Quasilinearization and curvature of Aleksandrov spaces

We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space is an Aleksandrov domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.      

On the number of moduli of plane sextics with six cusps

Let be the variety of irreducible sextics with six cusps as singularities. Let be one of irreducible components of . Denoting by the space of moduli of smooth curves of genus 4, we consider the rational map sending the general point [Γ] of Σ, corresponding to a plane curve , to the point of parametrizing the normalization curve of Γ. The number of moduli of Σ is, by definition the dimension of Π(Σ). We know that , where ρ(2, 4, 6) is the Brill–Noether number of linear series of dimension 2 and degree 6 on a curve of genus 4. We prove that both irreducible components of have number of moduli equal to seven.      

Semicrossed products of simple C*-algebras

Let and be C*-dynamical systems and assume that is a separable simple C*-algebra and that α and β are *-automorphisms. Then the semicrossed products and are isometrically isomorphic if and only if the dynamical systems and are outer conjugate. K. R. Davidson was partially supported by an NSERC grant. E. G. Katsoulis was partially supported by a summer grant from ECU     

On varieties parametrizing graded complex Lie algebras

We study spaces parametrizing graded complex Lie algebras from geometric as well as algebraic point of view. If R is a finite-dimensional complex Lie algebra, which is graded by a finite abelian group of order n, then a graded contraction of R, denoted by , is defined by a complex n × n-matrix , i, j = 1, . . . , n. In order for to be a Lie algebra, should satisfy certain homogeneous equations. In turn, these equations determine a projective variety X _R . We compute the first homology group of an irreducible component M of X _R , under some assumptions on M. We look into algebraic properties of graded Lie algebras where .      

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.      

An asymptotic version of Dumnicki’s algorithm for linear systems in $${\mathbb{CP}^2}$$

Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on . With this method we prove the lower bound for 10 general points on .      

Reduced Minimum Modulus Preserving in Banach Space

Let be the algebra of all bounded linear operators on a complex Banach space X and γ(T) be the reduced minimum modulus of operator . In this work, we prove that if , is a surjective linear map such that is an invertible operator, then , for every , if and only if, either there exist two bijective isometries and such that for every , or there exist two bijective isometries and such that for every . This generalizes for a Banach space the Mbekhta’s theorem [12].      

Dy namic of Abelia n Subgroups of GL( n, $$\mathbb{C}$$): A Structure Theorem

In this paper, we characterize the dynamic of every Abelian subgroups of , or . We show that there exists a -invariant, dense open set U in saturated by minimal orbits with a union of at most n -invariant vector subspaces of of dimension n−1 or n−2 over . As a consequence, has height at most n and in particular it admits a minimal set in . This work is supported by the research unit: systèmes dynamiques et combinatoire: 99UR15-15     

Chains and antichains in partial orderings

We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is or , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two sets. Our main result is that there is a computably axiomatizable theory K of partial orderings such that K has a computable model with arbitrarily long finite chains but no computable model with an infinite chain. We also prove the corresponding result for antichains. Finally, we prove that if a computable partial ordering has the feature that for every , there is an infinite chain or antichain that is relative to , then we have uniform dichotomy: either for all copies of , there is an infinite chain that is relative to , or for all copies of , there is an infinite antichain that is relative to .     

The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

Moduli spaces of quadratic complexes and their singular surfaces

We construct the coarse moduli space of quadratic line complexes with a fixed Segre symbol σ as well as the moduli space of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism . Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves.      

The Poisson boundary of lamplighter random walks on trees

Let be the homogeneous tree with degree q + 1 ≥ 3 and a finitely generated group whose Cayley graph is . The associated lamplighter group is the wreath product , where is a finite group. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first moment, then the probability space formed by this geometric boundary together with the limit distribution of the random walk is proved to be maximal, that is, the Poisson boundary. We also prove that the Dirichlet problem at infinity is solvable for continuous functions on the active part of the boundary, if the lamplighter “operates at bounded range”. Supported by ESF program RDSES and by Austrian Science Fund (FWF) P15577.     

Diameter preserving bijections between Grassmann spaces over Bezout domains

Let R, S be Bezout domains. Assume that n is an integer ≥ 3, 1 ≤ k ≤ n − 2. Denoted by the k-dimensional Grassmann space on . Let be a map. This paper proves the following are equivalent: (i) is an adjacency preserving bijection in both directions. (ii) is a diameter preserving bijection in both directions. Moreover, Chow’s theorem on Grassmann spaces over division rings is extended to the case of Bezout domains: If is an adjacency preserving bijection in both directions, then is induced by either a collineation or the duality of a collineation. Project 10671026 supported by National Natural Science Foundation of China.     

Parabolic twists for the linear algebras <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>−1</Subscript>

New solutions of twist equations for the universal enveloping algebras U (A_n−1) are found. These solutions can be represented as products of full chains of extended Jordanian twists Abelian factors (“rotations”) , and sets of quasi-Jordanian twists . The latter are generalizations of Jordanian twists (with carrier b²) for special deformed extensions of the Hopf algebra U (b²). The carrier subalgebra for the composition is a nonminimal parabolic subalgebra in A _n−1 such that . The parabolic twisting elements are obtained in an explicit form. Details of the construction are illustrated by considering the examples n = 4 and n = 11. Bibliography: 21 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 187–213.     

Numerically satisfactory solutions of Kummer recurrence relations

Pairs of numerically satisfactory solutions as for the three-term recurrence relations satisfied by the families of functions , , are given. It is proved that minimal solutions always exist, except when and z is in the positive or negative real axis, and that is minimal as whenever . The minimal solution is identified for any recurrence direction, that is, for any integer values of and . When the confluent limit , with fixed, is the main tool for identifying minimal solutions together with a connection formula; for , is the main tool to be considered.     

The $${\overline{\partial}}$$-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L ^p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.      

A perfect stratification of $${\mathcal M_ g}$$ for g ≤ 5

We find for g ≤ 5 a stratification of depth g − 2 of the moduli space of curves with the property that its strata are affine and the classes of their closures provide a -basis for the Chow ring of . The first property confirms a conjecture of one of us. The way we establish the second property yields new (and simpler) proofs of theorems of Faber and Izadi which, taken together, amount to the statement that in this range the Chow ring is generated by the λ-class.      

A group-theoretical finiteness theorem

We start with the universal covering space of a closed n-manifold and with a tree of fundamental domains which zips it . Our result is that, between T and , is an intermediary object, , obtained by zipping, such that each fiber of p is finite and admits a section.      

On the Circumference of 2-Connected $$\mathcal{P}_{3}$$-Dominated Graphs

Let G be a connected graph. For at distance 2, we define , and , if then . G is quasi-claw-free if it satisfies , and G is P ₃-dominated() if it satisfies , for every pair (x, y) of vertices at distance 2. Certainly contains as a subclass. In this paper, we prove that the circumference of a 2-connected P ₃-dominated graph G on n vertices is at least min or , moreover if then G is hamiltonian or , where is a class of 2-connected nonhamiltonian graphs.