In practice the hypothesis that x(uv) # ~(u’v) for some conjugate U’ of u is almost always satisfied. When x is faithful and u = v-r, then this just asserts that u does not belong to the center of G (though it is an easy exercise to give a direct proof of the theorem in this case). We shall derive Theorem 1 from a general result (Theorem 2) about common constituents of permutation characters. T...

In 1934, Whitney posed the problem of how to recognize whether a function f defined on a closed subset X of R is the restriction of a function of class C. Whitney himself solved the one-dimensional case (i.e., for n = 1) in terms of finite differences [W1, W2, W3], giving the classical Whitney’s extension theorem. A geometrical solution for the case C(R) was given by G. Glaeser [G], who introdu...

Spectral points of positive and negative type, and type π+ and type π− for closed linear operators and relations in Krein spaces are introduced with the help of approximative eigensequences. The main objective of the paper is to study these sign type properties in the non-selfadjoint case under various kinds of perturbations, e.g. compact perturbations and perturbations small in the gap metric....

This paper analyzes the generalization error of two-layer neural networks for computing ground state Schrödinger operator on add-dimensional hyperc...

In recent years there has been interest in a theorem on positive definite matrices known as Lyapunov's theorem. Several authors have proved generalizations of this theorem, (WIELANDT [29J, TAUSSKY [24J, [25J, [26J , OSTROWSKISCHNEIDER [20J, GIVENS [10J, CARLSON-SCHNEIDER [3J, CARLSON [4J) . Lyapunov's theorem and its generalizations have become known as inertia theorems. In this note we shall u...

0. Introduction We work over an algebraically closed field k of characteristic 0. 0.1. Statement. In this paper, we use techniques of toric geometry to reprove the following theorem: Theorem 0.1. Let X be a projective variety of finite type over k, and let Z ⊂ X be a proper closed subset. Let G ⊂ Aut k (Z ⊂ X) be a finite group. Then there is a G-equivariant modification r : X 1 → X such that X...

We investigate the existence of positive solutions a Riemann-Liouville fractional differential equation with sequential derivatives, parameter and nonnegative singular nonlinearity, supplemented integral-multipoint boundary conditions which contain derivatives various orders Riemann-Stieltjes integrals. Our general cover some symmetry cases for unknown function. In proof our main result, we use...

The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized Q-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein ...