Remark on a theorem of Lindström

Let Ω denote the set of all n by n doubly stochastic matrices and let m be a positive integer. Then the set $\text{Ω}{}_{\mathrm{m}}\text{= {A ? Ω : 0 ? a}{}_{\mathrm{ij}}\text{?}\text{1}\text{m}\text{, 1 ? i, j ≤ n}}$ is the convex hull of the matrices in $\text{Ω}{}_{\mathrm{m}}$ having exactly m entries equal to $\text{1}\text{m}$ in each row and column and the other entries equal to zero.     

Remark on a theorem of Aharonov and Walsh

The following theorem is proved: there is a functionf(z) analytic in |z|<1 and having the natural boundary |z|=1 such that for an infinite sequence of rational functions of degreen, r _n(z)=P_n(z)/q_n(z), the inequality 1 $$\left| {f(z) - r_n (z)} \right|< \varepsilon _n $$ holds in the closed unit circle |z|≦1. Here? ₁,? ₂,...,? _n is any sequence of positive numbers, tending to zero asn approaches infinity. This theorem is a refinement of a theorem of Aharonov and Walsh, who showed the existence of anf(z) satisfying (*) in |z|≦1 (with an infinite sequence {r _n(z)}) but having the natural boundary |z|=3.     

Remark on the central limit theorem for ergodic chains

We obtain sufficient conditions that should be imposed on a functionf in order that, for ergodic Markov chains, the sum $$\frac{1}{{\sqrt n }} \sum\limits_{k = 0}^{n - 1} { f(X_k )} $$ be asymptotically normal.     

Remark on Tono's theorem about cuspidal curves

We give an upper bound for the number of cusps of a plane affine or projective curve via its first Betti number.     

Remark on the Lyapunov theorem on vector measures

Kharkov State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 78–80, October–December, 1991.     

Remark on Wielandt's visibility theorem

A short proof is given of Wielandt's visibility theorem, using a special case of a theorem of Rédei, which was proved in an elementary way by Lóvasz and Schriver.     

A trace theorem for Dirichlet forms on fractals

We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on Sierpinski gaskets and Sierpinski carpets to their boundaries, where the boundaries are represented by triangles and squares that confine the gaskets and the carpets. As an application, we construct diffusion processes on a collection of fractals called fractal fields. These processes behave as an appropriate fractal diffusion within each fractal component of the field.     

A trace theorem and a linearization method for operator polynomials

Let p() = ⁿI - ^n-1 A₁ - A_n, where A_i are operators on a Banach space X with (1/i)^th power summable approximation numbers. Then the sum of the eigenvalues of the operator polynomial p() equals the trace of A₁. This is shown by a linearization method which is further used to generalize results on the completeness of the eigenvalues of p() under suitable conditions on the resolvent.     

On a theorem of existence for scaling problems

We study the question of the existence of the global minimum of the functional

Remark on the limit case of positive mass theorem for manifolds with inner boundary

In (Comm. Math. Phys. 188 (1997) 121–133) Herzlich proved a new positive mass theorem for Riemannian 3-manifolds (N,g) whose mean curvature of the boundary allows some positivity. In this paper we study what happens to the limit case of the theorem when, at a point of the boundary, the smallest positive eigenvalue of the Dirac operator of the boundary is strictly larger than one-half of the mean curvature (in this case the mass m(g) must be strictly positive). We prove that the mass is bounded from below by a positive constant c(g), m(g)c(g), and the equality m(g)=c(g) holds only if, outside a compact set, (N,g) is conformally flat and the scalar curvature vanishes. The constant c(g) is uniquely determined by the metric g via a Dirac-harmonic spinor.