Uniform Convergence Research Articles

Determining the time before or after a galaxy merger event

Aims. This work aims to reproduce the time before or after a merger event of merging galaxies from the IllustrisTNG cosmological simulation using machine learning. Methods. Images of merging galaxies were created in the u, g, r, and i bands from IllustrisTNG. The merger times were determined using the time difference between the last simulation snapshot where the merging galaxies were tracked as two galaxies and the first snapshot where the merging galaxies were tracked as a single galaxy. This time was then further refined using simple gravity simulations. These data were then used to train a residual network (ResNet50), a Swin Transformer (Swin), a convolutional neural network (CNN), and an autoencoder (using a single latent neuron) to reproduce the merger time. The full latent space of the autoencoder was also studied to see if it reproduces the merger time better than the other methods. This was done by reducing the latent space dimensions using Isomap, linear discriminant analysis (LDA), neighbourhood components analysis, sparse random projection, truncated singular value decomposition, and uniform manifold approximation and projection. Results. The CNN is the best of all the neural networks. The performance of the autoencoder was close to the CNN, with Swin close behind the autoencoder. ResNet50 performed the worst. The LDA dimensionality reduction performed the best of the six methods used. The exploration of the full latent space produced worse results than the single latent neuron of the autoencoder. For the test data set, we found a median error of 190 Myr, comparable to the time separation between snapshots in IllustrisTNG. Galaxies more than ≈625 Myr before a merger have poorly recovered merger times, as well as galaxies more than ≈125 Myr after a merger event.

A uniform semi-local limit theorem along sets of multiples for sums of i.i.d. random variables

A uniform semi-local limit theorem along sets of multiples for sums of i.i.d. random variables

Robust Average-Reward Reinforcement Learning

Robust Markov decision processes (MDPs) aim to find a policy that optimizes the worst-case performance over an uncertainty set of MDPs. Existing studies mostly have focused on the robust MDPs under the discounted reward criterion, leaving the ones under the average-reward criterion largely unexplored. In this paper, we develop the first comprehensive and systematic study of robust average-reward MDPs, where the goal is to optimize the long-term average performance under the worst case. Our contributions are four-folds: (1) we prove the uniform convergence of the robust discounted value function to the robust average-reward function as the discount factor γ goes to 1; (2) we derive the robust average-reward Bellman equation, characterize the structure of its solution set, and prove the equivalence between solving the robust Bellman equation and finding the optimal robust policy; (3) we design robust dynamic programming algorithms, and theoretically characterize their convergence to the optimal policy; and (4) we design two model-free algorithms unitizing the multi-level Monte-Carlo approach, and prove their asymptotic convergence

Explorations about the correlation between biological changes of meninges in periodontitis mice and cognitive impairment via single-cell RNA sequencing

Objective: To clarify the potential correlation between biological changes of meninges in periodontitis mice and cognitive impairment by analyzing the biological changes of meninges in periodontitis mice using single-cell RNA sequencing. Methods: Thirty C57BL/6 mice were divided into two groups by using random number table method (15 mice in each group). Mice in the control group were locally administered 2% carboxyl methyl cellulose (CMC) without Porphyromonas gingivalis (Pg) on both buccal sides. A mixture of Pg W83 and 2% CMC was applied on both buccal sides in the experimental group mice three times a week, lasting for 16 weeks in total. The absorption of alveolar bone, locomotor activity and cognitive function, the activation of microglia and astrocytes in the cortex were observed and assessed. The mRNA expression levels of Occludin in meninges and brain were detected in two groups. Single-cell RNA sequencing data of meninges were processed by uniform manifold approximation and projection (UMAP). Differential genes expressions of endothelial cells were processed by gene ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) enrichment analysis. In addition, real-time fluorescence quantitative PCR (RT-qPCR) was used to verify the expressions of transcription activating factor 3 (Atf3) and apolpoprotein L domain-containing 1 (Apold 1). Results: Methylene blue staining found the distances of buccal and palatal cement-enamel junction-alveolar bone crest in experimental mice [(185.60±17.60), (206.90±13.37) μm] increased significantly compared with the control group [(135.33±9.57), (163.05±14.98) μm] (t=5.02, P=0.002; t=4.37, P=0.005). Open field experiment showed the total distance and average speed of mice in the experimental group [(971.88±164.57) cm, (3.25±0.55) cm/s] were not statistically significant compared with the control group [(914.24±278.81) cm, (3.05±0.93) cm/s] (t=0.65, P=0.525; t=0.65, P=0.520). The recognition index of the experimental group [(48.02±16.92) %] was lower than the control group [(66.27±17.90) %] (t=2.40, P=0.027) by novel object recognition tests. Compared with the control group [(63.56±11.88) %], the alternation of experimental group [(50.99±14.17) %] was significantly decreased in Y maze tests (t=2.33, P=0.030). Immunohistochemistry results showed microglia and astrocytes were activated in the cortex of experimental mice. Compared with the control group (1.02±0.25, 1.04±0.31), the relative mRNA expressions of Occludin decreased significantly in the meninges and brain of periodontitis mice, respectively (0.61±0.10, 0.64±0.20) (t=3.47, P=0.010; t=2.66, P=0.024). By single-cell RNA sequencing, meninges cells were divided into 11 types, such as endothelial cells, fibroblasts, immune cells and so on. Endothelial cells were the main cell types in meninges [the control group: 26.47% (1 589/6 004), the experimental group: 26.26% (807/3 073)]. Compared with the control group [5.56% (334/6 004)], the percentage of granulocytes increased in the periodontitis mice [11.65% (358/3 073)]. Using clustering analysis to further focus on endothelial cells, GO enrichment analysis revealed differential genes were mainly related to angiogenesis, cell adhesion, apoptosis and so on. KEGG enrichment analysis revealed that differential genes were related to signaling pathways of interleukin-17, relaxin and so on. The relative mRNA expressions of Atf3 and Apold1 in meninges of periodontitis mice (0.42±0.24, 0.54±0.27) were significantly lower than the control group (1.03±0.26, 1.02±0.23) (t=3.88, P=0.005; t=3.02, P=0.017). Conclusions: The mice chronically infected with Pg W83 occurred memory impairment, neuroinflammation and changes of barrier function. In the meninges of periodontitis mice, there were infiltration of immune cells and down-regulation expressions of Atf3 and Apold1 by single-cell RNA sequencing. Meningeal immunity and changes of barrier function may play an important role in the cognitive impairment caused by periodontitis.

Uniform convergence of solutions to stochastic hybrid models of gene regulatory networks

In a recent paper by Kurasov et al. (Math Biosci 305:170–177, 2018), a hybrid gene regulatory network was proposed to model gene expression dynamics by using a stochastic system of coupled partial differential equations. In more recent work, the existence and strong convergence of the solutions to equilibrium were proven. In this article, we improve upon their result by showing that the convergence rate is independent of the initial state, therefore proving that the solutions converge not only strongly but even uniformly to equilibrium. To this end, we make use of a recent convergence theorem for stochastic, irreducible semigroups that contain partial integral operators.

Manifold learning-based UMAP method for geochemical anomaly identification

Geochemical data are usually high-dimensional data that could contain dozens of elements. Geochemical distribution patterns and anomalies related to mineralization and lithological features are always hidden in these high-dimensional data, which cannot be directly observed from the data. To solve this problem, a manifold learning-based uniform manifold approximation and projection (UMAP) method, was introduced to recognize mineralization-related geochemical anomalies from high-dimensional geochemical data in this study. The UMAP method is a nonlinear dimensionality reduction method, which is suitable for dimensionality reduction and visualization of high-dimensional data. A case study was conducted to demonstrate the advantages of the UMAP method for identifying ion-adsorbed rare-earth-element (REE) mineralization-related anomalies from high-dimensional data in the Nanling region, China. Factor analysis was used to determine ion-adsorbed REE mineralization-related element combination that consists of 10 elements. High-dimensional geochemical data were reduced to two dimensions based on the UMAP method. The results indicated that the UMAP method can effectively characterize the spatial distributions of ion-adsorbed REE mineralization-related anomalies by dimensionality reduction analysis and visualization analysis of high-dimensional geochemical data in the study area. To illustrate the superiority of the UMAP method, a comparative study was conducted between the UMAP and other three manifold learning methods, namely locally linear embedding (LLE), isometric feature mapping (Isomap) and t-distributed stochastic neighbor embedding (t-SNE). The performance of the four manifold learning methods was evaluated by receiver operating characteristic (ROC) curve and prediction-area (P-A) plot, showing that the performance of the UMAP method is superior to that of the LLE, Isomap and t-SNE methods in terms of recognizing ion-adsorbed REE mineralization-related anomalies and the spatial distributions of the REE-bearing geological bodies in the Nanling belt.

On relative uniform convergence of triple sequence of functions

This paper discusses relative uniform convergence of triple sequence of functions that are defined on a compact domain. Another central idea that is discussed is the regular relative uniform convergence and the Cauchy relative uniform convergence of triple sequence of functions. The idea that a continuous triple sequence defined on a compact domain is relative uniform convergent if and only if it is relative uniform Cauchy has been discussed and established. Subsequently, a glimpse into Cesaro summability of triple sequences and a theorem regarding triple Cesaro summability of bounded relative uniform triple sequence of functions have been introduced.

Safety and Mobility Performance Comparison of Two-Plus-One and Two-Lane Two-Way Roads: A Simulation Study

Two-plus-one (2+1) highways are a special configuration of two-lane two-way (TLTW) highways with a continuous center lane that is used to alternate passing lanes. The main objective of this paper is, therefore, to evaluate the suitability of the 2+1 design for Middle East conditions as a replacement for traditional TLTW roads with higher traffic volumes or as an interim solution before transforming TLTW roads into four-lane highways. In our analysis, we considered both safety and mobility performances by comparing the 2+1 and TLTW designs. The new suggested 2+1 designs were evaluated, with the first design prohibiting overtaking in the opposite direction, while the second design permitted it. Additionally, two-speed-limit strategies, uniform speed limit (USL), and differential speed limit (DSL) were also evaluated. The results showed that the 2+1 design, which prohibited overtaking in the opposite direction, was superior to TLTW in terms of mobility and safety, while the other design compromised safety compared to TLTW. The results provide valuable information to policymakers, urban planners, and transport authorities to guide evidence-based decisions on the integration of the 2+1 design as a viable solution for sustainable and efficient transportation.

Uniform convergence of the one-dimensional cubic Schrödinger equation

In this article, we investigate the uniform convergence problem of the one-dimensional cubic Schrödinger equation. By using the Kato smoothing estimate, the maximal function estimate and the dyadic mixed Lebesgue spaces, we establish the uniform convergence of the one-dimensional cubic Schrödinger equation in H s ( R ) ( s > 1 6 ) which is an alternative proof of Theorem 1.1 ( n = 1 , p = 3 ) of Compaan et al. [Pointwise convergence of the Schrödinger flow. Int Math Res Not. 2021;1:596–647].

Visualization strategies to aid interpretation of high-dimensional genotoxicity data.

This article describes a range of high-dimensional data visualization strategies that we have explored for their ability to complement machine learning algorithm predictions derived from MultiFlow® assay results. For this exercise, we focused on seven biomarker responses resulting from the exposure of TK6 cells to each of 126 diverse chemicals over a range of concentrations. Obviously, challenges associated with visualizing seven biomarker responses were further complicated whenever there was a desire to represent the entire 126 chemical data set as opposed to results from a single chemical. Scatter plots, spider plots, parallel coordinate plots, hierarchical clustering, principal component analysis, toxicological prioritization index, multidimensional scaling, t-distributed stochastic neighbor embedding, and uniform manifold approximation and projection are each considered in turn. Our report provides a comparative analysis of these techniques. In an era where multiplexed assays and machine learning algorithms are becoming the norm, stakeholders should find some of these visualization strategies useful for efficiently and effectively interpreting their high-dimensional data.

Numerical solutions to hyperbolic Maxwell quasi-variational inequalities in Bean-Kim model for type-II superconductivity

This paper is devoted to the finite element analysis for the Bean-Kim model governed by the full 3D Maxwell equations. Describing type-II superconductivity at the macroscopic level, this model leads to a challenging coupled system consisting of the Faraday equation and a hyperbolic quasi-variational inequality (QVI) of the second kind with $L^1$-type nonlinearity, that arises explicitly from the magnetic field dependency in the critical current. With the involved Maxwell coupling in the 3D $\H(\curl)$-setting, the hyperbolic QVI character poses the primary challenge in the numerical investigation. Two mixed finite element methods based on implicit Euler and leapfrog time-stepping are proposed. On the one hand, the implicit Euler method results in a nonstandard system of curl-curl elliptic QVI with a first-order curl-type nonlinearity. Though the well-posedness of this system is guaranteed, its numerical realization is not straightforward and requires the use of a two-stage iteration process of high computational complexity. On the other hand, by approximating the electric and magnetic fields at two different time step levels, the leapfrog method turns out to be more suitable as it naturally eliminates the notorious QVI structure. More importantly, utilizing suited subdifferential and optimization techniques, we are able to prove an efficiently computable explicit formula for its exact solution in terms of the electric field, which makes its numerical computation substantially more favorable than the Euler method. As further advantages, the leapfrog method applies to broad scenarios involving low regular data of bounded variation (BV) in time for both the applied current source and the temperature distribution. Through nonstandard technical arguments tailored to the BV data, our analysis proves the conditional stability and, eventually, the uniform convergence of the proposed leapfrog method. This paper is closed by 3D numerical tests showcasing the reasonable and efficient performance of the proposed numerical solution.

On Neutrosophic Fuzzy Metric Space and Its Topological Properties

The present research introduces a novel concept termed “neutrosophic fuzzy metric space”, which extends the traditional metric space framework by incorporating the notion of neutrosophic fuzzy sets. A thorough investigation of various structural and topological properties within this newly proposed generalization of metric space has been conducted. Additionally, counterparts of well-known theorems such as the Uniform Convergence Theorem and the Baire Category Theorem have been established for this generalized metric space. Through rigorous analysis, a detailed understanding of its fundamental characteristics has been attained, illuminating its potential applications and theoretical significance.

Long time existence of the non-isentropic slightly compressible Navier-Stokes equations with boundary conditions

We investigate the long time existence of smooth solutions to the initial boundary value problem for the non-isentropic slightly compressible Navier–Stokes equations with slip or non-slip boundary conditions on the velocity. We verify that the compressible Navier–Stokes equations with boundary conditions admit a unique smooth solution on the time interval where the smooth solution of the incompressible Navier–Stokes equations exists, when the Mach number is sufficiently small. Moreover, we obtain the uniform convergence of smooth solutions for the compressible system toward those for the corresponding incompressible system on that time interval.

Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0<ɛ<<1) is multiplied the highest order derivative term. The problem’s solution shows an exponential boundary layer on the right side of the spatial domain for small values of ɛ. The solution and its derivatives are discussed together with their properties and bounds. The Crank–Nicolson approach for time discretization and the fitted operator non-standard finite difference method for space discretization are used to solve the time fractional singularly perturbed time delay problem under consideration. The comparison principle and solution bound are used to examine and analyze the scheme’s stability. The scheme’s uniform convergence is examined and demonstrated. The proposed scheme has a uniform convergence order of Mx−1+(Δt)2−γ. Two numerical illustrations for various values of ɛ are taken into consideration to validate the theoretical analysis of the scheme. The constructed scheme provides a better order of convergence and more accurate results than methods found in the literature.

Explicit constants in the nonuniform local limit theorem for Poisson binomial random variables

In a recent paper the authors proved a nonuniform local limit theorem concerning normal approximation of the point probabilities P(S=k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$P(S=k)$\\end{document} when S=∑i=1nXi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$S=\\sum_{i=1}^{n}X_{i}$\\end{document} and X1,X2,…,Xn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X_{1},X_{2},\\ldots ,X_{n}$\\end{document} are independent Bernoulli random variables that may have different success probabilities. However, their main result contained an undetermined constant, somewhat limiting its applicability. In this paper we give a nonuniform bound in the same setting but with explicit constants. Our proof uses Stein’s method and, in particular, the K-function and concentration inequality approaches. We also prove a new uniform local limit theorem for Poisson binomial random variables that is used to help simplify the proof in the nonuniform case.

Baire 1 Functions and the Topology of Uniform Convergence on Compacta

Let X be a Tychonoff topological space, B1(X,R) be the space of real-valued Baire 1 functions on X and τUC be the topology of uniform convergence on compacta. The main purpose of this paper is to study cardinal invariants of (B1(X,R),τUC). We prove that the following conditions are equivalent: (1) (B1(X,R),τUC) is metrizable; (2) (B1(X,R),τUC) is completely metrizable; (3) (B1(X,R),τUC) is Čech-complete; and (4) X is hemicompact. It is also proven that if X is a separable metric space with a non isolated point, then the topology of uniform convergence on compacta on B1(X,R) is seen to behave like a metric topology in the sense that the weight, netweight, density, Lindelof number and cellularity are all equal for this topology and they are equal to c= |B1(X,R)|. We find further conditions on X under which these cardinal invariants coincide on B1(X,R).

ℏ 4 quantum corrections to semiclassical transmission probabilities

The combination of vibrational perturbation theory with the replacement of the harmonic oscillator quantization condition along the reaction coordinate with an imaginary action to be used in the uniform semiclassical approximation for the transmission probability has been shown in recent years to be a practical method for obtaining thermal reaction rates. To date, this theory has been developed systematically only up to second order in perturbation theory. Although it gives the correct leading order term in an ℏ2 expansion, its accuracy at lower temperatures, where tunneling becomes important, is not clear. In this paper, we develop the theory to fourth order in the action. This demands developing the quantum perturbation theory up to sixth order. Remarkably, we find that the fourth order theory gives the correct ℏ4 term in the expansion of the exact thermal rate. The relative magnitude of the fourth order correction as compared to the second order term objectively indicates the accuracy of the second order theory. We also extend the previous modified second order theory to the fourth order case, creating an ℏ2 modified potential for this purpose. The resulting theory is tested on the standard examples—symmetric and asymmetric Eckart potentials and a Gaussian potential. The modified fourth order theory is remarkably accurate for the asymmetric Eckart potential.

Relative uniform convergence of p-absolutely summable sequence of functions

In this article we introduce the notion of relative uniform convergence of p-absolutely summable sequence of functions. We have investigated its different properties such as solidness, symmetric, convergence-free, monotonic etc. We have established some inclusion relations involving this sequence space.

On two consequences of CH established by Sierpiński

AbstractWe study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.

Using single-cell RNA sequencing and bulk RNA sequencing data to reveal a correlation between smoking and neutrophil activation in esophageal carcinoma patients.

Cigarette smoking is considered as a major risk factor for esophageal carcinoma (ESCA) patients. Neutrophil activation plays a key role in cancer development and progression. However, the relationship between cigarette smoking and neutrophils in ESCA patients remained unclear. Single-cell RNA sequencing (scRNA-seq) and bulk RNA sequencing data were obtained from public databases. Uniform manifold approximation and projection (UMAP) was used to perform downscaling and clustering based on scRNA-seq data. The module genes associated with smoking in ESCA patients were filtered by weighted gene co-expression network analysis (WGCNA). Using the "AUCell" package, the enrichment of different cell subpopulations and gene collections were assessed. "CellChat" and "CellphoneDB" were used to infer the probability and significance of ligand-receptor interactions between different cell subpopulations. WGCNA was performed to screened module genes associated with smoking in ESCA patients from MEdarkquosie, MEturquoise, and MEgreenyellow. Next, eight cell clusters were identified, and using the AUCell score, we determined that neutrophil clusters were more active in the gene modules associated with smoking in ESCA patients. Two neutrophil subtypes, Neutrophils 1 and Neutrophils 2, exhibited greater enrichment in inflammatory response regulation, intercellular adhesion, and regulation of T cell activation. Furthermore, we found that neutrophils may pass through AMPT-(ITGA5 + ITGB1) and ICAM1-AREG in order to promote the development of ESCA, and that the expression levels of the receptor genes insulin-degrading enzyme and ITGB1 were significantly and positively correlated with cigarette smoking per day. Combining smoking-related gene modules and scRNA-seq, the current findings revealed the heterogeneity of neutrophils in ESCA and a tumor-promoting role of neutrophils in the tumor microenvironment of smoking ESCA patients.