A common framework for lattice-valued uniform spaces and probabilistic uniform limit spaces

We study a category of lattice-valued uniform convergence spaces where the lattice is enriched by two algebraic operations. This general setting allows us to view the category of lattice-valued uniform spaces as a reflective subcategory of our category, and the category of probabilistic uniform limit spaces as a coreflective subcategory

Initial Characterized L-spaces and Characterized L- topological Groups

In this research work, new topological notions are proposed and investigated. The notions are named initial characterized L-spaces and characterized L-topological groups. The properties of such notions are deeply studied. We show that the intitial characterized L-space for an characterized L-spaces exists. By this notion, the notions of characterized L-subspace and characterized product L-space are introduced and studied. More information can be found in the full paper

The Relationship Between Various Filter Notions on aGL-Monoid

AbstractThe notion of a generalised filter is extended to the setting of aGL-monoid. It is shown that there exists a one-to-one correspondence between the collection of generalised filters on a setXand the collection of strongly stratifiedL-filters onX. Specialising to the case whereLis the closed unit interval [0,c] viewed as a Heyting algebra, we show that any strongly stratified [0,c]-filter onXcan be uniquely identified with a saturated filter onIXwith characteristic valuec. In this way, the notion of a generalised filter unifies various filter notions. In particular, necessity measures and finitely additive probability measures are specific examples of generalised filters

On Initial and Final Characterized L- topological Groups

In this research work, new topological notions are proposed and investigated. The notions are named finalcharacterized L-spaces and initial and final characterized L-topological groups. The properties of such notionsare deeply studied. We show that all the final lefts and all the final characterized L-spaces are uniquely exist inthe category CRL-Sp and hence CRL-Sp is topological category over the category SET of all sets. By the notion offinal characterized L-space, the notions of characterized qoutien pre L-spaces and characterized sum L-spacesare introduced and studied. The characterized L-subspaces together with their related inclusion mappings andthe characterized quotient pre L-spaces together with their related canonical surjections are the equalizers andco-equalizers, respectively in CRL-Sp. Moreover, we show that the initial and final lefts and then the initial andfinal characterized L-topological groups uniquely exist in the category CRL-TopGrp. Hence, the category CRLTopGrpis topological category over the category Grp of all groups. By the notion of initial and finalcharacterized L-topological groups, the notions of characterized L-subgroups, characterized product Ltopologicalgroups and characterized L-topological quotient groups are introduced and studied., we show that thecategory CRL-TopGrp is concrete and co-concrete category of the category L-Top. More details can be found in the full paper

Multipurpose S-shaped solvable profiles of the refractive index: application to modeling of antireflection layers and quasi-crystals

A class of four-parameter solvable profiles of the electromagnetic admittance has recently been discovered by applying the newly developed Property & Field Darboux Transformation method (PROFIDT). These profiles are highly flexible. In addition, the related electromagnetic-field solutions are exact, in closed-form and involve only elementary functions. In this paper, we focus on those who are S-shaped and we provide all the tools needed for easy implementation. These analytical bricks can be used for high-level modeling of lightwave propagation in photonic devices presenting a piecewise-sigmoidal refractive-index profile such as, for example, antireflection layers, rugate filters, chirped filters and photonic crystals. For small amplitude of the index modulation, these elementary profiles are very close to a cosine profile. They can therefore be considered as valuable surrogates for computing the scattering properties of components like Bragg filters and reflectors as well. In this paper we present an application for antireflection layers and another for 1D quasicrystals (QC). The proposed S-shaped profiles can be easily manipulated for exploring the optical properties of smooth QC, a class of photonic devices that adds to the classical binary-level QC.Comment: 14 pages, 18 fi

On the auxiliary particle filter

In this article we study asymptotic properties of weighted samples produced by the auxiliary particle filter (APF) proposed by pitt and shephard (1999). Besides establishing a central limit theorem (CLT) for smoothed particle estimates, we also derive bounds on the Lp error and bias of the same for a finite particle sample size. By examining the recursive formula for the asymptotic variance of the CLT we identify first-stage importance weights for which the increase of asymptotic variance at a single iteration of the algorithm is minimal. In the light of these findings, we discuss and demonstrate on several examples how the APF algorithm can be improved.Comment: 26 page

Negative association, ordering and convergence of resampling methods

We study convergence and convergence rates for resampling schemes. Our first main result is a general consistency theorem based on the notion of negative association, which is applied to establish the almost-sure weak convergence of measures output from Kitagawa's (1996) stratified resampling method. Carpenter et al's (1999) systematic resampling method is similar in structure but can fail to converge depending on the order of the input samples. We introduce a new resampling algorithm based on a stochastic rounding technique of Srinivasan (2001), which shares some attractive properties of systematic resampling, but which exhibits negative association and therefore converges irrespective of the order of the input samples. We confirm a conjecture made by Kitagawa (1996) that ordering input samples by their states in $\mathbb{R}$ yields a faster rate of convergence; we establish that when particles are ordered using the Hilbert curve in $\mathbb{R}^d$, the variance of the resampling error is ${\scriptscriptstyle\mathcal{O}}(N^{-(1+1/d)})$ under mild conditions, where $N$ is the number of particles. We use these results to establish asymptotic properties of particle algorithms based on resampling schemes that differ from multinomial resampling.Comment: 54 pages, including 30 pages of supplementary materials (a typo in Algorithm 1 has been corrected