Directed inverse limits of spatial locales

In this note we consider spatiality of directed inverse limits of spatial locales. We give an example which shows that directed inverse limits of compact spatial locales are not necessarily spatial. This answers a question posed by John Isbell. We also give a condition which, if satisfied by the maps of a directed inverse system, implies that taking limits preserves local compactness and hence produces spatial locales.

     

Exactness of direct limits for abelian categories with an injective cogenerator

We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg–Moore category of a monad on the category of sets preserve regular epimorphisms.     

The algebra of harmonic functions for a matrix-valued transfer operator

We analyze matrix-valued transfer operators. We prove that the fixed points of transfer operators form a finite-dimensional C^∗-algebra. For matrix weights satisfying a low-pass condition we identify the minimal projections in this algebra as correlations of scaling functions, i.e., limits of cascade algorithms.     

On Discrete Limits of Sequences of Approximately Continuous Functions and tae-Continuous Functions

We show that the classes of all discrete limits of sequences of ap- proximately continuous functions, of all discrete limits of sequences of derivatives and of all discrete limits of sequences of Baire 1 functions are the same. We describe also the discrete limits of sequences of quasicontinuous functions, and of sequences of almost everywhere continuous functions, and we present anec- essary condition which must be satisfied by the discrete limits of sequences of T_ae -continuous functions.     

Retakh’s条件(M)与(LM)-空间的序列式回缩性

本文研究了（ＬＭ）－空间（即：可尺度局部凸空间序列的诱导极限）中,Ｒｅ－ｔａｋｈ’ｓ条件（Ｍ）与序列式回缩性之间的关系,给出了满足Ｒｅｔａｋｈ’ｓ条件（Ｍ）的（ＬＭ）－空间为序列式回缩的一系列特征条件．     

关于β-正则与一致β-正则诱导极限

本文举出反例说明文献［７］中关于一致β－正则的充要条件是错误的，并分别给出了诱导极限为β－正则与一致β－正则的正确的充要条件．     

Stochastic McKean-Vlasov equations

We prove the existence and uniqueness of solution to the nonlinear local martingale problems for a large class of infinite systems of interacting diffusions. These systems, which we call the stochastic McKean-Vlasov limits for the approximating finite systems, are described as stochastic evolutions in a space of probability measures onR ^d and are obtained as weak limits of the sequence of empirical measures for the finite systems, which are highly correlated and driven by dependent Brownian motions. Existence is shown to hold under a weak growth condition, while uniqueness is proved using only a weak monotonicity condition on the coefficients. The proof of the latter involves a coupling argument carried out in the context of associated stochastic evolution equations in Hilbert spaces. As a side result, these evolution equations are shown to be positivity preserving. In the case where a dual process exists, uniqueness is proved under continuity of the coefficients alone. Finally, we prove that strong continuity of paths holds with respect to various Sobolev norms, provided the appropriate stronger growth condition is verified. Strong solutions are obtained when a coercivity condition is added on to the growth condition guaranteeing existence.The research has been partially supported by NSERC, Canada.     

Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes

We present and analyze several ways of discretizing first-order Hamilton-Jacobi equations on unstructured meshes. We first discuss two Godunov-type Hamiltonians: the first one is an extension of a result by Bardi and Osher, where a particular decomposition of the initial condition is assumed, and we point out its practical limits; the other one arises from a particular decomposition of the Hamiltonian. Despite its complexity this decomposition enables us to construct a Lax-Friedrichs Hamiltonian. These schemes all share common properties: They are consistent, monotonic, and independent of the geometrical interpretation of the piecewise linear initial condition. Under these assumptions and classical ones on the mesh, we show these schemes are convergent. We describe their high-order extensions using the ENO technique and provide numerical illustrations. © 1996 John Wiley & Sons, Inc.     

Nonsmooth Characterizations of Asplund Spaces and Smooth Variational Principles

We show that the Asplund property of Banach spaces is not only sufficient but also a necessary condition for the fulfillment of some basic results in nonsmooth analysis involving Fréchet-like normals and subdifferentials as well as their sequential limits. In this way we obtain new characterizations of Asplund spaces within the framework of nonsmooth analysis. Then we study several versions of smooth variational principles in Asplund spaces, provide necessary and sufficient conditions for the validity of such principles, and establish their relationships with certain subdifferential properties of lower semicontinuous functions.     

Distributional Limits of Riemannian Manifolds and Graphs with Sublinear Genus Growth

In Benjamini and Schramm [BS01] introduced the notion of distributional limit of a sequence of graphs with uniformly bounded valence and studied such limits in the case that the involved graphs are planar. We investigate distributional limits of sequences of Riemannian manifolds with bounded curvature which satisfy a quasi-conformal condition. We then apply our results to somewhat improve Benjamini’s and Schramm’s original result on the recurrence of the simple random walk on limits of planar graphs. For instance, as an application we give a proof of the fact that for graphs in an expander family, the genus of each graph is bounded from below by a linear function of the number of vertices.     

Nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws

We consider the Riemann problem for a system of two decoupled, nonstrictly hyperbolic, Burgers-like conservation equations with added artificial viscosity. We analytically establish two different vanishing viscosity limits for the solution of this system, which correspond to the two cases where one of the viscosities vanishes much faster than the other. This is done without altering the initial condition as is necessary with travelling wave methods. Numerical evidence is then provided to show that when the two viscosities vanish at the same rate, the solution converges to a limit that lies strictly between the two previously established limits. Finally, we use control theory to explain the mechanism behind this nonuniqueness behavior, which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur.     

Compact perturbations of reflexive algebras

We study compact perturbations of reflexive algebras with commutative invariant subspace lattices. This leads us to study lattice isomorphisms that are uniform limits of spatially induced mappings and satisfy a compactness condition. We prove that two lattices that are nested or complemented are isomorphic in this way iff their corresponding perturbed algebras are unitary equivalent. While for complemented lattices this relation implies unitary equivalence, we prove that any two continuous nests are equivalent in this way.     

Strict Mittag–Leffler Conditions on Gorenstein Modules

In this article, we investigate the relations between Gorenstein projective modules and Gorenstein flat modules in terms of strict Mittag–Leffler condition. We give some conditions under which Gorenstein projectives are Gorenstein flat, and discuss when the direct limits of Gorenstein projective modules are Gorenstein flat. Moreover, we study the dual of Gorenstein injective modules with strict Mittag–Leffler condition.     

Probabilities of incidence between lines and a plane curve over finite fields

We study the probability for a random line to intersect a given plane curve, over a finite field, in a given number of points over the same field. In particular, we focus on the limits of these probabilities under successive finite field extensions. Supposing absolute irreducibility for the curve, we show how a variant of the Chebotarev density theorem for function fields can be used to prove the existence of these limits, and to compute them under a mildly stronger condition, known as simple tangency. Partial results have already appeared in the literature, and we propose this work as an introduction to the use of the Chebotarev theorem in the context of incidence geometry. Finally, Veronese maps allow us to compute similar probabilities of intersection between a given curve and random curves of given degree.     

M-Estimators Converging to a Stable Limit

We discuss the asymptotic linearization of multivariate M-estimators, when the limit distribution is stable. We consider two different types of kernels: VC and bracketing. When applied to the case of normal limits, our work improves the known results to obtain the limit distribution of M-estimators. We give weak conditions for the asymptotic normality of M-estimators over differentiable kernels. To obtain these results, we present an inequality on empirical processes satisfying a bracketing condition with respect to a norm smaller than the L₂ norm.     

Boundedly Compatible Webs and Strict Mackey Convergence

We look for characterizations of those locally convex spaces that satisfy the strict Mackey convergence condition within the context of spaces with webs. We will say that a locally convex space has a boundedly compatible web if it has a web of absolutely convex sets whose members behave like zero neighborhoods in a metrizable locally convex space. It will be shown that these locally convex spaces satisfy the strict Mackey convergence condition. One consequence of this result will be a characterization of boundedly retractive inductive limits. We will also prove that if E is locally complete and webbed, then the strict Mackey convergence condition is equivalent to E having a boundedly compatible web.     

Fluid models of integrated traffic and multipath routing

In this paper we consider a stochastic model describing the varying number of flows in a network. This model features flows of two types, namely file transfers (with fixed volume) and streaming traffic (with fixed duration), and extends the model of Key, Massoulié, Bain and Kelly [27] by allowing more general bandwidth allocation criteria. We analyse the dynamics of the system under a fluid scaling, and show Lyapunov stability of the fluid limits under a natural stability condition. We provide natural interpretations of the fixed points of these fluid limits. We then compare the fluid dynamics of file transfers under (i) balanced multipath routing and (ii) parallel, uncoordinated routing. We show that for identical traffic demands, parallel uncoordinated routing can be unstable while balanced multipath routing is stable. Finally, we identify multi-dimensional Ornstein-Uhlenbeck processes as second-order approximations to the first-order fluid limit dynamics.     

Moderate deviations and functional limits for random processes with stationary and independent increments

We first give a functional moderate deviation principle for random processes with stationary and independent increments under the Ledoux's condition. Then we apply the result to the functional limits for increments of the processes and obtain some Csorgo-Revesz type functional laws of the iterated logarithm.     

Global optimization numerical strategies for rate-independent processes

This paper presents an approach to numerical solution of problems posed in the framework of quasi-static rate-independent processes. As soon as a problem allows for an energetic formulation there are known methods of its time discretization by time incremental minimization problems, which demand for global optimization of a non-convex functional. Moreover the two-sided energy inequality, a necessary condition for optimization, can be formulated. Here we present an algorithm for finding solutions of rate-independent processes that verifies this condition and uses the strategy of backtracking if it is violated. We present the selectivity of the mentioned necessary condition in general and give numerical examples of the efficiency of such an algorithm, but also of situations that are beyond its limits. For those we propose a second strategy relying on wisely chosen combinations of spatial discretizations.     

A Poisson bridge between fractional Brownian motion and stable Lévy motion

We study a non-Gaussian and non-stable process arising as the limit of sums of rescaled renewal processes under the condition of intermediate growth. The process has been characterized earlier by the cumulant generating function of its finite-dimensional distributions. Here, we derive a more tractable representation for it as a stochastic integral of a deterministic function with respect to a compensated Poisson random measure. Employing the representation we show that the process is locally and globally asymptotically self-similar with fractional Brownian motion and stable Lévy motion as its tangent limits.