A generalization of Kneser's Addition Theorem

Let A=(A₁,…,Am) be a sequence of finite subsets from an additive abelian group G. Let Σ?(A) denote the set of all group elements representable as a sum of ? elements from distinct terms of A, and set . Our main theorem is the following lower bound:

     

Modules of derivations for toral arrangements

Let G be a connected semi-simple algebraic group over . Fix a maximal torus T in G with coordinate ring _T. Let Φ⁺ be the set of positive roots of G with respect to T. The pair (T, A), where A = {kerα}_α?φ⁺, is a toral arrangement. We show that if G is simply connected then the module of A-derivations D(A) is a free _T-module.     

Dynamics of composite functions meromorphic outside a small set

Let M denote the class of functions f meromorphic outside some compact totally disconnected set E=E(f) and the cluster set of f at any a∈E with respect to is equal to . It is known that class M is closed under composition. Let f and g be two functions in class M, we study relationship between dynamics of f○g and g○f. Denote by F(f) and J(f) the Fatou and Julia sets of f. Let U be a component of F(f○g) and V be a component of F(g○f) which contains g(U). We show that under certain conditions U is a wandering domain if and only if V is a wandering domain; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U unless U is a Siegel disk or Herman ring.     

Uniform stabilization of the wave equation on compact manifolds and locally distributed damping - a sharp result

Let (M,g) be an n-dimensional (n?2) compact Riemannian manifold with or without boundary where g denotes a Riemannian metric of class C^∞. This paper is concerned with the study of the wave equation on (M,g) with locally distributed damping, described by

     

Transportation cost inequalities on path and loop groups

Let G be a connected Lie group with the Lie algebra . The action of Cameron-Martin space on the path space P_e(G) introduced by L. Gross (Illinois J. Math. 36 (1992) 447) is free. Using this fact, we define the H-distance on P_e(G), which enables us to establish a transportation cost inequality on P_e(G). This method will be generalized to the path space over the loop group , so that we obtain a transportation cost inequality for heat measures on .     

On cohomology rings of infinite groups

Let R be any ring (with 1), G a torsion free group and RG the corresponding group ring. Let be the cohomology ring associated with the RG-module M. Let H be a subgroup of finite index of G. The following is a special version of our main Theorem: Assume the profinite completion of G is torsion free. Then an element is nilpotent (under Yoneda’s product) if and only if its restriction to is nilpotent. In particular this holds for the Thompson group.There are torsion free groups for which the analogous statement is false.     

A full row-rank system matrix generated by the strip-based projection model in discrete tomography

Let Cu = k be an underdetermined linear system generated by the strip-based projection model in discrete tomography, where C is row-rank deficient. In the case of one scanning direction the linear dependency of the rows of C is studied in this paper. An index set H is specified such that if all rows of C with row indices in H are deleted then the rows of resultant matrix F are maximum linearly independent rows of C. Therefore, the corresponding system is equivalent to Cu = k and consequently, the cost of an image reconstruction from is reduced.     

Pure-injective hulls of modules over valuation rings

If is the pure-injective hull of a valuation ring R, it is proved that is the pure-injective hull of M, for every finitely generated R-module M. Moreover , where (A_k)_1≤k≤n is the annihilator sequence of M. The pure-injective hulls of uniserial or polyserial modules are also investigated. Any two pure-composition series of a countably generated polyserial module are isomorphic.     

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.     

On primitive roots for rank one Drinfeld modules

Let k be a global function field over a finite field and let A be the ring of the elements in k regular outside a fixed place ∞. Let K be a global A-field of finite A-characteristic and let ? be a rank one Drinfeld A-module over K. Given any α∈K, we show that the set of places P of K for which α is a primitive root modulo P under the action of ? possesses a Dirichlet density. We also give conditions for this density to be positive.     

Partitions and sums with inverses in Abelian groups

Let G be an Abelian group and let be infinite. We construct a partition of A such that whenever (xn)_n<ω is a one-to-one sequence in A, g∈G and m<ω, one has

(g+FSI((xn)_n<ω))∩Am≠∅,     

Locally strongly convex hypersurfaces with constant affine mean curvature

Let be a locally strongly convex hypersurface, given by a strictly convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂An. We consider the Riemannian metric G# on M, defined by . In this paper we prove that if M is a locally strongly convex surface with constant affine mean curvature and if M is complete with respect to the metric G#, then M must be an elliptic paraboloid.     

On approximation of functions from Sobolev spaces on metric graphs

Some results on the approximation of functions from the Sobolev spaces on metric graphs by step functions are obtained. In particular, we show that the approximation numbers a_n of the embedding operator of the Sobolev space on a graph G of finite length |G| into the space , where μ is an arbitrary finite Borel measure on G, satisfy the inequality

     

Connections on modules over singularities of finite CM representation type

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection on M?We consider the maximal Cohen-Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections.Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a -Gorenstein singularity of dimension d≥2. We show that this condition is sufficient for the canonical module ω_A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω_A admits an integrable connection if and only if A is Gorenstein.     

On Fourier algebra homomorphisms

Let G be a locally compact group and let B(G) be the dual space of C∗(G), the group C∗ algebra of G. The Fourier algebra A(G) is the closed ideal of B(G) generated by elements with compact support. The Fourier algebras have a natural operator space structure as preduals of von Neumann algebras. Given a completely bounded algebra homomorphism we show that it can be described, in terms of a piecewise affine map with Y in the coset ring of H, as follows

     

Polynomials satisfied by two linked matrices

Polynomials in two variables, evaluated at A and with A being a square complex matrix and being its transform belonging to the set {A⁼, A^†, A^∗}, in which A⁼, A^†, and A^∗ denote, respectively, any reflexive generalized inverse, the Moore-Penrose inverse, and the conjugate transpose of A, are considered. An essential role, in characterizing when such polynomials are satisfied by two matrices linked as above, is played by the condition that the column space of A is the column space of . The results given unify a number of prior, isolated results.     

Note on integer powers in sumsets

Let k,m,n?2 be integers. Let A be a subset of {0,1,…,n} with 0∈A and the greatest common divisor of all elements of A is 1. Suppose that

     

Diameters of uniform subset graphs

Let r,s be positive integers with r>s, k a nonnegative integer, and n=2r−s+k. A uniform subset graph G(n,r,s) is a graph with vertex set [n]^r and where two r-subsets A,B∈[n]^r are adjacent if and only if |A∩B|=s. Let denote the diameter of a graph G.In this paper, we prove the following results: (1) If k>0, then if r≥2s+k+2, 2 if k≥s and 2s≤r≤s+k, or k<s and s+k≤r≤2s, and 3 otherwise; (2) If k=0, then . This generalizes a result in [M. Valencia-Pabon, J.-C. Vera, On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].