Net Convergence Research Articles

Differentiable, holomorphic, and analytic functions on complex Φ-algebras

Using the notion of order convergent nets, we develop an order-theoretic approach to differentiable functions on Archimedean complex Φ-algebras. Most notably, we improve the Cauchy-Hadamard formulas for universally complete complex vector lattices given by both authors in a previous paper in order to prove that analytic functions are holomorphic in this abstract setting.

Notes on noncommutative ergodic theorems

Given a semifinite von Neumann algebra M \mathcal M equipped with a faithful normal semifinite trace τ \tau , we prove that the spaces L 0 ( M , τ ) L^0(\mathcal M,\tau ) and R τ \mathcal R_\tau are complete with respect to pointwise—almost uniform and bilaterally almost uniform—convergences in L 0 ( M , τ ) L^0(\mathcal M,\tau ) . Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space L 1 ( M , τ ) L^1(\mathcal M,\tau ) can be extended to pointwise convergence of such nets in any fully symmetric space E ⊂ R τ E\subset \mathcal R_\tau , in particular, in any space L p ( M , τ ) L^p(\mathcal M,\tau ) , 1 ≤ p > ∞ 1\leq p>\infty . Some applications of these results in the noncommutative ergodic theory are discussed.

Biogeochemical Fluxes of Nickel in the Global Oceans Inferred From a Diagnostic Model

AbstractNickel (Ni) is a micronutrient that plays a role in nitrogen uptake and fixation in the modern ocean and may have affected rates of methanogenesis on geological timescales. Here, we present the results of a diagnostic model of global ocean Ni fluxes which addresses key questions about marine Ni cycling. Sparsely available observations of Ni concentration are first extrapolated into a global gridded climatology using tracers with better observational coverage such as macronutrients, and testing three different machine learning techniques. The physical transport of Ni is then estimated using the ocean circulation inverse model (OCIM2), revealing regions of net convergence or divergence. These diagnostics are not based on any assumption about Ni biogeochemical cycling, but their spatial patterns can be used to infer where biogeochemical processes such as biological Ni uptake and regeneration take place. Although Ni and silicate (Si) have similar concentration patterns in the ocean, we find that the spatial pattern of Ni uptake in the surface ocean is similar to phosphate (P) uptake but not to silicate (Si) uptake. This suggests that their similar distributions arise from different biogeochemical mechanisms, consistent with other evidence showing that Ni is not incorporated into diatom frustules. We find that Ni:P ratios at uptake do not decrease as Ni concentrations approach 2 nM, which challenges the hypothesis of a ∼2 nM pool of non‐bioavailable Ni in the surface ocean. Finally, we find that the net regeneration of Ni occurs deeper in the ocean than for P, though not as deeply as for Si.

Embedded unbounded order convergent sequences in topologically convergent nets in vector lattices

Embedded unbounded order convergent sequences in topologically convergent nets in vector lattices

Order boundedness and order continuity properties of positive operator semigroups

Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandic, Kramar-Fijavž, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of vector lattices, where no norm is present in general. In this article, we return to the more standard Banach lattice setting – where both ruc semigroups and C 0 -semigroups are well-defined concepts – and compare both notions. We show that the ruc semigroups are precisely those positive C 0 -semigroups whose orbits are order bounded for small times. We then relate this result to three different topics: (i) equality of the spectral and the growth bound for positive C 0 -semigroups; (ii) a uniform order boundedness principle which holds for all operator families between Banach lattices; and (iii) a description of unbounded order convergence in terms of almost everywhere convergence for nets which have an uncountable index set containing a co-final sequence.

Representative Functions, Variational Convergence and Almost Convexity

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal–cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space.

A New Notion of Fuzzy Function Ideal Convergence

P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class, L, on a non-empty set, X, consisting of triads (f,e,I), where f is a function from a non-empty set, D, to the set FP(X) of fuzzy points in X, which we call fuzzy function, e∈FP(X), and I is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology, δ, on X, such that (f,e,I)∈L iff fI-converges to e, relative to the fuzzy topology δ.

Demonstration of machine-learning-enhanced Bayesian quantum state estimation

Machine learning (ML) has found broad applicability in quantum information science in topics as diverse as experimental design, state classification, and even studies on quantum foundations. Here, we experimentally realize an approach for defining custom prior distributions that are automatically tuned using ML for Bayesian quantum state estimation methods that generally better conform to the physical properties of the underlying system than standard fixed prior distributions. Previously, researchers have looked to Bayesian quantum state tomography for advantages like uncertainty quantification, the return of reliable estimates under any measurement condition, and minimal mean-squared error. However, practical challenges related to long computation times and conceptual issues concerning how to incorporate prior knowledge most suitably can overshadow these benefits. Using both simulated and experimental measurement results, we demonstrate that ML-defined prior distributions reduce net convergence times and provide a natural way to incorporate both implicit and explicit information directly into the prior distribution. These results constitute a promising path toward practical implementations of Bayesian quantum state tomography.

NEW CHARACTERIZATIONS OF FUZZY TOPOLOGY

Following the generalization of Moore-Smith convergence of nets to fuzzy topological spaces which was given in Pu Pao-Ming, Liu Ying-Ming, \emph{Fuzzy topology I. Neighborhood structure of a fuzzy point and Moore-Smith convergence}, J. Math. Anal. Appl., \textbf{76} (1980), 571--599, a characterization theorem between fuzzy topologies and fuzzy convergence classes was introduced in Ying-Ming Liu, \emph{On fuzzy convergence classes}, Fuzzy Sets and Systems, \textbf{30} (1989), 47--51. Our goal in this paper is to provide modified versions of this characterization. Specifically, we will introduce the concept of fuzzy semi-convergence class to give an alternative characterization of fuzzy topology, in relation to the ordinary convergence of fuzzy nets, and then we will introduce the concept of fuzzy ideal convergence class to obtain analogous results, in relation to the ideal convergence of fuzzy nets.

On unbounded order convergence of operator nets

On unbounded order convergence of operator nets

Almost everywhere convergence of nets of operators and weak type maximal inequalities

Almost everywhere convergence of nets of operators and weak type maximal inequalities

Soft Np – Convergence of Soft Nets

The purpose of this article is to present a new concept of soft convergence in soft topological spaces, namely, soft Np – convergence of soft nets by using the notion of soft Np – open sets. Also, we study several properties of Np – limit soft points and Np – cluster soft point of soft nets.

Statistical convergence of nets on locally solid Riesz spaces

The statistical convergence is handled for sequences with the natural density, in general. In a recent paper, the statistical convergence for nets has been studied and investigated by developing topology-free techniques in Riesz spaces. In this paper, we introduce the statistically topological convergence for nets on locally solid Riesz spaces with solid topologies. Moreover, we give the notion of statistical continuity on locally solid Riesz spaces.

Net convergence structures with applications to vector lattices

Convergence is a fundamental topic in analysis that is most commonly modeled using topology. However, there are many natural convergences that are not given by any topology; e.g., convergence almost everywhere of a sequence of measurable functions and order convergence of nets in vector lattices. The theory of convergence structures provides a framework for studying more general modes of convergence. It also has one particularly striking feature: it is formalized using the language of filters. This paper develops a general theory of convergence in terms of nets. We show that it is equivalent to the filter-based theory and present some translations between the two areas. In particular, we provide a characterization of pretopological convergence structures in terms of nets. We also use our results to unify certain topics in vector lattices with general convergence theory.

Column Relative Humidity and Primary Condensation Rate as Two Useful Supplements to Atmospheric River Analysis

AbstractLandfalling atmospheric rivers (ARs) frequently trigger heavy and sometimes prolonged precipitation, especially in regions with favored orographic enhancement. The presence and strength of ARs are often described using the integrated water vapor (IWV) and the integrated vapor transport (IVT). However, the associated precipitation is not directly correlated with these two variables. Instead, the intensity of precipitation is mainly determined by the net convergence of moisture flux and the initial degree of saturation of the air column. In this study, a simple algorithm is proposed for estimating the heavy precipitation attributable to the IVT convergence. Bearing a strong resemblance to the Kuo‐Anthes parameterization scheme for cumulus convection, the proposed algorithm calculates the large‐scale primary condensation rate (PCR) as a proportion of the IVT convergence, with a reduction to account for the general moistening in the atmosphere. The amount of reduction is determined by the column relative humidity (CRH), which is defined as the ratio of IWV to its saturation counterpart. Our analysis indicates that the diagnosable PCR compares well to the forecast precipitation rate given by a numerical weather prediction model. It is also shown that the PCR in an air column with CRH 0.50 is negligibly small. The usefulness of CRH and PCR as two complements to standard AR analysis is illustrated in three case studies. The potential application of PCR to storm classification is also explored.

From Sugar to Flowers: A Transition of Shallow Cumulus Organization During ATOMIC

AbstractThe Atlantic Tradewind Ocean‐Atmosphere Mesoscale Interaction Campaign (ATOMIC) took place in January–February, 2020. It was designed to understand the relationship between shallow convection and the large‐scale environment in the trade‐wind regime. A Lagrangian large eddy simulation, following the trajectory of a boundary‐layer airmass, can reproduce a transition of trade cumulus organization from “sugar” to “flower” clouds with cold pools, observed on February 2–3. The simulation is driven with reanalysis large‐scale meteorology, and in‐situ aerosol data from ATOMIC and its joint field study EUREC4A. During the transition, large‐scale upward motion deepens the cloud layer. The total water path and optical depth increase, especially in the moist regions where flowers aggregate. This is due to mesoscale circulation that renders a net convergence of total water in the already moist and cloudy regions, strengthening the organization. An additional simulation shows that stronger large scale upward motion reinforces the mesoscale circulation and accelerates the organization process by strengthening the cloud‐layer mesoscale buoyant turbulence kinetic energy production.

Volume and Heat Transport in the South China Sea and Maritime Continent at Present and the End of the 21st Century

AbstractOcean transports through the Southeast Asian Seas connect the western tropical Pacific and Indian Oceans, thereby exerting an important role in regional and global climate. High‐resolution regional ocean model simulations over the South China Sea (SCS) and maritime continent are used to study the mean and seasonally varying thermohaline structure and volume transport through the straits surrounding the SCS. Diversity in the vertical structure of these straits is not only indicative of the role of widely varying bathymetry but also strong seasonality associated with monsoonal currents. The presence of a Pacific water mass in intermediate and deep layers of the Luzon Strait points to a key pathway between the Pacific and Indian Oceans. Further, examining a suite of global, high‐resolution model simulations reveals the projected changes in the regional upper ocean transports due to anthropogenic radiative forcing by the end of the 21st century. The global models predict an increase in heat and volume transport through the Luzon and Karimata Straits, and a decrease thereof through the Makassar and Lombok Straits by the end of the century. Overall, these changes impute additional net convergence of heat and volume in the SCS, a significant reduction of sea surface salinity and mixed layer depth, and an increase in the upper‐ocean heat content of the region. As the SCS serves as a regional heat capacitor and is impacted by the global thermohaline circulation locally via Indonesian Throughflow, these predicted changes have the potential to impact climate over the Indo‐Pacific region and globally.

Strong Whitney and strong uniform convergences on a bornology

For any two metric spaces (X,d), (Y,ρ) and a bornology B on X, we study the relationship between strong uniform convergence (introduced by Beer and Levi in [8]) and strong Whitney convergence (introduced by A. Caserta in [15]) on B on YX (and C(X,Y)). The relationships of these convergences with their classical counterparts, that is, Whitney and uniform convergences on B are also considered. Moreover, for any two bornologies B and B′ on X, we examine the relationship between strong Whitney (strong uniform) convergence on B′ and Whitney (uniform) convergence on B. Finally, we investigate the relation of strong Whitney convergence with the well-known continuous convergence of nets of functions.

Vector lattices with a Hausdorff uo-Lebesgue topology

We investigate the construction of a Hausdorff uo-Lebesgue topology on a vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal, and what the properties of the topologies thus obtained are. When the vector lattice has an order dense ideal with a separating order continuous dual, it is always possible to supply it with such a topology in this fashion, and the restriction of this topology to a regular sublattice is then also a Hausdorff uo-Lebesgue topology. A regular vector sublattice of L0(X,Σ,μ) for a semi-finite measure μ falls into this category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is then the convergence in measure on subsets of finite measure. When a vector lattice not only has an order dense ideal with a separating order continuous dual, but also has the countable sup property, we show that every net in a regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology always contains a sequence that is uo-convergent to the same limit. This enables us to give satisfactory answers to various topological questions about uo-convergence in this context.

Convergence structures and locally solid topologies on vector lattices of operators

For vector lattices E and F, where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators {mathscr{L}}_{mathrm{ob}}(E,F) from E into F. Using this, it follows that {mathscr{L}}_{mathrm{ob}}(E,F) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order convergence, unbounded order convergence, and—when applicable—convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on {mathscr{L}}_{mathrm{ob}}(E,F). Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We furthermore show that, in contrast to general order bounded operators, orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and—when applicable—convergence in the Hausdorff uo-Lebesgue topology as well.