Topology Of Convergence Research Articles

Convergence of the self-avoiding walk on random quadrangulations to SLE$_{8/3}$ on $\sqrt{8/3}$-Liouville quantum gravity

We prove that a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk (SAW) converges in the scaling limit to the metric gluing of two independent Brownian half-planes identified along their positive boundary rays. Combined with other work of the authors, this implies the convergence of the SAW on a random quadrangulation to SLE$_{8/3}$ on a certain $\sqrt{8/3}$-Liouville quantum gravity surface. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the local Gromov-Hausdorff topology to curve-decorated metric measure spaces. We also prove analogous scaling limit results for uniform infinite quadrangulations of the whole plane decorated by either a one-sided or two-sided SAW. Our proof uses only the peeling procedure for random quadrangulations and some basic properties of the Brownian half-plane, so can be read without any knowledge of SLE or LQG.

Multivariate isotropic random fields on spheres: Nonparametric Bayesian modeling and Lp fast approximations

We study multivariate Gaussian random fields defined over d-dimensional spheres. First, we provide a nonparametric Bayesian framework for modeling and inference on matrix-valued covariance functions. We determine the support (under the topology of uniform convergence) of the proposed random matrices, which cover the whole class of matrix-valued geodesically isotropic covariance functions on spheres. We provide a thorough inspection of the properties of the proposed model in terms of (a) first moments, (b) posterior distributions, and (c) Lipschitz continuities. We then provide an approximation method for multivariate fields on the sphere for which measures of Lp accuracy are established. Our findings are supported through simulation studies that show the rate of convergence when truncating a spectral expansion of a multivariate random field at a finite order. To illustrate the modeling framework developed in this paper, we consider a bivariate spatial data set of two 2019 NCEP/NCAR Flux Reanalyses.

Об устойчивости решений интегральных уравнений в классе измеримых функций

Consider the equation G(x)=(y,) ̃ where the mapping G acts from a metric space X into a space Y, on which a distance is defined, y ̃ ∈ Y. The metric in X and the distance in Y can take on the value ∞, the distance satisfies only one property of a metric: the distance between y,z ∈Y is zero if and only if y= z. For mappings X → Y the notions of sets of covering, Lipschitz property, and closedness are defined. In these terms, the assertion is obtained about the stability in the metric space X of solutions of the considered equation to changes of the mapping G and the element y ̃. This assertion is applied to the study of the integral equation f(t,∫_0^1▒K (t,s)x(s)ds,x(t))= y ̃(t),t ∈[0,1], with respect to an unknown Lebesgue measurable function x: [0,1] ∈R. Sufficient conditions are obtained for the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions f,K,(y.) ̃

Quasicontinuous functions and the topology of uniform convergence on compacta

Let X be a Hausdorff topological space, Q(X,R) be the space of all quasicontinuous functions on X with values in R and ?UC be the topology of uniform convergence on compacta. If X is hemicompact, then (Q(X,R), ?UC) is metrizable and thus many cardinal invariants, including weight, density and cellularity coincide on (Q(X,R), ?UC). We find further conditions on X under which these cardinal invariants coincide on (Q(X,R), ?UC) as well as characterizations of some cardinal invariants of (Q(X,R), ?UC). It is known that the weight of continuous functions (C(R,R), ?UC) is ?0. We will show that the weight of (Q(R,R), ?UC) is 2c.

Orbital chemistry of high valence band convergence and low-dimensional topology in PbTe

The tight-binding method provides insight into the orbital interactions that lead to the exceptional thermoelectric performance of PbTe. Using this framework, we can predict strategies to achieve enhanced thermoelectric performance in new alloys.

On H-sober spaces and H-sobrifications of T0 spaces

In this paper, we provide a uniform approach to d-spaces, sober spaces and well-filtered spaces, and develop a general framework for dealing with all these spaces. The concepts of irreducible subset systems (R-subset systems for short), H-sober spaces and super H-sober spaces for a general R-subset system H are introduced. It is proved that the product space of a family of T0 spaces is H-sober iff each factor space is H-sober, and if H has a natural property (called property M), then the super H-sobriety is a special type of H-sobriety, and hence the product space of a family of T0 spaces is super H-sober iff each factor space is super H-sober. Let Top0 be the category of all T0 spaces with continuous mappings. For a T0 space X and an H-sober space Y, we show that the function space Top0(X,Y) equipped with the topology of pointwise convergence is H-sober. Furthermore, if H has property M and Y is a super H-sober space, then the function space Top0(X,Y) equipped with the topology of pointwise convergence is super H-sober. One immediate corollary is that for a T0 space X and a well-filtered space Y, the function space Top0(X,Y) equipped with the topology of pointwise convergence is well-filtered. For an R-subset system H having property M, the Smyth power space of an H-sober space is not H-sober in general. But for the super H-sobriety, we prove that a T0 space X is super H-sober iff its Smyth power space PS(X) is super H-sober. A direct construction of the H-sobrifications and super H-sobrifications of T0 spaces is given. So the category of all H-sober spaces is reflective in Top0, and the category of all super H-sober spaces is also reflective in Top0 if H has property M.

Analogs of the Lebesgue Measure in Spaces of Sequences and Classes of Functions Integrable with Respect to These Measures

We examine translation-invariant measures on Banach spaces lp, where p ∈ [1,∞]. We construct analogs of the Lebesgue measure on Borel σ-algebras generated by the topology of pointwise convergence (σ-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not σ-finite. We also study spaces of functions integrable with respect to measures constructed and prove that these spaces are not separable. We consider various dense subspaces in spaces of functions that are integrable with respect to a translation-invariant measure. We specify spaces of continuous functions, which are dense in the functional spaces considered. We discuss Borel σ-algebras corresponding to various topologies in the spaces lp, where p ∈ [1,∞]. For p ∈ [1,∞), we prove the coincidence of Borel σ-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel σ-algebra corresponding to the topology of pointwise convergence. We also verify that the space l∞ does not possess similar properties.

On a class of Anosov diffeomorphisms on the infinite-dimensional torus

We study a quite natural class of diffeomorphisms on , where is the infinite-dimensional torus (the direct product of countably many circles endowed with the topology of uniform coordinatewise convergence). The diffeomorphisms under consideration can be represented as the sums of a linear hyperbolic map and a periodic additional term. We find some constructive sufficient conditions, which imply that any in our class is hyperbolic, that is, an Anosov diffeomorphism on . Moreover, under these conditions we prove the following properties standard in the hyperbolic theory: the existence of stable and unstable invariant foliations, the topological conjugacy to a linear hyperbolic automorphism of the torus and the structural stability of .

On the topological convergence of multi-rule sequences of sets and fractal patterns

In many cases occurring in the real world and studied in science and engineering, non-homogeneous fractal forms often emerge with striking characteristics of cyclicity or periodicity. The authors, for example, have repeatedly traced these characteristics in hydrological basins, hydraulic networks, water demand, and various datasets. But, unfortunately, today we do not yet have well-developed and at the same time simple-to-use mathematical models that allow, above all scientists and engineers, to interpret these phenomena. An interesting idea was firstly proposed by Sergeyev in 2007 under the name of “blinking fractals.” In this paper we investigate from a pure geometric point of view the fractal properties, with their computational aspects, of two main examples generated by a system of multiple rules and which are enlightening for the theme. Strengthened by them, we then propose an address for an easy formalization of the concept of blinking fractal and we discuss some possible applications and future work.

The Selectively Pseudocompact-Open Topology on C(X)

This paper studies the selectively pseudocompact-open topology on the set of the real-valued continuous functions on a Tychonoff space, and compares this topology with the $$\alpha $$ -open topology and the topology of uniform convergence on the elements of $$\alpha $$ where $$\alpha $$ is one of the following: $$\{X\}$$ , the finite subsets of X, the compact subsets of X, the countably compact subsets of X, and the pseudocompact subsets of X. Moreover, it also analyzes the almost cc- $$\omega $$ -bounded spaces.

Terrestrial laser scanning reveals convergence of tree architecture with increasingly dominant crown canopy position

AbstractTo fulfil their growth and reproductive functions, trees develop a three‐dimensional structure that is subject to both internal and external constraints. This is reflected by the unique architecture of each individual at a given time. Addressing the crown dimensions and topological structure of large tropical trees is challenging considering their complexity, size and longevity. Terrestrial laser scanning (TLS) technology offers a new opportunity for characterising and comparing these properties across a large number of individuals and species.In the present study, we specifically developed topology and geometry metrics of crown architecture from TLS data and investigated how they correlated with metrics of tree and crown form, crown position and shade tolerance.Fifty‐nine trees belonging to 14 coexisting canopy species in semideciduous forests of Cameroon were scanned with TLS and reconstructed using quantitative structural models (QSMs). The species belonged to different shade‐tolerance groups and were sampled in different crown positions. Crown‐form metrics and branch topology metrics were quantified from the TLS data, and principal component analysis (PCA) was used to study how the 59 sampled trees were distributed along axes of architectural diversity. Allometric scaling parameters derived from West Brown and Enquist (WBE) metabolic theory were also quantified from the QSMs, and their correlations with the PCA axes were evaluated.The results revealed that the branch topology and crown‐form metrics were not correlated since similar topologies could lead to contrasting crown forms. Crown form, but not branch topology, changed with tree shade tolerance, while convergence in tree topology and towards expected WBE parameters was observed for all trees reaching dominant crown positions independent of species shade tolerance.This convergence is interpreted as resulting from a liberation effect of canopy trees from side‐shading constraints, leading to crown development processes through sequential reiteration.A freePlain Language Summarycan be found within the Supporting Information of this article.

Generalized Multipliers for Left-Invertible Operators and Applications

Generalized multipliers for a left-invertible operator T, whose formal Laurent series U_x(z)=sum _{n=1}^infty (P_ET^{n}x)frac{1}{z^n}+sum _{n=0}^infty (P_E{T^{prime *n}}x)z^n, xin mathcal {H} actually represent analytic functions on an annulus or a disc are investigated. We show that they are coefficients of analytic functions and characterize the commutant of some left-invertible operators, which satisfies certain conditions in its terms. In addition, we prove that the set of multiplication operators associated with a weighted shift on a rootless directed tree lies in the closure of polynomials in z and frac{1}{z} of the weighted shift in the topologies of strong and weak operator convergence.

E-approximation properties

In this paper, we introduce the E-approximation property and the $$E^u$$ -approximation property which generalize Sinha and Karn’s p-approximation property. Characterizations of these properties are given parallel to Lima and Oja’s results on the strong approximation property and the weak bounded approximation property. Representation theorems for the dual of $${\mathcal {L}} (X, Y)$$ under the topology of uniform convergence on E-compact sets and $$E^u$$ -compact sets are also provided. As an application, the representations are used to generalize the main theorem of Choi and Kim in [1]. These results build up fundamental bases for future investigations of these properties.

Area anomaly in the rough path Brownian scaling limit of hidden Markov walks

We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.

Witnessing the lack of the Grothendieck property in C(K)-spaces via convergent sequences

Let K be a compact Hausdorff space and let C(K) be the space of all scalar-valued, continuous functions on K. We show that C(K) is an $$\ell _1(K)$$ -Grothendieck space but not a Grothendieck space exactly when the spaces $$C_p(K)$$ and $$C_p(K \oplus {\mathbb {N}}^{\#})$$ are not linearly isomorphic, where $${\mathbb {N}}^{\#}$$ is the one-point compactificiation of the discrete space of natural numbers. (That is, if C(K) contains a complemented copy of $$c_0$$ , then C(K) fails to be $$\ell _1(K)$$ -Grothendieck if and only if the topologies of pointwise convergence in $$C_p(K)$$ and $$C_p(K \oplus {\mathbb {N}}^{\#})$$ are linearly isomorphic.) Moreover, for infinite compact spaces K and L, there exists a compact space G that has a non-trivial convergent sequence and such that $$C_{p}(K\times L)$$ and $$C_{p}(G)$$ are linearly isomorphic. This extends a remarkable theorem of Cembranos and Freniche. Some examples illustrating the above results are provided.

Convergence, continuity, recurrence and Turing completeness in dynamic epistemic logic1

Abstract The paper analyses dynamic epistemic logic from a topological perspective. The main contribution consists of a framework in which dynamic epistemic logic satisfies the requirements for being a topological dynamical system thus interfacing discrete dynamic logics with continuous mappings of dynamical systems. The setting is based on a notion of logical convergence, demonstratively equivalent with convergence in Stone topology. Presented is a flexible, parametrized family of metrics inducing the Stone topology, used as an analytical aid. We show maps induced by action model transformations continuous with respect to the Stone topology and present results on the recurrent behaviour of said maps. Among the recurrence results, we show maps induced by finite action models may have uncountably many recurrent points, even when initiated on a finite input model. Several recurrence results draws on the class of action models being Turing complete, for which the paper provides proof in the postcondition-free case. As upper bounds, it is shown that either 1 atom, 3 agents and preconditions of modal depth 18 or 1 atom, 7 agents and preconditions of modal depth 3 suffice for Turing completeness.

A simpler description of the κ‐topologies on the spaces DLp,Lp,M1

AbstractFor the spaces , and , we consider the topology of uniform convergence on absolutely convex compact subsets of their (pre‐)dual space. Following the notation of J. Horváth's book we call these topologies κ‐topologies. They are given by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre‐)dual spaces. In many cases it is more convenient to work with a description of the topology by means of a family of semi‐norms defined by multiplication and/or convolution with functions and by classical norms. We give such families of semi‐norms generating the κ‐topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence‐space representation of the spaces equipped with the κ‐topology, which complements a result of J. Bonet and M. Maestre. As a byproduct, we give a characterisation of the compact subsets of the spaces , and .

Trotter–Kato Product Formulae in Normed Ideals

We show that for a certain class of Kato functions the Trotter–Kato product formulae converge in symmetrically-normed ideals of compact operators on a separable Hilbert space. The rate of convergence in topology of ideals inherits the error-bound estimate for the corresponding operator-norm convergence.

Notes on the Chernoff product formula

We revise the strong convergent Chernoff product formula and extend it, in a Hilbert space, to convergence in the operator-norm topology. Main results deal with the self-adjoint Chernoff product formula. The non-self-adjoint case concerns the quasi-sectorial contractions.

Analog fiber-wireless downlink transmission of IFoF/mmWave over in-field deployed legacy PON infrastructure for 5G fronthauling

We present a fixed mobile convergence topology for analog intermediate frequency over fiber (A-IFoF)/millimeter-wave (mmWave) transmission, benefiting from the reuse of the deployed passive optical network (PON) infrastructure, towards future mobile fronthaul architectures. Powerful fully programmable gate array boards located inside the access nodes convert the Ethernet-based traffic to orthogonal frequency-division multiplexing (OFDM)-modulated intermediate frequency (IF) waveforms, supporting the A-IFoF propagation through the optical legacy infrastructure. Coexistence of the 5G traffic with the residential legacy traffic for the field propagation is achieved through utilization of unused C-band channels and wavelength-division multiplexing. To this extent, we experimentally demonstrate the downlink operation of a converged A-IFoF/mmWave link, over Telecom Italia’s legacy infrastructure located at Turin. Four-quadrature amplitude modulation (QAM)-OFDM and 16QAM-OFDM IF signals with ∼ 200 M H z and 400 MHz bandwidth [considered within the 3rd Generation Partnership Project (3GPP) New Radio specifications] were generated through a radio frequency system-on-chip platform and optically multiplexed with the legacy fiber-to-the-home services. After propagation to the field, the A-IFoF stream was directly fed to a directional wireless link operating at 60 GHz. Successful PON/over-the-air transmission with error vector magnitude (EVM) values well below the 3GPP ( < 12.5 % ) requirements for 5G New Radio was demonstrated, with a 10.5% EVM for 16QAM-OFDM modulated with 400 MHz bandwidth.