Thin-type dense sets and related properties

We build on Gruenhage, Natkaniec, and Piotrowski?s study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that X² has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (GC) but not witnessed by a collection of finite sets. We show that the question of when the topological sum of two countable strongly irresolvable spaces satisfies (NC) is related to the Rudin-Keisler order on βω. We also introduce and study the concepts of <κ-thin and superslim dense sets.     

Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories

We introduce the concept of maximal (n−1)-orthogonal subcategories over Artin algebras and orders, and develop (n+1)-dimensional Auslander-Reiten theory on them. We give the n-Auslander-Reiten translation and the n-Auslander-Reiten duality, then show the existence of n-almost split sequences and n-fundamental sequences. We give some examples.     

Heat kernel analysis on infinite-dimensional Heisenberg groups

We introduce a class of non-commutative Heisenberg-like infinite-dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, {νt}_t>0, are also studied. We show that these heat kernel measures admit: (1) Gaussian like upper bounds, (2) Cameron-Martin type quasi-invariance results, (3) good Lp-bounds on the corresponding Radon-Nikodym derivatives, (4) integration by parts formulas, and (5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.     

Geometric constrains for global regularity of 2D quasi-geostrophic flows

We study the two-dimensional quasi-geostrophic equations (2D QG) in Sobolev spaces. We first prove a new analytic condition for global regularity which is both sufficient and necessary. We then prove several new results on the geometric constraints on the 2D QG active scalar which suppress the development of singularity from the nonlinear stretching mechanism. We focus mainly on the case with critical dissipation. Our results are also relevant to the inviscid case.     

Norm-attaining composition operators on the Bloch spaces

We prove that every composition operator C? on the Bloch space (modulo constant functions) attains its norm and characterize the norm-attaining composition operators on the little Bloch space (modulo constant functions). We also identify the extremal functions for ‖C?‖ in both cases.     

Group theory and unification of forces: Application to ‘non-universal’ gaugino masses

We will give an overview on how to embed the Standard Model (SM), based on SU(3) × SU(2) × U(1), within larger groups. We will review the different chains one follows when spontaneously breaking SO(10) down to the SM. Finally, We shall discuss the question of non-universal gaugino masses in supersymmetric SO(10) theories.     

Spin (7)-structures in principal fibre bundles over Riemannian manifolds withG 2-structure

We consider principal fibre bundles with one-dimensional fiber over manifolds withG ₂-structure. We define twoSpin(7)-structures on the total bundle space and find relations between the respective structures on the total space and the base. Finally we construct examples ofSpin(7)-structures using the results previously proved.     

Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory

Let (X_n) be a positive recurrent Harris chain on a general state space, with invariant probability measure π. We give necessary and sufficient conditions for the geometric convergence of λPⁿf towards its limit π(f), and show that when such convergence happens it is, in fact, uniform over f and in L¹(π)-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, ∝ π(dx)6Pⁿ(x,·)-π6 converges to zero geometrically fast. We also characterize the geometric ergodicity of (X_n) in terms of hitting time distributions. We show that here the so-called small sets act like individual points of a countable state space chain. We give a test function criterion for geometric ergodicity and apply it to random walks on the positive half line. We apply these results to non-singular renewal processes on [0,∞) providing a probabilistic approach to the exponencial convergence of renewal measures.     

Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) L^∞-a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times.     

A single-copy minimal-time simulation of a torus of automata by a ring of automata

We consider cellular automata on Cayley graphs and we simulate the behavior of a torus of n×m automata (nodes) by a ring of n·m automata (cells). Our simulation technique requires the neighborhood of the nodes to be preserved. We achieve this constraint by copying the contents of nodes on the cells. We consider the problem of minimizing the number of the copies. We prove that it is possible to simulate the behavior of a torus on a ring with a single copy on each cell if and only if n and m satisfy a given condition. In that case we propose a time-optimal algorithm. We thus improve a previous work done by Martin where two copies were requested. When the condition on n and m is not fulfilled one can use the previous algorithm.     

Nonholonomic (n + 1)-webs

We consider a nonholonomic (n + 1)-web NW on an n-dimensional manifold M, i.e., n + 1 codimension 1 distributions on M. We prove that a web NW on M is equivalent to a G-structure with structure group λE, the group of scalar matrices. We find the structure equations of a web NW and the integrability conditions of the distributions of a web NW. It is shown that on a manifold with nonholonomic (n + 1)-web an affine connection Γ arises naturally for which the distributions of the web are totally geodesic. We consider the case when the connection Γ has zero curvature and, in particular, when a web NW is defined by invariant distributions on a Lie group. In the case when all distributions of a web NW on a Lie group are integrable, we find the equations of this group in terms of local coordinates.     

A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices

We prove a general canonical factorization for meromorphic Herglotz functions on the unit disk whose notable elements are that there is no restriction (other than interlacing) on the zeros and poles for their Blaschke product to converge and there is no singular inner function. We use this result to provide a significant simplification in the proof of Killip-Simon (Ann. Math. 158 (2003) 253) of their result characterizing the spectral measures of Jacobi matrices, J, with J−J₀ Hilbert-Schmidt. We prove a nonlocal version of Case and step-by-step sum rules.     

Plateau-Stein manifolds

We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all ?∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.     

On the new examples of complete noncompact Spin(7)-holonomy metrics

We construct some complete Spin(7)-holonomy Riemannian metrics on the noncompact orbifolds that are ?⁴-bundles with an arbitrary 3-Sasakian spherical fiber M. We prove the existence of the smooth metrics for M = S ⁷ and M = SU(3)/U(1) which were found earlier only numerically.     

Hull determination and type decomposition for a generalized effect algebra

In this article, we show that certain generalized boolean subalgebras of the exocenter of a generalized effect algebra (GEA) determine hull systems on the GEA in a manner analogous to the determination of a hull mapping on an effect algebra (EA) by its set of invariant elements. We show that a hull system on a GEA E induces a hull mapping on each interval E[0, p] in E, and, using hull systems, we identify certain special elements of E (e.g., η-subcentral elements, η-monads, and η-dyads). We also extend the type-decomposition theory for EAs to GEAs.     

Approximation in quantale-enriched categories

Our work is a foundational study of the notion of approximation in Q-categories and in (U,Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Q- and (U,Q)-categories. We fully characterize continuous Q-categories (resp. (U,Q)-categories) among all cocomplete Q-categories (resp. (U,Q)-categories) in the same ways as continuous domains are characterized among all dcpos. By varying the choice of the quantale Q and the notion of ideals, and by further allowing the ultrafilter monad to act on the quantale, we obtain a flexible theory of continuity that applies to partial orders and to metric and topological spaces. We demonstrate on examples that our theory unifies some major approaches to quantitative domain theory.     

Inventory Control for Supply Chains with Service Level Constraints: A Synergy between Large Deviations and Perturbation Analysis

We consider a model of a supply chain consisting of n production facilities in tandem and producing a single product class. External demand is met from the finished goods inventory maintained in front of the most downstream facility (stage 1); unsatisfied demand is backlogged. We adopt a base-stock production policy at each stage of the supply chain, according to which the facility at stage i produces if inventory falls below a certain level w _i and idles otherwise. We seek to optimize the hedging vector w=(w ₁,...,w _n ) to minimize expected inventory costs at all stages subject to maintaining the stockout probability at stage 1 below a prescribed level (service level constraint). We make rather general modeling assumptions on demand and production processes that include autocorrelated stochastic processes. We solve this stochastic optimization problem by combining analytical (large deviations) and sample path-based (perturbation analysis) techniques. We demonstrate that there is a natural synergy between these two approaches.     

Spectral square means for period integrals of wave functions on real hyperbolic spaces

We study the period integrals of Laplace eigenfunctions on an arithmetic quotient X of the d-dimensional hyperbolic space along a fixed eigenfunction on an arithmetic quotient of (d−1)-dimensional hyperbolic space embedded in X. We introduce a certain counting function for period integrals and prove its asymptotic law.