On log canonical divisors that are log quasi-numerically positive

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK _X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.     

Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.     

Towards variational analysis in metric spaces: metric regularity and fixed points

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.     

<Emphasis Type="Italic">g</Emphasis>-metrizable spaces and the images of semi-metric spaces

In this paper, we prove that a space X is a g-metrizable space if and only if X is a weak-open, π and σ-image of a semi-metric space, if and only if X is a strong sequence-covering, quotient, π and mssc-image of a semi-metric space, where “semi-metric” can not be replaced by “metric”. This project was supported by NNSF of China (No. 10571151 and 10671173).     

The geodesic flow of a nonpositively curved graph manifold

We consider discrete cocompact isometric actions where X is a locally compact Hadamard space (following [B] we will refer to CAT(0) spaces — complete, simply connected length spaces with nonpositive curvature in the sense of Alexandrov — as Hadamard spaces) and G belongs to a class of groups (“admissible groups”) which includes fundamental groups of 3-dimensional graph manifolds. We identify invariants (“geometric data”) of the action which determine, and are determined by, the equivariant homeomorphism type of the action of G on the ideal boundary of X. Moreover, if are two actions with the same geometric data and is a G-equivariant quasi-isometry, then for every geodesic ray there is a geodesic ray (unique up to equivalence) so that . This work was inspired by (and answers) a question of Gromov in [Gr3, p. 136]. Submitted: May 2001.     

Almost locally minimal projections in finite dimensional Banach spaces

A projectionP on a Banach spaceX is called “almost locally minimal” if, for every α>0 small enough, the ballB(P,α) in the spaceL(X) of all operators onX contains no projectionQ with whereD is a constant. A necessary and sufficient condition forP to be almost locally minimal is proved in the case of finite dimensional spaces. This criterion is used to describe almost locally minimal projections on ℓ ₁ ⁿ . Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas, 1997. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

On a multi-channel change-point problem

A continuous change-point problem is studied in which N independent diffusion processes X _j are observed. Each process X _j is associated with a “channel”, each has an unknown piecewise constant drift and the unit diffusion coefficient. All the channels are connected only by a common change-point of drift. As the result, a change-point problem is defined in which the unknown and unidentifiable drift forms a 2N-dimensional nuisance parameter. The asymptotics of the minimax rate in estimating the change-point is studied as N → ∞. This rate is compared with the case of the known drift. This problem is a special case of an open change-point detection problem in the high-dimensional diffusion with nonparametric drift.      

Existence of a-fold Perfect （v, {5, 8}, 1）-Mendelsohn Designs

Let v be a positive integer and let K be a set of positive integers. A （v, K, 1）-Mendelsohn design, which we denote briefly by （v, K, 1）-MD, is a pair （X, B） where X is a v-set （of points） and B is a collection of cyclically ordered subsets of X （called blocks） with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B. If for all t =1, 2,..., r, every ordered pair of points of X are t-apart in exactly one block of B, then the （v, K, 1）-MD is called an r-fold perfect design and denoted briefly by an r-fold perfect （v, K, 1）-MD. If K = {k） and r = k - 1, then an r-fold perfect （v, （k）, 1）-MD is essentially the more familiar （v, k, 1）-perfect Mendelsohn design, which is briefly denoted by （v, k, 1）-PMD. In this paper, we investigate the existence of 4-fold perfect （v, （5, 8}, 1）-Mendelsohn designs.     

Monadic theory of order and topology, 1

We deal with the monadic theory of linearly ordered sets and topological spaces, disprove two of Shelah’s conjectures and prove some more results. In particular, if the Continuum Hypothesis holds, then there exist monadic formulae expressing the predicates “X is countable” and “X is meager” in the real line and in Cantor’s Discontinuum.     

Simple sheaves along dihedral Lagrangians

We prove the unicity of a complex of sheavesF whose microsupport is carried by a “dihedral” Lagrangian Λ ofT ^* X (X=a real manifold) and which is simple with a prescribed shift at a regular point of Λ. Our method consists in reducing Λ, by a real contact transformation, to the conormal bundle to aC ¹-hypersuface, and then in using [K-S 1, Prop. 6.2.1] in the variant of [D'A-Z 1]. This is similar to [Z 2] but more general, since complex contact transformations and calculations of shifts are not required. We then consider the case of a complex manifoldX, and obtain some vanishing theorems for the complex of “microfunctions along Λ” similar to those of [A-G], [A-H], [K-S 1] (cf. also [D'A-Z 3 5], [Z 2]).     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Some remarks on the study of good contractions

LetX be a complex projective variety with log terminal singularities admitting an extremal contraction in terms of Minimal Model Theory, i.e. a projective morphism φ:X→Z onto a normal varietyZ with connected fibers which is given by a (high multiple of a) divisor of the typeK _x+rL, wherer is a positive rational number andL is an ample Cartier divisor. We first prove that the dimension of anu fiberF of φ is bigger or equal to (r-1) and, if φ is birational, thatdimF≥r, with the equalities if and only ifF is the projective space andL the hyperplane bundle (this is a sort of “relative” version of a theorem of Kobayashi-Ochiai). Then we describe the structure of the morphism φ itself in the case in which all fibers have minimal dimension with the respect tor. If φ is a birational divisorial contraction andX has terminal singularities we prove that φ is actually a “blow-up”.     

Potential Theoretic Approach to Rendezvous Numbers

We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces (X,d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x ₁, …, x _n } ⊂ X, there exists some point x ∈ X with the average of the distances d(x,x _j ) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named “the magic number” of the metric space. Exploring related notions of general potential theory, as set up, e.g., in the fundamental works of Fuglede and Ohtsuka, we present an alternative, potential theoretic approach to rendezvous numbers.     

On the symmetric square Unstable twisted characters

We provide a purely local computation of the (elliptic) twisted (by “transpose-inverse”) character of the representationπ=I(1) of PGL(3) over ap-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads — see [FK] — to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals. The new case in [FK] is the unstable one. A direct local proof of the fundamental lemma is given in [V].     

Potential Theoretic Approach to Rendezvous Numbers

We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces (X,d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x ₁, …, x _n } ⊂ X, there exists some point x ∈ X with the average of the distances d(x,x _j ) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named “the magic number” of the metric space. Exploring related notions of general potential theory, as set up, e.g., in the fundamental works of Fuglede and Ohtsuka, we present an alternative, potential theoretic approach to rendezvous numbers.     

Dichotomies of the set of test measures of a Haar-null set

We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish group and every σ-compact subgroupG ofX there exists aG-invariantF _σ subset ofX which is neither prevalent nor Haar-null. Research supported by a grant of EPEAEK program “Pythagoras”.     

Independent sequences in Banach spaces

In every ∞-dimensional separable Banach spaceX there is a fundamental sequence such that no subsequence of it, which is fundamental inX, is independent (“{x _n} is fundamental inX” meansX=span {x _n}).     

Spectral representation of generalized transition kernels

Let (X, A) be a set with a countably σ-generated “Borel” field of subsets; letW be a “Borel” subset of the product of (X, A) with the real line ℝ and its Borel fieldB; and for eachx∈X let γ _x be a measure on the “slice”W _x={(w, t)∈W:w=x}. It is shown that, under reasonable conditions, the σ-field A⊗B|W can be generated by a real-valued functiong in such a way that, given any measurablef:W→ℝ,g can be chosen to be arbitrarily close tof and so that its “slice-integrals” coincide with those off. This theorem is the first step in a study of monotonic sequences of countably generated σ-fields.     

The symplectic Gram-Schmidt theorem and fundamental geometries for A-modules

Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf A is appropriately chosen) shows that symplectic A-morphisms on free A-modules of finite rank, defined on a topological space X, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if (ℰ, φ) is an A-module (with respect to a ℂ-algebra sheaf A without zero divisors) equipped with an orthosymmetric A-morphism, we show, like in the classical situation, that “componentwise” φ is either symmetric (the (local) geometry is orthogonal) or skew-symmetric (the (local) geometry is symplectic). Theorem 2.1 reduces to the classical case for any free A-module of finite rank.     

Soap films spanning rectangular prisms

In this paper, we consider soap films spanning rectangular prisms with regular n-gon bases. As the number of edges n varies, we show that there are significant changes in the qualitative properties of the spanning soap films as well as a change in the number of spanning soap films whose existence we can prove: We can find two nontrivial soap films for n = 3, 4, 5 but only one for n ≥ 6. We also prove some results concerning the interval of aspect ratios through which the soap films exist: The interval is finite if n = 3, 4, 5 and infinite if n ≥ 6. Furthermore, for n > 6, we have that the spanning soap film converges to a soap film spanning the vertical lines through the vertices of a regular n-gon as the aspect ratio goes to infinity. We can also make sense of the case n = ∞. Here, we discover some interesting singly and triply periodic soap films spanning singly and doubly periodic sets of vertical lines or spanning singly periodic sets of vertical line segments connected by pairs of parallel, horizontal lines. Finally, for n = 3, 4, 5, 6, we can derive parameterizations for the spanning soap films, and these parameterizations are explicit up to knowing the aspect ratio.