Number Of Arbitrary Functions Research Articles

Lie symmetry analysis of (2+1)-dimensional KdV equations with variable coefficients

ABSTRACTIn this paper, symmetry groups are used to obtain symmetry reductions of (2+1)-dimensional KdV equations with variable coefficients. Despite the fact that these equations emerge in a nonlocal form, by using suitable transformations, they can be written as systems of partial differential equations, and in potential form, as fourth-order partial differential equations. We show that the point symmetries of the potential equation involve a large number of arbitrary functions. Moreover, these symmetries are used to transform the fourth-order partial differential equations into (1+1)-dimensional fourth-order differential equations. Furthermore, we have determined all two-dimensional solvable symmetry subalgebras, under certain restrictions, which the potential equation admits. Finally, by way of example, taking into account a two-dimensional abelian subalgebra, we obtain a direct reduction of the potential equation to an ordinary differential equation.

Verification of Programs with Mutual Recursion in the Pifagor Language

In the article, we consider verification of programs with mutual recursion in the data driven functional parallel language Pifagor. In this language the program could be represented as a data flow graph, that has no control connections, and has only data relations. Under these conditions it is possible to simplify the process of formal verification, since there is no need to analyse resource conflicts, which are present in the systems with ordinary architectures. The proof of programs correctness is based on the elimination of mutual recursions by program transformation. The universal method of mutual recursion of an arbitrary number of functions elimination consists in constructing the universal recursive function that simulates all the functions in the mutual recursion. A natural number is assigned to each function in mutual recursion. The universal recursive function takes as its argument the number of a function to be simulated and the arguments of this function. In some cases of the indirect recursion it is possible to use a simpler method of program transformation, namely, the merging of the functions code into a single function. To remove mutual recursion of an arbitrary number of functions, it is suggested to construct a graph of all connected functions and transform this graph by removing functions that are not connected with the target function, then by merging functions with indirect recursion and finally by constructing the universal recursive function. It is proved that in the Pifagor language such transformations of functions as code merging and universal recursive function construction do not change the correctness of the initial program. An example of partial correctness proof is given for the program that parses a simple arithmetic expression. We construct the graph of all connected functions and demonstrate two methods of proofs: by means of code merging and by means of the universal recursive function.

On deformations of one-dimensional Poisson structures of hydrodynamic type with degenerate metric

We provide a complete list of two- and three-component Poisson structures of hydrodynamic type with degenerate metric, and study their homogeneous deformations. In the non-degenerate case any such deformation is trivial, that is, can be obtained via Miura transformations. We demonstrate that in the degenerate case this class of deformations is non-trivial, and depends on a certain number of arbitrary functions. This shows that the second Poisson–Lichnerowicz cohomology group does not vanish.

A model of the expanding plasmasphere. 2. Spatial distribution of the electric field

UDC 533.95+537.84+537.814 We study further a model of the expanding plasmasphere. Possible structures of the spatial distribution of the electric current, which can exist within the framework of this model, are found. The obtained results are used to demonstrate the necessity of refinement of the model. The simplest variant of such a refinement, which consists in the transition from the two-layer magnetosphere to the three-layer one, is presented. It is shown for this variant that the structure of the spatial distribution of the electric current depends on some arbitrary functions whose form can be chosen using qualitative considerations. The results of using some methods of choosing such functions are demonstrated. In the first part of this work [1], we developed a model of the expanding plasmasphere for the Earth and other planets having magnetic fields to continue analysis of the problems considered in [2–4]. Partially, the model was described in the conference proceedings [5]. The self-consistent axisymmetric problem about of plasma configurations around a rotating magnetized sphere was solved in the assumption of azimuthal and radial motions of its plasma shell consisting of two parts: the inner region confined by the shell L = L∗ (L = r/(R sin 2 ϑ), in which R is the sphere radius), in which the pressure is isotropic and the motion is a purely azimuthal solid-state rotation, and the outer region, in which the radial motion of the medium takes place along with the differential azimuthal rotation, the plasma pressure is anisotropic, and viscosity is insignificant. This second part of the work deals with search for the structure of the spatial distribution of the electric current on the basis of the solution found in the first part. In this case, the obtained results make it necessary to introduce additional modifications to the model in order to ensure the applicability of the chosen simplifications. The transition from the two-layer plasma shell to the three-layer one with a purely radial motion in the outer region is considered the simplest variant of such changes. Possible structures of the spatial current distribution, which were found for this model, are determined by a number of arbitrary functions and cannot be specified uniquely within the framework of this model, but can be chosen using qualitative considerations. The results corresponding to the simplest methods of choosing these functions are presented. 2. FORM OF THE MODEL SOLUTION IN THE TWO-LAYER PLASMASPHERE

Geometric Rényi divergence: A comparative measure with applications to atomic densities

An alternative one-parameter measure of divergence is proposed, quantifying the discrepancy among general probability densities. Its main mathematical properties include (i) comparison among an arbitrary number of functions, (ii) the possibility of assigning different weights to each function according to its relevance on the comparative procedure, and (iii) ability to modify the relative contribution of different regions within the domain. Applications to the study of atomic density functions, in both conjugated spaces, show the versatility and universality of this divergence.

Generalized Quantum Similarity Index: Applications in atoms

A Generalized Quantum Similarity Index is defined, quantifying the similarity among density functions. The generalization includes, as new features (i) comparison among an arbitrary number of functions, (ii) its ability to modify the relative contribution of different regions within the domain, and (iii) the possibility of assigning different weights to each function according to its relevance on the comparative procedure. The similarity among atomic one-particle densities in both conjugated spaces, and neutral–cation similarity in ionization processes are analyzed. The results are interpreted attending to shell-filling patterns, and also in terms of experimentally accessible quantities of relevance in ionization processes.

Arbitrariness of the general solution of the partial differential equations and its applications

Using the framework of formal theory of partial differential equations, we consider a method of computation of the bi-Hilbert polynomial (i.e. Hilbert polynomial in two variables). Furthermore, present an approach to compute the number of arbitrary functions of positive differential order in the general solution. Then, under the "AC=BD" model for mathematics mechanization developed by Hong-qing ZHANG, we present a method to reduce an overdetermined system to a well-determined one. As applications, the Maxwell equations and weakly overdetermined equations are considered.

Painlevé test for integrability for a combination of Yang’s self-dual equations for SU (2) gauge fields and Charap’s equations for chiral invariant model of pion dynamics and a comparative discussion among the three

Painleve test for integrability for the combined equations generated from Yang’s self-dual equations for SU (2) gauge fields and Charap’s equations for chiral invariant model of pion dynamics faces some peculiar situations that allow none of the stages (leading order analysis, resonance calculation and checking of the existence of the requisite number of arbitrary functions) to be conclusive. It is also revealed from a comparative study with the previous results that the existence of abnormal behaviour at any of the stated stages may have a correlation with the existence of chaotic property or some other properties that do not correspond to solitonic behaviour.

Some properties of the inertia groups of Boolean bijunctive functions and an inductive method of generating such functions

AbstractThe class of bijunctive Boolean functions consists of the functions representable by the 2-CNF. The problem of enumeration of such function of arbitrary number of variables has not been solved. In the paper, we consider properties of inertia groups ofbijunctive functions in several groups and give an inductive method of generating all distinct representatives of the classes of geometric equivalence of bijunctive functions. By this method we calculate the numbers of bijunctive functions of 5, 6, and 7 variables.

Transformations of mechanical systems with cyclic coordinates and new integrable problems

We show that the well-known Routhian procedure of ignoring cyclic coordinates is far more than a tool for obtaining equations of motion of reduced order in a Lagrangian form. A transformation is introduced that involves a number of arbitrary functions or additional parameters in the system while preserving the Routhian equations of motion. Although these parameters invoke new physical effects in the transformed system, the solution of the latter is always obtained in a simple way from that of the original system. In particular, from any integrable system with k cyclic degrees of freedom we obtain a family of systems integrable on a fixed level of the cyclic integrals and physically generalizing that system through the inclusion of k arbitrary functions that depend only on the noncyclic coordinates. In many problems of physical interest, a general integrable case can also be generalized to an integrable case for arbitrary initial conditions. The method is applied to some problems of rigid body dynamics. Four new integrable problems are obtained as generalizations of known cases by including certain additional combinations of gravitational and electromagnetic forces. The new cases are presented in an explicit way that enables direct verification of the constancy of the integrals using the equations of motion. Explicit time solution of the new cases is discussed. Physical interpretation is given for two cases.

Some integrable hierarchies in (2+1) dimensions and their twistor description

The twistor space construction, due to Mason and Sparling, for solutions of the nonlinear Schrödinger and Korteweg–deVries hierarchies is generalized, resulting in families of integrable models in (2+1) dimensions. Soliton solutions to one such hierarchy [associated with the gauge group SU(2)] are constructed by adapting the methods used to construct instanton and monopole solutions. These integrable models have the feature that their solutions depend on a number of arbitrary functions as well as arbitrary constants, in contrast to the closely related Davey–Stewartson hierachy.

Painlev� analysis for a nonlinear schr�dinger equation in three dimensions

A Painleve analysis is performed for the nonlinear Schrodinger equation in (2+1) dimensions following the methodology of Weiss et al. simplified in the sense of Kruskal. At least for one branch it is found that the required number of arbitrary functions (as demanded by the Cauchy-Kovalevskaya theorem) exists, signalling complete integrability.

Conformal extensions of the Galilei group and their relation to the Schrödinger group

Various authors have considered a conformal extension CG0 of the Galilei group which in some sense is the nonrelativistic limit of the conformal extension of the Poincaré group, and have also established an invariance group for the free-particle Schrödinger equation, the ’’Schrödinger group.’’ Here we establish the most general conformal extension CG of the Galilei group, which is found to be identical to the group of the most general coordinate transformations that permit the use of noninertial frames of reference and of curvilinear coordinates in Galilei-invariant theories, which was considered by one of us some time ago, and is a gauge group containing a number of arbitrary functions. Both CG0 and the Schrödinger group are subgroups of CG containing the Galilei group, but otherwise they do not overlap. The Hamilton–Jacobi and Schrödinger equations for particles which are free or interact via inverse-square potentials are shown to be invariant under the Schrödinger group, and a further invariance of the Hamilton–Jacobi equation is established.

Conformally invariant solution of quantum field theory equations (1)

The exact conformally-invariant solution of renormalized quantum field theory equations is considered. Dynamical equations form an infinite system of equations for vertices. This system does not determine the vertices uniquely; it must be supplemented by axiomatic requirements (locality, crossing symmetry etc.). The latter were not taken into consideration in the course of the solution of equations, therefore the solution obtained contains a number of arbitrary functions. The role of the mentioned additional requirements for the construction of the unique solution is discussed. The vertices are considered, containing conserved external lines (the current and the energy-momentum tensor). It is shown that the generalized Ward identity makes it possible to obtain these vertices explicitly up to an arbitrary transverse part.

Enumeration of the True Observables in Gauge-Invariant Theories

By making use of invariance arguments, we shall show that the total number of true observables which can serve as "coordinates" in a gauge-invariant theory is $N\ensuremath{-}2n$ where $N$ is the number of field variables and $n$ is the number of arbitrary functions needed to specify an element of the gauge group.