A note on the volume flux group of four-manifolds

We show that closed orientable smooth four-manifolds with non-trivial volume flux group and fundamental group of subexponential growth type are finitely covered by a manifold homeomorphic to S³×S¹, S²×T² or a nil-manifold. We also show that if a compact complex surface has non-trivial volume flux group then it has zero minimal volume.     

Decoupling of Modes for the Elastic Wave Equation in Media of Limited Smoothness

We establish a decoupling result for the P and S waves of linear, isotropic elasticity, in the setting of twice-differentiable Lamé parameters. Precisely, we show that the P?S components of the wave propagation operator are regularizing of order one on L ² data, by establishing the diagonalization of the elastic system modulo a L ²-bounded operator. Effecting the diagonalization in the setting of twice-differentiable coefficients depends upon the symbol of the conjugation operator having a particular structure.     

Left mean-ergodicity,fixed points,and invariant means

Recently Lau [15] generalized a result of Yeadon [25]. In the present paper we generalize Yeadon's result in another direction recasting it as a theorem of ergodic type. We call the notion of ergodicity required left mean-ergodicity and show how it relates to the mean-ergodicity of Nagel [21]. Connections with the existence of invariant means on spaces of continuous functions on semitopological semigroups S are made, connections concerning, among other things, a fixed point theorem of Mitchell [20] and Schwartz's property P of W¹-algebras [22]. For example, if $\text{M}$(S) is a certain subspace of C(S) (which was considered by Mitchell and is of almost periodic type, i.e., the right translates of a member of $\text{M}$(S) satisfy a compactness condition), then the assumption that $\text{M}$(S) has a left invariant mean is equivalent to the assumption that every representation of S of a certain kind by operators on a linear topological space X is left mean-ergodic. An analog involving the existence of a (left and right) invariant mean on $\text{M}$(S) is given, and we show our methods restrict in the Banach space setting to give short direct proofs of some results in [4], results involving the existence of an invariant mean on the weakly almost periodic functions on S or on the almost periodic functions on S. An ergodic theorem of Lloyd [16] is generalized, and a number of examples are presented.     

On polyhedra induced by point sets in space

Given a set S of n?3 points in the plane (not all on a line) it is well known that it is always possible to polygonize S, i.e., construct a simple polygon P such that the vertices of P are precisely the given points in S. For example, the shortest circuit through S is a simple polygon. In 1994, Grünbaum showed that an analogous theorem holds in R³. More precisely, if S is a set of n?4 points in R³ (not all of which are coplanar) then it is always possible to polyhedronize S, i.e., construct a simple (sphere-like) polyhedron P such that the vertices of P are precisely the given points in S. Grünbaum's constructive proof may yield Schönhardt polyhedra that cannot be triangulated. In this paper several alternative algorithms are proposed for constructing such polyhedra induced by a set of points, which may always be triangulated, and which enjoy several other useful properties as well. Such properties include polyhedra that are star-shaped, have Hamiltonian skeletons, and admit efficient point-location queries. We show that polyhedronizations with a variety of such useful properties can be computed efficiently in O(nlogn) time. Furthermore, we show that a tetrahedralized, xy-monotonic, polyhedronization of S may be computed in time O(n^1+ε), for any ε>0.     

On harmonic tori in compact rank one symmetric spaces

In this paper we prove that, in contrast with the Sn and CPn cases, there are harmonic 2-tori into the quaternionic projective space HPn which are neither of finite type nor of finite uniton number; we also prove that any harmonic 2-torus in a compact Riemannian symmetric space which can be obtained via the twistor construction is of finite type if and only it is constant; in particular, we conclude that any harmonic 2-torus in CPn or Sn which is simultaneously of finite type and of finite uniton number must be constant.     

Plane Partitions and Their Pedestal Polynomials

For a linear extension P of a partially ordered set S, we consider a generating multivariate polynomial of certain reverse partitions on S, called P-pedestals. We establish a remarkable property of this polynomial: it does not depend on the choice of P. For S a Young diagram, we show that this polynomial generalizes the hook polynomial.     

Homotopy type of gauge groups of quaternionic line bundles over spheres

Let P be a principal S³-bundle over a sphere Sn, with n?4. Let GP be the gauge group of P. The homotopy type of GP when n=4 was studied by A. Kono in [A. Kono, A note on the homotopy type of certain gauge groups, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991) 295-297]. In this paper we extend his result and we study the homotopy type of the gauge group of these bundles for all n?25.     

Cells in any simple polygon formed by a planar point set

Let P be a finite point set in general position in the plane. We consider empty convex subsets of P such that the union of the subsets constitute a simple polygon S whose dual graph is a path, and every point in P is on the boundary of S. Denote the minimum number of the subsets in the simple polygons S's formed by P by f_p(P), and define the maximum value of f_p(P) by F_p(n) over all P with n points. We show that ⌈(4n-17)/15⌉?F_p(n)?⌊n/2⌋.     

On harmonic and 2-stein spaces of Iwasawa type

We study geometric properties of solvable metric Lie groups S of Iwasawa type; in particular harmonicity and the 2-stein condition. One restriction we obtain is that harmonic spaces of Iwasawa type have algebraic rank one, that is, the commutator subgroup of S has codimension one.We show that among Carnot solvmanifolds the only harmonic spaces are the Damek–Ricci spaces. Moreover, this rigidity result remains valid if harmonicity is replaced by the weaker 2-stein condition. As an application, we show that a harmonic Lie group of Iwasawa type with nonsingular 2-step nilpotent commutator subgroup is, up to scaling, a Damek–Ricci space.     

The structure of 0-bisimple strongly E *-unitary inverse monoids

We introduce a class of strongly E ^*-unitary inverse semigroups S _i (G, P) (i = 1,2) determined by a group G and a submonoid P of G and give an embedding theorem for S _i (G, P). Moreover we characterize 0-bisimple strongly E ^*-unitary inverse monoids and 0-bisimple strongly F ^*-inverse monoids by using S _i (G, P).     

S(n)-θ-closed spaces

A P-space is called P-θ-closed if it is θ-closed in every P-space in which it is embedded. S(n)-θ-closed spaces are characterized in terms of special covers and filters, and a categorical property of S(n)-θ-closed spaces is established. Examples are given to distinguish S(n)-θ-closed and S(n)-closed spaces and to show that the property of being S(n)-θ-closed is not productive.     

Tail Asymptotics for the Supremum of a Random Walk when the Mean Is not Finite

We consider the sums S _n =ξ₁+?+ξ _n of independent identically distributed random variables. We do not assume that the ξ's have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability P{M>x} as x→∞, provided that M=sup {S _n ,n≥1} is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that the subexponentiality of distribution F does not imply the subexponentiality of its integrated tail distribution F ^I.     

Compressing thin spheres in the complement of a link

Let L be a link in S³ that is in thin position but not in bridge position and let P be a thin level sphere with compressing disk D. We introduce the idea of alternating level spheres for D and show that all such spheres are thin and their widths are monotone decreasing. This allows us to generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks P can have in terms of the width of P, including identifying thin spheres with unique compressing disks. We also give conditions under which P must be incompressible on some side or be weakly incompressible. In particular we show that the thin level sphere of second lowest width is weakly incompressible. If P is strongly compressible we describe how a pair of compressing disks must lie relative to the link.     

On the height of 2-component links

Let p be a prime and let L be a 2-component link in S³. We define a numerical invariant, called p-height of L, using a tower of successive p-fold branched cyclic coverings of L, and show, in particular, 2-height is algorithmically determined for any 2-component link. Some relationships between p-height and known link type invariants are also established.     

Orbital stability for periodic standing waves of the Klein–Gordon–Zakharov system and the beam equation

The existence and stability of spatially periodic waves ${\left(e^{i{\omega}t}\varphi_\omega, \psi_\omega\right)}$ in the Klein–Gordon–Zakharov (KGZ) system are studied. We show a local existence result for low regularity initial data. Then, we construct a one-parameter family of periodic dnoidal waves for (KGZ) system when the period is bigger than ${\sqrt{2}\pi}$ . We show that these waves are stable whenever an appropriate function satisfies the standard Grillakis–Shatah–Strauss (Grillakis et al. J Funct Anal 74(1):160–197, 1987; Grillakis et al. J Funct Anal 94(2):308–348, 1990) type condition. We compute the intervals for the parameter ω explicitly in terms of L and by taking the limit L → ∞ we recover the previously known stability results for the solitary waves in the whole line case. For the beam equation, we show the existence of spatially periodic standing waves and show that orbital stability holds if an appropriate functional satisfies Grillakis–Shatah–Strauss type condition.     

Classification of some metric automorphisms defined by Standish

Let F be a finite set with a probability distribution {P_i: i?F} and (Ω $\text{F}$, P) denote the product space of countably many copies of (F, P). A permutation (bijection) φ of the integers induces an invertible measure preserving transformation T_φ on (Ω $\text{F}$, P) given by the equation (T_φw)_i = w_φ(j). Such metric automorphisms we call S-automorphisms.We show in this paper that S-automorphisms are ergodic if and only they are Bernoulli shifts and two ergodic S-automorphisms are isomorphic if and only if their associated permutations are conjugate.We also show that S-automorphisms have discrete spectrum if and only if they have zero entropy and every S-automorphism is either a Bernoulli shift, has discrete spectrum, or is a product of a Bernoulli shift and an automorphism with discrete spectrum.S-automorphism with discrete spectrum are those whose associated permutations contain only cycles of finite length. These automorphisms are studied according to the number of such finite cycles. Those whose permutations have infinitely many finite cycles with unbounded lengths are shown to be antiperiodic and those whose permutations have infinitely many finite cycles of bounded length are periodic with almost no fixed points. An example is given of two automorphisms of this latter type which are isomorphic but whose permutations are not conjugate.A complete isomorphism invariant is given for S-automorphisms whose associated permutations consist of finitely many finite cycles. Using this invariant we show that if φ is either a product of k disjoint cycles of prime power p^α, or a single cycle of length pq where p and q are primes, or a product of k disjoint cycles of prime lengths p₁ < p₂ < ··· < p_kand if ψ is a permutation such that T_ψand T_φ are isomorphic then ψ is conjugate to φ.     

The birational type of the moduli space of even spin curves

We determine the Kodaira dimension of the moduli space Sg of even spin curves for all g. Precisely, we show that Sg is of general type for g>8 and has negative Kodaira dimension for g<8.     

Chow groups of surfaces with h2,0⩽1

We will investigate the geometry of rational equivalence classes of points on a surface S. We will show that if S is a general projective K3 surface then these equivalence classes are dense in the complex topology. We will also show that if S has the property that these equivalence classes are Zariski dense, then h^2,0(S)?1. To cite this article: C. Maclean, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.