions are of fundamental importance to anyone interested in justifying that the Cauchy problem makes sense globally. They are also closely
Eva Lowen-Colebunders
1982-09-01
Full Text Available A family C of filters on a set X is uniformizable if there is a uniformity on X such that C is its collection of Cauchy filters. Using the theory of completions and Cauchy continuous maps for Cauchy spaces, we obtain characterizations of uniformizable Cauchy spaces. In particular, given a Cauchy structure C on X we investigate under what conditions the filter u(C=Ã¢Â‹Â‚FÃ¢ÂˆÂˆCFÃƒÂ—F is a uniformity and C is its collection of Cauchy filters. This problem is treated using Cauchy covering systems.
THE CAUCHY PROBLEM FOR THE SYSTEM OF EQUATIONS OF THERMOELASTICITY IN E^n
Ikbol E. Niyozov
2014-05-01
Full Text Available ABSTRACT: In this paper we consider the problem of analytical continuation of solutions to the system of equations of thermoelasticity in a bounded domain from their values and values of their strains on a part of the boundary of this domain, i.e., we study the Cauchy problem. ABSTRAK: Di dalam kajian ini, kami menyelidiki masalah keselanjaran analitik bagi penyelesaian-penyelesaian terhadap sistem persamaan-persamaan termoelastik di dalam domain bersempadan berdasarkan nilai-nilainya dan nilai tegasannya bagi sebahagian daripada sempadan domain tersebut, iaitu kami mengkaji masalah Cauchy.
The Cauchy problem for a modified Camassa-Holm equation with analytic initial data
Gorsky, Jennifer M.
2004-01-01
We show that the periodic Cauchy problem for a modified Camassa-Holm equation with analytic initial data is analytic in the space variable $x$ for time near zero. By differentiating the equation and the initial condition with respect to $x$ we obtain a sequence of initial-value problems of KdV-type equations. These, written in the form of integral equations, define a mapping on a Banach space whose elements are sequences of functions equipped with a norm expressing the Cauchy e...
An optimal iterative algorithm to solve Cauchy problem for Laplace equation
Majeed, Muhammad Usman
2015-05-25
An optimal mean square error minimizer algorithm is developed to solve severely ill-posed Cauchy problem for Laplace equation on an annulus domain. The mathematical problem is presented as a first order state space-like system and an optimal iterative algorithm is developed that minimizes the mean square error in states. Finite difference discretization schemes are used to discretize first order system. After numerical discretization algorithm equations are derived taking inspiration from Kalman filter however using one of the space variables as a time-like variable. Given Dirichlet and Neumann boundary conditions are used on the Cauchy data boundary and fictitious points are introduced on the unknown solution boundary. The algorithm is run for a number of iterations using the solution of previous iteration as a guess for the next one. The method developed happens to be highly robust to noise in Cauchy data and numerically efficient results are illustrated.
A class of neutral functional differential equations and the abstract Cauchy problem
In this paper we establish the basic equivalence between the generalized solutions of a certain class of Neutral Functional Differential Equations and the trajectories of the associated abstract Cauchy problem. These results have applications in several fields including Mathematical Biology, Ecology and Control Theory. (author)
The Cauchy problem for a model of immiscible gas flow with large data
Sande, Hilde
2008-12-15
The thesis consists of an introduction and two papers; 1. The solution of the Cauchy problem with large data for a model of a mixture of gases. 2. Front tracking for a model of immiscible gas flow with large data. (AG) refs, figs
Global 1 Estimation of the Cauchy Problem Solutions to the Navier-Stokes Equation
Asset Durmagambetov; Leyla Fazilova
2014-01-01
The analytic properties of the scattering amplitude are discussed, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. The paper also describes the time blowup of classical solutions for the Navier-Stokes equations by the smoothness assumption.
The Cauchy Problem for Schrödinger Equations with Time-Dependent Hamiltonian
Massimo Cicognani
2014-12-01
Full Text Available We consider the Cauchy problem for a Schrödinger equation with an Hamiltonian depending also on the time variable and that may vanish at t = 0. We find optimal Levi conditions for well-posedness in Sobolev and Gevrey spaces.
Cauchy Problem of Some Doubly Degenerate Parabolic Equations with Initial Datum a Measure
Hui Jun FAN
2004-01-01
This paper discusses the Cauchy problem of the equation ut=△·（|△um|p-2△um）-uq with initial datum a measure. Under the assumption of the parameters, one proves the existence and non-existence of the non-negative generalized solution.
On the quasilinear Cauchy problem for a hyperbolic functional differential equation
Elżbieta Puźniakowska-Gałuch
2015-01-01
Full Text Available The Cauchy problem for hyperbolic functional differential equations is considered. Volterra and Fredholm dependence are considered. A theorem on the local existence of generalized solutions defined on the Haar pyramid is proved. A result on differentiability of a solution with respect to initial data is proved.
The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan
Rubinstein, Yanir A
2012-01-01
We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge-Ampere equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the C^3 lifespan of the HCMA in terms of analytic continuation of Hamiltonian mechanics and intersection of complex time characteristics. We use a conservation law type argument to prove uniqueness of solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy problem is ill-posed in C^3, in the sense that there exists a dense set of C^3 Cauchy data for which there exists no C^3 solution even for a short time. In the real domain we show that the HRMA is equivalent to a Hamilton--Jacobi equation, and use the equivalence to prove that any differentiable weak solution is smooth, so that the differentiable li...
Numerical analysis of Cauchy-type problem arising in electrical engineering
Dücker, Monika
2010-01-01
Inverse problems arise often in physical and technical processes. In this dissertation we consider problems of Cauchy-type based on the application of hybrid insulation. The experimental setup takes place at the NTNU Trondheim which placed at the disposal the measurements for the calculation. In this case we search Dirichlet or Neumann control for given Neumann measurements. After describing the problem of hybrid insulation and introducing the basic principals of functional analysis we analys...
A robust data completion method for 2D Laplacian Cauchy problems
The purpose is to propose an improved regularization method for data completion problems. This method is presented on the Cauchy problem for the Laplace equation in 2D situations. Many numerical simulations using finite element method highlight the efficiency of this new approach. In particular, it gives reconstructions with an increased accuracy, it is stable with respect to strong perturbations on the data and is able to deblur noisy data.
A robust data completion method for 2D Laplacian Cauchy problems
Delvare, F [Laboratoire Energetique Explosions Structures and Institut PRISME, Universite d' Orleans and ENSI de Bourges, 88 Boulevard Lahitolle, 18020 Bourges Cedex (France); Cimetiere, A [Laboratoire de Metallurgie Physique, Universite de Poitiers and ENSMA Poitiers, Boulevard Marie et Pierre CURIE, Teleport 2, BP 30179, 86962 Futuroscope Chasseneuil Cedex (France)], E-mail: franck.delvare@ensi-bourges.fr, E-mail: alain.cimetiere@univ-poitiers.fr
2008-11-01
The purpose is to propose an improved regularization method for data completion problems. This method is presented on the Cauchy problem for the Laplace equation in 2D situations. Many numerical simulations using finite element method highlight the efficiency of this new approach. In particular, it gives reconstructions with an increased accuracy, it is stable with respect to strong perturbations on the data and is able to deblur noisy data.
Global existence and decay of solutions of the Cauchy problem in thermoelasticity with second sound
Kasimov, Aslan R.
2013-06-04
We consider the one-dimensional Cauchy problem in non-linear thermoelasticity with second sound, where the heat conduction is modelled by Cattaneo\\'s law. After presenting decay estimates for solutions to the linearized problem, including refined estimates for data in weighted Lebesgue-spaces, we prove a global existence theorem for small data together with improved decay estimates, in particular for derivatives of the solutions. © 2013 Taylor & Francis.
The Cauchy problem for the homogeneous Monge-Ampere equation, I. Toeplitz quantization
Rubinstein, Yanir A
2010-01-01
The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow defined by the Cauchy data, analytically continuing the quantization, and then taking a kind of logarithmic classical limit. In this article, we prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) this ``quantum analytic continuation potential" coincides with the well-known Legendre transform potential, and hence solves the equation as long as it is smooth. In the sequel we prove that the Legendre transform potential ceases to solve the HRMA after that time.
The Fujita exponent for the Cauchy problem in the hyperbolic space
Bandle, Catherine; Pozio, Maria Assunta; Tesei, Alberto
It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita's phenomenon. To have the same situation as for the Cauchy problem in R, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles.
On the Cauchy problem for some fractional order partial differential equations
Moustafa, O L
2003-01-01
In the present paper, we study the Cauchy problem for a fractional order partial differential equation of the form D sub t supalpha((partial deriv u(x,t))/(partial deriv t))-SIGMA sub a sub b sub s sub ( sub q sub ) sub = sub 2 sub m a sub q (x,t)D sub x sup q u(x,t)=SIGMA sub a sub b sub s sub ( sub q sub ) subapprox sub 2 sub m b sub q (x,t)D sub x sup q u(x,t) where 0
On the Cauchy problem for some fractional order partial differential equations
Moustafa, Osama L. E-mail: olmoustafa@yahoo.com
2003-09-01
In the present paper, we study the Cauchy problem for a fractional order partial differential equation of the form D{sub t}{sup {alpha}}(({partial_derivative}u(x,t))/({partial_derivative}t))-{sigma}{sub abs(q)=2m}a{sub q}(x,t)D{sub x}{sup q}u(x,t)={sigma}{sub abs(q){approx}}{sub 2m}b{sub q}(x,t)D{sub x}{sup q}u(x,t) where 0<{alpha}{<=}1 and {sigma}{sub abs(q)=2m}a{sub q}(x,t)D{sub x}{sup q} is a uniformly elliptic partial differential operator. The existence and uniqueness of the solution of the considered Cauchy problem is studied, also a continuity property is considered.
Global existence for weak solutions of the Cauchy problem in a model of radiation hydrodynamics
Ducomet, B.; Nečasová, Šárka
2014-01-01
Roč. 420, č. 1 (2014), s. 464-482. ISSN 0022-247X R&D Projects: GA ČR GAP201/10/1920; GA ČR GA13-00522S Institutional support: RVO:67985840 Keywords : Cauchy problem * Navier - Stokes -Fourier system * radiation hydrodynamics Subject RIV: BA - General Mathematics Impact factor: 1.120, year: 2014 http://www.sciencedirect.com/science/article/pii/S0022247X14004855
2008-01-01
In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||f|||h and
Initial surface disturbance on a shear current: the Cauchy--Poisson problem with a twist
Ellingsen, Simen Å.
2014-01-01
We solve for the first time the classical linear Cauchy--Poisson problem for the time evolution an initial surface disturbance when a uniform shear current is present beneath the surface. The solution is general, including the effects of gravity, surface tension and constant finite depth. The particular case of an initially Gaussian disturbance of width $b$ is studied for different values of three system parameters: a "shear Froude number" $S\\sqrt{b/g}$ ($S$ is the uniform vorticity), the Bon...
On the Blow-up Phenomena of Cauchy Problem for the Camassa- Holm Equation
LIU Yongqin; WANG Weike
2006-01-01
We focus on the blow-up phenomena of Cauchy problem for the Camassa-Holm equation. Blow-up can occur only in the form of wave-breaking, i.e. the solution is bounded but its slope becomes unbounded in finite time. We proved that there is such a point that its slope becomes infinite exactly at breaking time. We also gave the precise blow-up rate and the blow-up set.
Probability representations of Cauchy problem solutions for quantum mechanics equations
Probability representations of quantum mechanics equations in a number of cases allow to see a new interpretation of the generally accepted notions; they also serve as a theoretical basis of the numerical solution of a problems. Probability representations of the solutions of Schroedinger equations and Dirac-Pauli vector equations are presented. In addition, the Heisenberg picture of quantum mechanics equation is considered
Li, Jing; Liang, Zhilei
2013-01-01
This paper concerns the Cauchy problem of the barotropic compressible Navier-Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier-Stokes equations admits a unique local strong solution provided the initial den...
Efficient numerical method for solving Cauchy problem for the Gamma equation
Koleva, Miglena N.
2011-12-01
In this work we consider Cauchy problem for the so called Gamma equation, derived by transforming the fully nonlinear Black-Scholes equation for option price into a quasilinear parabolic equation for the second derivative (Greek) Γ = VSS of the option price V. We develop an efficient numerical method for solving the model problem concerning different volatility terms. Using suitable change of variables the problem is transformed on finite interval, keeping original behavior of the solution at the infinity. Then we construct Picard-Newton algorithm with adaptive mesh step in time, which can be applied also in the case of non-differentiable functions. Results of numerical simulations are given.
GENERAL CAUCHY PROBLEM FOR THE LINEAR SHALLOW -WATER EQUATIONS ON AN EQUATORIAL BETA-PLANE
SHEN Chun; SHI Wei-hui
2006-01-01
Based on the theory of stratification, the well-posedness of the initial value problem for the linear shallow-water equations on an equatorial beta-plane was discussed. The sufficient and necessary conditions of the existence and uniqueness for the local solution of the equations were presented and the existence conditions for formal solutions of the equations were also given. For the Cauchy problem on the hyper-plane, the local analytic solution were worked out and a special case was discussed. Finally, an example was used to explain the variety of formal solutions for the ill-posed problem.
A Series Solution of the Cauchy Problem for Turing Reaction-diffsion Model
L. Päivärinta; Shidfar, A.; A. Molabahrami
2011-01-01
In this paper, the series pattern solution of the Cauchy problem for Turing reaction-diffusion model is obtained by using the homotopy analysis method (HAM). Turing reaction-diffusion model is nonlinear reaction-diffusion system which usually has power-law nonlinearities or may be rewritten in the form of power-law nonlinearities. Using the HAM, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a series of functions which converges ra...
A remark on the Cauchy problem for the generalized Benney-Luke equation
Quintero, José Raúl
2008-01-01
In this article, we address the well posedness of the Cauchy problem associated with the generalized Benney--Luke equation in $\\mathbb R^{1+2}:$ \\begin{multline*} \\Phi_{tt} - \\Delta \\Phi + a \\Delta^2 \\Phi - b \\Delta \\Phi_{tt} + \\theta\\Big ( \\Phi_t \\big [\\partial_{x} \\big [ \\big (\\partial_{x} \\Phi \\big )^{p} \\big ]+ \\partial_{y} \\big [ \\big (\\partial_{y}\\Phi \\big )^{p} \\big ] \\big ] \\\\ + 2 \\big [ \\big (\\partial_{x} \\Phi \\big )^{p}\\Phi_{xt}+ \\big (\\partial_{y} \\Phi \\big )^{p}\\Phi...
On the Cauchy problem for nonlinear Schrödinger equations with rotation
Antonelli, Paolo
2011-10-01
We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superuid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in [11, 12]. Moreover, we find that the rotation term has a considerable in fiuence in proving finite time blow-up in the focusing case.
Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation
Arkeryd, Leif; Nouri, Anne
2015-01-01
A fully non-linear kinetic Boltzmann equation for anyons is studied in a periodic 1d setting with large initial data. Strong L 1 solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness and stabililty. We use the Bony functional, the two-dimensional velocity frame specific for anyons, and an initial layer analysis that moves the solution away from a critical value. 1 Anyons and the Boltzmann equation. Let us first recall the definition of anyon. Con...
The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform
Rubinstein, Yanir A
2010-01-01
We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA). In the prequel a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article---that uses only tools of convex analysis and can be read independently---we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable.
The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation
Molinet, Luc
2000-01-01
We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers (KP-Burgers) equation in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts which makes it quite particular. After showing local existence by contraction principle for initial data $ \\varphi\\in H^s(\\mathbb R^2) $ such that $ {\\mathcal F}^{-1} (\\frac{k_2}{k_1} \\widehat{\\varphi})\\in H^r(\\mathbb R^2) $, $ 0{\\leqslant} r {\\leqslant} s- 1 $, we try to extend the s...
Two-point boundary value and Cauchy formulations in an axisymmetrical MHD equilibrium problem
In this paper we present two equilibrium solvers for axisymmetrical toroidal configurations, both based on the expansion in poloidal angle method. The first one has been conceived as a two-point boundary value solver in a system of coordinates with straight field lines, while the second one uses a well-conditioned Cauchy formulation of the problem in a general curvilinear coordinate system. In order to check the capability of our moment methods to describe equilibrium accurately, a comparison of the moment solutions with analytical solutions obtained for a Solov'ev equilibrium has been performed. (author)
A Series Solution of the Cauchy Problem for Turing Reaction-diffusion Model
L. Päivärinta
2011-12-01
Full Text Available In this paper, the series pattern solution of the Cauchy problem for Turing reaction-diffusion model is obtained by using the homotopy analysis method (HAM. Turing reaction-diffusion model is nonlinear reaction-diffusion system which usually has power-law nonlinearities or may be rewritten in the form of power-law nonlinearities. Using the HAM, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a series of functions which converges rapidly to the exact solution of the problem. The efficiency of the approach will be shown by applying the procedure on two problems. Furthermore, the so-called homotopy-Pade technique (HPT is applied to enlarge the convergence region and rate of solution series given by the HAM.
A dimension decomposition approach based on iterative observer design for an elliptic Cauchy problem
Majeed, Muhammad Usman
2015-07-13
A state observer inspired iterative algorithm is presented to solve boundary estimation problem for Laplace equation using one of the space variables as a time-like variable. Three dimensional domain with two congruent parallel surfaces is considered. Problem is set up in cartesian co-ordinates and Laplace equation is re-written as a first order state equation with state operator matrix A and measurements are provided on the Cauchy data surface with measurement operator C. Conditions for the existence of strongly continuous semigroup generated by A are studied. Observability conditions for pair (C, A) are provided in infinite dimensional setting. In this given setting, special observability result obtained allows to decompose three dimensional problem into a set of independent two dimensional sub-problems over rectangular cross-sections. Numerical simulation results are provided.
On the Regularity of the Solutions for Cauchy Problem of Incompressible 3D Navier-Stokes Equation
Lin, Qun
2013-01-01
In this paper we will prove that the vorticity belongs to $L^{\\infty}(0, T; L^2(R^3))$ for the Cauchy problem of 3D incompressible Navier-Stokes equation, then the existence of a global smooth solution is obtained. Our approach is to construct a set of auxiliary problems to approximate the original one for vorticity equation.
Cauchy problem for multiscale conservation laws: Application to structured cell populations
Shang, Peipei
2010-01-01
In this paper, we study a vector conservation law that models the growth and selection of ovarian follicles. During each ovarian cycle, only a definite number of follicles ovulate, while the others undergo a degeneration process called atresia. This work is motivated by a multiscale mathematical model starting on the cellular scale, where ovulation or atresia result from a hormonally controlled selection process. A two-dimensional conservation law describes the age and maturity structuration of the follicular cell populations. The densities intersect through a coupled hyperbolic system between different follicles and cell phases, which results in a vector conservation law and coupling boundary conditions. The maturity velocity functions possess both a local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with bounded initial and boundary data.
Milena Netka
2009-01-01
Full Text Available The paper is concerned with weak solutions of a generalized Cauchy problem for a nonlinear system of first order differential functional equations. A theorem on the uniqueness of a solution is proved. Nonlinear estimates of the Perron type are assumed. A method of integral functional inequalities is used.
dos Santos, J. P. C.; Vijayakumar, V.; R. Murugesu
2013-01-01
In this article, we study the existence of mild solutions for the nonlocal Cauchy problem for a class of abstract fractional neutral integro-differential equations with infinite delay. The results are obtained by using the theory of resolvent operators. Finally, an application is given to illustrate the theory.
Nguyen Manh Hung
2009-01-01
Full Text Available The main goal of this paper is to obtain the regularity of weak solutions of Cauchy-Neumann problems for the second-order general Schrödinger equations in domains with conical points on the boundary of the bases.
The exact solutions of the Cauchy problem for two equations which are slight generalization of the so-called linear vectorial Fokker-Planck equation are found using the disentangling techniques of Feynman and algebraic (operational) methods. This approach may be considered as a generalization of the Suzuki's method for solving the one-dimensional linear Pokker-Planck equation
The Cauchy problem for metric-affine f(R)-gravity in presence of perfect-fluid matter
Capozziello, S
2009-01-01
The Cauchy problem for metric-affine f(R)-gravity `a la Palatini and with torsion, in presence of perfect fluid matter acting as source, is discussed following the well-known Bruhat prescriptions for General Relativity. The problem results well-formulated and well-posed when the perfect-fluid form of the stress-energy tensor is preserved under conformal transformations. The key role of conservation laws in Jordan and in Einstein frame is also discussed.
Initial surface disturbance on a shear current: the Cauchy--Poisson problem with a twist
Ellingsen, Simen Å
2014-01-01
We solve for the first time the classical linear Cauchy--Poisson problem for the time evolution an initial surface disturbance when a uniform shear current is present beneath the surface. The solution is general, including the effects of gravity, surface tension and constant finite depth. The particular case of an initially Gaussian disturbance of width $b$ is studied for different values of three system parameters: a "shear Froude number" $S\\sqrt{b/g}$ ($S$ is the uniform vorticity), the Bond number and the depth relative to the initial perturbation width. Different phase and group velocity in different directions yield very different wave patterns in different parameter regimes when the shear is strong, and the well known pattern of diverging ring waves in the absence of shear can take on very different qualitative behaviours. For a given shear Froude number, both finite depth and nonzero capillary effects are found to weaken the influence of the shear on the resulting wave pattern. The various patterns are...
Well-posedness of the Cauchy problem for models of large amplitude internal waves
We consider in this paper the 'shallow-water/shallow-water' asymptotic model obtained in Choi and Camassa (1999 J. Fluid Mech. 396 1–36), Craig et al (2005 Commun. Pure. Appl. Math. 58 1587–641) (one-dimensional interface) and Bona et al (2008 J. Math. Pures Appl. 89 538–66) (two-dimensional interface) from the two-layer system with rigid lid, for the description of large amplitude internal waves at the interface of two layers of immiscible fluids of different densities. For one-dimensional interfaces, this system is of hyperbolic type and its local well-posedness does not raise serious difficulties, although other issues (blow-up, loss of hyperbolicity, etc) turn out to be delicate. For two-dimensional interfaces, the system is nonlocal. Nevertheless, we prove that it conserves some properties of 'hyperbolic type' and show that the associated Cauchy problem is locally well posed in suitable Sobolev classes provided some natural restrictions are imposed on the data. These results are illustrated by numerical simulations with emphasis on the formation of shock waves
The paper is concerned with the Cauchy problem for a nonlinear system of partial differential equations with parameters. This system describes the two-layer quasi-solenoidal Lorenz model for a baroclinic atmosphere on a rotating two-dimensional sphere. The right-hand side of the system is perturbed by white noise, and random initial data is considered. This system is shown to be uniquely solvable, and an estimate for the continuous dependence of the solution on the set of random initial data and the right-hand side is established on a finite time interval. In passing, an estimate for the continuous dependence on the set of parameters, the initial data, and the right-hand side is obtained on a finite time interval for the solution of the Cauchy problem with deterministic initial data and deterministic right-hand side. Bibliography: 32 titles.
Deng Yinbin; Yang Fen
2008-01-01
This article is contributed to the Cauchy problem u/t = △u + K(ㄧxㄧ)up in Rn x (0,T), u(x,0) =(ψ)(x) in Rn; with initial function(ψ)≠0. The stability of positive radial steady state, which are positive solutions of △u + K(ㄧxㄧ)up =0, is obtained when p is critical for general K(ㄧxㄧ).
We solve the Cauchy problem for the Korteweg–de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finite-gap potentials under the assumption that the respective spectral bands either coincide or are disjoint
Campos Palomino, Juvitsa Milena
2009-01-01
El propósito de este trabajo es abordar la buena colocación en los espacios de Sobolev Hs(T2) para s≥1 del problema de Cauchy asociado a una ecuación del tipo Kuramoto-Sivashisky bidimensional periódica; que modela fenómenos físicos que ocurren en películas delgadas y plasma. / Abstract. The purpose of this work is to establish the well-posedness in the spaces Sobolev spaces Hs(T2) para s≥1 for the Cauchy problem associated for the Kuramoto-Sivashisky two-dimensional equation in the periodica...
Diffuse interface methods for inverse problems: case study for an elliptic Cauchy problem
Burger, Martin; Løseth Elvetun, Ole; Schlottbom, Matthias
2015-12-01
Many inverse problems have to deal with complex, evolving and often not exactly known geometries, e.g. as domains of forward problems modeled by partial differential equations. This makes it desirable to use methods which are robust with respect to perturbed or not well resolved domains, and which allow for efficient discretizations not resolving any fine detail of those geometries. For forward problems in partial differential equations methods based on diffuse interface representations have gained strong attention in the last years, but so far they have not been considered systematically for inverse problems. In this work we introduce a diffuse domain method as a tool for the solution of variational inverse problems. As a particular example we study ECG inversion in further detail. ECG inversion is a linear inverse source problem with boundary measurements governed by an anisotropic diffusion equation, which naturally cries for solutions under changing geometries, namely the beating heart. We formulate a regularization strategy using Tikhonov regularization and, using standard source conditions, we prove convergence rates. A special property of our approach is that not only operator perturbations are introduced by the diffuse domain method, but more important we have to deal with topologies which depend on a parameter \\varepsilon in the diffuse domain method, i.e. we have to deal with \\varepsilon -dependent forward operators and \\varepsilon -dependent norms. In particular the appropriate function spaces for the unknown and the data depend on \\varepsilon . This prevents the application of some standard convergence techniques for inverse problems, in particular interpreting the perturbations as data errors in the original problem does not yield suitable results. We consequently develop a novel approach based on saddle-point problems. The numerical solution of the problem is discussed as well and results for several computational experiments are reported. In
The Cauchy problem for metric-affine f(R)-gravity in the presence of perfect-fluid matter
The Cauchy problem for metric-affine f(R)-gravity in the manner of Palatini and with torsion, in the presence of perfect fluid matter acting as a source, is discussed following the well-known Bruhat prescriptions for general relativity. The problem results in being well formulated and well posed when the perfect-fluid form of the stress-energy tensor is preserved under conformal transformations and the set of viable f(R)-models is not empty. The key role of conservation laws in the Jordan and in the Einstein frame is also discussed.
The Cauchy problem for metric-affine f(R)-gravity in the presence of perfect-fluid matter
Capozziello, S.; Vignolo, S.
2009-09-01
The Cauchy problem for metric-affine f(R)-gravity in the manner of Palatini and with torsion, in the presence of perfect fluid matter acting as a source, is discussed following the well-known Bruhat prescriptions for general relativity. The problem results in being well formulated and well posed when the perfect-fluid form of the stress-energy tensor is preserved under conformal transformations and the set of viable f(R)-models is not empty. The key role of conservation laws in the Jordan and in the Einstein frame is also discussed.
The Cauchy problem for metric-affine f(R)-gravity in the presence of perfect-fluid matter
Capozziello, S [Dipartimento di Scienze Fisiche, Universita ' Federico II' di Napoli and INFN Sez. di Napoli, Compl. Univ. Monte S. Angelo Ed. N, via Cinthia, I-80126 Napoli (Italy); Vignolo, S [DIPTEM Sez. Metodi e Modelli Matematici, Universita di Genova, Piazzale Kennedy, Pad. D-16129 Genova (Italy)
2009-09-07
The Cauchy problem for metric-affine f(R)-gravity in the manner of Palatini and with torsion, in the presence of perfect fluid matter acting as a source, is discussed following the well-known Bruhat prescriptions for general relativity. The problem results in being well formulated and well posed when the perfect-fluid form of the stress-energy tensor is preserved under conformal transformations and the set of viable f(R)-models is not empty. The key role of conservation laws in the Jordan and in the Einstein frame is also discussed.
We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified
Addona, Davide, E-mail: d.addona@campus.unimib.it [Università degli Studi di Milano Bicocca, (MILANO BICOCCA) Dipartimento di Matematica (Italy)
2015-08-15
We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.
苗长兴
2003-01-01
In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data in the homogeneous spaces. We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in Cσ,s,p and Lq([O, T);Hs,p) by introducing the concept of both admissiblequintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic typeequations. For the small data, we prove that these results can be extended globally in time. We also study theregularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in Cσ,s,p. Asan application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneousSobolev spaces.
Lv, Boqiang; Shi, Xiaoding; Zhong, Xin
2015-01-01
We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D Cauchy problem of the density-dependent Navier-Stokes equations on the whole space $\\mathbb{R}^2$ admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compa...
Exponential Decay for the Semilinear Cauchy-Ventcel Problem with Localized Damping
Mohamed Medjden
2004-11-01
Full Text Available Le but de ce travail est d’ étudier la décroissance exponentielle del’énergie des solutions losque le temps tend vers l’inﬁni du problème aux limites de Cauchy-Ventcel semi-linéaire dissipatif dans un domaine borné. On donne des conditions suﬃsantes sur les non linéarités de f et g pour avoir la décroissance exponentielle de l’énergie. Ce problème décrit les vibrations d’un corps élastiqueavec un raidisseur mince sur le bord. La méthode de démonstration est basée sur les techniques de multiplicateurs et un principe de continuation unique qui permettent d’estimer l’énergie totale des solutions.
Many-dimensional non-strictly hyperbolic systems of conservation laws with a radially degenerate flux function are considered. For such systems the set of entropies is defined and described, the concept of generalized entropy solution of the Cauchy problem is introduced, and the properties of generalized entropy solutions are studied. The class of strong generalized entropy solutions is distinguished, in which the Cauchy problem in question is uniquely soluble. A condition on the initial data is described that ensures that the generalized entropy solution is strong and therefore unique. Under this condition the convergence of the 'vanishing viscosity' method is established. An example presented in the paper shows that a generalized entropy solution is not necessarily unique in the general case
Lantsman, L
2006-01-01
We show that manifest superfluid properties of the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac may be described in the framework of the Cauchy problem to the Gribov ambiguity equation. The latter equation specifies the ambiguity in choosing the covariant Coulomb (transverse) gauge for Yang-Mills fields represented as topological Dirac variables, may be treated as solutions to the Gauss law constraint at the removal of temporal components of these fields. We demonstrate that the above Cauchy problem comes just to fixing the covariant Coulomb gauge for topological Dirac variables in the given initial time instant $t_0$ and finding the solutions to the Gribov ambiguity equation in the shape of vacuum BPS monopoles and excitations over the BPS monopole vacuum referring to the class of multipoles. The next goal of the present study will be specifying the look of Gribov topological multipliers entering Dirac variables in the Minkowskian Higgs model quantized by Dirac, especially at the spat...
Huang, Xiangdi; Li, Jing
2012-01-01
For smooth initial data, we establish the global existence and uniqueness of strong and classical solutions to the Cauchy problem for the barotropic compressible Navier-Stokes equations in two spatial dimensions with vacuum state as far field and with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is $\\lambda = \\rho^{\\beta}$ with $\\beta>4/3$.
Marzena Pytel-Kudela
2006-01-01
Full Text Available The analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations in Hilbert spaces are studied using theory of bi-linear forms in respectively rigged Hilbert spaces triples. Theorems specifying the existence of a dissolving operator for a class of adiabatically perturbed nonautonomous partial differential equations are stated. Some applications of the results obtained are discussed.
Cauchy problems for fifth-order KdV equations in weighted Sobolev spaces
Eddye Bustamante
2015-05-01
Full Text Available In this work we study the initial-value problem for the fifth-order Korteweg-de Vries equation $$ \\partial_{t}u+\\partial_{x}^{5}u+u^k\\partial_{x}u=0, \\quad x,t\\in \\mathbb{R}, \\; k=1,2, $$ in weighted Sobolev spaces $H^s(\\mathbb{R}\\cap L^2(\\langle x \\rangle^{2r}dx$. We prove local and global results. For the case $k=2$ we point out the relationship between decay and regularity of solutions of the initial-value problem.
On analytic continuability of the missing Cauchy datum for Helmholtz boundary problems
Karamehmedovic, Mirza
2015-01-01
We relate the domains of analytic continuation of Dirichlet and Neumann boundary data for Helmholtz problems in two or more independent variables. The domains are related à priori, locally and explicitly in terms of complex polyrectangular neighbourhoods of planar pieces of the boundary. To this...
成军祥; 尚海锋
2013-01-01
研究了方程ut－div（｜Dum｜p－2 Dum ）＝t－σuq的Cauchy问题，其中m＞0， p＞1。 此方程为双非线性抛物方程，根据参数m， p的不同取值，方程主部会发生双重退化或双重奇异性。 本文利用先验估计和紧性方法以及对参数m， p作精细划分，得到了测度初值解的最优存在性。%The Cauchy problem of the equation.
On analytic continuability of the missing Cauchy datum for Helmholtz boundary problems
Karamehmedovic, Mirza
2015-01-01
We relate the domains of analytic continuation of Dirichlet and Neumann boundary data for Helmholtz problems in two or more independent variables. The domains are related à priori, locally and explicitly in terms of complex polyrectangular neighbourhoods of planar pieces of the boundary. To this end we identify and characterise a special subspace of the standard pseudodifferential operators with real-analytic symbols. The result is applicable in the estimation of the domain of analytic contin...
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2005-01-01
[1]Oleinik, O. A., Samokhin,V. N., Mathematical Models in Boundary Layer Theorem, Boca Raton; Chapman and Hall/CRC, 1999.[2]Volpert, A.I., Hudjaev, S.I., On the problem for quasilinear degenerate parabolic equations of second order(Russian), Mat. Sb., 1967, 3: 374-396.[3]Zhao, J., Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeastern Math. J., 1985,1(2): 153-165.[4]Wu, Z., Yin, J., Some properties of functions in BVx and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeastern Math. J., 1989,5(4): 395-422.[5]Brezis, H., Crandall, M.G., Uniqueness of solutions of the initial value problem for ut- △ψ(u) = 0, J.Math. Pures et Appl., 1979, 58: 153-163.[6]Kruzkov, S.N., First order quasilinear equations in several independent varaiables, Math. USSR-Sb., 1970, 10:217-243.[7]Cockburn, B., Gripenberg, G., Continious dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Diff. Equatiaons, 1999, 151: 231-251.[8]Volpert, A.I., BV space and quasilinear equations, Mat. Sb., 1967, 73: 255-302.[9]Volpert, A.I., Hudjave, S.I., Analysis of class of discontinuous functions and the equations of mathematical physics (Russian), Izda. Nauka Moskwa, 1975.[10]Evans, L.C., Weak convergence methods for nonlinear partial differential equations, Conference Board of the Mathematical Sciences, Regional Conferences Series in Mathematics Number 74, 1998.[11]Wu, Z., Zhao, J., Yin, J., et al., Nonlinear Diffusion Equations, Singapore: Word Scientific, 2001.
Kazeykina, Anna
2011-01-01
In the present paper we are concerned with the Novikov--Veselov equation at negative energy, i.e. with the $(2 + 1)$--dimensional analog of the KdV equation integrable by the method of inverse scattering for the two--dimensional Schr\\"odinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non--singular scattering data behaves asymptotically as $\\frac{\\const}{t^{3/4}}$ in the uniform norm at large times $t$. We also present some arguments which indicate that this asymptotics is optimal.
Morassi, Antonino; Vessella, Sergio
2010-01-01
We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequality is derived, is proved in an elementary and direct way based on standard integration by parts.
D. C. Kent
1985-01-01
Full Text Available This paper is concerned with the notion of ordered Cauchy space which is given a simple internal characterization in Section 2. It gives a discription of the category of ordered Cauchy spaces which have ordered completions, and a construction of the fine completion functor on this category. Sections 4 through 6 deals with certain classes of ordered Cauchy spaces which have ordered completions; examples are given which show that the fine completion does not preserve such properties as uniformizability, regularity, or total boundedness. From these results, it is evident that a further study of ordered Cauchy completions is needed.
On the comparison of Cauchy mean values
Losonczi László
2002-01-01
Suppose that and exist, with , on . Then there is (moreover if ) such that where denotes the divided difference of at the points . This is the Cauchy Mean Value Theorem for divided differences (see e.g. [4]). If the function is invertible then is a mean value of . It is called the Cauchy mean of the numbers and will be denoted by ). Here we completely solve the comparison problem of Cauchy means in the special cases and . In the general case we find necessary conditions (which ar...
Faugeras, Blaise; Blum, Jacques; Boulbe, Cedric
2010-01-01
A numerical method for the computation of the magnetic flux in the vacuum surrounding the plasma in a Tokamak is investigated. It is based on the formulation of a Cauchy problem which is solved through the minimization of a constitutive law error functional. Several numerical experiments are conducted which show the efficiency of the method.
A formal asymptotic expansion of a solution of the initial problem for a singularly perturbed differential-operational nonlinear equation in a small parameter has been constructed in the critical case. Splash functions of and boundary functions have been estimated of found and assessment of the residual member of the expansion has been obtained
Ghorbani, Mohammad
2013-01-01
In this paper we introduce an instance of the well-know Neyman–Scott cluster process model with clusters having a long tail behaviour. In our model the offspring points are distributed around the parent points according to a circular Cauchy distribution. Using a modified Cramér-von Misses test...... statistic and the simulated pointwise envelopes it is shown that this model fits better than the Thomas process to the frequently analyzed long-leaf pine data-set....
Cosmology is a discipline that encompasses many diverse aspects of physics and astronomy. This is part of its attraction, but also a reason why it is difficult for new researchers to acquire sufficient grounding to enable them to make significant contributions early in their careers. For this reason there are many cosmology textbooks aimed at the advanced undergraduate/beginning postgraduate level. Physical Foundations of Cosmology by Viatcheslav Mukhanov is a worthy new addition to this genre. Like most of its competitors it does not attempt to cover every single aspect of the subject but chooses a particular angle and tries to unify its treatment around that direction. Mukhanov has chosen to focus on the fundamental principles underlying modern cosmological research at the expense of some detail at the frontiers. The book places great emphasis on the particle-astrophysics interface and issues connected with the thermal history of the big-bang model. The treatment of big-bang nucleosynthesis is done in much more detail than in most texts at a similar level, for example. It also contains a very extended and insightful discussion of inflationary models. Mukhanov makes great use of approximate analytical arguments to develop physical intuition rather than concentrating on numerical approaches. The book is quite mathematical, but not in a pedantically formalistic way. There is much use of 'order-of-magnitude' dimensional arguments which undergraduate students often find difficult to get the hang of, but which they would do well to assimilate as early as possible in their research careers. The text is peppered with problems for the reader to solve, some straightforward and some exceedingly difficult. Solutions are not provided. The price to be paid for this foundational approach is that there is not much about observational cosmology in this book, and neither is there much about galaxy formation or large-scale structure. It also neglects some of the trendier recent
THE CAUCHY-KOVALEVSKAYA THEOREM-OLD AND NEW
W. Tutschke
2005-01-01
The paper surveys interactions between complex and functional-analytic methods in the Cauchy-Kovalevskaya theory. For instance, the behaviour of the derivative of a bounded holomorphic function led to abstract versions of the Cauchy-Kovalevskaya Theorem.Recent trends in the Cauchy-Kovalevskaya theory are based on the concept of associated differential operators. Since an evolution operator may posses several associated operators, initial data may be decomposed into components belonging to different associated spaces. This technique makes it also possible to solve ill-posed initial value problems.
蒋毅; 成和平; 孟宪良; 蒲成林
2006-01-01
For the Cauchy problem for the nonlinear Klein-Gordon equation with potential,we define new stable and unstable sets for the initial data.We prove that if during the evolution enters into the unstable set,the solution blows up in finite time.If during the evolution enters into the stable set,the solution is global.By using scaling argument,we also answer the question of how small the initial data are the global solution of the Cauchy problem exists.%对带势的非线性Klein-Gordon方程柯西问题,我们定义了新的对于初值的稳定和不稳定集.我们证明了如果发展进入了不稳定集,解在有限时间内爆破;如果发展进入了稳定集,解整体存在.运用势并讨论,我们回答了当初值为多少时,柯西问题的整体解存在.
Identifiability of surface sources from Cauchy data
We explore the reconstruction of surface sources from single Cauchy data on the potential. Identifiability is first proven when the location of these sources is known and their density is to be sought. We also investigate the ill-posedness degree to establish that the problem is severely ill-posed, and exhibit the necessary orthogonality conditions fulfilled by exact Cauchy data, for which the process of recovering the density function of the sources succeeds. We later turn to the more difficult reconstruction of the location, the shape and the density function of the sources. Identifiability cannot be obtained for general geometries. Nevertheless, under some acceptable assumptions on the size and the location of these sources, we are able to state identifiability results
Conservation law for the Cauchy-Navier equation of elastodynamics wave via Fourier transform
Van Vinh, Nguyen; Minh, Nguyen Tuan
2012-01-01
In this paper, we use the method of Fourier analysis to derive the formula of the total energy for the Cauchy problem of the Cauchy-Navier elastodynamics wave equation describing the motion of an isotropic elastic body. The conservation law of the total energy is obtained and consequently, the global uniqueness of the solution to the problem is implied.
郝兴文; 王钦
2011-01-01
Kuznetsov方法是估计双曲型方程熵解与它的数值解之间误差的有效方法,文中应用该方法证明一个二阶退化抛物-双曲方程熵解的唯一性,并用双变量方法得到了熵解在初值的L1连续性.%Kuznetsov＇s method is a useful tool to establish the error estimate between numerical solutions and entropy weak solution for hyperbolic equation.This paper uses this method to prove the uniqueness to the Cauchy problem of a special example of second order degenerate parabolic-hyperbolic equation,Meanwhile,the paper obtained that entropy solutions are continuous at t=0 in the sense of L1 norm by the double variable device.
The regularity series of a Cauchy space
Darrell C. Kent
1984-03-01
Full Text Available This study extends the notion of regularity series from convergence spaces to Cauchy spaces, and includes an investigation of related topics such as that T2 and T3 modifications of a Cauchy space and their behavior relative to certain types of quotient maps. These concepts are applied to obtain a new characterization of Cauchy spaces which have T3 completions.
On Bimodal Offset Cauchy Distribution
Girija S.V.S.
2013-05-01
Full Text Available The bivariate Cauchy distribution has received applications in many areas, including biological analyses, clinical trials, stochastic modeling of decreasing failure rate life components, study of labour turnover, queuing theory and reliability (Nayak (1987 and Lee and Gross (1991. In the study of biological analyses, clinical trials and reliability circular distributions will yield suitable results. Circular data arises in a number of different areas such as geological, meteorological, biological and industrial sciences. It is not suggestive to use standard statistical techniques to model circular data, due to the circular geometry of the sample space (p.2 Jammalamadaka and Sen Gupta (2001. It is possible to construct a circular model by transforming a bivariate linear random variate to just its directional component and the resultant model is called ‘offset distribution’. In the literature most of the available circular models were constructed by wrapping a linear model. In recent years some wrapped models were constructed by Dattatreya Rao et al (2007. Here an attempt is made to exploit method of offsetting on Bivariate Cauchy distribution to construct a circular model named by us “OFFSET CAUCHY DISTRIBUTION (OC”. The characteristic function of the Offset Cauchy model is derived and its characteristics are discussed.
Control of Cauchy System for an Elliptic Operator
G.MASSENGO MOPHOU; O.NAKOULIMA
2009-01-01
The control of a Cauchy system for an elliptic operator seems to be globally an open problem. In this paper, we analyze this problem using a regularization method which consists in viewing a singular problem as a limit of a family of well-posed problems. Following this analysis and assuming that the interior of considered convex is non-empty, we obtain a singular optimality system (S.O.S.) for the considered control problem.
Variational approach for restoring blurred images with cauchy noise
Sciacchitano, Federica; Dong, Yiqiu; Zeng, Tieyong
2015-01-01
model, we add a quadratic penalty term, which guarantees the uniqueness of the solution. Due to the convexity of our model, the primal dual algorithm is employed to solve the minimization problem. Experimental results show the effectiveness of the proposed method for simultaneously deblurring and...... denoising images corrupted by Cauchy noise. Comparison with other existing and well-known methods is provided as well....
Uniqueness theorems for Cauchy integrals
Melnikov, Mark; Volberg, Alexander
2007-01-01
If $\\mu$ is a finite complex measure in the complex plane $\\C$ we denote by $C^\\mu$ its Cauchy integral defined in the sense of principal value. The measure $\\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\\mu=0$ at $\\mu$-almost every point. We show that if $\\mu$ is reflectionless and its Cauchy maximal function $C^\\mu_*$ is summable with respect to $|\\mu|$ then $\\mu$ is trivial. An example of a reflectionless measure whose maximal function belongs to the "weak" $L^1$ is also constructed, proving that the above result is sharp in its scale. We also give a partial geometric description of the set of reflectionless measures on the line and discuss connections of our results with the notion of sets of finite perimeter in the sense of De Giorgi.
Cakalli, Huseyin
2012-01-01
An ideal $I$ is a family of subsets of positive integers $\\textbf{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_n)$ of real numbers is said to be $I$-convergent to a real number $L$, if for each \\;$ \\varepsilon> 0$ the set $\\{n:|x_{n}-L|\\geq \\varepsilon\\}$ belongs to $I$. We introduce $I$-ward compactness of a subset of $\\textbf{R}$, the set of real numbers, and $I$-ward continuity of a real function in the senses that a subset $E$ of $\\textbf{R}$ is $I$-ward compact if any sequence $(x_{n})$ of points in $E$ has an $I$-quasi-Cauchy subsequence, and a real function is $I$-ward continuous if it preserves $I$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called to be $I$-quasi-Cauchy when $(\\Delta x_{n})$ is $I$-convergent to 0. We obtain results related to $I$-ward continuity, $I$-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, $\\delta$-ward continuity, and slowly oscillating continuity.
Complex integration and Cauchy's theorem
Watson, GN
2012-01-01
This brief monograph by one of the great mathematicians of the early twentieth century offers a single-volume compilation of propositions employed in proofs of Cauchy's theorem. Developing an arithmetical basis that avoids geometrical intuitions, Watson also provides a brief account of the various applications of the theorem to the evaluation of definite integrals.Author G. N. Watson begins by reviewing various propositions of Poincaré's Analysis Situs, upon which proof of the theorem's most general form depends. Subsequent chapters examine the calculus of residues, calculus optimization, the
Myhaylo O. Stashenko
2004-01-01
Full Text Available It is proved convergence of solution in cumulant expansions of the initial value problem for BBGKY chain of equations of non-symmetrical one-dimensional system of particles which interact via a short-range potential in the space \\(E_{\\xi}\\ of the sequences of continuous bounded functions.
The log-convexity of the poly-Cauchy numbers
Komatsu, Takao; Zhao, Feng-Zhen
2016-01-01
In 2013, Komatsu introduced the poly-Cauchy numbers, which generalize Cauchy numbers. Several generalizations of poly-Cauchy numbers have been considered since then. One particular type of generalizations is that of multiparameter-poly-Cauchy numbers. In this paper, we study the log-convexity of the multiparameter-poly-Cauchy numbers of the first kind and of the second kind. In addition, we also discuss the log-behavior of multiparameter-poly-Cauchy numbers.
Katz, Mikhail G.; Tall, David
2012-01-01
The Dirac delta function has solid roots in 19th century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his infinitesimal definition of delta.
Mitrinović, Dragoslav S
1993-01-01
Volume 1, i. e. the monograph The Cauchy Method of Residues - Theory and Applications published by D. Reidel Publishing Company in 1984 is the only book that covers all known applications of the calculus of residues. They range from the theory of equations, theory of numbers, matrix analysis, evaluation of real definite integrals, summation of finite and infinite series, expansions of functions into infinite series and products, ordinary and partial differential equations, mathematical and theoretical physics, to the calculus of finite differences and difference equations. The appearance of Volume 1 was acknowledged by the mathematical community. Favourable reviews and many private communications encouraged the authors to continue their work, the result being the present book, Volume 2, a sequel to Volume 1. We mention that Volume 1 is a revised, extended and updated translation of the book Cauchyjev raeun ostataka sa primenama published in Serbian by Nau~na knjiga, Belgrade in 1978, whereas the greater part ...
Pérez, Javier; Aizpuru, Antonio
1999-01-01
En este artículo presentamos un estudio contextualizado de Cours d’Analyse de Cauchy, analizando su significado e importancia. Presentamos especial atención al grado de elaboración teórica de límites, continuidad, series, números reales funciones y series completas, relacionando las aportaciones de Cauchi del nivel conceptual anterior a esta ahora.
蒋毅
2011-01-01
For the Cauchy problem of the nonlinear wave equation with linear damping and potential terms, we define new stable and unstable sets for the initial data. We prove that if during the evolution enters into the unstable set, the solution blows up in finite time. If during the evolution enters into the stable set, the solution is global. By using scaling argument, we also answer the question of how small the initial data are, the global solution of the Cauchy oroblem exists.%本文考虑带线性坍塌项和竞争势的非线性波动方程柯西问题．定义了新的稳定集和不稳定集，证明了如果初值进入不稳定集，则解在有限时间爆破；如果初值进入稳定集，则整体解存在．运用势井讨论，回答了当初值在多么小的时候，该柯西问题的整体解存在．
Cauchy-characteristic evolution and waveforms
This paper investigates a new method for computing waveforms generated by independent sources. A global spacetime evolution algorithms based on the Cauchy initial-value approach is used for the bounded interior area and the same algorithm based on the characteristic hypersurfaces is used for the exterior region. Both methods are equated at the common interface
Cauchy-characteristic evolution and waveforms
Bishop, N.T. [Univ. of South Africa, Pretoria (South Africa); Holvorcem, P.R.; Matzner, R.A. [Univ. of Texas, Austin, TX (United States)]|[Pennsylvania State Univ., University Park, PA (United States)] [and others
1997-09-01
This paper investigates a new method for computing waveforms generated by independent sources. A global spacetime evolution algorithms based on the Cauchy initial-value approach is used for the bounded interior area and the same algorithm based on the characteristic hypersurfaces is used for the exterior region. Both methods are equated at the common interface.
Dirac Equation with External Potential and Initial Data on Cauchy Surfaces
Deckert, D -A
2014-01-01
With this paper we provide a mathematical review on the initial-value problem of the one-particle Dirac equation on space-like Cauchy hypersurfaces for compactly supported external potentials. We, first, discuss the physically relevant spaces of solutions and initial values in position and mass shell representation; second, review the action of the Poincar\\'e group as well as gauge transformations on those spaces; third, introduce generalized Fourier transforms between those spaces and prove convenient Paley-Wiener- and Sobolev-type estimates. These generalized Fourier transforms immediately allow the construction of a unitary evolution operator for the free Dirac equation between the Hilbert spaces of square-integrable wave functions of two respective Cauchy surfaces. With a Picard-Lindel\\"of argument this evolution map is generalized to the Dirac evolution including the external potential. For the latter we introduce a convenient interaction picture on Cauchy surfaces. These tools immediately provide anothe...
Moving Heat Source Reconstruction from the Cauchy Boundary Data
Nilson C. Roberty
2010-01-01
Full Text Available We consider the problem of reconstruction of an unknown characteristic transient thermal source inside a domain. By introducing the definition of an extended dirichlet-to-Neumann map in the time-space cylinder and the adoption of the anisotropic Sobolev-Hilbert spaces, we can treat the problem with methods similar to those used in the analysis of the stationary source reconstruction problem. Further, the finite difference θ scheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic star-shape source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula, we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support.
External Field QED on Cauchy Surfaces
Deckert, D -A
2015-01-01
The Shale-Stinespring Theorem (1965) together with Ruijsenaar's criterion (1977) provide a necessary and sufficient condition for the implementability of the evolution of external field quantum electrodynamics between constant-time hyperplanes on standard Fock space. The assertion states that an implementation is possible if and only if the spacial components of the external electromagnetic four-vector potential $A_\\mu$ are zero. We generalize this result to smooth, space-like Cauchy surfaces and, for general $A_\\mu$, show how the second-quantized Dirac evolution can always be implemented as a map between varying Fock spaces. Furthermore, we give equivalence classes of polarizations, including an explicit representative, that give rise to those admissible Fock spaces. We prove that the polarization classes only depend on the tangential components of $A_\\mu$ w.r.t. the particular Cauchy surface, and show that they behave naturally under Lorentz and gauge transformations.
Molecular dynamics at constant Cauchy stress
Miller, Ronald E.; Tadmor, Ellad B.; Gibson, Joshua S.; Bernstein, Noam; Pavia, Fabio
2016-05-01
The Parrinello-Rahman algorithm for imposing a general state of stress in periodic molecular dynamics simulations is widely used in the literature and has been implemented in many readily available molecular dynamics codes. However, what is often overlooked is that this algorithm controls the second Piola-Kirchhoff stress as opposed to the true (Cauchy) stress. This can lead to misinterpretation of simulation results because (1) the true stress that is imposed during the simulation depends on the deformation of the periodic cell, (2) the true stress is potentially very different from the imposed second Piola-Kirchhoff stress, and (3) the true stress can vary significantly during the simulation even if the imposed second Piola-Kirchhoff is constant. We propose a simple modification to the algorithm that allows the true Cauchy stress to be controlled directly. We then demonstrate the efficacy of the new algorithm with the example of martensitic phase transformations under applied stress.
The random conductance model with Cauchy tails
Barlow, Martin T
2009-01-01
We consider a random walk in an i.i.d. Cauchy-tailed conductances environment. We obtain a quenched functional CLT for the suitably rescaled random walk, and, as a key step in the arguments, we improve the local limit theorem for $p^\\om_{n^2 t}(0,y)$ in [BD09, Theorem 5.14] to a result which gives uniform convergence for $p^\\om_{n^2 t}(x,y)$ for all $x, y$ in a ball.
The Southampton Cauchy-characteristic matching project
The Southampton Numerical Relativity Group have set up a long term project concerned with investigating Cauchy-characteristic matching (CCM) codes in numerical relativity. The CCM approach has two distinct features. Firstly, it dispenses with an outer boundary condition and replaces this with matching conditions at an interface residing in the vacuum between the Cauchy and characteristic regions. A successful CCM code leads to a transparent interface and so avoids the spurious reflections which plague most codes employing outer boundary conditions. Secondly, by employing a compactified coordinate, it proves possible to generate global solutions. This means that gravitational waves can be identified unambiguously at future null infinity. To date, cylindrical codes have been developed which have been checked against the exact solutions of Weber-Wheeler, Safier-Stark-Piran and Xanthopoulos. In addition, a cylindrical code has been constructed for investigating dynamic cosmic strings. Recently a master vacuum axi-symmetric CCM code has been completed which consists of four independent modules comprising an interior Cauchy code, an exterior characteristic code together with injection and extraction codes. The main goal of this work is to construct a 3 dimensional code possessing the characteristic, injection and extraction modules which can be attached to an interior code based on a finite grid. Such a code should lead to the construction of more accurate templates which are needed in the search for gravitational waves. (author)
Hamada, Yûsaku
2000-01-01
In our preceding article, by applying the results of Bieberbach, Fatou and Picard, we have studied the domain of holomorphy of the solution of the Cauchy problem for the differential operator with coefficients of entire functions. In this article, by employing the results of the modular function and its ordinary differential equation, we give a remark on the domain of holomorphy of the solution of the Cauchy problem for the differential operator of principal part with polynomial coefficients.
Proof of linear instability of the Reissner-Nordstr\\"om Cauchy horizon under scalar perturbations
Luk, Jonathan
2015-01-01
It has long been suggested that solutions to linear scalar wave equation $$\\Box_g\\phi=0$$ on a fixed subextremal Reissner-Nordstr\\"om spacetime with non-vanishing charge are generically singular at the Cauchy horizon. We prove that generic smooth and compactly supported initial data on a Cauchy hypersurface indeed give rise to solutions with infinite nondegenerate energy near the Cauchy horizon in the interior of the black hole. In particular, the solution generically does not belong to $W^{1,2}_{loc}$. This instability is related to the celebrated blue shift effect in the interior of the black hole. The problem is motivated by the strong cosmic censorship conjecture and it is expected that for the full nonlinear Einstein-Maxwell system, this instability leads to a singular Cauchy horizon for generic small perturbations of Reissner-Nordstr\\"om spacetime. Moreover, in addition to the instability result, we also show as a consequence of the proof that Price's law decay is generically sharp along the event horiz...
Structure of the Cauchy horizon singularity
Burko, L M
1999-01-01
We study the Cauchy horizon (CH) singularity of a spherical charged black hole perturbed nonlinearly by a self-gravitating massless scalar field. We show numerically that the singularity is weak both at the early and at the late sections of the CH, where the focusing of the area coordinate $r$ is strong. In the early section the metric perturbations vanish, and the fields behave according to perturbation analysis. We find exact analytical expressions for the gradients of $r$ and of the scalar field, which are valid at both sections. We then verify these analytical results numerically.
Application of geometric algebra to electromagnetic scattering the Clifford-Cauchy-Dirac technique
Seagar, Andrew
2016-01-01
This work presents the Clifford-Cauchy-Dirac (CCD) technique for solving problems involving the scattering of electromagnetic radiation from materials of all kinds. It allows anyone who is interested to master techniques that lead to simpler and more efficient solutions to problems of electromagnetic scattering than are currently in use. The technique is formulated in terms of the Cauchy kernel, single integrals, Clifford algebra and a whole-field approach. This is in contrast to many conventional techniques that are formulated in terms of Green's functions, double integrals, vector calculus and the combined field integral equation (CFIE). Whereas these conventional techniques lead to an implementation using the method of moments (MoM), the CCD technique is implemented as alternating projections onto convex sets in a Banach space. The ultimate outcome is an integral formulation that lends itself to a more direct and efficient solution than conventionally is the case, and applies without exception to all types...
Self-accelerating Massive Gravity: Superluminality, Cauchy Surfaces and Strong Coupling
Motloch, Pavel; Joyce, Austin; Motohashi, Hayato
2015-01-01
Self-accelerating solutions in massive gravity provide explicit, calculable examples that exhibit the general interplay between superluminality, the well-posedness of the Cauchy problem, and strong coupling. For three particular classes of vacuum solutions, one of which is new to this work, we construct the conformal diagram for the characteristic surfaces on which isotropic stress-energy perturbations propagate. With one exception, all solutions necessarily possess spacelike characteristics, indicating perturbative superluminality. Foliating the spacetime with these surfaces gives a pathological frame where kinetic terms of the perturbations vanish, confusing the Hamiltonian counting of degrees of freedom. This frame dependence distinguishes the vanishing of kinetic terms from strong coupling of perturbations or an ill-posed Cauchy problem. We give examples where spacelike characteristics do and do not originate from a point where perturbation theory breaks down and where spacelike surfaces do or do not inte...
Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces
Choonkil BAAK
2006-01-01
Let X, Y be vector spaces. It is shown that if a mapping f: X → Y satisfiesf(x+y/2+z)+f(x-y/2+z)=f(x)+2f(z), (0.1)f(x+y/2+z)-f(x-y/2+z)= f(y), (0.2)or2f(x+y/2+z)=f(x)+f(y)+2f(z) (0.3)for all x, y, z ∈ X, then the mapping f: X → Y is Cauchy additive.Furthermore, we prove the Cauchy-Rassias stability of the functional equations (0.1), (0.2) and (0.3) in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebra.
Abstract Description of Internet Traffic of Generalized Cauchy Type
Ming Li
2012-01-01
. Therefore, there is a limitation for fGn to accurately model traffic. Recently, the generalized Cauchy (GC process was introduced to model traffic with the flexibility to separately measure the fractal dimension DGC and the Hurst parameter HGC of traffic. However, there is a fundamental problem whether or not there exists the generality that the GC model is more conformable with real traffic than single parameter models, such as fGn, irrelevant of traffic traces used in experimental verification. The solution to that problem remains unknown but is desired for model evaluation in traffic theory or for model selection against specific issues, such as queuing analysis relating to the autocorrelation function (ACF of arrival traffic. The key contribution of this paper is our solution to that fundamental problem (see Theorem 3.17 with the following features in analysis. (i Set-valued analysis of the traffic of the fGn type. (ii Set-valued analysis of the traffic of the GC type. (iii Revealing the generality previously mentioned by comparing metrics of the traffic of the fGn type to that of the GC type.
Cauchy Based Matched Filter for Retinal Vessels Detection
Zolfagharnasab, Hooshiar; Naghsh-Nilchi, Ahmad Reza
2014-01-01
In this paper, a novel matched filter based on a new kernel function with Cauchy distribution is introduced to improve the accuracy of the automatic retinal vessel detection compared with other available matched filter-based methods, most notably, the methods built on Gaussian distribution function. Several experiments are conducted to pick the best values of the parameters for the new designed filter, including both Cauchy function parameters as well as the matched filter parameters such as ...
Cauchy-Kowalevski and polynomial ordinary differential equations
Roger J. Thelwell
2012-01-01
Full Text Available The Cauchy-Kowalevski Theorem is the foremost result guaranteeing existence and uniqueness of local solutions for analytic quasilinear partial differential equations with Cauchy initial data. The techniques of Cauchy-Kowalevski may also be applied to initial-value ordinary differential equations. These techniques, when applied in the polynomial ordinary differential equation setting, lead one naturally to a method in which coefficients of the series solution are easily computed in a recursive manner, and an explicit majorization admits a clear a priori error bound. The error bound depends only on immediately observable quantities of the polynomial system; coefficients, initial conditions, and polynomial degree. The numerous benefits of the polynomial system are shown for a specific example.
Nonlocal Cauchy problem for nonlinear mixed integrodifferential equations
H.L. Tidke
2010-12-01
Full Text Available The present paper investigates the existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition. The results obtained by using Schauder fixed point theorem and the theory of semigroups.
ON CAUCHY-POMPEIU FORMULA FOR FUNCTIONS WITH VALUES IN A UNIVERSAL CLIFFORD ALGEBRA
无
2003-01-01
This paper obtains the Cauchy-Pompeiu formula on certain distinguishedboundary for functions with values in a universal Clifford algebra. This formula is just anextension of the Cauchy's integral formula obtained in [11].
A Cauchy-Davenport theorem for linear maps
Herdade, Simao; Kim, John; Kopparty, Swastik
2015-01-01
We prove a version of the Cauchy-Davenport theorem for general linear maps. For subsets $A,B$ of the finite field $\\mathbb{F}_p$, the classical Cauchy-Davenport theorem gives a lower bound for the size of the sumset $A+B$ in terms of the sizes of the sets $A$ and $B$. Our theorem considers a general linear map $L: \\mathbb{F}_p^n \\to \\mathbb{F}_p^m$, and subsets $A_1, \\ldots, A_n \\subseteq \\mathbb{F}_p$, and gives a lower bound on the size of $L(A_1 \\times A_2 \\times \\ldots \\times A_n)$ in ter...
On cosmic censorship: do compact Cauchy horizons imply symmetry?
The basic idea of Cosmic Censorship is that, in a physically reasonable spacetime, an observer should not encounter any naked singularities. The authors discuss some new results which provide strong support for one of the statements of Cosmic Censorship: Strong Cosmic Censorship says that the maximal spacetime development of a set of Cauchy data on a spacelike initial surface (evolved via the vacuum Einstein equations, the Einstein-Maxwell equations, or some other 'reasonable' set) will not be extendible across a Cauchy horizon. (Auth.)
Area theorem and smoothness of compact Cauchy horizons
Minguzzi, E
2014-01-01
We obtain an improved version of the area theorem for not necessarily differentiable horizons which, in conjunction with a recent result on the completeness of generators, allows us to prove that under the null energy condition every compactly generated Cauchy horizon is smooth and compact. We explore the consequences of this result for time machines, topology change, black holes and cosmic censorship. For instance, it is shown that compact Cauchy horizons cannot form in a non-empty spacetime which satisfies the stable dominant energy condition wherever there is some source content.
Area Theorem and Smoothness of Compact Cauchy Horizons
Minguzzi, E.
2015-10-01
We obtain an improved version of the area theorem for not necessarily differentiable horizons which, in conjunction with a recent result on the completeness of generators, allows us to prove that under the null energy condition every compactly generated Cauchy horizon is smooth and compact. We explore the consequences of this result for time machines, topology change, black holes and cosmic censorship. For instance, it is shown that compact Cauchy horizons cannot form in a non-empty spacetime which satisfies the stable dominant energy condition wherever there is some source content.
UNCONDITIONAL CAUCHY SERIES AND UNIFORM CONVERGENCE ON MATRICES
A. AIZPURU; A. GUTIERREZ-DAVILA
2004-01-01
The authors obtain new characterizations of unconditional Cauchy series in terms of separation properties of subfamilies of p(N), and a generalization of the Orlicz-Pettis Theorem is also obtained. New results on the uniform convergence on matrices and a new version of the Hahn-Schur summation theorem are proved. For matrices whose rows define unconditional Cauchy series, a better sufficient condition for the basic Matrix Theorem of Antosik and Swartz, new necessary conditions and a new proof of that theorem are given.
First order differential operator associated to the Cauchy-Riemann operator in a Clifford algebra
The complex differentiation transforms holomorphic functions into holomorphic functions. Analogously, the conjugate Cauchy-Riemann operator of the Clifford algebra transforms regular functions into regular functions. This paper determines more general first order operator L (matrix-type) for which Lu is regular provided u is regular. For such operator L, the initial value problem ∂u / ∂t = L (t, x, u, ∂u / ∂x) (1) u(0, x) = φ(x) (2) is solvable for an arbitrary regular function φ and the solution is regular in x for each t. (author)
Hamilton-Jacobi Theory in Cauchy Data Space
Campos, CéAdric M.; de Leóan, Manuel; de Diego, David Martín; Vaquero, Miguel
2015-12-01
Recently, M. de LeóAn et al. [8] have developed a geometric Hamilton-Jacobi theory for classical fields in the setting of multisymplectic geometry. Our purpose in the current paper is to establish the corresponding Hamilton-Jacobi theory in the Cauchy data space, and relate both approaches.
Multiple elliptic hypergeometric series --An approach from the Cauchy determinant--
Kajihara, Yasushi; Noumi, Masatoshi
2003-01-01
A multiple generalization of elliptic hypergeometric series is investigated and a duality transformation for multiple hypergeometric series is proposed. Our duality transformation obtained from an identity arising from the Cauchy determinant formula for Weierstrass sigma functions, by spcialization of a particular form.
Unit Root Testing in Heteroscedastic Panels Using the Cauchy Estimator
Demetrescu, Matei; Hanck, Christoph
2012-01-01
The Cauchy estimator of an autoregressive root uses the sign of the first lag as instrumental variable. The resulting IV t-type statistic follows a standard normal limiting distribution under a unit root case even under unconditional heteroscedasticity, if the series to be tested has no deterministi
Cauchy Tetrahedron Argument Applied to Higher Contact Interactions
dell'Isola, F.; Madeo, A.; Seppecher, P.
2016-03-01
Second gradient theories are nowadays used in many studies in order to describe in detail some transition layers which may occur in micro-structured materials and in which physical properties are sharply varying. Sometimes higher order theories are also evoked. Up to now these models have not been based on a construction of stresses similar to the one due to Cauchy, which has been applied only for simple materials. It has been widely recognized that the fundamental assumption by Cauchy that the traction depends only on the normal of the dividing surface cannot be maintained for higher gradient theories. However, this observation did not urge any author, to our knowledge, to revisit the Cauchy construction in order to adapt it to a more general conceptual framework. This is what we do in this paper for a continuum of grade N (also called N-th gradient continuum). Our construction is very similar to the one due to Cauchy; based on the tetrahedron argument, it does not introduce any argument of a different nature. In particular, we avoid invoking the principle of virtual work. As one should expect, the balance assumption and the regularity hypotheses have to be adapted to the new framework and tensorial computations become more complex.
Hamada, Yûsaku
2002-01-01
In Y. Hamada, Tohoku Math. J., 52 (2000), 79-94, we have given a remark on the domain of holomorphy of the solution of the Cauchy problem for the differential operator with polynomial coefficients. In this article, we give some complements to the results of Y. Hamada, Tohoku Math. J., 52 (2000), 79-94.
Differential operators associated to the Cauchy-Riemann operator in a quaternion algebra
This paper deals with the initial value problem of the type φw / φt = L (t, x, w, φw / φxi) (1) w(0, x) = φ(x) (2) where t is the time, L is a linear first order operator (matrix-type) in a Quaternion algebra and φ is a regular function. The article proves necessary and sufficient conditions on the coefficients of operator L under which L is associated to the Cauchy-Riemann operator of Quarternion algebra. This criterion makes it possible to construct the operator L for which the initial problem (1),(2) is solvable for an arbitrary initial regular function φ and the solution is also regular for each t. (author)
Regularity of Cauchy horizons in S{sup 2} x S{sup 1} Gowdy spacetimes
Hennig, Joerg [Max Planck Institute for Gravitational Physics, Am Muehlenberg 1, D-14476 Golm (Germany); Ansorg, Marcus, E-mail: pjh@aei.mpg.d, E-mail: marcus.ansorg@helmholtz-muenchen.d [Institute of Biomathematics and Biometry, Helmholtz Zentrum Muenchen, Ingolstaedter Landstr. 1, D-85764 Neuherberg (Germany)
2010-03-21
We study general S{sup 2} x S{sup 1} Gowdy models with a regular past Cauchy horizon and prove that a second (future) Cauchy horizon exists, provided that a particular conserved quantity J is not zero. We derive an explicit expression for the metric form on the future Cauchy horizon in terms of the initial data on the past horizon and conclude the universal relation A{sub p}A{sub f} = (8piJ){sup 2} where A{sub p} and A{sub f} are the areas of past and future Cauchy horizons respectively.
On the generalized Cauchy function and new Conjecture on its exterior singularities
Theodore Yaotsu Wu
2011-01-01
This article studies on Cauchy's function f(z) and its integral, (2πi)J[f(z)] ≡ Φc f(t)dt/(t - z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D+ bounded by C and the open domain D- outside C. ( 1 ) With f(z) assumed to be Cn (n ＜ ∞-times continuously differentiable) (Λ)z ∈ D+ and in a neighborhood of C, f(z) and its derivatives f(n)(z) are proved uniformly continuous in the closed domain D+ = [D+ + C].(2) Cauchy's integral formulas and their derivatives (V)z ∈ D+(or (V)z ∈ D-) are proved to converge uniformly in D+ (or in D- = [D- + C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f(z) and J[f(z)]) are shown extended to hold for the complement function F(z), defined to be Cn(V)z ∈ D- and about C. (4) The uniform convergence theorems for f(z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four generalized Hilbert-type integral transforms are proved. (5) Further,the singularity distribution of f(z) in D- is elucidated by considering the direct problem exemplified with several typical singularities prescribed in D-. (6) A comparative study is made between generalized integral formulas and Plemelj's formulas on their differing basic properties. (7) Physical significances of these formulas are illustrated with applications to nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f(z) in domain D-, based on the continuous numerical value of f(z)(V)z ∈ D+ = [D+ + C], is presented for resolution as a conjecture.
α-times Integrated Regularized Cosine Functions and Second Order Abstract Cauchy Problens
张寄洲; 陶有山
2001-01-01
In this paper, α -times integrated C-regularized cosine functions and mild α-times integrated C-existence families of second order are introduced. Equivalences are proved among α -times integrated C-regularized cosine function for a linear operator A, C-wellposed of (α + 1)-times abstract Cauchy problem and mild α -times integrated C-existence family of second order for A when the commutable condition is satisfied. In addition, if A = C-1AC, they are also equivalent to A generating the α -times integrated C-regularized cosine finction.The characterization of an exponentially botnded mild α -times integrated C-existence family of second order is given out in terms of a Laplace transform.
Cauchy inequality and uncertainty relations for mixed states
Cauchy inequality (CI) relates scalar products of two vectors and their norms. I point out other similar inequalities (SI). Starting with CI Schroedinger derived his uncertainty relation (UR). By using SI other various UR can be obtained. It is shown that they follow from the Schroedinger UR. Two generalizations of CI are obtained for mixed states described by density matrices. Using them two generalizations of UR for mixed states are derived. Both differ from the UR generalization known from the literature. The discussion of these generalizations is given
Implementing Cauchy Reed-Solomon Utilizing OpenCL
Karlsson, Tim
2013-01-01
In this paper the performance of executing Cauchy Reed-Solomon (CRS) coding on the GPU is evaluated and compared with execution on the CPU. Three different prototypes are developed. One is sequential and developed in C++, the other two are developed utilizing C++ and OpenCL. The measurements are done by comparing the execution time of different data block sizes ranging from 16KB up to 256MB with two different encoding ratios, 9/12 and 10/16. The measurements are done on an Intel CPU with 4 co...
Structure of the black hole's Cauchy horizon singularity
Burko, L M
1997-01-01
We study the Cauchy horizon (CH) singularity of a spherical charged black hole perturbed nonlinearly by a self-gravitating massless scalar field. We show numerically that the singularity is weak both at the early and at the late sections of the CH, where the focusing of the area coordinate $r$ is strong. In the early section the metric is almost Reissner-Nordström, and the fields behave according to perturbation analysis. We find exact analytical expressions for the gradients of $r$ and of the scalar field, which are valid at both sections. We then verify these analytical results numerically.
Cauchy, Sturm et les racines des équations
Benis Sinaceur, Hourya
1992-01-01
International audience Relying both on chronological data and textual analysis, this article seeks to determine the mutual influence between A.-L. Cauchy's and Ch.-F. Sturm's research from 1829 to around 1840 on the roots of algebraic equations. The first aim is to specify the elements of a micro-history, strictly limited in time and centered on a well-defined question, yet bound up with the endeavour to build up a mathematical physics at the beginning of the 19th century. The second aim p...
On a Higher Order Cauchy-Pompeiu Formula for Functions with Values in a Universal Clifford Algebra
Zhongxiang, Zhang
2007-01-01
By constructing suitable kernel functions, a higher order Cauchy-Pompeiu formula for functions with values in a universal Clifford algebra is obtained, leading to a higher order Cauchy integral formula.
Hyperelastic bodies under homogeneous Cauchy stress induced by non-homogeneous finite deformations
Mihai, L Angela
2016-01-01
We discuss whether homogeneous Cauchy stress implies homogeneous strain in isotropic nonlinear elasticity. While for linear elasticity the positive answer is clear, we exhibit, through detailed calculations, an example with inhomogeneous continuous deformation but constant Cauchy stress. The example is derived from a non rank-one convex elastic energy. Connections to conforming and non-conforming finite element implementations are drawn.
Gravitational wave extraction based on Cauchy-characteristic extraction and characteristic evolution
Babiuc, M; Hawke, I; Zlochower, Y; Babiuc, Maria; Szilagyi, Bela; Hawke, Ian; Zlochower, Yosef
2005-01-01
We implement a code to find the gravitational news at future null infinity by using data from a Cauchy code as boundary data for a characteristic code. This technique of Cauchy-characteristic Extraction (CCE) allows for the unambiguous extraction of gravitational waves from numerical simulations.
External Field QED on Cauchy Surfaces for Varying Electromagnetic Fields
Deckert, D.-A.; Merkl, F.
2016-08-01
The Shale-Stinespring Theorem (J Math Mech 14:315-322, 1965) together with Ruijsenaar's criterion (J Math Phys 18(4):720-737, 1977) provide a necessary and sufficient condition for the implementability of the evolution of external field quantum electrodynamics between constant-time hyperplanes on standard Fock space. The assertion states that an implementation is possible if and only if the spatial components of the external electromagnetic four-vector potential {A_μ} are zero. We generalize this result to smooth, space-like Cauchy surfaces and, for general {A_μ}, show how the second-quantized Dirac evolution can always be implemented as a map between varying Fock spaces. Furthermore, we give equivalence classes of polarizations, including an explicit representative, that give rise to those admissible Fock spaces. We prove that the polarization classes only depend on the tangential components of {A_μ} w.r.t. the particular Cauchy surface, and show that they behave naturally under Lorentz and gauge transformations.
Chatnugrob Sangsawang
2016-06-01
Full Text Available This paper addresses a problem of the two-stage flexible flow shop with reentrant and blocking constraints in Hard Disk Drive Manufacturing. This problem can be formulated as a deterministic FFS|stage=2,rcrc, block|Cmax problem. In this study, adaptive Hybrid Particle Swarm Optimization with Cauchy distribution (HPSO was developed to solve the problem. The objective of this research is to find the sequences in order to minimize the makespan. To show their performances, computational experiments were performed on a number of test problems and the results are reported. Experimental results show that the proposed algorithms give better solutions than the classical Particle Swarm Optimization (PSO for all test problems. Additionally, the relative improvement (RI of the makespan solutions obtained by the proposed algorithms with respect to those of the current practice is performed in order to measure the quality of the makespan solutions generated by the proposed algorithms. The RI results show that the HPSO algorithm can improve the makespan solution by averages of 14.78%.
Moving Heat Source Reconstruction from Cauchy Boundary Data: The Cartesian Coordinates Case
Nilson C. Roberty
2011-01-01
Full Text Available We consider the problem of reconstruction of an unknown characteristic interval and block transient thermal source inside a domain. By exploring the definition of an Extended Dirichlet to Neumann map in the time space cylinder that has been introduced in Roberty and Rainha (2010a, we can treat the problem with methods similar to that used in the analysis of the stationary source reconstruction problem. Further, the finite difference θ-scheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic interval and parallelepiped source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support. Numerical experiment for capture of an interval and an rectangular parallelepiped characteristic source inside a cubic box domain from boundary data are presented in threedimensional and one-dimensional implementations. The problem of centroid determination is addressed and questions are discussed from an computational points of view.
Noise Kernel for Reissner Nordstrom Metric: Results at Cauchy Horizon
Satin, Seema
2015-01-01
We obtain the Noise Kernel, which defines quantum stress tensor fluctuations for Reissner Nordstrom metric modeled as gravitationally collapsing spacetime, after transforming to coordinates suitable for the evolution.The quantum stress tensor fluctuations at the final stage of collapse namely naked singularity or a black hole are then analysed.The behavior of this entity which is central to semiclassical stochastic gravity at the Cauchy Horizon shows markedly different behaviour from earlier obtained result for self similar Tolman Bondi case showing a very unique divergence. It is known that the quantum stress tensor itself diverges at CH for both Tolman Bondi and Reissner Nordstrom metric. In contrast, it can now be seen that the correlations of the fluctuations of the stress tensor behave differently for the two cases. We give a discussion and further directions of investigations for this interesting behaviour in the two cases (regarding the collapse scenario).
On Cauchy conditions for asymmetric mixed convection boundary layer flows
Amaouche, Mustapha [Laboratoire de Physique Theorique, Universite de Bejaia (Algeria); Kessal, Mohand [Departement Transport et Equipement Petrolier, Faculte des Hydrocarbures et de la Chimie, Universite de Boumerdes, 35000, Boumerdes (Algeria)
2003-06-01
The fundamental question of how and where does an asymmetric mixed convection boundary layer flow around a heated horizontal circular cylinder begin to develop is raised. We first transform the classical boundary layer equations by using an integral method of Karman-Pohlhausen type and obtain two coupled equations governing the evolutions of the dynamic and thermal boundary layers. Because of its global character, the implemented method allows to bypass the difficulty of downstream-upstream interactions. Cauchy conditions characterizing the starting of the boundary layers are found; they are obtained in a surprisingly simple manner for the limiting cases corresponding to Pr=1, Pr{yields}0 and Pr{yields}{infinity}. Otherwise, these conditions can be found by using a prediction correction algorithm. Some numerical experiments are finally performed in order to illustrate the theory. (authors)
Completion of a Cauchy space without the T2-restriction on the space
Nandita Rath
2000-08-01
Full Text Available A completion of a Cauchy space is obtained without the T2 restriction on the space. This completion enjoys the universal property as well. The class of all Cauchy spaces with a special class of morphisms called s-maps form a subcategory CHY' of CHY. A completion functor is defined for this subcategory. The completion subcategory of CHY' turns out to be a bireflective subcategory of CHY'. This theory is applied to obtain a characterization of Cauchy spaces which allow regular completion.
Spectral Cauchy Characteristic Extraction of strain, news and gravitational radiation flux
Handmer, Casey J; Winicour, Jeffrey
2016-01-01
We present a new approach for the Cauchy-characteristic extraction of gravitational radiation strain, news function, and the flux of the energy-momentum, supermomentum and angular momentum associated with the Bondi-Metzner-Sachs asymptotic symmetries. In Cauchy-characteristic extraction, a characteristic evolution code propagates numerical data on an inner worldtube supplied by a Cauchy evolution code to obtain the space-time metric in a neighborhood of null infinity. The metric is first determined in a scrambled form in terms of coordinates determined by the Cauchy formalism. In prior treatments, the waveform is first extracted from this metric and then transformed into an asymptotic inertial coordinate system. This procedure provides the physically proper description of the waveform and the radiated energy but it does not generalize to determine the flux of angular momentum or supermomentum. Here we formulate and implement a new approach which transforms the full metric into an asymptotic inertial frame and...
Analysis of linear waves near the Cauchy horizon of cosmological black holes
Hintz, Peter
2015-01-01
We show that linear scalar waves are bounded and continuous up to the Cauchy horizon of Reissner-Nordstr\\"om-de Sitter and Kerr-de Sitter spacetimes, and in fact decay exponentially fast to a constant along the Cauchy horizon. We obtain our results by modifying the spacetime beyond the Cauchy horizon in a suitable manner, which puts the wave equation into a framework in which a number of standard as well as more recent microlocal regularity and scattering theory results apply. In particular, the conormal regularity of waves at the Cauchy horizon - which yields the boundedness statement - is a consequence of radial point estimates, which are microlocal manifestations of the blue-shift and red-shift effects.
Gravitational wave extraction based on Cauchy-characteristic extraction and characteristic evolution
Babiuc, M.; Szilagyi, B; Hawke, I.; Zlochower, Y.
2005-01-01
We implement a code to find the gravitational news at future null infinity by using data from a Cauchy code as boundary data for a characteristic code. This technique of Cauchy-characteristic extraction (CCE) allows for the unambiguous extraction of gravitational waves from numerical simulations. We first test the technique on non-radiative spacetimes: Minkowski spacetime, perturbations of Minkowski spacetime and static black hole spacetimes in various gauges. We show the convergence and limi...
Ortiz, Néstor
2013-01-01
A spherical dust cloud which is initially at rest and which has a monotonously decaying density profile collapses and forms a shell-focussing singularity. Provided the density profile is not too flat, meaning that its second radial derivative is negative at the center, this singularity is visible to local, and sometimes even to global observers. According to the strong cosmic censorship conjecture, such naked singularities should be unstable under generic, nonspherical perturbations of the initial data or when more realistic matter models are considered. In an attempt to gain some understanding about this stability issue, in this work we initiate the analysis of a simpler but related problem. We discuss the stability of test fields propagating in the vicinity of the Cauchy horizon associated to the naked central singularity. We first study the high-frequency limit and show that the fields undergo a blueshift as they approach the Cauchy horizon. However, in contrast to what occurs at inner horizons of black ho...
Cauchy graph embedding based diffusion model for salient object detection.
Tan, Yihua; Li, Yansheng; Chen, Chen; Yu, Jin-Gang; Tian, Jinwen
2016-05-01
Salient object detection has been a rather hot research topic recently, due to its potential applications in image compression, scene classification, image registration, and so forth. The overwhelming majority of existing computational models are designed based on computer vision techniques by using lots of image cues and priors. Actually, salient object detection is derived from the biological perceptual mechanism, and biological evidence shows that the spread of the spatial attention generates the object attention. Inspired by this, we attempt to utilize the emerging spread mechanism of object attention to construct a new computational model. A novel Cauchy graph embedding based diffusion (CGED) model is proposed to fulfill the spread process. Combining the diffusion model and attention prediction model, a salient object detection approach is presented through perceptually grouping the multiscale diffused attention maps. The effectiveness of the proposed approach is validated on the salient object dataset. The experimental results show that the CGED process can obviously improve the performance of salient object detection compared with the input spatial attention map, and the proposed approach can achieve performance comparable to that of state-of-the-art approaches. PMID:27140886
Who Gave you the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus
Borovik, Alexandre; 10.1007/s10699-011-9235-x
2011-01-01
Cauchy's contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Bjorling, an infinitesimal definition of the criterion of uniform convergence. Cauchy's foundational stance is hereby reconsidered.
Cauchy´s stress theorem for stresses represented by measures
Šilhavý, Miroslav
2008-01-01
Roč. 20, č. 2 (2008), s. 75-96. ISSN 0935-1175 Institutional research plan: CEZ:AV0Z10190503 Keywords : Cauchy stress theorem * divergence measure vectorfield * Newton homotopy Subject RIV: BA - General Mathematics Impact factor: 1.552, year: 2008
Convergence and the Cauchy Property of Sequences in the Setting of Actual Infinity
Shipman, Barbara A.
2013-01-01
Traditional definitions, language, and visualizations of convergence and the Cauchy property of sequences convey a sense of the sequence as a potentially infinite process rather than an actually infinite object. This has a deep-rooted influence on how we think about and teach concepts on sequences, particularly in undergraduate calculus and…
TIAN Dongyan; JIN Ming; DUI Guansuo
2006-01-01
A new approach for the derivation of the principal invariants of the stretch tensor with respect to the right Cauchy Green tensor is presented in this paper. According to the definition of the derivation of tensor function, the three first-order derivatives for the principal invariants of the stretch tensor are obtained through derivation directly to the right Cauchy-Green tensor by incremental method. Then the three second-order derivatives are yielded by the derivation to the right Cauchy-Green strain tensor directly. Furthermore, an explicit expression of the tangent modulus of the general Varga material is given as an example.
Ansorg, Marcus; Hennig, Jörg
2009-06-01
We study the interior electrovacuum region of axisymmetric and stationary black holes with surrounding matter and find that there exists always a regular inner Cauchy horizon inside the black hole, provided the angular momentum J and charge Q of the black hole do not vanish simultaneously. In particular, we derive an explicit relation for the metric on the Cauchy horizon in terms of that on the event horizon. Moreover, our analysis reveals the remarkable universal relation (8piJ);{2}+(4piQ;{2});{2}=A;{+}A;{-}, where A+ and A- denote the areas of event and Cauchy horizon, respectively. PMID:19658851
Yang, Jerry Zhijian
2014-01-01
Irving and Kirkwood formulism (IK formulism) provides a way to compute continuum mechanics quantities at certain location in terms of molecular variables. To make the approach more practical in computer simulation, Hardy proposed to use a spacial kernel function that couples continuum quantities with atomistic information. To reduce irrational fluctuations, Murdoch proposed to use a temporal kernel function to smooth the physical quantities obtained in Hardy's approach. In this paper, we generalize the original IK formulism to systematically incorporate both spacial and temporal average. The Cauchy stress tensor is derived in this generalized IK formulism (g-IK formulism). Analysis is given to illuminate the connection and difference between g-IK formulism and traditional temporal post-process approach. The relationship between Cauchy stress and first Piola-Kirchhoff stress is restudied in the framework of g-IK formulism. Numerical experiments using molecular dynamics are conducted to examine the analysis res...
The generalized Cauchy relation (gCR) of epoxy/silica nano-composites does not show either the chemically induced sol-gel transition or the chemically induced glass transition in the course of polymerization. Astonishingly, by varying the silica nanoparticles' concentration between 0 and 25 vol% in the composites, the Cauchy parameter A of the gCR remains universal and can be determined from the pure epoxy's elastic moduli. Air-filled porous silica glasses are considered as models for percolated silica particles. A longitudinal modulus versus density representation evidences the aforementioned transition phenomena during polymerization of the epoxy/silica nanocomposites. The existence of optically and mechanically relevant interphases is discussed.
Sobre la equivalencia entre sucesiones con límite finito y sucesiones de Cauchy
Claros, Francisco Javier; Sánchez, María Teresa; Coriat, Moisés
2009-01-01
Estudiamos, desde perspectivas simbólica y fenomenológica, diferencias y analogías existentes entre dos definiciones: la de límite finito de una sucesión y la de sucesión de Cauchy. Las diferencias entre una y otra definición parecen acentuarse en el aspecto fenomenológico, ya que observamos fenómenos distintos en cada una de ellas.
Jian-Guo Zheng; Chao-Qun Zhang; Yong-Quan Zhou
2015-01-01
Artificial bee colony (ABC) algorithm is a popular swarm intelligence technique inspired by the intelligent foraging behavior of honey bees. However, ABC is good at exploration but poor at exploitation and its convergence speed is also an issue in some cases. To improve the performance of ABC, a novel ABC combined with grenade explosion method (GEM) and Cauchy operator, namely, ABCGC, is proposed. GEM is embedded in the onlooker bees’ phase to enhance the exploitation ability and accelerate c...
Dukkipati, Ambedkar; Murty, Narasimha M; Bhatnagar, Shalabh
2004-01-01
Boltzmann selection is an important selection mechanism in evolutionary algorithms as it has theoretical properties which help in theoretical analysis. However, Boltzmann selection is not used in practice because a good annealing schedule for the `inverse temperature' parameter is lacking. In this paper we propose a Cauchy annealing schedule for Boltzmann selection scheme based on a hypothesis that selection-strength should increase as evolutionary process goes on and distance between two sel...
The tortoise coordinates and the cauchy problem in the stable study of the Schwarzschild black hole
Tian, G; Zhong, S; Tian, Gui-hua; Wang, Shi-kun; Zhong, Shuquan
2006-01-01
Generally, the Schwarzschild black hole was proved stable through two different methods: the mode-decomposition method and the integral method. In the paper, we show the integral method can only apply to the initial data vanishing at both the horizon and the spatial infinity. It can not treat the initial data only vanishing at the spatial infinity. We give an example to show the misleading information caused by the use of the tortoise coordinates in the perturbation equations. Subsequently, the perturbation equation in the Schwarzschild coordinates is shown not sufficient for the stable study.
Approach to a Cauchy Problem in Stability Study of the Schwarzschild Black Hole
TIAN Gui-Hua; WANG Shi-Kun; ZHONG Shu-Quan
2007-01-01
Generally, the Schwarzschild black hole is proven to be stable by two different methods: the mode-decomposition method and the integral method. We show that the integral method can only apply to the initial data vanishing at both the horizon and the spatial infinity. It can not treat the initial data only vanishing at the spatial infinity. We give an example to show the misleading information caused by the use of tortoise coordinates in the perturbation equations. Subsequently, the perturbation equations in the Schwarzschild coordinates are shown to be insufficient for the stability study.
The Cauchy Problem for the p-Laplacian Equation with a Nonlinear Source
LEI Pei-dong
2001-01-01
In this paper we study the existence and uniqueness of positive solutions for the p-Laplacian equation with nonlinear sourceu/ t = div(｜ Du ｜p-2Du) + u-q, p ＞ 2, 0 ＜ q ＜ ∞ in the class of functions with some prescribed growth rate as ｜ x ｜→ ∞. We also give a description of thelarge time behaviour and show that it is determined by the competition between the diffusion and the source.
MAXIMAL SUBSPACES FOR SOLUTIONS OF THE SECOND ORDER ABSTRACT CAUCHY PROBLEM
无
2007-01-01
For a continuous, increasing function ω: R+ → R+\\{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R+, and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.
Global smooth solution of the Cauchy problem for a model of a radiative flow
Ducomet, B.; Nečasová, Šárka
2015-01-01
Roč. 14, č. 1 (2015), s. 1-36. ISSN 0391-173X R&D Projects: GA ČR GA201/08/0012 Institutional support: RVO:67985840 Keywords : Navier-Stokes-Fourier system * radiative equilibrium Subject RIV: BA - General Mathematics Impact factor: 0.921, year: 2014 http://annaliscienze.sns.it/index.php?page=Article&id=332
On the Cauchy Problem of Evolution p-Laplacian Equation with Nonlinear Gradient Term
Mingyu CHEN; Junning ZHAO
2009-01-01
The authors study the existence of solution to p-Laplacian equation with non-linear forcing term under optimal assumptions on the initial data,which are assumed to be measures.The existence of local solution is obtained.
Cauchy problem for a generalized weakly dissipative periodic two-component Camassa-Holm system
Wenxia Chen
2014-05-01
Full Text Available In this article, we study a generalized weakly dissipative periodic two-component Camassa-Holm system. We show that this system can exhibit the wave-breaking phenomenon and determine the exact blow-up rate of strong solution to the system. In addition, we establish a sufficient condition for having a global solution.
100 great problems of elementary mathematics their history and solution
Dorrie, Heinrich
2013-01-01
Problems that beset Archimedes, Newton, Euler, Cauchy, Gauss, Monge and other greats, ready to challenge today's would-be problem solvers. Among them: How is a sundial constructed? How can you calculate the logarithm of a given number without the use of logarithm table? No advanced math is required. Includes 100 problems with proofs.
Symmetries of 2-lattices and second order accuracy of the Cauchy--Born Model
Van Koten, Brian
2012-01-01
We show that the Cauchy--Born model of a single-species 2-lattice is second order if the atomistic and continuum kinematics are connected in a novel way. Our proof uses a generalization to 2-lattices of the point symmetry of Bravais lattices. Moreover, by identifying similar symmetries in multi-species pair interaction models, we construct a new stored energy density, using shift-gradients but not strain gradients, that is also second order accurate. These results can be used to develop highly accurate continuum models and atomistic/continuum coupling methods for materials such as graphene, hcp metals, and shape memory alloys.
Initial Value Problems for Wave Equations on Manifolds
We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hörmander
Initial Value Problems for Wave Equations on Manifolds
Bär, Christian, E-mail: baer@math.uni-potsdam.de [Universität Potsdam, Institut für Mathematik (Germany); Wafo, Roger Tagne, E-mail: rtagnewafo@yahoo.com [University of Douala, Faculty of Science, Department of Mathematics and Computer Science (Cameroon)
2015-12-15
We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hörmander.
Navier-Stokes Equations—Millennium Prize Problems
Asset A. Durmagambetov; Leyla S. Fazilova
2015-01-01
In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We also describe the loss of smoothness of classical solutions for the Navier-Stokes equations.
$q$-poly-Bernoulli numbers and $q$-poly-Cauchy numbers with a parameter by Jackson's integrals
Komatsu, Takao
2015-01-01
We define $q$-poly-Bernoulli polynomials $B_{n,\\rho,q}^{(k)}(z)$ with a parameter $\\rho$, $q$-poly-Cauchy polynomials of the first kind $c_{n,\\rho,q}^{(k)}(z)$ and of the second kind $\\widehat c_{n,\\rho,q}^{(k)}(z)$ with a parameter $\\rho$ by Jackson's integrals, which generalize the previously known numbers and polynomials, including poly-Bernoulli numbers $B_n^{(k)}$ and the poly-Cauchy numbers of the first kind $c_n^{(k)}$ and of the second kind $\\widehat c_n^{(k)}$. We investigate their p...
Jian-Guo Zheng
2015-01-01
Full Text Available Artificial bee colony (ABC algorithm is a popular swarm intelligence technique inspired by the intelligent foraging behavior of honey bees. However, ABC is good at exploration but poor at exploitation and its convergence speed is also an issue in some cases. To improve the performance of ABC, a novel ABC combined with grenade explosion method (GEM and Cauchy operator, namely, ABCGC, is proposed. GEM is embedded in the onlooker bees’ phase to enhance the exploitation ability and accelerate convergence of ABCGC; meanwhile, Cauchy operator is introduced into the scout bees’ phase to help ABCGC escape from local optimum and further enhance its exploration ability. Two sets of well-known benchmark functions are used to validate the better performance of ABCGC. The experiments confirm that ABCGC is significantly superior to ABC and other competitors; particularly it converges to the global optimum faster in most cases. These results suggest that ABCGC usually achieves a good balance between exploitation and exploration and can effectively serve as an alternative for global optimization.
The Cauchy-Lagrangian method for numerical analysis of Euler flow
Podvigina, O; Frisch, U
2015-01-01
A novel semi-Lagrangian method is introduced to solve numerically the Euler equation for ideal incompressible flow in arbitrary space dimension. It exploits the time-analyticity of fluid particle trajectories and requires, in principle, only limited spatial smoothness of the initial data. Efficient generation of high-order time-Taylor coefficients is made possible by simple recurrence relations that follow from the Cauchy invariants formulation of the Euler equations (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430). Truncated time-Taylor series of very high order allow the use of time steps vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the accuracy of the solution. Tests performed on the two-dimensional Euler equation indicate that the Cauchy-Lagrangian method is more --- and occasionally much more --- efficient and less prone to instability than Eulerian Runge-Kutta methods and less prone to rapid growth of rounding errors than the high-order Eulerian time-Taylor algor...
S Baskar; Phoolan Prasad
2006-02-01
For the structure of a sonic boom produced by a simple aerofoil at a large distance from its source we take a physical model which consists of a leading shock (LS), a trailing shock (TS) and a one-parameter family of nonlinear wavefronts in between the two shocks. Then we develop a mathematical model and show that according to this model the LS is governed by a hyperbolic system of equations in conservation form and the system of equations governing the TS has a pair of complex eigenvalues. Similarly, we show that a nonlinear wavefront originating from a point on the front part of the aerofoil is governed by a hyperbolic system of conservation laws and that originating from a point on the rear part is governed by a system of conservation laws, which is elliptic. Consequently, we expect the geometry of the TS to be kink-free and topologically different from the geometry of the LS. In the last section we point out an evidence of kinks on the LS and kink-free TS from the numerical solution of the Euler’s equations by Inoue, Sakai and Nishida [5].
无
2009-01-01
The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied, and the suffcient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given. In the end, concrete examples are constructed to justify the effectiveness of the criterion.
Alatancang; WU DeYu
2009-01-01
The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied,and the sufficient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given.In the end,concrete examples are constructed to justify the effectiveness of the criterion.
Taylor, Nicholas W; Reisswig, Christian; Scheel, Mark A; Chu, Tony; Kidder, Lawrence E; Szilagyi, Bela
2013-01-01
We extract gravitational waveforms from numerical simulations of black hole binaries computed using the Spectral Einstein Code. We compare two extraction methods: direct construction of the Newman-Penrose (NP) scalar $\\Psi_4$ at a finite distance from the source and Cauchy-characteristic extraction (CCE). The direct NP approach is simpler than CCE, but NP waveforms can be contaminated by near-zone effects---unless the waves are extracted at several distances from the source and extrapolated to infinity. Even then, the resulting waveforms can in principle be contaminated by gauge effects. In contrast, CCE directly provides, by construction, gauge-invariant waveforms at future null infinity. We verify the gauge invariance of CCE by running the same physical simulation using two different gauge conditions. We find that these two gauge conditions produce the same CCE waveforms but show differences in extrapolated-$\\Psi_4$ waveforms. We examine data from several different binary configurations and measure the domi...
Newton`s iteration for inversion of Cauchy-like and other structured matrices
Pan, V.Y. [Lehman College, Bronx, NY (United States); Zheng, Ailong; Huang, Xiaohan; Dias, O. [CUNY, New York, NY (United States)
1996-12-31
We specify some initial assumptions that guarantee rapid refinement of a rough initial approximation to the inverse of a Cauchy-like matrix, by mean of our new modification of Newton`s iteration, where the input, output, and all the auxiliary matrices are represented with their short generators defined by the associated scaling operators. The computations are performed fast since they are confined to operations with short generators of the given and computed matrices. Because of the known correlations among various structured matrices, the algorithm is immediately extended to rapid refinement of rough initial approximations to the inverses of Vandermonde-like, Chebyshev-Vandermonde-like and Toeplitz-like matrices, where again, the computations are confined to operations with short generators of the involved matrices.
A Modified Differential Evolution Algorithm with Cauchy Mutation for Global Optimization
Ali, Musrrat; Pant, Millie; Singh, Ved Pal
Differential Evolution (DE) is a powerful yet simple evolutionary algorithm for optimization of real valued, multi modal functions. DE is generally considered as a reliable, accurate and robust optimization technique. However, the algorithm suffers from premature convergence, slow convergence rate and large computational time for optimizing the computationally expensive objective functions. Therefore, an attempt to speed up DE is considered necessary. This research introduces a modified differential evolution (MDE), a modification to DE that enhances the convergence rate without compromising with the solution quality. In Modified differential evolution (MDE) algorithm, if an individual fails in continuation to improve its performance to a specified number of times then new point is generated using Cauchy mutation. MDE on a test bed of functions is compared with original DE. It is found that MDE requires less computational effort to locate global optimal solution.
Modelling mortality of a stored grain insect pest with fumigation: probit, logistic or Cauchy model?
Shi, Mingren; Renton, Michael
2013-06-01
Computer simulation models can provide a relatively fast, safe and inexpensive means to judge and weigh the merits of various pest control management options. However, the usefulness of such simulation models relies on the accurate estimation of important model parameters, such as the pest mortality under different treatments and conditions. Recently, an individual-based simulation model of population dynamics and resistance evolution has been developed for the stored grain insect pest Rhyzopertha dominica, based on experimental results showing that alleles at two different loci are involved in resistance to the grain fumigant phosphine. In this paper, we describe how we used three generalized linear models, probit, logistic and Cauchy models, each employing two- and four-parameter sub-models, to fit experimental data sets for five genotypes for which detailed mortality data was already available. Instead of the usual statistical iterative maximum likelihood estimation, a direct algebraic approach, generalized inverse matrix technique, was used to estimate the mortality model parameters. As this technique needs to perturb the observed mortality proportions if the proportions include 0 or 1, a golden section search approach was used to find the optimal perturbation in terms of minimum least squares (L2) error. The results show that the estimates using the probit model were the most accurate in terms of L2 errors between observed and predicted mortality values. These errors with the probit model ranged from 0.049% to 5.3%, from 0.381% to 8.1% with the logistic model and from 8.3% to 48.2% with the Cauchy model. Meanwhile, the generalized inverse matrix technique achieved similar results to the maximum likelihood estimation ones, but is less time consuming and computationally demanding. We also describe how we constructed a two-parameter model to estimate the mortalities for each of the remaining four genotypes based on realistic genetic assumptions. PMID:23473941
On an inverse problem for scalar conservation laws
We study in what sense one can determine the flux functions k = k(x) and f = f(u), k piecewise constant, in the scalar hyperbolic conservation law ut + (k(x)f(u))x = 0 by observing the solution u(t, ·) of the Cauchy problem with suitable piecewise constant initial data u|t=0 = uo. (paper)
N{sup ±}-integrals and boundary values of Cauchy-type integrals of finite measures
Aliev, R. A., E-mail: aliyevrashid@hotmail.ru, E-mail: alievrashid@box.az [Baku State University (Azerbaijan)
2014-07-31
Let Γ be a simple closed Lyapunov contour with finite complex measure ν, and let G{sup +} be the bounded and G{sup −} the unbounded domains with boundary Γ. Using new notions (so-called N-integration and N{sup +}- and N{sup −}-integrals), we prove that the Cauchy-type integrals F{sup +}(z), z∈G{sup +}, and F{sup −}(z), z∈G{sup −}, of ν are Cauchy N{sup +}- and N{sup −}-integrals, respectively. In the proof of the corresponding results, the additivity property and the validity of the change-of-variable formula for the N{sup +}- and N{sup −}-integrals play an essential role. Bibliography: 21 titles. (paper)
Taylor, Nicholas W.; Boyle, Michael; Reisswig, Christian; Scheel, Mark A.; Chu, Tony; Kidder, Lawrence E.; Szilágyi, Béla
2013-12-01
We extract gravitational waveforms from numerical simulations of black hole binaries computed using the Spectral Einstein Code. We compare two extraction methods: direct construction of the Newman-Penrose (NP) scalar Ψ4 at a finite distance from the source and Cauchy-characteristic extraction (CCE). The direct NP approach is simpler than CCE, but NP waveforms can be contaminated by near-zone effects—unless the waves are extracted at several distances from the source and extrapolated to infinity. Even then, the resulting waveforms can in principle be contaminated by gauge effects. In contrast, CCE directly provides, by construction, gauge-invariant waveforms at future null infinity. We verify the gauge invariance of CCE by running the same physical simulation using two different gauge conditions. We find that these two gauge conditions produce the same CCE waveforms but show differences in extrapolated-Ψ4 waveforms. We examine data from several different binary configurations and measure the dominant sources of error in the extrapolated-Ψ4 and CCE waveforms. In some cases, we find that NP waveforms extrapolated to infinity agree with the corresponding CCE waveforms to within the estimated error bars. However, we find that in other cases extrapolated and CCE waveforms disagree, most notably for m=0 “memory” modes.
Grid-size dependence of Cauchy boundary conditions used to simulate stream-aquifer interactions
Mehl, S.; Hill, M.C.
2010-01-01
This work examines the simulation of stream–aquifer interactions as grids are refined vertically and horizontally and suggests that traditional methods for calculating conductance can produce inappropriate values when the grid size is changed. Instead, different grid resolutions require different estimated values. Grid refinement strategies considered include global refinement of the entire model and local refinement of part of the stream. Three methods of calculating the conductance of the Cauchy boundary conditions are investigated. Single- and multi-layer models with narrow and wide streams produced stream leakages that differ by as much as 122% as the grid is refined. Similar results occur for globally and locally refined grids, but the latter required as little as one-quarter the computer execution time and memory and thus are useful for addressing some scale issues of stream–aquifer interactions. Results suggest that existing grid-size criteria for simulating stream–aquifer interactions are useful for one-layer models, but inadequate for three-dimensional models. The grid dependence of the conductance terms suggests that values for refined models using, for example, finite difference or finite-element methods, cannot be determined from previous coarse-grid models or field measurements. Our examples demonstrate the need for a method of obtaining conductances that can be translated to different grid resolutions and provide definitive test cases for investigating alternative conductance formulations.
Kabal`skii, M.M.
1994-06-05
A method is established for constructing a functional of the variational problem associated with a given nonlinear differential equation of higher order. The Cauchy problem for strongly nonlinear differential equations is solved. The application of this method to the solution of the problem of the resistance of materials is examined. 2 refs.
THE INTERACTION PROBLEM BETWEEN THE ELASTIC LINE INCLUSIONS
陶昉敏; 张明焕; 汤任基
2002-01-01
Using the engineering model of elastic line inclusion and the basic solutions of a single inclusion, the interaction problem between line inclusions in an elastic solid was investigated. A set of standard Cauchy-type singular equations of the problem was presented. The stress intensity factors at points of inclusions and the interface stresses of two sides of the inclusion were calculated. Several numerical examples were given. The results could be regarded as a reference to engineering.
Criterion of positivity for semilinear problems with applications in biology
Duprez, Michel; Perasso, Antoine
2016-01-01
The goal of this article is to provide an useful criterion of positivity and well-posedness for a wide range of infinite dimensional semilinear abstract Cauchy problems. This criterion is based on some weak assumptions on the non-linear part of the semilinear problem and on the existence of a strongly continuous semigroup generated by the differential operator. To illustrate a large variety of applications, we exhibit the feasibility of this criterion through three examples in mathematical bi...
Huang, Jingchi; Wang, Chao
2014-01-01
For periodic initial data with initial density, we establish the global existence and uniqueness of strong and classical solutions for the two-dimensional compressible Navier-Stokes equations with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is $\\lam=\\rho^{\\b}$ with $\\b>1.$
Rahali, Radouane
2013-03-01
In this paper, we investigate the decay property of a Timoshenko system in thermoelasticity of type III in the whole space where the heat conduction is given by the Green and Naghdi theory. Surprisingly, we show that the coupling of the Timoshenko system with the heat conduction of Green and Naghdi\\'s theory slows down the decay of the solution. In fact we show that the L-2-norm of the solution decays like (1 + t)(-1/8), while in the case of the coupling of the Timoshenko system with the Fourier or Cattaneo heat conduction, the decay rate is of the form (1 + t)(-1/4) [25]. We point out that the decay rate of (1 + t)(-1/8) has been obtained provided that the initial data are in L-1 (R) boolean AND H-s (R); (s >= 2). If the wave speeds of the fi rst two equations are di ff erent, then the decay rate of the solution is of regularity-loss type, that is in this case the previous decay rate can be obtained only under an additional regularity assumption on the initial data. In addition, by restricting the initial data to be in H-s (R) boolean AND L-1,L-gamma (R) with gamma is an element of [0; 1], we can derive faster decay estimates with the decay rate improvement by a factor of t(-gamma/4).
Young, Brent
2010-01-01
Recently, M.K.-H. Kiessling and A.S. Tahvildar-Zadeh proved that a unique global classical solution to the relativistic Vlasov-Poisson system exists whenever the positive, integrable initial datum is spherically symmetric, compactly supported in momentum space, vanishes on characteristics with vanishing angular momentum, and for $\\beta \\ge 3/2$ has $\\mathfrak{L}^{\\beta}$-norm strictly below a positive, critical value $\\mathcal{C}_{\\beta}$. Everything else being equal, data leading to finite t...
LogCauchy, log-sech and lognormal distributions of species abundances in forest communities
Yin, Z.-Y.; Peng, S.-L.; Ren, H.; Guo, Q.; Chen, Z.-H.
2005-01-01
Species-abundance (SA) pattern is one of the most fundamental aspects of biological community structure, providing important information regarding species richness, species-area relation and succession. To better describe the SA distribution (SAD) in a community, based on the widely used lognormal (LN) distribution model with exp(-x2) roll-off on Preston's octave scale, this study proposed two additional models, logCauchy (LC) and log-sech (LS), respectively with roll-offs of simple x-2 and e-x. The estimation of the theoretical total number of species in the whole community, S*, including very rare species not yet collected in sample, was derived from the left-truncation of each distribution. We fitted these three models by Levenberg-Marquardt nonlinear regression and measured the model fit to the data using coefficient of determination of regression, parameters' t-test and distribution's Kolmogorov-Smirnov (KS) test. Examining the SA data from six forest communities (five in lower subtropics and one in tropics), we found that: (1) on a log scale, all three models that are bell-shaped and left-truncated statistically adequately fitted the observed SADs, and the LC and LS did better than the LN; (2) from each model and for each community the S* values estimated by the integral and summation methods were almost equal, allowing us to estimate S* using a simple integral formula and to estimate its asymptotic confidence internals by regression of a transformed model containing it; (3) following the order of LC, LS, and LN, the fitted distributions became lower in the peak, less concave in the side, and shorter in the tail, and overall the LC tended to overestimate, the LN tended to underestimate, while the LS was intermediate but slightly tended to underestimate, the observed SADs (particularly the number of common species in the right tail); (4) the six communities had some similar structural properties such as following similar distribution models, having a common
Generalized Moisil-ThÃƒÂ©odoresco Systems and Cauchy Integral Decompositions
Ricardo Abreu Blaya
2008-05-01
Full Text Available Let Ã¢Â„Â0,m+1(s be the space of s-vectors (0Ã¢Â‰Â¤sÃ¢Â‰Â¤m+1 in the Clifford algebra Ã¢Â„Â0,m+1 constructed over the quadratic vector space Ã¢Â„Â0,m+1, let r,p,qÃ¢ÂˆÂˆÃ¢Â„Â• with 0Ã¢Â‰Â¤rÃ¢Â‰Â¤m+1,Ã¢Â€Â‰0Ã¢Â‰Â¤pÃ¢Â‰Â¤q, and r+2qÃ¢Â‰Â¤m+1, and let Ã¢Â„Â0,m+1(r,p,q=Ã¢ÂˆÂ‘j=pqÃ¢Â¨ÂÃ¢Â€Â‰Ã¢Â„Â0,m+1(r+2j. Then, an Ã¢Â„Â0,m+1(r,p,q-valued smooth function W defined in an open subset ÃŽÂ©Ã¢ÂŠÂ‚Ã¢Â„Âm+1 is said to satisfy the generalized Moisil-ThÃƒÂ©odoresco system of type (r,p,q if Ã¢ÂˆÂ‚xW=0 in ÃŽÂ©, where Ã¢ÂˆÂ‚x is the Dirac operator in Ã¢Â„Âm+1. A structure theorem is proved for such functions, based on the construction of conjugate harmonic pairs. Furthermore, if ÃŽÂ© is bounded with boundary ÃŽÂ“, where ÃŽÂ“ is an Ahlfors-David regular surface, and if W is a Ã¢Â„Â0,m+1(r,p,q-valued HÃƒÂ¶lder continuous function on ÃŽÂ“, then necessary and sufficient conditions are given under which W admits on ÃŽÂ“ a Cauchy integral decomposition W=W++WÃ¢ÂˆÂ’.
A unified approach to problems of scattering of surface water waves by vertical barriers
Chakrabarti, A; Banerjea, S; Mandal, BN; Sahoo, T
1997-01-01
A unified analysis involving the solution of multiple integral equations via a simple singular integral equation with a Cauchy type kernel is presented to handle problems of surface water wave scattering by vertical barriers. Some well known results are produced in a simple and systematic manner.
Some improperly posed problems of mathematical physics
Lavrentiev, M M
1967-01-01
This monograph deals with the problems of mathematical physics which are improperly posed in the sense of Hadamard. The first part covers various approaches to the formulation of improperly posed problems. These approaches are illustrated by the example of the classical improperly posed Cauchy problem for the Laplace equation. The second part deals with a number of problems of analytic continuations of analytic and harmonic functions. The third part is concerned with the investigation of the so-called inverse problems for differential equations in which it is required to determine a dif ferential equation from a certain family of its solutions. Novosibirsk June, 1967 M. M. LAVRENTIEV Table of Contents Chapter I Formu1ation of some Improperly Posed Problems of Mathematic:al Physics § 1 Improperly Posed Problems in Metric Spaces. . . . . . . . . § 2 A Probability Approach to Improperly Posed Problems. . . 8 Chapter II Analytic Continuation § 1 Analytic Continuation of a Function of One Complex Variable fro...
Sedrakian, D M; Khachatrian, A Z
2000-01-01
It is shown that the problem on scattering of an arbitrary polarized plane wave to be incident on the plate of a one-dimensional medium can be formulated as a Cauchy problem for wave equations for s and p waves.
STABILITY OF DISPLACEMENT TO THE SECOND FUNDAMENTAL PROBLEM IN PLANE ELASTICITY
Juan LIN; Jinyuan DU
2014-01-01
In this article, by using the stability of Cauchy type integral when the smooth perturbation for integral curve and the Sobolev type perturbation for kernel density hap-pen, we discuss the stability of the second fundamental problem in plane elasticity when the smooth perturbation for the boundary of the elastic domain (unit disk) and the Sobolev type perturbation for the displacement happen. And the error estimate of the displacement between the second fundamental problem and its perturbed problem is obtained.
The borderlines of the invisibility and visibility for Calderon's inverse problem
Astala, Kari; Lassas, Matti; Paivarinta, Lassi
2011-01-01
We consider the determination of a conductivity function in a two-dimensional domain from the Cauchy data of the solutions of the conductivity equation on the boundary. We prove uniqueness results for this inverse problem, posed by Calderon, for conductivities that are degenerate, that is, they may not be bounded from above or below. In particular, for scalar conductivities we solve the inverse problem in a class which is larger than $L^\\infty$. Also, we give new counterexamples for the uniqu...
M. G. Crandall
1999-07-01
Full Text Available We study existence of continuous weak (viscosity solutions of Dirichlet and Cauchy-Dirichlet problems for fully nonlinear uniformly elliptic and parabolic equations. Two types of results are obtained in contexts where uniqueness of solutions fails or is unknown. For equations with merely measurable coefficients we prove solvability of the problem, while in the continuous case we construct maximal and minimal solutions. Necessary barriers on external cones are also constructed.
Pérez Cantor, Edwin Leonardo
2014-01-01
El propósito de este trabajo es estudiar el buen planteamiento en los espacios de Sobolev Hs(T2) para s 1> del problema de Cauchy asociado a una generalización de la ecuación de Kuramoto-Sivashinsky bidimensional periódica, que modela fenómenos físicos que ocurren en películas delgadas
El problema de cauchy asociado a una ecuación del tipo kuramoto-sivashinsky bidimensional periódica
Campos, Juvitsa; Duque, Omar; Rodríguez-Blanco, Guillermo
2012-01-01
El propósito de este trabajo es abordar el buen planteamiento en los espacios de Sobolev $H^{s}(\\mathbb{T}^2)$ para $s\\geq1$ del problema de Cauchy asociado a una ecuación del tipo Kuramoto-Sivashinsky bidimensional periódica, que modela fenómenos físicos que ocurren en películas delgadas.
Abanin, A.V.; Khoi, Le Hai
2016-01-01
Let $\\Omega$ be a Carathéodory domain in the complex plane $\\mathbb C$, $A^{-\\infty}(\\Omega)$ the space of functions that are holomorphic in $\\Omega$ with polynomial growth near the boundary $\\partial\\Omega$, and $A^\\infty(\\complement\\Omega)$ the space of holomorphic functions in the interior of $\\complement\\Omega:=\\overline{\\mathbb C}\\setminus\\Omega$, vanishing at infinity and being in $C^\\infty(\\complement\\Omega)$. We prove that the Cauchy transformation of analytic fun...
Finding Solutions to the Picard Boundary Value Problem via Homotopy Method%用同伦方法求Picard边值问题的解
李兰; 徐旭
2008-01-01
This paper deals with the problem of finding solutions to the Picard boundary problem. In our approach, by means of the homotopy method, the equation considered is linked to a simpler equation by introducing a parameter. We first find the solutions of the simpler equation, and give a priori estimates of" the equa tion we considered, and then one can obtain the solutions of Picard boundary problem by following the path of solutions of Cauchy problem.
Miyata, Y.; Suzuki, T.; Takechi, M.; Urano, H.; Ide, S. [Japan Atomic Energy Agency, Naka, Ibaraki 311-0193 (Japan)
2015-07-15
For the purpose of stable plasma equilibrium control and detailed analysis, it is essential to reconstruct an accurate plasma boundary on the poloidal cross section in tokamak devices. The Cauchy condition surface (CCS) method is a numerical approach for calculating the spatial distribution of the magnetic flux outside a hypothetical surface and reconstructing the plasma boundary from the magnetic measurements located outside the plasma. The accuracy of the plasma shape reconstruction has been assessed by comparing the CCS method and an equilibrium calculation in JT-60SA with a high elongation and triangularity of plasma shape. The CCS, on which both Dirichlet and Neumann conditions are unknown, is defined as a hypothetical surface located inside the real plasma region. The accuracy of the plasma shape reconstruction is sensitive to the CCS free parameters such as the number of unknown parameters and the shape in JT-60SA. It is found that the optimum number of unknown parameters and the size of the CCS that minimizes errors in the reconstructed plasma shape are in proportion to the plasma size. Furthermore, it is shown that the accuracy of the plasma shape reconstruction is greatly improved using the optimum number of unknown parameters and shape of the CCS, and the reachable reconstruction errors in plasma shape and locations of strike points are within the target ranges in JT-60SA.
Existence and regularity of solutions for hyperbolic functional differential problems
Zdzisław Kamont
2014-01-01
Full Text Available A generalized Cauchy problem for quasilinear hyperbolic functional differential systems is considered. A theorem on the local existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions for this system is proved by using a method of successive approximations. We show a theorem on the differentiability of solutions with respect to initial functions which is the main result of the paper.
Validity of the Cauchy-Born rule applied to discrete cellular-scale models of biological tissues
Davit, Y.
2013-04-30
The development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations, although formal relationships between the discrete and continuum frameworks remain to be established. For crystal mechanics, the discrete-to-continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient, an assumption known as the Cauchy-Born rule (CBR). Although the CBR does not hold exactly for noncrystalline materials, it may still be used as a first-order approximation for analytic calculations of effective stresses or strain energies. In this work, our goal is to investigate numerically the applicability of the CBR to two-dimensional cellular-scale models by assessing the mechanical behavior of model biological tissues, including crystalline (honeycomb) and noncrystalline reference states. The numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic position of internal cells. The position of internal cells is then compared with the prediction of the CBR and an average deviation is calculated in the strain domain. For center-based cell models, we show that the CBR holds exactly when the deformation gradient is relatively small and the reference stress-free configuration is defined by a honeycomb lattice. We show further that the CBR may be used approximately when the reference state is perturbed from the honeycomb configuration. By contrast, for vertex-based cell models, a similar analysis reveals that the CBR does not provide a good representation of the tissue mechanics, even when the reference configuration is defined by a honeycomb lattice. The paper concludes with a discussion of the implications of these results for concurrent discrete and continuous modeling, adaptation of atom-to-continuum techniques to biological
Domoshnitsky Alexander
2009-01-01
Full Text Available We obtain the maximum principles for the first-order neutral functional differential equation where , and are linear continuous operators, and are positive operators, is the space of continuous functions, and is the space of essentially bounded functions defined on . New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.
EXPLICIT FORMULATION FOR THE DIRICHLET PROBLEM FOR PARABOLIC-HYPERBOLIC CONSERVATION LAWS
Andreianov, Boris; Karimou Gazibo, Mohamed
2015-01-01
We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definiti...
Ricciardi, Carlo Alberto; Ricciardi, Elisabetta; Visconti, Luca; Lacquaniti, Antonio; Savica, Vincenzo; Santoro, Domenico; Buemi, Michele; Ricciardi, Biagio
2016-02-01
The Study of urine from the outset has always aroused the interest of scientists and physicians all over the world, from ancient Greeks and Romans to Hindus , Hulcos in Mexico, Australian native etc. The urine in such case was considered not only as a waste product but also as a therapeutic product. In the late XIX century scientific knowledge had already identified the function of substances that favor the increase of urinary output, and physicians over the centuries have always tried to analyze urine in various ways. In Cauchis work in 1933 all chemistry and pathophysiological knowledge of the time was condensed. Cauchi signed the preface as Member of the medical council of Malta. He was a medical doctor of the early20thcentury, He wrote about the physiopathology of urine ranging from chemical and physical behavior, to the analysis of sediments and the special reactions of the urine in various pathologies. In particular Cauchi emphasizes the main diseases of the time combines the behavior of the reaction of urine as a diagnostic and prognostic instrument, stressing the importance of the urine test and describing the method used for analysis at the time. The analyses of the text in the issue seems to belong to archaic medicine, and it is difficult to think today, that what was presented as very up-to-date- science at that time, took place only 80 years ago. Reading the full original text with today experience we are led to consider the increasing importance that scientific community gave in the past, and still gives to urine test. PMID:26913892
Gerhardt, Claus
2016-01-01
In a recent paper we quantized the interaction of gravity with a Yang-Mills and Higgs field and obtained as a result a gravitational wave equation in a globally hyperbolic spacetime. Assuming that the Cauchy hypersurfaces are compact we proved a spectral resolution for the wave equation by applying the method of separation of variables. In this paper we extend the results to the case when the Cauchy hypersurfaces are non-compact by considering a Gelfand triplet and applying the nuclear spectral theorem.
Solution of the generalized Saint Venant problem
A well-known problem in the mathematical theory of elasticity about the torsional rigidity P(Ω) of a bar whose cross-section is an arbitrary simply connected domain Ω is considered. It is shown that P(Ω) is equivalent to the moment of inertia of the domain relative to its boundary. Thus, a new interpretation of the well-known Coulomb's formula is suggested, and on this basis the following problem, which has its origins in works of Cauchy and Saint Venant, is solved: find a geometric parameter equivalent to the torsional rigidity coefficient of elastic bars with simply connected cross-sections. The proof is based on the definition of the torsional rigidity as the norm of a certain embedding operator in a Sobolev space and on the theory of conformal maps. In particular, some conformally invariant inequalities are established
Choquet-Bruhat, Yvonne; Chrusciel, Piotr; Martin-garcia, Jose
2010-01-01
Представлен класс характеристических релятивистски обобщенных исходных данных, удовлетворяющих условию почти закругленности на вершине светового конуса. Показано, что для любых подобных аналитических данных существует соответствующее решение для вакуумных уравнений Эйнштейна, определенное в окрестности вершины в световом конусе будущего.A class of characteristic general relativistic initial data satisfying a near-roundness condition at the tip of a light-cone is introduced. It is shown that f...
The program library IVP++ has been described and tested on the simulation of the hydrodynamics of a one-dimensional channel. Numerical integration methods are briefly reviewed. The calculation-theoretical analysis of the application of implicit high-order integration methods to the simulation of one-dimensional hydro and gas dynamics has been performed
ANALYSIS OF THE SINGULARITY FOR THE VERTICAL CONTACT PROBLEM OF LINE CRACK-INCLUSION
沈安; 陶昉敏; 汤任基
2001-01-01
Taking the short-fiber composite materials as engineering back-ground, utilizing the existing basic solutions of single inclusion and single crack, the plane problem of vertical contact interactions between line crack and rigid line inclusion in infinite plane (matrix) from the viewpoint of crack fracture mechanics is studied. According to boundary conditions, a set of standard Cauchy-type singular integral equations of the problem is obtainable. Besides, singular indexes, stresses and stress intensity factors around the contact point are expressed. Numerical examples are given to provide references to engineering.
De Lacroix a Cauchy: La fundamentación del cálculo infinitesimal en José Mariano Vallejo (1807-1832
Ausejo, Elena
2015-12-01
Full Text Available This paper studies the foundations of infinitesimal calculus in the work of José Mariano Vallejo (1779-1846, the most influential mathematician in Spanish during the first half of the 19th century. For this purpose, the analytical foundations of his three main works related to infinitesimal calculus are considered: Memoria sobre la curvatura de las líneas en sus diferentes puntos, sobre el radio de curvatura y sobre las evolutas (1807, and the two editions (1813 and 1832 of the volume of his Tratado Elemental de Matemáticas devoted to calculus. The study shows the evolutionary dynamics of Vallejo’s thought, following the European leading references in the development of mathematical analysis in his time, and how Vallejo combines the successive influence of Lagrange, Lacroix, and Cauchy, according to criteria of educational adequacy and mathematical rigor. In particular, the influences and discrepancies on infinitesimal calculus between Vallejo’s Treatise (1813 and Lacroix’ Traité élémentaire de calcul différentiel et de calcul intégral (1802 are determined, as well as the scope of the introduction of Cauchy’s ideas, that were published for the first time in Spain in the second edition of the volume on infinitesimal calculus of Vallejo’s Treatise (1832.Este artículo estudia los aspectos relativos a la fundamentación del cálculo infinitesimal en la obra de José Mariano Vallejo (1779-1846, el matemático de mayor difusión en lengua española de la primera mitad del siglo XIX. Para ello, se consideran los fundamentos analíticos de su Memoria sobre la curvatura de las líneas en sus diferentes puntos, sobre el radio de curvatura y sobre las evolutas (1807 y de las dos ediciones (1813 y 1832 del volumen de su Tratado Elemental de Matemáticas dedicado al cálculo infinitesimal. El estudio muestra la dinámica evolutiva del pensamiento de Vallejo al ritmo de las referencias europeas —especialmente francesas– que encabezan el
The Scattering Problem for a Noncommutative Nonlinear Schrödinger Equation
Bergfinnur Durhuus
2010-06-01
Full Text Available We investigate scattering properties of a Moyal deformed version of the nonlinear Schrödinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has solitary wave solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.
Edwards, P. J.; Huang, X.; Li, Y. Q. (Editor); Wang, Y. Z. (Editor)
1996-01-01
We briefly review quantum mechanical and semi-classical descriptions of experiments which demonstrate the macroscopic violation of the three Cauchy-Schwarz inequalities: g(sup 2)(sub 11)(0) greater than or equal to 1; g(sup 2)(sub 11)(0) greater than or equal to g(sup 2)(sub 11)(t), (t approaches infinity); (the absolute value of g(sup 2)(sub 11)(0))(exp 2) less than or equal to g(sup 2)(sub 11)(0) g(sup 2)(sub 11)(0). Our measurements demonstrate the violation, at macroscopic intensities, of each of these inequalities. We show that their violation, although weak, can be demonstrated through photodetector current covariance measurements on correlated sub-Poissonian Poissonian, and super Poissonian light beams. Such beams are readily generated by a tandem array of infrared-emitting semiconductor junction diodes. Our measurements utilize an electrically coupled array of one or more infrared-emitting diodes, optically coupled to a detector array. The emitting array is operated in such a way as to generate highly correlated beams of variable photon Fano Factor. Because the measurements are made on time scales long compared with the first order coherence time and with detector areas large compared with the corresponding coherence areas, first order interference effects are negligible. The first and second inequalities are violated, as expected, when a sub-Poissonian light beam is split and the intensity fluctuations of the two split beams are measured by two photodetectors and subsequently cross-correlated. The third inequality is violated by bunched (as well as anti-bunched) beams of equal intensity provided the measured cross correlation coefficient exceeds (F - 1)/F, where F is the measured Fano Factor of each beam. We also investigate the violation for the case of unequal beams.
[Population problem, comprehension problem].
Tallon, F
1993-08-01
Overpopulation of developing countries in general, and Rwanda in particular, is not just their problem but a problem for developed countries as well. Rapid population growth is a key factor in the increase of poverty in sub-Saharan Africa. Population growth outstrips food production. Africa receives more and more foreign food, economic, and family planning aid each year. The Government of Rwanda encourages reduced population growth. Some people criticize it, but this criticism results in mortality and suffering. One must combat this ignorance, but attitudes change slowly. Some of these same people find the government's acceptance of family planning an invasion of their privacy. Others complain that rich countries do not have campaigns to reduce births, so why should Rwanda do so? The rate of schooling does not increase in Africa, even though the number of children in school increases, because of rapid population growth. Education is key to improvements in Africa's socioeconomic growth. Thus, Africa, is underpopulated in terms of potentiality but overpopulated in terms of reality, current conditions, and possibilities of overexploitation. Africa needs to invest in human resources. Families need to save, and to so, they must refrain from having many children. Africa should resist the temptation to waste, as rich countries do, and denounce it. Africa needs to become more independent of these countries, but structural adjustment plans, growing debt, and rapid population growth limit national independence. Food aid is a means for developed countries to dominate developing countries. Modernization through foreign aid has had some positive effects on developing countries (e.g., improved hygiene, mortality reduction), but these also sparked rapid population growth. Rwandan society is no longer traditional, but it is also not yet modern. A change in mentality to fewer births, better quality of life for living infants, better education, and less burden for women must occur
Sakharova, L V; Zhukov, M Yu
2013-01-01
The mathematical model describing the natural textrm{pH}-gradient arising under the action of an electric field in an aqueous solution of ampholytes (amino acids) is constructed and investigated. This paper is the second part of the series papers \\cite{Part1,Part3,Part4} that are devoted to pH-gradient creation problem. We present the numerical solution of the stationary problem. The equations system has a small parameter at higher derivatives and the turning points, so called stiff problem. To solve this problem numerically we use the shooting method: transformation of the boundary value problem to the Cauchy problem. At large voltage or electric current density we compare the numerical solution with weak solution presented in Part 1.
Stability analysis of the inverse transmembrane potential problem in electrocardiography
Burger, Martin; Mardal, Kent-André; Nielsen, Bjørn Fredrik
2010-10-01
In this paper we study some mathematical properties of an inverse problem arising in connection with electrocardiograms (ECGs). More specifically, we analyze the possibility for recovering the transmembrane potential in the heart from ECG recordings, a challenge currently investigated by a growing number of groups. Our approach is based on the bidomain model for the electrical activity in the myocardium, and leads to a parameter identification problem for elliptic partial differential equations (PDEs). It turns out that this challenge can be split into two subproblems: the task of recovering the potential at the heart surface from body surface recordings; the problem of computing the transmembrane potential inside the heart from the potential determined at the heart surface. Problem (1), which can be formulated as the Cauchy problem for an elliptic PDE, has been extensively studied and is well known to be severely ill-posed. The main purpose of this paper is to prove that problem (2) is stable and well posed if a suitable prior is available. Moreover, our theoretical findings are illuminated by a series of numerical experiments. Finally, we discuss some aspects of uniqueness related to the anisotropy in the heart.
王丽萍; 彭维玲; 肖卓峰
2008-01-01
In the first part of this paper, we discuss the Holder continuity of the cauchy integral operator for regular functions and the relation between ‖T[f] ‖α and ‖f‖α. In the second part of this paper, we introduce the modified cauchy integral operator T for regular functions. Firstly,we prove that the operator T has a unique fixed point by the Banach's Contraction Mapping Principle. Secondly, we give the Mann iterative sequence, and then we show the iterative sequence strongly converges to the fixed point of the operator T.
$delta$-Quasi Cauchy Sequences
Cakalli, Huseyin
2010-01-01
Recently, a concept of forward continuity and a concept of forward compactness are introduced in the senses that a function $f$ is forward continuous if $\\lim_{n\\to\\infty} \\Delta f(x_{n})=0$ whenever $\\lim_{n\\to\\infty} \\Delta x_{n}=0$,\\; and a subset $E$ of $\\textbf{R}$ is forward compact if any sequence $\\textbf{x}=(x_{n})$ of points in $E$ has a subsequence $\\textbf{z}=(z_{k})=(x_{n_{k}})$ of the sequence $\\textbf{x}$ such that $\\lim_{k\\to \\infty} \\Delta z_{k}=0$ where $\\Delta z_{k}=z_{k+1}...
Interactions between discontinuities for binary mixture separation problem and hodograph method
Elaeva, M S; Yu, Zhukov M
2016-01-01
The Cauchy problem for first-order PDE with the initial data which have a piecewise discontinuities localized in different spatial points is completely solved. The interactions between discontinuities arising after breakup of initial discontinuities are studied with the help of the hodograph method. The solution is constructed in analytical implicit form. To recovery the explicit form of solution we propose the transformation of the PDEs into some ODEs on the level lines (isochrones) of implicit solution. In particular, this method allows us to solve the Goursat problem with initial data on characteristics. The paper describes a specific problem for zone electrophoresis (method of the mixture separation). However, the method proposed allows to solve any system of two first-order quasilinear PDEs for which the second order linear PDE, arising after the hodograph transformation, has the Riemann-Green function in explicit form.
Explaining the Mind: Problems, Problems
Harnad, Stevan
2001-01-01
The mind/body problem is the feeling/function problem: How and why do feeling systems feel? The problem is not just "hard" but insoluble (unless one is ready to resort to telekinetic dualism). Fortunately, the "easy" problems of cognitive science (such as the how and why of categorization and language) are not insoluble. Five books (by Damasio, Edelman/Tononi...
... you're not getting enough air. Sometimes mild breathing problems are from a stuffy nose or hard ... conditions such as asthma, emphysema or pneumonia cause breathing difficulties. So can problems with your trachea or ...
The Dirichlet problem for the two-dimensional Helmholtz equation for an open boundary
Hayashi, Y.
1973-01-01
Development of a complete theory of the two-dimensional Dirichlet problem for an open boundary. It is shown that the solution of the Dirichlet problem for an open boundary requires the solution of a Fredholm integral equation of the first kind. Although a Fredholm integral equation of the first kind usually has no solution if the kernel is continuous, owing to the logarithmic singularity of the kernel, the equation in this case is converted to a singular integral equation with a Cauchy kernel. It is proven that the homogeneous adjoint equation of the singular integral equation has no nonzero solution. By virtue of this result, and with the aid of an existence theorem known in the theory of singular integral equations, the existence of solutions of the singular integral equation, and then of the unique solution of the Fredholm integral equation of the first kind is proved.
Jordan Hristov
2016-01-01
Full Text Available The article addresses a reappraisal of the famous Ward–Tordai equation describing the equilibrium of surfactants at air/liquid interfaces under diffusion control. The new derivation is entirely developed in the light of fractional calculus. The unified approach demonstrates that this equation can be clearly reformulated as a nonlinear ordinary time-fractional equation of order 1/2. The work formulates versions with different isotherms. A simple solution of the case with the Henry’s isotherm and a discussion of a Cauchy problem involving the Freundlich isotherm are provided.
Christensen, Anders Bøggild; Rasmussen, Tove; Bundesen, Peter;
Sociale problemer kan betragtes som selve udgangspunktet for socialt arbejde, hvor ambitionen er at råde bod på problemerne og sikre, at udsatte borgere får en bedre tilværelse. Det betyder også, at diskussionen af sociale problemer er afgørende for den sociale grundfaglighed. I denne bog sætter en...... række fagfolk på tværs af det danske socialfaglige felt fokus på sociale problemer. Det diskuteres, hvad vi overhovedet forstår ved sociale problemer, hvordan de opstår, hvilke konsekvenser de har, og ikke mindst hvordan man som fagprofessionel håndterer sociale problemer i det daglige arbejde. Bogen er...... skrevet som lærebog til professionsuddannelser, hvor sociale problemer udgør en dimension, bl.a. socialrådgiver-, pædagog- og sygeplejerskeuddannelserne....
Skovhus, Randi Boelskifte; Thomsen, Rie
2016-01-01
This article introduces a method to critical reviews and explores the ways in which problems have been formulated in knowledge production on career guidance in Denmark over a 10-year period from 2004 to 2014. The method draws upon the work of Bacchi focussing on the ‘What's the problem represented...... provides a constructive basis for a critical analysis and discussion of the collective empirical knowledge production on career guidance, stimulating awareness of problems and potential solutions among the career guidance community....
An approach combining boundary integral and finite element methods is introduced for the solution of three-dimensional inverse electromagnetic medium scattering problems. Based on the equivalence principle, unknown equivalent electric and magnetic surface current densities on a closed surface are utilized to decompose the inverse medium problem into two parts: a linear radiation problem and a nonlinear cavity problem. The first problem is formulated by a boundary integral equation, the computational burden of which is reduced by employing the multilevel fast multipole method (MLFMM). Reconstructed Cauchy data on the surface allows the utilization of the Lorentz reciprocity and the Poynting's theorems. Exploiting these theorems, the noise level and an initial guess are estimated for the cavity problem. Moreover, it is possible to determine whether the material is lossy or not. In the second problem, the estimated surface currents form inhomogeneous boundary conditions of the cavity problem. The cavity problem is formulated by the finite element technique and solved iteratively by the Gauss–Newton method to reconstruct the properties of the object. Regularization for both the first and the second problems is achieved by a Krylov subspace method. The proposed method is tested against both synthetic and experimental data and promising reconstruction results are obtained
... in Chinese 繁體中文 ) What Are Knee Problems? (in Korean 한국어 ) What Are Knee Problems? (in Vietnamese bằng ... the knee. Ultrasound. A technique that uses sound waves to produce images of the soft tissue structures ...