Number Of Arbitrary Constants Research Articles

Напряженно-деформированное состояние прямоугольных пластин на основе уточненной теории

Two variants of a refined theory for calculation of the rectangular orthotropic plates stress-strain state are represented. The plate's state equations are presented in the form three-dimensional equations of elasticity theory. The components of the plate's stress-strain state are received as the polynomial func- tions on the coordinate which is normal to the middle plane. These functions are one or two degree higher than in the Kirchhoff-Love theory are used. The virtual displacements principle is applied to obtain the two-dimensional equations and its natural conditions. The modified conditions for standard cases of the plate mounting are formulated. Calculation of plate stress-strain is carried out by using Laplace transform, and then the number of arbitrary constants in the integration of differential equations systems thus is twice reduced. One of the refined theory distinctive features consist in direct integration of the three dimensional elasticity problems equilibrium equations at transverse normal and tangential stresses determination. As an example, the paper considers the calculation of a rectangular isotropic plate's stress-strain under a local load. The results obtained by the refined theories and by the classical theory are compared. The essential contribution of normal transverse stress of type boundary layer to the general stress- strain state of a plate is shown. The received results can be used in calculations and at tests for strength and durability of aviation and space-rocket and also engineering structures of different destination

On the relationship between control theory and nonholonomic mechanics

This paper concerns transitions of a mechanical system, in a given time, from one state, in which the generalized coordinates and velocities are given, into another one, in which the system should have the required coordinates and velocities. It is assumed that such a transition may be effected using a single control force. It is shown that if one determines the force using the Pontryagin maximum principle (from the minimality condition of the time integral of the force squared during the time of motion), then a nonholonomic high-order constraint occurs in the defined process of motion of the system. As a result, this problem can be attacked by the theory of motion of nonholonomic systems with high-order constraints. According to this theory, in the set of different motions with a constraint of the same order, the optimal motion is the one for which the generalized Gauss principle is fulfilled. Thus, a control force that is chosen from the set of forces that provide the transition of a mechanical system from one state to another during a given time can be specified both on the basis of the Pontryagin maximum principle and on the basis of the generalized Gauss principle. Particular attention is paid to the comparison of the results that are obtained by these two principles. This is illustrated using the example of a horizontal motion of a pendula cart system to which a required force is applied. To obtain the control force without jumps at the beginning and the end of the motion, an extended boundary problem is formulated in which not only the coordinates and velocities are given at these times, but also the derivatives of the coordinates with respect to time up to the order n ≥ 2. This extended boundary-value problem cannot be solved via the Pontryagin maximum principle, because in this case the number of arbitrary constants will be smaller than the total number of the boundary-value conditions. At the same time, the generalized Gauss principle is capable of solving the problem, because in this case it is only required to increase the order of the principle to a value that agrees with the number of given boundary-value conditions. The results of numerical calculations are presented.

Analytical approximation of the exterior gravitational field of rotating neutron stars

It is known that Bäcklund transformations can be used to generate stationary axisymmetric solutions of Einstein's vacuum field equations with any number of constants. We will use this class of exact solutions to describe the exterior vacuum region of numerically calculated neutron stars. Therefore, we study how an Ernst potential given on the rotation axis and containing an arbitrary number of constants can be used to determine the metric everywhere. Then, we review two methods to determine those constants from a numerically calculated solution. Finally, we compare the metric and physical properties of our analytic solution with the numerical data and find excellent agreement even for a small number of parameters.

The application of a novel computational approach to physically important nonlinear evolution equations of higher order

A new algebraic procedure is introduced to find new classes of solutions of (1+1)-nonlinear partial differential equations (nPDEs) of physical relevance. The proposed new computational algebraic approach leads to traveling-wave solutions and moreover a new class of explicit one-soliton-solutions can be obtained. The crucial step of the method is the basic assumption that unknown solutions satisfy an ordinary differential equation (ODE) of first order that can be integrated easily. A further important aspect of this paper, however, is the fact that one is able to calculate a distinctive class of solutions which cannot be found in the current literature. By increasing the number of arbitrary constants the approach allows one to handle problems occurring in solving nonlinear algebraic systems closely related with nPDEs. It is worth mentioning that the benefit of the given approach is surprisingly the easiness in deriving different classes of solutions; so it can therefore be applied in the physical sciences, biology, chemistry and related domains alike. On the contrary it is worth stressing the necessity of such sophisticated methods since a general theory of nPDEs does not exist.

Solution of Ordinary Differential Equations with a Large Lie Symmetry Group

The group method of solving ODEs without any quadrature goes back to Lie. In order to apply it, the number of symmetries admitted by a given ODE has to be greater by one than the number of arbitraryconstants in the general solution of the equation. In this paper, we use thetechnique of canonical Lie–Backlund symmetries that makes theproof of the statement concerning integrals of ODEs more evident. The methodis extended to the solution of system of ODEs with a small parameter of arbitrary order.

Toward an Infinite-component Field Theory with a Double Symmetry: Free Fields

We begin a study of possibilities of describing hadrons in terms of monolocal fields which transform as proper Lorentz group representations decomposable into an infinite direct sum of finite-dimensional irreducible representations. The additional requirement that the free-field Lagrangians be invariant under the secondary symmetry transformations generated by the polar or the axial four-vector representation of the orthochronous Lorentz group provides an effective mechanism for selecting the class representations considered and eliminating an infinite number of arbitrary constants allowed by the relativistic invariance of the Lagrangians.

Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution

We present several algorithms, executable in a finite number of steps, which have been implemented in the symbolic language maple. The standard form algorithm reduces a system of PDEs to a simplified standard form which has all of its integrability conditions satisfied (i.e. is involutive). The initial data algorithm uses a system's standard form to calculate a set of initial data that uniquely determines a local solution to the system without needing to solve the system. The number of arbitrary constants and arbitrary functions in the general solution to the system is directly calculable from this set. The taylor algorithm uses a system's standard form and initial data set to determine the Taylor series expansion of its solution about any point to any given finite degree. All systems of linear PDEs and many systems of nonlinear PDEs can be reduced to standard form in a finite number of steps. Our algorithms have simple geometric interpretations which are illustrated through the use of diagrams. The standard form algorithm is generally more efficient than the classical methods due to Janet and Cartan for reducing systems of PDEs to involutive form.

Minimum-time three-dimensional turn to a point of supersonic aircraft

The objective of this paper is to present real-time, on-line, minimum-time three-dimensional turn to a point of supersonic aircraft/The analytic aspect of the theory of optimal trajectories is emphasized. This includes the general properties of optimal trajectories consisting of the integrals of motion and the characteristic features in engine and aerodynamic controls. By using the integrals of motion the totality of the optimal trajectories can be obtained as a family of curves depending on a certain number of arbitrary constants. The optimal control is obtained by geometrical method through the domain of maneuverability. This makes explicit the switching characteristics of the optimal control, in particular when singular or chattering control is involved. The optimality of singular thrust control and the optimal junction of different subarcs are analyzed. The computation of the optimal trajectories is carried out using the aerodynamic and engine characteristics of a lightweight, high thrust-to-weight ratio supersonic fighter. By using normalized control variables, i.e., the thrust-to-weight ratio, the bank angle, and the load factor, results can be applied to any supersonic aircraft.

Synthesis of Finite State Algorithms in a Galois Field GF[pn

This correspondence describes a method for achieving synthesis of finite state algorithms by the use of a set of logic elements that execute field operations from the Galois field GF[pn]. The method begins with a definition of the algorithm to be synthesized in a completely specified finite state flow table form. A polynomial expansion of this flow table function is derived. A canonical sequential circuit corresponding to this polynomial expansion is defined. Subsequently, the given algorithm is synthesized using the canonical circuit by specification of a number of arbitrary constants in the canonical circuit. A mechanical method for deriving constants used in the canonical circuits is given. Finally, some estimates on complexity of the given circuit structure are stated assuming the most fundamental logic element structures.

A general integral of the axially symmetric stationary Einstein equations

The Ernst function of an axially symmetric stationary asymptotically flat space-time involving an arbitrary harmonic function and an arbitrary number of constants is presented and discussed.

Backlund transformations of axially symmetric stationary gravitational fields

A generation theorem for solutions of Einstein's equations is presented. It consists mainly of algebraic steps. With its aid, one obtained from an 'old' solution (e.g. from the Minkowski space) 'new' solutions with an arbitrary number of constants. The method of repeated application of potential and coordinate transformations considered by Geroch (1972) and Kinnerley (1977) is included.

A proof of the cut-elimination theorem in simple type theory

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.Inductive definition of types.0 and 1 are types.If τ1, …, τn are types, then (τ1, …, τn) is a type.Basic symbols.a1τ, a2τ, … for free variables of type τ.x1τ, x2τ, … for bound variables of type τ.An arbitrary number of constants of certain types.An arbitrary number of function symbols with certain argument places.

Electrical Units and Electromagnetic Field Vectors

This paper attempts to present more clearly the principles underlying the definition of a system of electrical units. First, it recalls briefly how fundamental and derived quantities are defined and how fundamental quantities are chosen. The main laws of electromagnetism are then written in a general form using the least number of arbitrary constants, namely three, compatible with the assumption that all the electrical units are derived in a consistent manner from the units of charge and current. It is shown how a system of units is selected and how this choice determines the values and dimensions of the three constants, two of which are just ?o and ?o. These considerations lead us to a discussion of the more or less fundamental nature of the four vectors describing the electromagnetic field and of the different ways in which these vectors can be associated.

Multipole expansions for stationary fields in general relativity

Stationary solutions of the Einstein field equations with no spatial symmetries are considered in order to discover the number of arbitrary constants, corresponding to the moments of the source in the Newtonian field, which appear in the solution. It is shown that the stationary field contains twice the number of moments which appear in the static field, apart from the mass term which appears singly. These additional moments are essentially angular momentum terms, which in Newtonian theory make no contribution to the field.

Dynamic programming and the Hamilton-Jacobi method of classical mechanics

The conventional dynamic programming method for analytically solving a variational problem requires the determination of a particular solution, the optimal value function or return function, of the fundamental partial differential equation. Associated with it is another function, the optimal policy function. At each point, this function yields the value of the slope of the optimal curve to that point (or from that point, depending on the method of solution). The optimal curve itself can then be found by integration. In this paper, dynamic programming concepts and principles are used to develop two alternatives to the conventional method of solution. In the first method, a particular solution of two simultaneous partial differential equations is used to generate optimal curves by differentiations and solution of simultaneous equations. In the second method, any solution of the fundamental equation containing an appropriate number of arbitrary constants is sought. It is shown how such a function yields directly, by differentiations and solution of simultaneous equations, the optimal curve for a given problem. While the derivations to follow are new, the results are equivalent to those of a method due to Hamilton and its modification due to Jacobi.

Study of a Class of Time Dependent Systems

ABSTRACT In this paper the study of a class of linear second-order time dependent systems, wherein the law of time dependency is specified either deterministic- cally or stochastically, has been undertaken. The responses of such systems have been obtained analytically and these exact solutions have the necessary number of arbitrary constants to fit the initial conditions. Also, the statistical properties of the response have been presented for the case of systems with randomly varying parameters.

On unions of chains of models

(*) IIxi, • • • , x„Syi, • • • , ym v where v is quantifier-free, if and only if K is closed under unions of chains of models.1 We have adopted most of the notational features of Tarski [7], in which can be found the definitions of relational systems, subsystems, and unions of a chain of systems; we have assumed familiarity with the symbolism of [7]. We have also adopted the convention of giving definitions and theorems only for a similarity class of systems of the type (A, S) where 5 is a ternary relation over A. All the results we have obtained here can be carried over trivially for similarity classes of systems in which there are a finite number of finitary relations. It will be apparent to the reader that this paper exploits a special method and a special theorem. The method is that of graph-diagrams first introduced in Robinson [6], and the theorem is the completeness theorem of Godel [3]. We shall use the completeness theorem in the generalized form (by Malcev and Henkin) where we allow an arbitrary number of constants in the first-order predicate calculus. The plan of the paper is first to give a mathematical characterization of those classes K which are determined by sentences of the form (*), then to use this characterization to prove our main result.2 At the end, we shall apply our result to convex arithmetical classes.

Complete Set of Dispersion Relations for a Class of Fixed-Source Meson Theories

The structure of the transition matrices for all processes that can occur for a class of fixed-source meson theories is studied. The model consists of a scalar meson field coupled to an extended source in such a way that any finite number of quanta can be emitted or absorbed at a given time (multiple vertices), but that all interactions are restricted to be $S$ wave in nature. The general reaction matrix element for $m$ incident and $n$ emergent particles contains many terms describing sequences of independent processes, which must be removed before one obtains a proper object for studies of a dispersion-theoretic nature. It is shown that the ratio of the residual matrix element to a suitable product of source functions possesses those analytic properties, as a function of the total energy of the system, which permit dispersion relations to be stated. Other than for the elastic scattering amplitude the latter make reference to values of the amplitude in a nonphysical energy region. In conjunction with a suitably generalized unitarity condition, however, the scheme, when viewed as a set of coupled integral equations, can be solved by successive approximations in terms of a number of arbitrary constants, essentially equal to the number of coupling constants in the original Hamiltonian (actually one fewer). Nevertheless, it is pointed out that the scheme does not admit a unique solution, and this is illustrated physically by exhibiting an extended class of Hamiltonians which yield the same dispersion relations, but which, as a class, contain more coupling constants than make their appearance in the dispersion relations. Physically, these are connected with the occurrence of resonance scattering.

A theoretical determination of the elastic constants of ice

A simplified model of the ice crystal, equivalent to Barnes's, was used. By applying the symmetry operations of this model to the dynamical matrix, it was made to depend on six arbitrary constants. By assuming that the tetrahedra of oxygen atoms which form the lattice are regular, and hence applying a further symmetry transformation to a smaller unit of the crystal, the number of arbitrary constants was reduced to two. The elastic constants were then found in terms of these two atomic constants. Two of the experimental measurements of the elastic constants of polycrystalline (quasiamorphous) ice were used to calculate the atomic constants and hence the elastic constants for single crystals.

Applications of the Operational Calculus to the Theory of Structures

The differential equations governing the deflection of structural elements such as loaded cords and beams freely suspended or resting on an elastic foundation are solved operationally. The operational method simplifies the analysis because it reduces the number of arbitrary constants to be determined and enables the effect of concentrated loads or applied couples to be determined directly.