Cube structures and intersection bundles

We define the notion of a hypercube structure on a functor between two commutative Picard categories which generalizes the notion of a cube structure on a G_m-torsor over an abelian scheme. We prove that the determinant functor of a relative scheme X/S of relative dimension n is canonically endowed with a (n+2)-cube structure. We use this result to define the intersection bundle I_X/S(L₁,…,L_n+1) of n+1 line bundles on X/S and to construct an additive structure on the functor I_X/S:PIC(X/S)ⁿ⁺¹→PIC(S). Then, we construct the resultant of n+1 sections of n+1 line bundles on X, and the discriminant of a section of a line bundle on X. Finally we study the relationship between the cube structures on the determinant functor and on the discriminant functor, and we use it to prove a polarization formula for the discriminant functor.     

Subcompactness and domain representability in GO-spaces on sets of real numbers

In this paper we explore a family of strong completeness properties in GO-spaces defined on sets of real numbers with the usual linear ordering. We show that if τ is any GO-topology on the real line R, then (R,τ) is subcompact, and so is any Gδ-subspace of (R,τ). We also show that if (X,τ) is a subcompact GO-space constructed on a subset X⊆R, then X is a Gδ-subset of any space (R,σ) where σ is any GO-topology on R with τ=σX_|. It follows that, for GO-spaces constructed on sets of real numbers, subcompactness is hereditary to Gδ-subsets. In addition, it follows that if (X,τ) is a subcompact GO-space constructed on any set of real numbers and if τS is the topology obtained from τ by isolating all points of a set S⊆X, then (X,τS) is also subcompact. Whether these two assertions hold for arbitrary subcompact spaces is not known.We use our results on subcompactness to begin the study of other strong completeness properties in GO-spaces constructed on subsets of R. For example, examples show that there are subcompact GO-spaces constructed on subsets X⊆R where X is not a Gδ-subset of the usual real line. However, if (X,τ) is a dense-in-itself GO-space constructed on some X⊆R and if (X,τ) is subcompact (or more generally domain-representable), then (X,τ) contains a dense subspace Y that is a Gδ-subspace of the usual real line. It follows that (Y,τY_|) is a dense subcompact subspace of (X,τ). Furthermore, for a dense-in-itself GO-space constructed on a set of real numbers, the existence of such a dense subspace Y of X is equivalent to pseudo-completeness of (X,τ) (in the sense of Oxtoby). These results eliminate many pathological sets of real numbers as potential counterexamples to the still-open question: “Is there a domain-representable GO-space constructed on a subset of R that is not subcompact”? Finally, we use our subcompactness results to show that any co-compact GO-space constructed on a subset of R must be subcompact.     

Invariant Palm and related disintegrations via skew factorization

Let G be a measurable group with Haar measure ??, acting properly on a space S and measurably on a space T. Then any ??-finite, jointly invariant measure M on S?× T admits a disintegration ${\nu \otimes \mu}$ into an invariant measure ?? on S and an invariant kernel ?? from S to T. Here we construct ?? and??? by a general skew factorization, which extends an approach by Rother and Z?hle for homogeneous spaces S over G. This leads to easy extensions of some classical propositions for invariant disintegration, previously known in the homogeneous case. The results are applied to the Palm measures of jointly stationary pairs (??, ??), where ?? is a random measure on S and ?? is a random element in T.     

The relaxed game chromatic index of k-degenerate graphs

The (r,d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r,d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with Δ(G)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ+k-1 and d?2k²+4k.     

Induced Hamiltonian maps on the symplectic quotient

Let (M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map fH. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH.     

Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D. Let C_φ,M_ψ and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces.     

Secant varieties and Hirschowitz bound on vector bundles over a curve

For a vector bundle V of rank n over a curve X and for each integer r in the range 1 ≤ r ≤ n ? 1, the Segre invariant s _r is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we generalize Lange and Narasimhan’s results on rank 2 bundles which related the invariant s ₁ to the secant varieties of the curve inside certain extension spaces. For any n and r, we find a way to get information on the invariant s _r from the secant varieties of certain subvariety of a scroll over X. Using this geometric picture, we obtain a new proof of the Hirschowitz bound on s _r .     

Local and 2-local derivations on noncommutative Arens algebras

The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ^ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ^ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ^ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.     

Some aspects of groups acting on finite posets

Let P be a finite poset and G a group of automorphisms of P. The action of G on P can be used to define various linear representations of G, and we investigate how these representations are related to one another and to the structure of P. Several examples are analyzed in detail, viz., the symmetric group $\text{G}$_n acting on a boolean algebra, GL_n(q) acting on subspaces of an n-dimensional vector space over GF(q), the hyperoctahedral group B_n acting on the lattice of faces of a cross-polytope, and $\text{G}$_n acting on the lattice Π_n of partitions of an n-set. Several results of a general nature are also proved. These include a duality theorem related to Alexander duality, a special property of geometric lattices, the behavior of barycentric subdivision, and a method for showing that certain sequences are unimodal. In particular, we give what seems to be the simplest proof to date that the q-binomial coefficient $\text{k+l}\text{k}$ has unimodal coefficients.     

Actions of large semigroups and random walks on isometric extensions of boundaries

Let G be a real algebraic semi-simple group, X an isometric extension of the flag space of G by a compact group C. We assume that G is topologically transitive on X. We consider a closed sub-semigroup T of G and a probability measure μ on T such that T is Zariski-dense in G and the support of μ generates T. We show that there is a finite number of T-invariant minimal subsets in X and these minimal subsets are the supports of the extremal μ-stationary measures on X. We describe the structure of these measures, we show the conditional equidistribution on C of the μ-random walk and we calculate the algebraic hull of the corresponding cocycle. A certain subgroup generated by the “spectrum” of T can be calculated and plays an essential role in the proofs.     

Some inequalities for the Hadamard product and the Fan product of matrices

If A and B are nonsingular M-matrices, a sharp lower bound on the smallest eigenvalue τ(A★B) for the Fan product of A and B is given, and a sharp lower bound on τ(A°B^-1) for the Hadamard product of A and B^-1 is derived. In addition, we also give a sharp upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B.     

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

On the existence of an arrow and a Bergson-Samuelson social welfare function

Let Σ be the set of all possible preferences over a given set of alternatives A. Let Ω be a proper subset of Σ and let P?Ωⁿ be a fixed profile of preferences. P is heterogeneous in Ω if for all a,b,c?A and Q?Ωⁿ, there exist three alternatives x,y,z?A such that Q(a,b,c)=P(x,y,z) where Q(B) denotes the subprofile over a set of alternatives B?A. An Arrow SWF ? is a function ?:Ωⁿ→Σ satisfying the conditions Pareto and IIA. A Bergson-Samuelson SWF is a function ?:P→Σ satisfying Pareto and Independence+Neutrality. The paper shows that (a) there exist a neutral nondictatorial Arrow SWF on Ω if and only if there exist a neutral nondictatorial Bergson-Samuelson SWF on P. (b) There exist a nondictatorial n person Bergson-Samuelson SWF on P if and only if there exists a 3 person Bergson-Samuelson SWF on P. (c) There exists a nondictatorial Arrow SWF on Ω if and only if there exists a nondictatorial Bergson-Samuelson SWF on P.     

A principal function problem and the field of meromorphic functions

Let R be an open Riemann surface and A an open subset of R. For a given meromorphic function s on A, we show, under an additional condition on A, that there exists a meromorphic function q on R such that both q/s and s/q are bounded functions on A.     

On the syntomic regulator for K 1 of a surface

We consider elements of K ₁(S), where S is a proper surface over a p-adic field with good reduction, which are given by a formal sum ??(Z _i , f _i ) with Z _i curves in S and f _i rational functions on the Z _i in such a way that the sum of the divisors of the f _i is 0 on S. Assuming compatibility of pushforwards in syntomic and motivic cohomologies, our result computes the syntomic regulator of such an element, interpreted as a functional on H _dR ² (S), when evaluated on the cup product ????[??] of a holomorphic form ?? by the first cohomology class of a form of the second kind ??. The result is ?? _i ??F _?? , log(f _i ); F _?? ??_gl,Z _i , where F _?? and F _?? are Coleman integrals of ?? and ??, respectively, and the symbol in brackets is the global triple index, as defined in our previous work.     

Solving polynomial optimization problems via the truncated tangency variety and sums of squares

Let f,g_i,i=1,…,l,h_j,j=1,…,m, be polynomials on Rⁿ and S?{x∈Rⁿ∣g_i(x)=0,i=1,…,l,h_j(x)≥0,j=1,…,m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.     

The interval function of a connected graph and road systems

Let V be a finite nonempty set. In this paper, a road system on V (as a generalization of the set of all geodesics in a connected graph G with V(G)=V) and an intervaloid function on V (as a generalization of the interval function (in the sense of Mulder) of a connected graph G with V(G)=V) are introduced. A natural bijection of the set of all intervaloid functions on V onto the set of all road systems on V is constructed. This bijection enables to deduce an axiomatic characterization of the interval function of a connected graph G from a characterization of the set of all geodesics in G.     

Large orbits of subgroups of solvable linear groups

Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be a nilpotent subgroup of G; then H has at least three regular orbits on V ⊕ V. Let H be a subgroup of G and 3 ? |H|; then H has at least three regular orbits on V ⊕ V. Let H be a subgroup of G and assume the Sylow 2-subgroups of the semidirect product HV are abelian; then H has at least two regular orbits on V ⊕ V.     

Left Reductive Congruences on Semigroups

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ? a congruence on S. Consider the sequence ? ⁽⁰⁾?? ⁽¹⁾???? ⁽ⁿ⁾?? of congruences on S, where ? ⁽⁰⁾=? and, for an arbitrary non-negative integer n, ? ⁽ⁿ⁺¹⁾ is defined by (a,b)∈? ⁽ⁿ⁺¹⁾ if and only if (xa,xb)∈? ⁽ⁿ⁾ for all x∈S. We show that $\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )$ for an arbitrary congruence ? on a semigroup S, where lrc(?) denotes the least left reductive congruence on S containing ?. We focuse our attention on congruences ? on semigroups S for which the congruence $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is left reductive. We prove that, for a congruence ? on a semigroup S, $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is a left reductive congruence of S if and only if $\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}$ is a left reductive congruence on the factor semigroup S/? (here ι _(S/?) denotes the identity relation on S/?). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S ⁿ =S ⁿ⁺¹ is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.     

Polynomial indexing of integer lattice-points I. General concepts and quadratic polynomials

Denoting the nonnegative integers by N and the signed integers by Z, we let S be a subset of Z^m for m = 1, 2,… and f be a mapping from S into N. We call f a storing function on S if it is injective into N, and a packing function on S if it is bijective onto N. Motivation for these concepts includes extendible storage schemes for multidimensional arrays, pairing functions from recursive function theory, and, historically earliest, diagonal enumeration of Cartesian products. Indeed, Cantor's 1878 denumerability proof for the product N² exhibits the equivalent packing functions $\text{f}{}_{\mathrm{<mtext>Cantor</mtext>}}\text{(x, y) = {}\text{either}\text{x}\text{or}\text{y} +}\text{(x + y)(x + y + 1)}\text{2}$ on the domain N², and a 1923 Fueter-Pólya result, in our terminology, shows f_Cantor the only quadratic packing function on N². This paper extends the preceding result. For any real-valued function f on S we define a density $\text{S ÷ f =}\text{lim}{}_{\mathrm{n\to \infty }}\text{(}\text{1}\text{n}\text{)\#{S ? f}{}^{\mathrm{?1}}\text{([?n, +n])}}$, and for any packing function f on S we observe the fact S ÷ f = 1. Using properties of this density, and invoking Davenport's lemma from geometric number theory, we find all polynomial storing functions with unit density on N, and exclude any polynomials with these properties on Z, then find all quadratic storing functions with unit density on N², and exclude any quadratics with these properties on Z × N, Z². The admissible quadratics on N² are all nonnegative translates of f_Cantor. An immediate sequel to this paper excludes some higher-degree polynomials on subsets of Z².