Majorization in de Branges spaces I. Representability of subspaces

In this series of papers we study subspaces of de Branges spaces of entire functions which are generated by majorization on subsets D of the closed upper half-plane. The present, first, part is addressed to the question which subspaces of a given de Branges space can be represented by means of majorization. Results depend on the set D where majorization is permitted. Significantly different situations are encountered when D is close to the real axis or accumulates to i∞.     

Convergence to equilibrium of critical branching particle systems and superprocesses,and related nonlinear partial differential equations

We consider a class of multitype particle systems in ^d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions

We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain. We show the existence of a weak solution for arbitrarily large data for the pressure law p(ϱ, ϑ) ∼ ϱ ^γ + ϱϑ if γ > 1 and p(ϱ, ϑ) ∼ ϱ ln ^α (1 + ϱ) + ϱϑ if γ = 1, α > 0, depending on the model for the heat flux.     

Characterization of Pinched Ricci Curvature by Functional Inequalities

In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, \(L^p\)-inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can be used in particular to characterize general geometric flow and Ricci flow by functional inequalities.     

On Gradient Solitons of the Ricci–Harmonic Flow

In this paper, we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the soliton is compact, we get a classification result. We also discuss the relation with quasi-Einstein manifolds.     

Existence of non-trivial harmonic functions on Cartan–Hadamard manifolds of unbounded curvature

The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.      

Global estimates of fundamental solutions for higher-order Schr?dinger equations

In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schr?dinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related L ^p ?CL ^q estimates on the Schr?dinger solution. These estimates extend known results from the literature and are sharp. This result was lately already generalized to a degenerate case (cf. [4]).     

Computational study of singlet and triplet sulfonylnitrenes insertion into 1,3‐butadienes: 1,2‐ or 1,4‐cycloaddition?

The formation of N‐trifluoromethylsulfonyl‐2‐vinylaziridine and N‐trifluoromethylsulfonyl‐3‐pyrroline by the reaction of the singlet and triplet trifluoromethanesulfonylnitrenes with s‐cis‐ and s‐trans‐1,3‐butadienes was studied theoretically at the B3LYP/6‐311++G(d,p) and M06‐2X/6‐311++G(d,p) levels of theory. The singlet trifluoromethanesulfonylnitrene adds to s‐cis‐ and s‐trans‐1,3‐butadiene exothermally in one step to give the product of 1,2‐cycloaddition, N‐trifluoromethylsulfonyl‐2‐vinylaziridine, the energy decreasing by 88.5 and 86.2 kcal/mol at the B3LYP level and by 105.2 and 103.0 kcal/mol at the M06‐2X level, respectively. The formed 2‐vinylaziridine can undergo rotation about the C(2)–C_sp2 bond with the barrier not exceeding 3.5 kcal/mol and to rearrange into N‐trifluoromethylsulfonyl‐3‐pyrroline. The triplet trifluoromethanesulfonylnitrene reacts with s‐cis‐ and s‐trans‐1,3‐butadiene in two steps. The first exothermic step is the formation of the triplet diradical adducts. The second step is the spin inversion with the energy raising by 5.8 and 17.8 kcal/mol at the B3LYP level and by 11.0 and 20.8 kcal/mol at the M06‐2X level for the adducts to s‐cis‐ and s‐trans‐1,3‐butadiene, respectively. Recombination of the radical centers occurs selectively to give N‐trifluoromethylsulfonyl‐2‐vinylaziridine that is exothermally rearranged into N‐trifluoromethylsulfonyl‐3‐pyrroline with the energy barrier of 40 kcal/mol at the B3LYP level and of 50 kcal/mol at the M06‐2X level. Copyright © 2014 John Wiley & Sons, Ltd.     

Monte Carlo simulation of brain sensing by optical diffuse spectroscopy

We report on development of adapted Monte Carlo based algorithm and code for simulation of light propagation in turbid media with complex geometry aimed for simulation of optical diffuse spectroscopy signal in noninvasive brain sensing. Simulation will allow to determine optimal characteristics of a prototype device for optical diffuse brain sensing. The developed Monte Carlo code can be efficiently parallelized both for SMP and distributed memory systems. We show that the speed-up of the developed algorithm almost linearly depends on the number of nodes/threads in a utilized system.