We establish universality limits for measures on a smooth closed contour Γ in the plane. Assume that μ is a regular measure on Γ, in the sense of Stahl, Totik, and Ullmann. Let Γ1 be a closed subarc of Γ, such that μ is absolutely continuous in an open arc containing Γ1, and μ′ is positive and continuous in that open subarc. Then universality for μ holds in Γ1, in the sense that the reproducing...

The stationary problem for the heat convection of compressible fluid is considered around the equilibrium solution with the external forces in the horizontal strip domain z0 < z < z0 + 1 and it is proved that the solution exists uniformly with respect to z0 ≥ Z0 . The limit system as z0 → +∞ is the Oberbeck-Boussinesq equations. Mathematics Subject Classification (2010). Primary 35Q35; Secondar...

Can we assure math computations by automatic verifying floating-point accuracy? We define fast arithmetic (based on Dekker [1]) over twofold approximations z ≈ z0 + z1, such that z0 is standard result and z1 assesses inaccuracy ∆z0 = z − z0. We propose on-fly tracking z1, detecting if ∆z0 appears too high. We believe permanent tracking is worth its cost. C++ test code for Intel AVX available vi...

For z0 ∈ C and r > 0, let D(z0, r) := {z ∈ C : |z − z0| < r} . In this paper we show that a polynomial p of the form

for all z0 ∈ D(A). Here T (t) is the C0-semigroup generated by A. A system (1) that satisfies the above conditions will be denoted by Σ(A,C). The admissibility of C, eqn. (2), implies that we can extend the mapping z0 → CT (·)z0 to a bounded linear mapping from Z to L2(0,∞). We denote this mapping by C. Thus we have that for any initial condition z0 the solution of (1) is given by z(t) = T (t)z...

Let P = PtP2 + 2 be a linear partial differential operator on R^ with Pt and P2, of orders mx and ra2, respectively, strictly hyperbolic with respect to the first variable and Q of order ml + m2 2. Although the characteristic variety of P may have self-intersections, the hyperbolicity of Px and P2 implies local solvability for Pu = ƒ; indeed the Cauchy problem for P is locally solvable. In this...

We establish universality limits for measures on a subarc of the unit circle. Assume that μ is a regular measure on such an arc, in the sense of Stahl, Totik, and Ullmann, and is absolutely continuous in an open arc containing some point z0 = e0 . Assume, moreover, that μ′ is positive and continuous at z0. Then universality for μ holds at z0, in the sense that the reproducing kernel Kn (z, t) f...

We prove that for ??(0,8), if (?1,?2) is a 2-SLE?pair in simply connected domain D with an analytic boundary point z0, then as r?0+, P[dist(z0,?j)0, which called the two-curve Green’s function. The exponent ? equals 12 ??1 or 2(12 ??1) depending on whether z0 one of endpoints ?1 ?2. also find convergence rate and exact formula function up multiplica...