Respect Tox Research Articles

Interpolating them-th power ofx at the zeros of then-th Chebyshev-polynomial yields an almost best Chebyshev-approximation

LetX={x1,x2,..., n }eI=[−1, 1] and\(\varphi (x) = \prod\limits_{j = 1}^n {(x - x_j )} \). Forf∈C1(I) definef* byf−p f =φf*, wherep f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf →f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT n (x) and obtain ∥x m −pxm∥≤8eEn−1(x m ), whereEn−1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE n (f) that omits theassumptionf(n+1)>0 in a similar estimate of Meinardus.

Lipschitzian solutions of the implicit Cauchy problemg(x′)=f(t,x),x(0)=0, withf discontinuous inx

In this paper, we study the existence of lipschitzian solutions of the implicit Cauchy problemg(x′)=f(t,x),x(0)=0, without assuming that the functionf is continuous with respect tox. After establishing our main results, we also present several counterexamples to possible improvement of them.

On regularization of right hand sides of differential relations

SynopsisLet the values ofFbe convex compact subsets ofRnand letFbe upper semicontinuous with respect tox. There are two ways known of replacingFby a more regular map so that the set of solutions of (2) remains unchanged. We prove that both ways lead to the same more regular map and extend the results to the case whereRnis replaced by a separable Banach space.

On the generalized invertibility of Wiener-Hopf operators in Banach spaces

It is proved that a Wiener-Hopf operator Tp (A) on a Banach space PX is generalized invertible iff A has a cross factorization with respect toX and P. IfX is a separable Hilbert space, then a criterion for the weak factorization of A can be concluded.

A commutativity theorem of partial differential operators

Letu=u(x, t) be a function ofx andt, andu i =D i u,D=d/dx,i=0, 1, 2, ..., be its derivatives with respect tox. Denote byW n the set {f|f=f(u,u1, ...,u n ), (∂/∂u n )f≠0}, wheref(u, ...,u n ) are polynomials ofu i with constant coefficients. To any\(f \in W = \mathop \cup \limits_{n = 2}^\infty W_n \), we relate it with an operator . In this paper we prove that: ℳ(f) commutes with ℳ(g) if they commute respectively with ℳ(h), providedf,g,heW. Relating to this commutativity theorem, we prove that, if an evolution equationu t =f(u, ...,u n ) possesses nontrivial symmetries (or conservation laws for a class of polynomialsf), thenf=Cu n +f1(u, ...,u r ), whereC=const, andr<n. In the end of this paper, we state a related open problem whose solution would be of much value to the theory of soliton.

Gaussian integers with small prime factors

Let ψG(xt, x) denote the number of Gaussian integers with norm not exceeding x2t whose Gaussian prime factors have norm not exceeding x2. Previous estimates have required restrictions on the parameter t with respect to x. The purpose of this note is to present asymptotic estimates for ψG(xt, x) for all ranges of the parameter t with respect to x.

An implicit function theorem

Suppose thatF:D⊂Rn×Rm→Rn, withF(x0,y0)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ∂1F(x0,y0) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.

Uniqueness for ordinary differential equations

Various criteria are known for assuring uniqueness of the solution of a system ofn ordinary differential equations,x′ = f(t, x), with initial conditionx(t0) = x0. Most of these involve some sort of relaxed Lipschitz condition onf(t, x), with respect tox, valid on an open setD ⊂ R1+n which contains the point (t0, x0). The present paper generalizes (and unifies) a number of known uniqueness criteria to cover cases when (t0, x0) lies on the boundary ofD.

Legendre functions with both parameters large

By application of the theory for second-order linear differential equations with two turning points developed in the preceding paper, some new asymptotic approximations are obtained for the associated Legendre functions when both the degree n and order m are large. The approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid with respect to x ∈ ( − 1 , 1 ) and m / ( n + 1 2 ) ∈ [ δ , 1 + Δ ] where δ and ∆ are arbitrary fixed numbers such that 0 &lt; δ &lt; 1 and ∆ &gt; 0. The values of m and n + ½ are either both real, or both purely imaginary. In all cases explicit bounds are supplied for the error terms associated with the approximations.

Runge Kutta processes with multiple nodes

This paper studies generalisations of the classicalRunge Kutta processes. The generalisation is in such a way, that not only the functionsf(x, y) are evaluated at some intermediate points, but in addition the functionsDf, D2f, ...,Dmf, where D is the differential operator defined by (1.3) and corresponds to a differentiation with respect tox. Since, in recent years, many authors have found recursion formulas for an automatic evaluation ofDkf (k=1, 2, ...) the methods in this paper can be implemented in a convenient way. In Part 1 we develop the general form of conditions for the coefficients of these processes extending thereby results ofJ. C. Butcher [1]. These equations are even more complicated than those for classical processes.

On the integrability cases of the equation of motion for a satellite in an axially symmetric gravitational field

The projection of an axially symmetric satellite's orbit on a plane perpendicular to the rotation axis (z=const.) is given by the second-order differential equation. $$\frac{{y''}}{{1 + y'^2 }} = \bar \Psi _y - y'\bar \Psi _{x,}$$ where the prime denotes the derivative with respect tox and\(\bar \Psi (x,y)\) is a known function. Two integrability cases have been investigated and it has been shown that for these two cases the integration can be carried out either by quadratures or reduced to a first-order differential equation. Analytical and physical properties are expressed, and it is shown that the equation can be derived from the calssical plane eikonal equation of geometric optics.

Generalized reaction forces from potential scattering

The forces exerted by momentum reaction from the potential scattering of nonrelativistic particles are investigated. By application of the adiabatic principle to the variation of a parameterx, the average generalized force\(\bar F_x \) is expressed as a surface integral of the wave function and its variation. The surface integral is reduced to a form involving particle-flux amplitudes, and derivatives of the collision matrix with respect tox. In this form the results are the same as for radiation-pressure forces in electromagnetism or acoustics. Some implications arising from an identification of the reaction forces with radiation pressures of a Schrodinger field are outlined.

Generalized reaction forces from potential scattering

The forces exerted by momentum reaction from the potential scattering of nonrelativistic particles are investigated. By application of the adiabatic principle to the variation of a parameterx, the average generalized force $$\bar F_x $$ is expressed as a surface integral of the wave function and its variation. The surface integral is reduced to a form involving particle-flux amplitudes, and derivatives of the collision matrix with respect tox. In this form the results are the same as for radiation-pressure forces in electromagnetism or acoustics. Some implications arising from an identification of the reaction forces with radiation pressures of a Schrodinger field are outlined.

Magnetohydrodynamic waves in rotating liquids

The directional distribution and attenuation of waves created by a harmonically oscillating point source placed at the axis OX of an incompressible, inviscid, unbounded, electrically conducting liquid has been studied by using a technique developed by Lighthill. The undisturbed motion of the liquid consists of a rigid rotation Ω, about OX and a uniform velocity of translation U along OX in the presence of a uniform external magnetic field H 0 along OX . The effect of U is to make wave propagation asymmetrical with respect to x . The rotation causes radial expansion of the Alfvén waves and the magnetic field produces a radial contraction of the Taylor waves.* The interaction of U, Ω, H 0 and ω is the main theme of this paper. A novel feature of the interaction is the splitting of Taylor waves by the magnetic field.

Note on the solution of a certain boundary-value problem

A simple algorithm is described for inverting the operatorDxDy (Dx andDy here and subsequently denote partial differentiation with respect tox andy respectively) which occurs in the iterative solution of the equationDxDyf (x, y)=g (x, y, f, Dxf, Dx2f,DxDyf, Dy2f) when boundary values off(x,y) are given along the sides of the rectangle in thexy-plane whose corners are at the points (a,b); (a+(n+1)k,b); (a+(n+1)k,b+(n+1)h); (a,b+(n+1)h).

The temperature distribution along a radiating gas stream in which heat is being liberated by a chemical reaction

As a first approximation, to calculate the variation of flame temperature ( Y ) with distance ( X ) along a slowly burning flame, the flame is taken to consist of a central stream or jet of fuel which enters at the temperature ( T ) of the heat sink and entrains combustion air at a rate constant with respect to X . This entrained air is assumed to react rapidly with the fuel stream and the products of the reactions remain in the fuel stream, so that the temperature ( Y ) of the latter rises at a rate dY/dX which falls off as the heat capacity of this stream increases. When there is no heat loss from the fuel jet the temperature-distance curve is shown to be a rectangular hyperbola. The curvature at any point of the hyperbola increases as ( q ), the ratio of the heat capacity of the initial fuel stream to that of the final combustion products, decreases. In other cases heat transfer is supposed to take place by convection (α [ Y ─ T ]) orradiation (α [ Y 4 ─ T 4 ]) between the fuel jet and the heat sink with a heat-transfer coefficient which is assumed to be constant for a cylindrical flame and proportional to distance from the inlet for a conical flame. It is shown that in the case of the cylindrical flame the flame temperature must increase monotonically until combustion is complete, whereas the temperature in the conical flame can begin to fall off at an earlier stage. In the case of convection-heat transfer the shape of the temperature-distance curve is dependent only on ( q ) and on the ratio L/L 0 (where L ═ length for all combustion air to be entrained and L 0 ═ length in which all the combustion energy would be transferred to the surroundings if the flame remained at the adiabatic combustion temperature T a ). With radiative heat transfer the shape of the curves depends on ( q ) and L/L 0 but also on the ratio T/T a .

Finite strain in elastic problems

The mathematical theory of Elasticity, as at present developed, is based on the assumption that the displacements that have to be considered in elastic solid bodies are so small that the squares and products of the first differential coefficients of m, v, w with respect to x, y, z can be neglected in comparison with their first powers. That is why we cannot use it to get a satisfactory solution of many a problem in elasticity in which the displacement is finite and the strain produced is not small enough to justify the above assumption. For example, we can bend any rectangular plate in the form of a cylinder by couples applied to the edges only. As far as I know there exists no exact solution for this simple problem. In Section III I have attempted to give its solution based on the theory of finite strain. The first step towards the solution of the type of problems we have in mind is to get the components of strain corresponding with any displacement. Like the body-stress equations, these should be referred to the actual position of a point P of the material in the strained condition, and not to the position of a point considered before strain. The importance of this point, overlooked by various authors, cannot be exaggerated. Apparently Filon and Coker were the first to notice it and stress its importance. To a first approximation it is immaterial which method of reference is adopted, for the values of the strain components in the two cases differ only in the second order terms.

On the transformation of integrals

§ 1. It is known that, if x = x ( u , v ), y = y ( u , v ), be one-valued continuous functions of ( u , v ), possessing continuous differential coefficients, while f ( x , y ) is any continuous function of ( x , y ), ∬ c f ( x , y ) dx dy = ∫ c a ∫ d b f { x ( u , v ), y ( u , v )}∂( x , y )/∂( u , v ) du dv , where the integration on the left-hand side is taken over the area of the plane curve, C, which is the image in the ( x , y )-plane of the rectangle ( a , b ; c , d ) in the { u , v )-plane. Here it is tacitly assumed that C divides the plane into two distinct parts, a limitation which, however, disappears when we employ the definition of integration over an area which I have found it necessary to introduce into analysis. 2. If we denote by F ( x,y ) one of the indefinite integrals of f ( x, y ) with respect to x , and by x = x (t) , y= y (t),