On uniqueness of an entire function and its derivatives

In this paper, we study the growth of solutions of a first order linear differential equation. From this we verify that a conjecture given by Brück is true under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its first derivative sharing a nonzero constant CM. The results in this paper also improve some known results. Some examples show that the results in this paper are best possible. Project supported by the NSFC (NO. A0324617) and the RFDP (NO. 20060422049). Received: 20 April 2006 Revised: 4 February 2007     

Calculus of functors and model categories

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [B. Chorny, W.G. Dwyer, Homotopy theory of small diagrams over large categories, preprint, 2005]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for a classification of polynomial and homogeneous functors. In the n-homogeneous model structure, the nth derivative is a Quillen functor to the category of spectra with Σn-action. After taking into account only finitary functors—which may be done in two different ways—the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T.G. Goodwillie [T.G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003) 645-711 (electronic)].     

Local algebras in the representation of MV-algebras

In this paper we present several results about local MV-algebras, extending existing results given for MV-chains. The role of local MV-algebras in sheaf representation and weak boolean product is stressed and the relationship of local MV-algebras with varieties of MV-algebras is analyzed. Presented by S. Pulmannova. Received November 11, 2005; accepted in final form December 20, 2005.     

Invariant measures and regularity properties of perturbed Ornstein-Uhlenbeck semigroups

This paper deals with perturbations of the Ornstein-Uhlenbeck operator on L²-spaces with respect to a Gaussian measure μ. We perturb the generator of the Ornstein-Uhlenbeck semigroup by a certain unbounded, non-linear drift, and show various properties of the perturbed semigroup such as compactness and positivity. Strong Feller property, existence and uniqueness of an invariant measure are discussed as well.     

A large deviation principle for 2D stochastic Navier–Stokes equation

In this paper one specifies the ergodic behavior of the 2D-stochastic Navier–Stokes equation by giving a Large Deviation Principle for the occupation measure for large time. It describes the exact rate of exponential convergence. The considered random force is non-degenerate and compatible with the strong Feller property.     

On Genuine Bernstein–Durrmeyer Operators

We continue the studies on the so–called genuine Bernstein–Durrmeyer operators U _n by establishing a recurrence formula for the moments and by investigating the semigroup T(t) approximated by U _n . Moreover, for sufficiently smooth functions the degree of this convergence is estimated. We also determine the eigenstructure of U _n , compute the moments of T(t) and establish asymptotic formulas. Received: January 26, 2007.     

A non-isothermal Ginzburg–Landau model in superconductivity: Existence,uniqueness and asymptotic behaviour

In this paper we study a Ginzburg–Landau model which describes the behaviour of a superconducting material including thermal effects. We extend the traditional formulation of the problem, by introducing the temperature as an additional state variable. Accordingly, together with the Gor’kov–Eliashberg system, we introduce an evolution equation for the absolute temperature. We examine in detail the case which allows only variations of the concentration of superconducting electrons and of the temperature, neglecting the electromagnetic field. For this problem existence and uniqueness of the solution are shown. Finally we analyze the asymptotic behaviour of the solutions, proving that the system possesses a global attractor.     

Fire containment in grids of dimension three and higher

We consider a deterministic discrete-time model of fire spread introduced by Hartnell [Firefighter! an application of domination, Presentation, in: 20th Conference on Numerical Mathematics and Computing, University of Manitoba in Winnipeg, Canada, September 1995] and the problem of minimizing the number of burnt vertices when a fixed number of vertices can be defended by firefighters per time step. While only two firefighters per time step are needed in the two-dimensional lattice to contain any outbreak, we prove a conjecture of Wang and Moeller [Fire control on graphs, J. Combin. Math. Combin. Comput. 41 (2002) 19-34] that 2d-1 firefighters per time step are needed to contain a fire outbreak starting at a single vertex in the d-dimensional square lattice for d?3; we also prove that in the d-dimensional lattice, d?3, for each positive integer f there is some outbreak of fire such that f firefighters per time step are insufficient to contain the outbreak. We prove another conjecture of Wang and Moeller that the proportion of elements in the three-dimensional grid P_n×P_n×P_n which can be saved with one firefighter per time step when an outbreak starts at one vertex goes to 0 as n gets large. Finally, we use integer programming to prove results about the minimum number of time steps needed and minimum number of burnt vertices when containing a fire outbreak in the two-dimensional square lattice with two firefighters per time step.     

The essential Lagrangian-Grassmannian and the homotopy type of the Fredholm Lagrangian-Grassmannian

Let H be a separable infinite dimensional Hilbert space endowed with a symplectic structure and let L₀⊂H be a Lagrangian subspace. Using the results of [A. Abbondandolo, P. Majer, Infinite dimensional Grassmannians, math.AT/0307192], we show that the Fredholm Lagrangian-Grassmannian FL₀(Λ) has the homotopy type of Gc(L₀), the Grassmannian of all Lagrangian subspaces of H that are compact perturbations of L₀. It is well known that the latter has the homotopy type of the quotient U(∞)/O(∞). As a corollary, we recover a result by B. Booss-Bavnbek and K. Furutani (see [B. Booss-Bavnbek, K. Furutani, Symplectic functional analysis and spectral invariants, Contemp. Math. 242 (1999) 53-83; K. Furutani, Fredholm-Lagrangian-Grassmannian and the Maslov index, J. Geom. Phys. 51 (2004) 269-331]) that the L₀-Maslov index is an isomorphism between the fundamental group of FL₀(Λ) and the integers.     

Second order evolution equations with time-dependent subdifferentials

We study an abstract second order nonlinear evolution equation in a real Hilbert space. We consider time-dependent convex functions and their subdifferentials operating on the first derivative of the unknown function. Introducing appropriate assumptions on the convex functions and other data, we prove the existence and uniqueness of a strong solution, and give some applications of the abstract theorem to hyperbolic variational inequalities with time-dependent constraints.