Topology of Definable Hausdorff Limits

Let A ^n+r be a set definable in an o-minimal expansion S of the real field, let A ^r be its projection, and assume that the non-empty fibers A_a ⁿ are compact for all a A and uniformly bounded, i.e. all fibers are contained in a ball of fixed radius B(0,R). If L is the Hausdorff limit of a sequence of fibers A_ai, we give an upper-bound for the Betti numbers b_k(L) in terms of definable sets explicitly constructed from a fiber A_a. In particular, this allows us to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the relative closure to construct the o-minimal structure S_Pfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure (X,Y)₀ in the special case where Y=.     

The Direct Limits Of The Banach-mazur Compacta

Let 1 p . For each n-dimensional Banach space E = (E, || ·||), we define a norm || · ||_p on E x R as follows: [formula] It is shown that the correspondence (E, || · ||) (Ex R, || · ||_p) defines a topological embedding of oneBanach–Mazur compactum, BM(n), into another, BM(n 1),and hence we obtain a tower of Banach–Mazur compacta:BM(1) BM(2) BM(3) ···. Let BM_p be thedirect limit of this tower. We prove that BM_p is homeomorphicto Q = dir lim Qⁿ, where Q = [0, 1] is the Hilbert cube. 1991Mathematics Subject Classification 46B04, 46B20, 52A21, 57N20,54H15.     

Limits of Degenerate Parabolic Quadratic Rational Maps

We investigate the closure in moduli space of the set of quadratic rational maps which possess a degenerate parabolic fixed point.     

On the Topology Induced by Mixtures of Exponentials

It is shown that if X is a real vector space of uncountable dimension then the coarsest topology on X for which the function is continuous whenever is a measure on the dual space X ^* integrating the integrands is strictly coarser than the finest locally convex topology. This is derived from an inequality relating the averages of such a 'mixture of exponentials' on the vertices and facet midpoints, respectively, of a 'generalized octahedron' in a finite-dimensional space (Lemma 1).     

Direct Limits and Inverse Limits of SHR Semigroups

In this paper, it is proved that the direct limit (inverse limit) of an SHR direct (respectively, inverse) family of SHR semigroups is an SHR semigroup.AMS Subject Classifications Primary 20N20 Secondary 20M99     

The Limits of Porous Materials in the Topology Optimization of Stokes Flows

We consider a problem concerning the distribution of a solid material in a given bounded control volume with the goal to minimize the potential power of the Stokes flow with given velocities at the boundary through the material-free part of the domain.We also study the relaxed problem of the optimal distribution of the porous material with a spatially varying Darcy permeability tensor, where the governing equations are known as the Darcy–Stokes, or Brinkman, equations. We show that the introduction of the requirement of zero power dissipation due to the flow through the porous material into the relaxed problem results in it becoming a well-posed mathematical problem, which admits optimal solutions that have extreme permeability properties (i.e., assume only zero or infinite permeability); thus, they are also optimal in the original (non-relaxed) problem. Two numerical techniques are presented for the solution of the constrained problem. One is based on a sequence of optimal Brinkman flows with increasing viscosities, from the mathematical point of view nothing but the exterior penalty approach applied to the problem. Another technique is more special, and is based on the “sizing” approximation of the problem using a mix of two different porous materials with high and low permeabilities, respectively. This paper thus complements the study of Borrvall and Petersson (Internat. J. Numer. Methods Fluids, vol. 41, no. 1, pp. 77–107, 2003), where only sizing optimization problems are treated.     

On Inverse Limits and Direct Limits for Fuzzy Topological Spaces

Li has introduced the concepts of inverse system and direct system for fuzzy topological spaces and studied inverse limits and direct limits on such spaces by presenting the explicit constructions of these limits.In this paper some important concepts of fuzzy topology,such as,product fuzzy topology,quotient fuzzy topology,fuzzy continuity etc.,are used for further study of inverse limits and direct limits for fuzzy topological spaces.     

Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob [11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see [10, Theorem 3.1] or [11,1.XII.23] – and the second author [15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihland Seidel [6], Schneider [21], Berman [7], and Armitage andNelson [4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans [12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ[–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l[–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l[–, +]: for each neighbourhood N of l in [–,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.     

Derivations of Direct Limits of Lie Superalgebras

We describe the derivations of a direct limit 𝔏 of Lie superalgebras 𝔏ⁱ (i ∈ I) in an 𝔏-module 𝔲 as the inverse limit of the derivations of 𝔏ⁱ's in 𝔲. Using this, in case the first cohomology group of each 𝔏ⁱ with coefficients in 𝔲 is zero, we describe the derivations of 𝔏 in 𝔲 as the inverse limit of 𝔲/𝔲𝔏ⁱ (i ∈ I). This then allows us to compute the derivations of direct limits of finite-dimensional basic classical simple Lie superalgebras.     

Minimal Cantor Systems and Unimodal Maps

We consider the restriction of unimodal maps f to the omega-limit set y ( c ) of the critical point for certain cases where y ( c ) is a Minimal Cantor set. We investigate the relation of these minimal systems to enumeration scales (generalized adding machines), to Vershik adic transformations on ordered Bratelli diagrams and to substitution shifts. Sufficient conditions are given for ( y ( c ), f ) to be uniquely ergodic.     

Factorization Systems Induced by Weak Distributive Laws

We relate weak distributive laws in the bicategory of spans to strictly associative (but not strictly unital) pseudoalgebras of the 2-monad ( − ) ² on Cat. The corresponding orthogonal factorization systems are characterized by a certain bilinearity property.     

On Direct and Inverse Limits of Retractive Spectra

We prove the universal equivalence of the direct and inverse limits of retractive spectra of universal algebras and give a few consequences of this assertion.     

Continuous Limits of Classical Repeated Interaction Systems

We consider the physical model of a classical mechanical system (called “small system”) undergoing repeated interactions with a chain of identical small pieces (called “environment”). This physical setup constitutes an advantageous way of implementing dissipation for classical systems; it is at the same time Hamiltonian and Markovian. This kind of model has already been studied in the context of quantum mechanical systems, where it was shown to give rise to quantum Langevin equations in the limit of continuous time interactions (Attal and Pautrat in Ann Henri Poincaré 7:59–104, 2006), but it has never been considered for classical mechanical systems yet. The aim of this article is to compute the continuous limit of repeated interactions for classical systems and to prove that they give rise to particular stochastic differential equations (SDEs) in the limit. In particular, we recover the usual Langevin equations associated with the action of heat baths. In order to obtain these results, we consider the discrete-time dynamical system induced by Hamilton’s equations and the repeated interactions. We embed it into a continuous-time dynamical system and compute the limit when the time step goes to 0. This way, we obtain a discrete-time approximation of SDE, considered as a deterministic dynamical system on the Wiener space, which is not exactly of the usual Euler scheme type. We prove the L ^p and almost sure convergence of this scheme. We end up with applications to concrete physical examples such as a charged particle in a uniform electric field or a harmonic interaction. We obtain the usual Langevin equation for the action of a heat bath when considering a damped harmonic oscillator as the small system.     

Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry

We find lower bounds on the topology of the fibers F^-1(y) ì X{F^{-1}(y)\subset X} of continuous maps F : X → Y in terms of combinatorial invariants of certain polyhedra and/or of the cohomology algebras H*(X). Our exposition is conceptually related to but essentially independent of Part 1 of the paper.     

Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere

We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic laminations of the disk. Received: December 2006, Revision: August 2007, Accepted: October 2007     

Operator Algebraic Quotient Structures Induced by Direct Graphs II

The main purpose of this paper is to investigate the operator algebraic quotient structures induced by directed graphs. We enlarge our study of Cho (Compl Anal Oper Theory, 2008) to the general case. This can be done by constructing new graphs from given graphs called the pull-back graphs. We consider the corresponding groupoids, and von Neumann algebras of pull-back graphs.     

Sums and Limits of Generalized Direct Families of Algebras

In this paper we introduce a generalization of direct families of algebras and we study their limits and sums. In the case of generalized direct families of algebras carried by idempotent algebras we investigate some subdirect decompositions of their sums. The results that we obtain generalize various results given by J.L. Chrislock and T. Tamura [2], M. iri and S. Bogdanovi [3-7], H. Mitsch [13], M. Petrich [14-16], B.M. Schein [23-24] and others.Supported by Grant 04M03B of RFNS through Math. Inst. SANU