Equations Of Kind Research Articles

On Kinetic Equations in Heavy-Ion Physics

We present the theoretical foundations of the dynamical approaches used in heavy-ion collisions at beam energies of some tens of MeV per nucleon. After a brief review of mean field (TDHF) and Intra nuclear Cascade approaches we consider Boltzmann-like kinetic equations, adapted to the nuclear context. Simulations of these equations (BUU, VUU, ...) are widely used in this field of physics. We show how to obtain this kind of equations in a nuclear context and we discuss the formal difficulties raised by these equations. In a second part we consider alternative descriptions, such as stochastic extensions of these kinetic equations or molecular dynamics models. We present several models (MD,QMD, ...) of this kind and discuss their limitations. In a last part we discuss the ability of the approaches introduced in the previous parts to fulfill Pauli exclusion principle. We show the difficulties raised by this principle, as well in the case of simulations of kinetic equations as in molecular dynamics models. We conclude by introducing fermionic molecular dynamics (FMD, AMD) methods, which are directly linked to TDHF, and a stochastic extension of TDHF (STDHF).

A Spatial Decay Estimate for the Hyperbolic Heat Equation

In this paper we establish a spatial decay estimate for the hyperbolic heat equation similar to other known estimates for the parabolic equations. Alternatively, the results may be viewed as theorems of Phragmen–Lindelof type [J. B. Conway, Functions of One Complex Variable, Graduate Texts in Mathematics, 2nd ed., Springer-Verlag, New York, 1978], [R. Quintanilla, Publ. Mat., 37 (1993), pp. 443–463], [C. O. Horgan and L. E. Payne, Arch. Rational Mech. Anal., 122 (1993), pp. 123–144] for this kind of equation. We conclude the paper by extending the results for a type of semilinear wave equation.

Simultaneous heat and mass transfer model for spray tower design: Application on VOCs removal

Our paper is devoted to a general approach for spray tower design. From a macroscopic point of view, this model includes the presence of a continuous gas phase, the nature of the spray nozzle, the dispersion of the droplets as well as the apparition of a film along the column wall. From a microscopic point of view, it includes the simultaneous heat and mass transfer between the various phases. The model is described by an algebro differential set of equations. This DAE system with boundary conditions at each extremity is solved by decoupling the two kinds of equations. The numerical results of our model are compared with experimental values obtained on a pilot plant. The experiments concern the absorption of a mixture of alcohols, ketones and aldehydes in air by water, and confirm the validity between our theorical approach and experimentation.

A Fast Method for Finding the Basis of Non-negative Solutions to a Linear Diophantine Equation

We present a complete characterization of the set of minimal solutions of a single linear Diophantine equation in three unknowns over the natural numbers. This characterization, for which we give a geometric interpretation, is based on well-known properties of congruences and we use it as the foundation of direct algorithms for solving this particular kind of equation. These direct algorithms and an enumeration procedure are then put together to build an algorithm for solving the general case of a Diophantine equation over the naturals. We also put forth a statistical method for comparing algorithms for solving Diophantine equations which is more sound than comparisons based on times observed for small sets of equations. From an extensive comparison with algorithms described by other authors it becomes clear that our algorithm is the fastest known to date for a class of equations. Typically the equations in this class have a small number of unknowns in one side, the maximum value for their coefficients being greater than 3.

Dimensionless critical shear stress evaluation from flume experiments using different gravel beds

AbstractFlume experiments were conducted using four different gravel beds (D50 + 12–39 mm) and a range of marked particles (10–65 mm). The shear stresses were evaluated from friction velocities, when initial movement of marked particles occurred. Two kinds of equations were produced: first for the threshold of initial movement, and second for generalized movement. Equations of the type 0c + a(Di/D50)b, as proposed by Andrews (1983) are applicable even if the material is relatively well sorted. However, the values of a and b are lower (respectively 0·050 and ‐0·70) for initial movement. Generalized movement requires a higher shear stress (a + 0·068 and b + ‐0·80).D90 of the bed material and y0 (the bed roughness parameter) were also used as reference values in place of D50. They produced lower values than in natural streams, mainly owing to the fact that the material used in the flume is better sorted: clusters are less well developed and the bed roughness is lower.

An approximate solution of nonlinear wave equation

A uniformly valid approximate solution of a kind of nonlinear wave equations is studied. The research results indicate that the solution of this kind of equations can be represented by Airy function approximately. The usually, used W. K. B. approximation is the first order approximation of the present result in the region far away from the turning point of refractivity. At the turning point of refractivity, the present result is still valid.

The theoretical foundation of the Adomian method

In this paper, we work out the theoretical basis of the Adomian method. This powerful method that G. Adomian has developed in the beginning of the 1980's supplies analytical approximations to the solution of many kinds of equations. Thanks to our decomposition theory, we justify the practical method and we generalize the convergence results obtained by Y. Cherruault. Afterward, we explain how to solve rigorously equations with linear, nonlinear or composed operator and also equations with several terms or with a second member.

Reasoning about equations and functional dependencies on complex objects

Virtually all semantic or object-oriented data models assume that objects have an identity separate from any of their parts, and allow users to define complex object types in which part values may be any other objects. This often results in a choice of query language in which a user can express navigating from one object to another by following a property value path. We consider a constraint language in which one may express equations and functional dependencies over complex object types. The language is novel in the sense that component attributes of individual constraints may correspond to property paths. The kind of equations we consider are also important, because they are a natural abstraction of the class of conjunctive queries for query languages that support property value navigation. In our introductory comments, we give an example of such a query and outline two applications of the constraint theory to problems relating to a choice of access plan for the query. We present a sound and complete axiomatization of the constraint language for the case in which interpretations are permitted to be infinite, where interpretations themselves correspond to a form of directed labeled graph. Although the implication problem for our form of equational constraint alone over arbitrary schema is undecidable, we present decision procedures for the implication problem for both kinds of constraints when the problem schema satisfies a stratification condition, and when all input functional dependencies are keys.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Inward and outward integral equations and the KKR method for photons

In the case of electromagnetic waves it is necessary to distinguish between inward and outward on-shell integral equations. Both kinds of equation are derived. A correct implementation of the photonic KKR method then requires the inward equations and it follows directly from them. A derivation of the KKR method from variational principles is also outlined. Rather surprisingly, the variational KKR method cannot be entirely written in terms of surface integrals unless permeabilities are piecewise constant. Both kinds of photonic KKR method use the standard structure constants of the electronic KKR method and hence allow for a direct numerical application. As a by-product, matching rules are obtained for derivatives of fields on different sides of the discontinuity of permeabilities.

Higher order methods for solving Volterra integrodifferential equations of the first kind

A highly accurate, diagonally implicit Runge-Kutta method (DIRK) is used for the numerical solution of linear Volterra integrodifferential equations of the first kind after converting the equations into equations of the second kind. The numerical method shows very good stability, and, as opposed to the first kind of equations, the second kind of equations are not ill-posed.

On the quantum theory of measurements continuous in time

A quantum theory of measurements continuous in time has been developed at various levels of generality and by means of various mathematical tools. Here, only some points of this theory are presented. By using the notion of convolution semigroup of instruments one can give a prescription for extracting from quantum mechanics a consistent set of finite-dimensional probabilities (multi-time joint probabilities for some family of observables) describing the continuous measurement. Some results on the structure of such semigroups are given. Moreover, one can go beyond finite- dimensional distributions by using more elaborate families of instruments which can be constructed via certain stochastic differential equations for processes with values in the trace-class operators (related to the notion of a posteriori states). This kind of equations gives a detailed description of the behaviour of the quantum system and allows for generalizations of the theory (such generalizations, which allow to handle certain memory effects, are not described here); moreover, independently from continuous measurement theory, these stochastic equations are now used also for numerical simulations of master equations in quantum optics. A higher level of description, involving quantum stochastic calculus, dilations of quantum dynamical semigroups, quantum stochastic processes, …, is left out from this paper.

A Method for Measuring Depth Using Fuzzy Reasoning and a Modified Implicit Function

Many scientists and engineers are interested in three-dimensional (3-D) vision for robots. The focus of this paper is to describe a newly developed measuring technique for 3-D robot vision which employs a blurred image method to obtain depth perception. The measuring system consists of only one camera and an image processor. Several basic experiments were done using a checkered 3-D object and a new type of modified fuzzy reasoning. A new membership function is defined from two kinds of equations using a modified implicit function. As a result, the relationship between the measured distance and the estimated distance was found to be linear. This blurred image method together with fuzzy reasoning was found to be very effective in depth measurement of 3-D objects.

Two-evolution equations for nonlinear subpicosecond pulses in optical fibers and their equivalent solitary solutions

Abstract By means of the multiscale singular perturbation method and directly from the Maxwell vector wave equation containing the nonlinear polarization term, we have obtained two kinds of equations describing the propagation of nonlinear sub-picosecond optical pulse in optical fibers. These two equations are the nonlinear Schrodinger equations with different linear combinations of the higher order terms (the higher order dispersion term, the self-steepening term, and the delayed nonlinear response term). The approximate coefficients of the equations are derived under the weakly guiding condition. Using a perturbation method, we show that these two equations share a solitary wave solution.

A New Constant Pressure Model for N-Branch Junctions

Branch junctions are very frequently found in the intake and exhaust systems of multi-cylinder engines and they represent some of the most complex boundary conditions for wave action models. This article deals with the way of modelling a junction of N branches. Firstly, a completely new view of the general problem is presented, stating the number of closing equations that are required to obtain a well-posed problem and what kind of equations they should be. Secondly, the constant pressure theory proposed by R. S. Benson is revised from this new standpoint, showing that besides the assumption that the pressure is the same at every branch end, a second set of closing equations has to be added to the global system, and how the selection of this set is arbitrary. Then two new approaches for this second set of closing equations are proposed, analysed and compared with Benson's original theory. Finally, one of them—the assumption that the total enthalpy for all the outgoing flows is the same—is proposed to close the problem, constituting with the constant pressure hypothesis a new constant pressure model for N-branch junctions.

Of lines and planes

Abstract A professional inquiry from a surveyor as to how to compute the angle between two inclined planes in space led to the presentation in this paper of different kinds of equations for lines and planes. Their relationships are discussed, and it is shown how the parameters are found. The emphasis is on coordinate transformation from one reference system to another by means of a rotation matrix. The rotation matrix, together with the two sets of three-dimensional coordinates, allows the users to retain their conventional perceptions of space. An example is included of striking a plane through a number of data points, so as to determine the location and the perpendicular offset of those points with respect to the plane. May 1991

Stability analysis for two kinds of equations in two-species population dynamics

In this paper we study the stability for equilibrium points of equations in two-population dynamics. We discuss two predator-prey-patch models. Model 1 is described by a differential equation. Model 2 is described by an integral differential equation. We obtain the conditions for the stability of their equilibrium points. The results show that the overall population stability despite local extinction is realizable.

Lagrange equation of another class of nonholonomic systems

Using the method of [1], the present paper derives the Lagrange equation without multipliers for another class of first-order nonholonomic dynamical systems by means of variational principle. This kind of equations is also new.

Lagrange equation of a class of nonholonomic systems

Making use of conclusions from [1]: (1) d-δ operations are commutative; (2) the Appell-Chetaev condition restricting virtual displacements is superfluous, the present paper derives the Lagrange equation without multipliers for a class of first-order nonlinear nonholonomic dynamical systems by means of variational principle. This kind of equations is new.

An ergodic control problem arising from the principal eigenfunction of an elliptic operator

Let us consider the following second order quasi-linear partial differential equation: -1/2Δvα+H(x,⊇vα)αvα=0 with a quadratic growth nonlinear term H(x,⊇vα) on ⊇vα, where α is a positive constant. Such kinds of equations on bounded regions with periodic or Neumann boundary conditions have been studied in connection with ergodic control problems, where the asymptotic behaviour of the solution vα as α tends to 0 is investigated. In the present article we specialize the equation above to the case where H(x,⊇vα)=1/2|⊇vα| 2 -V(x) but treat it on whole Euclidean space R n

General equations of motion for test particles in space-time with torsion

The momentum and spin equations of motion for test particles possessing different spins in space-time with torsion are derived from the most general functional form of ℒ M . The same kinds of equations in general relativity and in Kibble's gauge theory of gravitation are special cases of our equations.