Curvature Element Transformations Which Preserve Integrable Fields

Curvature Element Transformations Which Preserve Integrable Fields.

I Shohat, J., Bull. Am. Math. Soc., 41, 51 (1935). 2 Haar, A., Math. Ann., 78, 121-136 (1918). 3 Young, W. H., Comptes Rendus, 165, 696-699 (1917). 4Szego, G., Math. Zeit., 12, 61-94 (1922). b A general treatment, which sets forth necessary and sufficient conditions for equivalence of expansions in terms of orthogonal functions, is given by Walsh, J. L., and Wiener, N., Jour. Math. and Phys. of...

Lineal Element Transformations Which Preserve the Dual-Isothermal Character.

THEOREM. In order that a system S of o I curves represents the additivemultiplicative trajectories ofa given field, it is necessary (but not sufficient) that S possesses the dynamical Property I; this may be stated simply in either of these alternative geometric forms: Property Ia: For the o I curves of S passing through any lineal element E of the plane, construct the osculating parabolas at E...

Graph Transformations Which Preserve the Multiplicity of an Eigenvalue

Graph transformations which preserve the multiplicity of the eigenvalue zero in the spectrum are known since 1970s and are of importance in chemical applications. We now show that analogous transformations hold for all graph eigenvalues that are of the form 2 cos rn, where Y is a rational number, 0 < r < 1.

Transformations that Preserve Learnability

On certain semigroups of transformations that preserve double direction equivalence

Let TX be the full transformation semigroups on the set X. For an equivalence E on X, let TE(X) = {α ∈ TX : ∀(x, y) ∈ E ⇔ (xα, yα) ∈ E}It is known that TE(X) is a subsemigroup of TX. In this paper, we discussthe Green's *-relations, certain *-ideal and certain Rees quotient semigroup for TE(X).