On stability of solutions of integral equations in the class of measurable functions

Consider the equation G(x)=(y,) ̃ where the mapping G acts from a metric space X into a space Y, on which a distance is defined, y ̃ ∈ Y. The metric in X and the distance in Y can take on the value ∞, the distance satisfies only one property of a metric: the distance between y,z ∈Y is zero if and only if y= z. For mappings X → Y the notions of sets of covering, Lipschitz property, and closedness are defined. In these terms, the assertion is obtained about the stability in the metric space X of solutions of the considered equation to changes of the mapping G and the element y ̃. This assertion is applied to the study of the integral equation f(t,∫_0^1▒K (t,s)x(s)ds,x(t))= y ̃(t),t ∈[0,1], with respect to an unknown Lebesgue measurable function x: [0,1] ∈R. Sufficient conditions are obtained for the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions f,K,(y.) ̃

On a class of operators in the hyperfinite $\mathrm{II}_1$ factor

Let $R$ be the hyperfinite $\mathrm {II}_1$ factor and let $u$, $v$ be two generators of $R$ such that $u^*u=v^*v=1$ and $vu=e^{2\pi i\theta } uv$ for an irrational number $\theta$. In this paper we study the class of operators $uf(v)$, where $f$ is a bounded Lebesgue measurable function on the unit circle $S^1$. We calculate the spectrum and Brown spectrum of operators $uf(v)$, and study the invariant subspace problem of such operators relative to $R$. We show that under general assumptions the von Neumann algebra generated by $uf(v)$ is an irreducible subfactor of $R$ with index $n$ for some natural number $n$, and the $C^*$-algebra generated by $uf(v)$ and the identity operator is a generalized universal irrational rotation $C^*$-algebra.

Some Admissible Estimators in Extreme Value Densities

Let X be a random variable having the extreme value density of the form(1)where r is assumed to be a positive Lebesgue measurable function of x and the function q is defined byfor all θ in Ω = (0, ∞). It is further assumed that q(θ) approaches zero as θ → ∞.In this note we are concerned with estimating parametric functions g(θ) of the form [1/q(θ)]α, α any real number. The loss function is assumed to be squared error and the estimators are assumed to be functions of a single observation X.

On Singular Normal Linear Integral Equations

In this work we consider the equation1where K(x, y) is singular in the sense that it does not properly belong to L2 and f(x) is an arbitrary L2 function.A Lebesgue measurable function K(x, y) of two variables, having real values on [0.1] × [0.1] is called a singular normal kernel of(i)There exists approximating kernels Km(x, y) satisfying(ii)(iii)(iv)