Commuting dual Toeplitz operators with pluriharmonic symbols

In this paper we completely characterize commuting dual Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball. We show that for φ and ψ pluriharmonic, SφSψ=Sψφ on only in the trivial case. Here the trivial case is φ or holomorphic.     

The class semigroup of local one-dimensional domains

Let R be a local one-dimensional domain. We investigate when the class semigroup S(R) of R is a Clifford semigroup. We make use of the Archimedean valuation domains which dominate R, as a main tool to study its class semigroup. We prove that if S(R) is Clifford, then every element of the integral closure of R is quadratic. As a consequence, such an R may be dominated by at most two distinct Archimedean valuation domains, and coincides with their intersection. When S(R) is Clifford, we find conditions for S(R) to be a Boolean semigroup. We derive that R is almost perfect with Boolean class semigroup if, and only if R is stable. We also find results on S(R), through examination of [V/P:R/M] and v(M), where V dominates R, and P, M are the respective maximal ideals.     

The existence and upper bound for two types of restricted connectivity

In this paper, we study two types of restricted connectivity: κ_k(G) is the cardinality of a minimum vertex cut S such that every component of G−S has at least k vertices; is the cardinality of a minimum vertex cut S such that there are at least two components in G−S of order at least k. In this paper, we give some sufficient conditions for the existence and upper bound of κ_k(G) and/or , and study some properties of these two parameters.     

Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions

Let S be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices. Let J be an open subset of R. If there exists a self-adjoint extension S₀ of S such that J is contained in the resolvent set of S₀ and the associated Weyl function of the pair {S,S₀} is monotone with respect to J, then for any self-adjoint operator R there exists a self-adjoint extension such that the spectral parts and R_J are unitarily equivalent. It is shown that for any extension of S the absolutely continuous spectrum of S₀ is contained in that one of . Moreover, for a wide class of extensions the absolutely continuous parts of and S are even unitarily equivalent.     

Sequences of integers with missing quotients

Let S be any set of natural numbers, and A be a given set of rational numbers. We say that S is an A-quotient-free set if x,y∈S implies y/x∉A. Let and , where the supremum is taken over all A-quotient-free sets S, and are the upper and lower asymptotic densities of S respectively. Let ρ(A)=sup_Sδ(S), where the supremum is taken over all A-quotient-free sets S such that δ(S) exists. In this paper we study the properties of , and ρ(A).     

Generalized Haar integral

The aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) such that for each f∈A(S) and for any f∈C(Ω,R) and s∈S, , where .     

Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups

Let S be a closed, connected, orientable surface of genus at least 3, be the complex of curves on S and be the extended mapping class group of S. We prove that a simplicial map, , preserves nondisjointness (i.e. if α and β are two vertices in and i(α,β)≠0, then i(λ(α),λ(β))≠0) iff it is induced by a homeomorphism of S. As a corollary, we prove that if K is a finite index subgroup of and is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of .     

On a convex operator for finite sets

Let S be a finite set with m elements in a real linear space and let J_S be a set of m intervals in R. We introduce a convex operator co(S,J_S) which generalizes the familiar concepts of the convex hull, , and the affine hull, , of S. We prove that each homothet of that is contained in can be obtained using this operator. A variety of convex subsets of with interesting combinatorial properties can also be obtained. For example, this operator can assign a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For two types of families J_S we give two different upper bounds for the number of vertices of the polytopes produced as co(S,J_S). Our motivation comes from a recent improvement of the well-known Gauss-Lucas theorem. It turns out that a particular convex set co(S,J_S) plays a central role in this improvement.     

On the normal completion of a Boolean algebra

A familiar construction for a Boolean algebra A is its normal completion, given by its normal ideals or, equivalently, the intersections of its principal ideals, together with the embedding taking each element of A to its principal ideal. In the classical setting of Zermelo-Fraenkel set theory with Choice, is characterized in various ways; thus, it is the unique complete extension of A in which the image of A is join-dense, the unique essential completion of A, and the injective hull of A.Here, we are interested in characterizing the normal completion in the constructive context of an arbitrary topos. We show among other things that it is, even at this level, the unique join-dense, or alternatively, essential completion. En route, we investigate the functorial properties of and establish that it is the reflection of A, in the category of Boolean homomorphisms which preserve all existing joins, to the complete Boolean algebras. In this context, we make crucial use of the notion of a skeletal frame homomorphism.     

The Lichtenbaum-Hartshorne theorem for modules which are finite over a ring homomorphism

Let φ:(R,m)→S be a flat ring homomorphism such that mS≠S. Assume that M is a finitely generated S-module with dim_R(M)=d. If the set of support of M has a special property, then it is shown that if and only if for each prime ideal satisfying , we have . This gives a generalization of the Lichtenbaum-Hartshorne vanishing theorem for modules which are finite over a ring homomorphism. Furthermore, we provide two extensions of Grothendieck’s non-vanishing theorem. Applications to connectedness properties of the support are given.     

The doubles of a numerical semigroup

Let S be a numerical semigroup and let p be a positive integer. Then the quotient is also a numerical semigroup. When p=2 we say that is half of the numerical semigroup S. Dually, we say that S is a double of the numerical semigroup . We characterize the set of all doubles of a numerical semigroup. We also give some alternative proofs and improvements for some results that we find in previous papers.     

On the index of minimal zero-sum sequences over finite cyclic groups

Let G be a cyclic group of order n?2 and a sequence over G. We say that S is a zero-sum sequence if and that S is a minimal zero-sum sequence if S is a zero-sum sequence and S contains no proper zero-sum sequence.The notion of the index of a minimal zero-sum sequence (see Definition 1.1) in G has been recently addressed in the mathematical literature. Let l(G) be the smallest integer t∈N such that every minimal zero-sum sequence S over G with length |S|?t satisfies index(S)=1. In this paper, we first prove that for n?8. Secondly, we obtain a new result about the multiplicity and the order of elements in long zero-sumfree sequences.     

Prevalence and structure of adding machines for cellular automata

Consider the collection of left permutive cellular automata Φ with no memory, defined on the space S of all doubly infinite sequences from a finite alphabet. There exists , a dense subset of S, such that is topologically conjugate to an odometer for all so long as Φm is not the identity map for any m. Moreover, Φ generates the same odometer for all . The set is a dense Gδ subset with full measure of a particular subspace of S.     

Hecke curves and Hitchin discriminant

Let C be a smooth projective curve of genus g?4 over the complex numbers and be the moduli space of stable vector bundles of rank r with a fixed determinant of degree d. In the projectivized cotangent space at a general point E of , there exists a distinguished hypersurface SE consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety CE consisting of vectors tangent to Hecke curves in through E. Our main result establishes that the hypersurface SE and the variety CE are dual to each other. As an application of this duality relation, we prove that any surjective morphism , where C^′ is another curve of genus g, is biregular. This confirms, for , the general expectation that a Fano variety of Picard number 1, excepting the projective space, has no non-trivial self-morphism and that morphisms between Fano varieties of Picard number 1 are rare. The duality relation also gives simple proofs of the non-abelian Torelli theorem and the result of Kouvidakis-Pantev on the automorphisms of .     

Asymptotic bounds for permutations containing many different patterns

We say that a permutation σ∈Sn contains a permutation π∈Sk as a pattern if some subsequence of σ has the same order relations among its entries as π. We improve on results of Wilf, Coleman, and Eriksson et al. that bound the asymptotic behavior of pat(n), the maximum number of distinct patterns of any length contained in a single permutation of length n. We prove that by estimating the amount of redundancy due to patterns that are contained multiple times in a given permutation. We also consider the question of k-superpatterns, which are permutations that contain all patterns of a given length k. We give a simple construction that shows that Lk, the length of the shortest k-superpattern, is at most . This may lend evidence to a conjecture of Eriksson et al. that .     

The Dirac-Ramond operator on loops in flat space

In this paper, a rigorous construction of the S¹-equivariant Dirac operator (i.e., Dirac-Ramond operator) on the space of (mean zero) loops in is given and its equivariant L²-index computed. Essential use is made of infinite tensor product representations of the canonical anticommutation relations algebra.     

On the realization of double occurrence words

Let S be a double occurrence word, and let M_S be the word’s interlacement matrix, regarded as a matrix over . Gauss addressed the question of which double occurrence words are realizable by generic closed curves in the plane. We reformulate answers given by Rosenstiehl and by de Fraysseix and Ossona de Mendez to give new graph-theoretic and algebraic characterizations of realizable words. Our algebraic characterization is especially pleasing: S is realizable if and only if there exists a diagonal matrix D_S such that M_S+D_S is idempotent over .     

Fixed points of dual quantum operations

Let ?A be a normal completely positive map on B(H) with Kraus operators . Denote M the subset of normal completely positive maps by . In this note, the relations between the fixed points of ?A and are investigated. We obtain that , where K(H) is the set of all compact operators on H and is the dual of ?A∈M. In addition, we show that the map is a bijection on M.     

Extremal and Barabanov semi-norms of a semigroup generated by a bounded family of matrices

Let S={Si}_i∈I be an arbitrary family of complex n-by-n matrices, where 1?n<∞. Let denote the joint spectral radius of S, defined as