Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

Fredholm differential operators with unbounded coefficients

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both and and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u′(t)=A(t)u(t) with, generally, unbounded operators , the operator G is a closure of the operator . Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg.     

Function spaces and one point extensions for the construct of metered spaces

The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M.     

A remark on conservative cocompletions of categories

For a set F of small categories, F-conservative cocompletions of a category are cocompletions preserving all existing colimits of type F. We prove that every category has a free F-conservative cocompletion. However, unless F is trivial, this cocompletion fails in general to be locally small.     

Projective sup-algebras: a general view

We begin with the notion of K-flat projectivity. For each sup-algebra L we then introduce a binary relation L_? on it. The K-flat projective sup-algebras are exactly such sup-algebras with each element a approximated by the element x, xL_?a and the relation L_? being stable with respect to the operations on L. Further on, we introduce the notion of a K-comonad and characterize K-flat projective sup-algebras as such sup-algebras having a coalgebra structure for the K-comonad.     

On a 3D isothermal model for nematic liquid crystals accounting for stretching terms

In the present contribution, we study a PDE system describing the evolution of a nematic liquid crystals flow under kinematic transports for molecules of different shapes. More in particular, the evolution of the velocity field u is ruled by the Navier–Stokes incompressible system with a stress tensor exhibiting a special coupling between the transport and the induced terms. The dynamics of the director field d is described by a variation of a parabolic Ginzburg–Landau equation with a suitable penalization of the physical constraint |d| = 1. Such equation accounts for both the kinematic transport by the flow field and the internal relaxation due to the elastic energy. The main aim of this contribution is to overcome the lack of a maximum principle for the director equation and prove (without any restriction on the data and on the physical constants of the problem) the existence of global in time weak solutions under physically meaningful boundary conditions on d and u.     

Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term χtt, χ being the order parameter, which is linearly coupled with an evolution equation for the (relative) temperature ?. The latter can be of hyperbolic type if the Cattaneo-Maxwell heat conduction law is assumed. The state variables and the chemical potential are subject to the homogeneous Neumann boundary conditions. We first provide conditions which ensure the well-posedness of the initial and boundary value problem. Then, we prove that the corresponding dynamical system is dissipative and possesses a global attractor. Moreover, assuming that the nonlinear potential is real analytic, we establish that each trajectory converges to a single steady state by using a suitable version of the ?ojasiewicz-Simon inequality. We also obtain an estimate of the decay rate to equilibrium.     

Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

Exponentiation for unitary structures

Motivated by the observation that both pretopologies and preapproach limits can be characterized as those convergence relations which have a unit for a suitable composition, we introduce the category Algu(T;V) of reflexive and unitary lax algebras, for a symmetric monoidal closed lattice V and a Set-monad T=(T,e,m). For T=U the ultrafilter monad, we characterize exponentiable morphisms in Algu(U;V). Further, we give a sufficient condition for an object to be exponentiable in the category Alg(U;V) of reflexive and transitive lax algebras. This specializes to known and new results for pretopological, preapproach and approach spaces.     

Energy dissipation for weak solutions of incompressible MHD equations

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that stems from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.     

A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness

We scrutinize both from the physical and the analytical viewpoint the equations ruling the paramagnetic-ferromagnetic phase transition in a rigid three-dimensional body. Starting from the order structure balance, we propose a non-isothermal phase-field model which is thermodynamically consistent and accounts for variations in space and time of all fields (the temperature θ, the magnetic field vector H and the magnetization vector M). In particular, we are able to establish a well-posedness result for the resulting coupled system.     

Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem

We investigate singular and degenerate behavior of solutions of the unstable free boundary problem

Δu=−χ{u>0}.     

Exponentiation in V-categories

For a Heyting algebra V which, as a category, is monoidal closed, we obtain characterizations of exponentiable objects and morphisms in the category of V-categories and apply them to some well-known examples. In the case these characterizations of exponentiable morphisms and objects in the categories (P)Met of (pre)metric spaces and non-expansive maps show in particular that exponentiable metric spaces are exactly the almost convex metric spaces, while exponentiable complete metric spaces are the complete totally convex ones.     

Gradient based and least squares based iterative algorithms for matrix equations AXB + CXD = F

This paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation AXB + CX^TD = F. The basic idea is to decompose the matrix equation (system) under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving Ax = b and AXB = F. The analysis shows that when the matrix equation has a unique solution (under the sense of least squares), the iterative solution converges to the exact solution for any initial values. A numerical example verifies the proposed theorems.     

Orbifolds and groupoids

We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases on (Pre-Orb) and Morita equivalence in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.     

Some explicit solutions of the additive Deligne-Simpson problem and their applications

In this paper we construct three infinite series and two extra triples (E₈ and ) of complex matrices B, C, and A=B+C of special spectral types associated to Simpson's classification in Amer. Math. Soc. Proc. 1 (1992) 157 and Magyar et al. classification in Adv. Math. 141 (1999) 97. This enables us to construct Fuchsian systems of differential equations which generalize the hypergeometric equation of Gauss-Riemann. In a sense, they are the closest relatives of the famous equation, because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions. In all the cases except for E₈, we also explicitly construct scalar products such that A, B, and C are self-adjoint with respect to them. In the context of Fuchsian systems, these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions.When the eigenvalues of A, B, and C are real, the matrices and the scalar products become real as well. We find inequalities on the eigenvalues of A, B, and C which make the scalar products positive-definite.As proved by Klyachko, spectra of three hermitian (or real symmetric) matrices B, C, and A=B+C form a polyhedral convex cone in the space of triple spectra. He also gave a recursive algorithm to generate inequalities describing the cone. The inequalities we obtain describe non-recursively some faces of the Klyachko cone.     

On the exact rate of convergence of frequencies of digits in self-similar sets

We study the exact rate of convergence of frequencies of digits of “normal” points of a self-similar set. Our results have applications to metric number theory. One particular application gives the following surprising result: there is an uncountable family of pairwise disjoint and exceptionally big subsets of ?^d that do not obey the law of the iterated logarithm. More precisely, we prove that there is an uncountable family of pairwise disjoint and exceptionally big sets of points x in ?^d—namely, sets with full Hausdorff dimension—for which the rate of convergence of frequencies of digits in the N-adic expansion of x is either significantly faster or significantly slower than the typical rate of convergence predicted by the law of the iterated logarithm.     

On the universal coefficients formula for shape homology

In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.     

Unique solvability of the stationary Fokker-Planck equation in a class of positive functions

We study the stationary Focker-Planck equation Δu ? div(u f) = 0 with a given vector field f of the class C ₀ ^∞ (R ⁿ ) on the basis of a fixed point principle that generalizes the contraction mapping method. Next, we introduce a parameter in the equation and prove the unique solvability of the equation Δu ? div(uγ f) = 0 with the parameter in the class of positive slowly increasing functions. We reveal the analytic dependence of the positive solution u on the parameter γ. Pointwise estimates for positive solutions are proved.