Constrained inhomogeneous spherical equations: average-case hardness

Iterative solution of homogeneous integral equations

In particular, they showed that for a consistent approximation stability and convergence are equivalent. The results presented here have to do with the problem of approximating solutions of the inhomogeneous and quasi-linear equations which are obtained when terms of the form f(t) or g(t, u(t)) are added to equations such as the one above. Finite difference schemes for specific types of inhomogeneous equations and quasi-linear equations have been studied many times (see, for example, [1], [2], [4], and [7]). An extensive bibliography is included in [2]. In [7] Strang deals with linear inhomogeneous parabolic and hyperbolic partial differential equations; he shows that here, too, with appropriate definitions, stability and convergence are equivalent. Fritz John [4] deals with parabolic equations and includes a section on quasi-linear equations. John's paper is particularly remarkable in that he does not assume existence of solutions for the differential equations but proves existence using properties of the difference approximations. Frequently a difference approximation for an inhomogeneous equation or a quasi-linear equation is a simple modification of an approximation for the associated homogeneous equation. In this paper it will be shown that, it is often sufficient to consider the problem of convergence only for the homogeneous equation. That is, the conditions for convergence for linear homogeneous equations, which are provided by the Lax-Richtmnyer theory, are sufficient for the more general equations. Duhamel's method is used to show that a convergent approximation for a linear homogeneous equation

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The inhomogeneous Milne equation. Schwartz, Sandra Abstract The mathematical analysis of the transfer of integrated radiation through a semi-infinite grey atmosphere stratified in plane layers leads to the inhomogeneous Milne equation: J(r) - E1( t - rI)J(t)dt = k(r) where J(r) is the source function and k (r) is - ~1 times the divergence of the flux. In the study of monochromatic radiation transfer, whether or not the atmosphere is grey, a similar equation will appear, containing an additional parameter which corresponds to the frequency of the radiation. This will not affect the form of the solution. The divergence of the integrated flux is unequal to zero in those parts of the atmosphere which are not in radiative equilibrium: e.g., the regions where convection or energy generation takes place, or where external illumination is received. The divergence of the net monochromatic flux is also a function of the depth in the atmosphere since the atmosphere cannot be expected to be in radiative equilibrium with the radiation of each frequency. Another problem yielding a non-vanishing flux divergence term arises when one analyzes the flow of radiation in terms of the diffusion of photons. On approaching the stellar surface the flux of photons continually increases because of the breakup of high-frequency photons into photons of lower frequency. The physical theories of photon diffusion and radiative transfer corresponding to the case of a non-vanishing inhomogeneous term have not yet been developed to the point where an explicit form for that term can be found. However, general considerations indicate that at least two types of expansions of k (r) will be appropriate to these problems: one, in terms of decreasing exponentials (Problem A); the other, in terms of Laguerre polonomials of order 2m, multiplied by the function e-r/,rm (Problem B). The procedure that is used to solve for the explicit form of the source function corresponding to problems A and B is modeled after the solution of the homogeneous equation and begins with the derivation of the Laplace transform of the source function, i.e., the limb darkening function, for the assumed expansions of k (T). The source function itself is then found by inverting the Laplace transform. Observations of the limb darkening together with an assumed expansion for k (r) will suffice to find the values of the coefficients in the expansion as well as the source function itself. It is found that the function H (i ) which gives the limb darkening for integrated radiation in the case of strict radiative equilibrium also appears as a factor in the limb-darkening functions corresponding to problems A and B. Due to the generality of the assumed expansions, and the myriad transfer problems they can represent, the presence of this function can be seen to arise simply from the mathematical similarity of the homogeneous and inhomogeneous equations and does not appear to have any physical significance, contrary to a surmise made by some authors. In principle, it should be possible to apply this analysis to the observed limb darkening in the sun, and using the assumption of local thermodynamic equilibrium, obtain more exact knowledge of the regions of convection. Vale University Observatory, New Haven, Coun. Publication: The Astronomical Journal Pub Date: 1956 DOI: 10.1086/107384 Bibcode: 1956AJ.....61Q..12S full text sources ADS | data products SIMBAD (2)

Aiming at the problem of low global convergence and local convergence rate of trust region interior points of bounded variable constrained nonlinear equations, a trust region interior point algorithm for bounded variable constrained nonlinear equations under edge calculation is designed. By constructing the basic function form of nonlinear equations constrained by bounded variables, the boundary of nonlinear equations is determined by Gauss Newton iterative process to ensure the global convergence of changes; solve the original objective function, analyze the trust region subproblem of the unconstrained optimization problem, and generate an acceptable region. The region is generated through two-dimensional example interpretation. The interior points in the acceptable region are determined by the primal dual interior point method, and the interior points in the acceptable region are optimized. With the help of edge calculation, the trust region interior point programming model of bounded variable constrained nonlinear equations is designed to realize the algorithm design. The experimental results show that the designed algorithm can improve the trust region interior point global convergence and local convergence rate of nonlinear equations with bounded variable constraints.

Employing the theory of differentiable manifolds we give a geometric coordinate-free description of constrained differential equations (CDE), which are usually thought of as systems of simultaneous differential and algebraic equations. We regard the algebraic constraints as defining a differentiable manifold, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma</tex> , and regard solutions of the CDE as curves in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma</tex> . Our main contribution in this paper is to characterize geometrically a particular class of singular constrained differential equations for which consistency forces solutions to lie in a proper subset of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Sigma</tex> . Constrained differential equations describing electric circuits having capacitor-only loops and/or inductor-only cutsets are shown to be of the above type.

INHOMOGENEOUS FUNCTIONAL AND OPERATIONAL DIFFERENTIAL EQUATIONS

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic quotient by an abelian normal subgroup, over number fields.

Matrix algorithms for solving (in)homogeneous bound state equations

This paper presents an approach for finding the solution of partial differential equation describing the motion of transverse vibrations of rectangular plates of unidirectional linear varying thickness. The original partial differential equation consists of three operators: fourth-order spatial-dependent, second-order spatial-dependent, and second-order time-dependent. Using the method of multiple scales, the partial differential equation has been reduced to two simpler partial differential equations which can be analytically solved and which represent two levels of approximation. The first partial differential equation was a homogeneous equation and consisted of two operators, the fourth-order spatial-dependent and second-order time-dependent. The solution of this equation was found using the factorization method. This solution was zeroth-order approximation of the exact solution. The second partial differential equation was an inhomogeneous equation. The solution of this equation was also found and led to first-order approximation of the exact solution of the original problem. This way the first-order approximations of the natural frequencies and mode shapes are found. Various boundary conditions can be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

This paper presents a method for the supervision of an industrial plant. This supervision is aimed at guaranteeing the respect of a maximal duration constraint for some specific processing, and is addressed by considering a discrete event system model for this industrial plant. In this associated model, the time constraint is reduced to elementary constraints whose contributions are taken into account in the state equation of the system, yielding a constrained state equation for the plant. Supervisors are then synthesized by looking for solutions of this constrained state equation.

Control of discrete event systems with respect to strict duration: Supervision of an industrial manufacturing plant

1. In studying the structure of abstract groups, one is often confronted with the seemingly trivial question: when do we consider a group (or class of groups) as known? The question has many aspects, and is usually a very complex one. Of course, the group is formally known as soon as its multiplication table is determined; but in most cases, little further information can be obtained from the multiplication table itself on the structural elements of the group, such as subgroups, normal subgroups, automorphisms, etc. If one is however, interested in the classification of groups belonging to a certain class, then the problem is concentrated on the enumeration of groups having some common property, and the structural elements of the individual groups are of a secondary importance. In this case, the problem can be formulated in the following way: Find some finite algebraical procedure, depending on some integral or more general algebraical parameters, which enables us to construct, in an abstract or symbolic way, each group belonging to the class under consideration. A group is of course constructed if its elements are known, and the groupproduct of any two elements can be calculated by means of the defining procedure. It is desirable that the parameters, called invariants of the group, should be characteristic of the group in the sense that no two distinct systems of operations should define the same group. A classical example for such an interpretation of the structure problem is furnished by the fundamental theorem of finite Abelian groups. In this paper we are concerned with the enumeration of certain finite metabelian groups. (Metabelian is a group with Abelian commutator subgroup.) The problem in all its generality is a very difficult one, and far from being solved in the above sense. However, special classes of metabelian groups are accessible to investigation, and such investigations have an interest in so far as they clear more aspects of the general problem. In studying the structure of metabelian groups, the method of group extensions is often found to be the most natural tool. As an extensive use will be made of this method in this paper, I want to formulate it in such a way as it seems most suitable for the case discussed here. A similar but formally different formulation was given by A. Scholz1. See also W. Magnus2. Let ( be a finite metabelian group, additively written, 2t an Abelian normal subgroup of (M such that the quotient group e = (5/Wt is Abelian. Elements

In this paper, we consider infinitesimal (n. m.) first-order deformations of single-connected regular surfaces in three-dimensional Euclidean space. The search for the vector field of this deformation is generally reduced to the study and solution of a system of four equations (among them there are differential equations) with respect to seven unknown functions. To avoid uncertainty, the following restriction is imposed on a given surface: the Ricci tensor is stored (mainly) on the surface. A mathematical model of the problem is created: a system of seven equations with respect to seven unknown functions. Its mechanical content is established. It is shown that each solution of the obtained system of equations will determine the field of displacement n. m. deformation of the first order of the surface of nonzero Gaussian curvature, which will be an unambiguous function (up to a constant vector). It is proved that each regular surface of nonzero Gaussian and mean curvatures allows first-order n. m. deformation with a stationary Ricci tensor. The tensor fields are found explicitly and depend on two functions, which are the solution of a linear inhomogeneous second-order differential equation with partial derivatives. The class of rigid surfaces in relation to the specified n. m. deformations. Assuming that one of the functions is predetermined, the obtained differential equation in the General case will be a inhomogeneous differential Weingarten equation, and an equation of elliptical type. The geometric and mechanical meaning of the function that is the solution of this equation is found. The following result was obtained: any surface of positive Gaussian and nonzero mean curvatures admits n. m of first-order deformation with a stationary Ricci tensor in the region of a rather small degree. Tensor fields will be represented by a predefined function and some arbitrary regular functions. Considering the Dirichlet problem, it is proved that the simply connected regular surface of a positive Gaussian and nonzero mean curvatures under a certain boundary condition admits a single first-order deformation with a stationary Ricci tensor. The strain tensors are uniquely defined.

Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such equations usually do not have extremely large solutions. Therefore in several applications it is useful to have a fast algorithm to calculate the “small” solutions of these equations. Under “small” solutions we mean the solutions, say, with absolute values or sizes $\leq 10^{100}$. Such algorithms were formerly constructed for Thue equations, relative Thue equations. The relative and inhomogeneous Thue equations have applications in solving index form equations and certain resultant form equations. It is also known that certain “totally real” relative Thue equations can be reduced to absolute Thue equations (equations over $\mathbb{Z}$). As a common generalization of the above results, in our paper we develop a fast algorithm for calculating “small” solutions (say with sizes $\leq 10^{100}$) of inhomogeneous relative Thue equations, more exactly of certain inequalities that generalize those equations. We shall show that in the “totally real” case these can similarly be reduced to absolute inhomogeneous Thue inequalities. We also give an application to solving certain resultant equations in the relative case.

This paper discusses a 1D one-dimensional mathematical model for the thermoelasticity problem in a two-layer plate. Basic equations in dimensionless form contain both temperature and displacement. General solutions of homogeneous equations (displacement and temperature equations) are assumed to be a linear combination of Trefftz functions. Particular solutions of these equations are then expressed with appropriately constructed sums of derivatives of general solutions. Next, the inverse operators to those appearing in homogeneous equations are defined and applied to the right-hand sides of inhomogeneous equations. Thus, two systems of functions are obtained, satisfying strictly a fully coupled system of equations. To determine the unknown coefficients of these linear combinations, a functional is constructed that describes the error of meeting the initial and boundary conditions by approximate solutions. The minimization of the functional leads to an approximate solution to the problem under consideration. The solutions for one layer and for a two-layer plate are graphically presented and analyzed, illustrating the possible application of the method. Our results show that increasing the number of Trefftz functions leads to the reduction of differences between successive approximations.