Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Some remarks on the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for powers of the Ahlfors–Beurling operator

We provide a proof of sharp lower L ^p bounds for powers of the Ahlfors–Beurling operator T and improve the numerical constant in their asymptotic estimates. We also discuss the possibilities of applying de Branges’ theorem in the p − 1 problem.     

Properties of distance functions on convex surfaces and applications

If X is a convex surface in a Euclidean space, then the squared intrinsic distance function dist²(x, y) is DC (d.c., delta-convex) on X×X in the only natural extrinsic sense. An analogous result holds for the squared distance function dist²(x,F) from a closed set F ⊂ X. Applications concerning r-boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.     

The Kramer–Mesner method with tactical decompositions: some new unitals on 65 points

We propose a combination of two known computational methods for the construction of designs with prescribed groups of automorphisms: the Kramer–Mesner method and the method of tactical decompositions. This combined method is used to construct new unitals with parameters 2‐(65, 5, 1). © 2011 Wiley Periodicals, Inc. J Combin Designs 19:290‐303, 2011     

Primitive symmetric designs with up to 2500 points

By this article we conclude the construction of all primitive ( v, k,λ ) symmetric designs with v < 2500 , up to a few unsolved cases. Complementary to the designs with prime power number of points published previously, here we give 55 primitive symmetric designs with v ≠ p ^m , p prime and m positive integer, together with the analysis of their full automorphism groups. The research involves programming and wide‐range computations. We make use of the software package GAP and the library of primitive groups which it contains. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:463‐474, 2011     

A transformation of certain infinite series based on the non-local derangement identity

The aim of the paper is a new transformation of certain infinite series based on an identity from Sylvester also known as the non-local derangement identity. Surprisingly, the result found can be applied to evaluate certain types of the generalized hypergeometric function _p F _q (z) or infinite series involving the combination of hyperbolic functions, irrational functions and polynomials like

On mixed joint discrete universality for a class of zeta-functions. II

We investigate the mixed joint discrete value distribution and the mixed joint discrete universality for the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with transcendental parameter. The common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are used.

     

Turbulent Prandtl Number in Two Spatial Dimensions: Two-Loop Renormalization Group Analysis

Theoretical and Mathematical Physics - Using the field theory renormalization group technique in the framework of the so-called double-expansion scheme, which takes additional divergences that...     

On quasi-contraction mappings of Ćirić and Fisher type via ω-distance

Kada, Suzuki, and Takahashi introduced and studies the concept of ω- distance in fixed point theory. In this paper, we generalize and unify ?iri?’ and Fisher fixed points results for quasi-contractions on metric space to ω-distance on complete metric spaces. We also extend some results of Kada, Suzuki and Takahashi, and Suzuki. Our methods of proofs are new and even simpler than the corresponding methods in metric spaces.     

On Joint Universality of the Riemann Zeta-Function

Mathematical Notes - A theorem is obtained on the approximation of a collection of analytic functions in short intervals by a collection of shifts of the Riemann zeta-function...