Uniform Convergence Research Articles

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.

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 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 Atf3 and Apold1. Results: Methylene blue staining found the distances of buccal and palatal cement-enamel junction-alveolar bone crest (CEJ-ABC) 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 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 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.

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.

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.

Exercise Mitigates Angiotensin II-Mediated Sympathetic Nerve-Macrophage Interaction in Aortic Adventitia Leading to Vascular Remodeling

Large artery stiffness is a significant risk factor for cardiovascular diseases. Our previous study suggested that angiotensin II (Ang II) stimulates sympathetic innervation in aortic adventitia (AA) leading to elevated mean arterial blood pressure and increased arterial fibrosis, while voluntary wheel-running (VWR) reduces them. The aim of this study is to explore mechanism under Ang II-induced the aortic remodeling through sympathetic nerves and immune cell interactions using transcriptomics. We isolated cells in AA from Ang II (1.5 mg/kg/day) vs. Vehicle (Saline) infused C57BL/6 male mice (n=3 of each group), and we performed single-cell RNA-sequencing (scRNAseq). We profiled 23,419 cells after quality control and filtering steps and annotated 9 cell types in the AA employing a uniform manifold approximation and projection (UMAP) to visualize the remaining cells and the Leiden cluster algorithm. Firstly, we identified the Adrb2 gene for β2-adrenergic receptor (β2-ADR) was predominantly expressed in macrophages/monocytes as compared to other cell types, and significantly more expressed in macrophages vs. monocytes (adjusted p-value, &lt;0.05). We validated the transcriptomic results using immunostaining for β2-ADR and synapsin with CD68 in AA. In addition, Ang II increased AA macrophage proportion which displays M1 phenotype (pro-inflammatory) (adjusted p-value, &lt;0.05). These findings suggest that Ang II promotes M1 like macrophages which interact with sympathetic nerves in AA. Next, we performed Gene Ontology (GO) enrichment test in macrophages and fibroblasts. Ang II Macrophages were enriched in GO terms with inflammation and cytokines, and Ang II fibroblasts were enriched in GO terms involved in extracellular matrix deposition. The GO results indicate that Ang II induced inflammatory macrophages activates fibroblasts to accumulate ECM deposition in AA. Congruent with the scRNAseq results, we found that number of macrophages, ECM accumulation, and vascular stiffness measured by pulse wave velocity were increased with Ang II vs. Vehicle (all, p &lt;0.05), whereas interestingly VWR ameliorated the Ang II effects. This study provides the evidence that Ang II promotes aortic remodeling by interaction between sympathetic nerves and macrophage which results in ECM deposition in AA using transcriptomics approach, while VWR countermeasure to these detrimental effects. This is the full abstract presented at the American Physiology Summit 2024 meeting and is only available in HTML format. There are no additional versions or additional content available for this abstract. Physiology was not involved in the peer review process.

Convergence order of one point large deviations rate functions for backward Euler method of stochastic delay differential equations with small noise

The present work focuses on the convergence order of one point large deviations rate functions for the backward Euler method of stochastic delay differential equations (SDDEs) with small noise. The drift and diffusion coefficients of SDDEs are allowed to grow super-linearly with respect to both the state variables and the delay variables. It is shown that the backward Euler method satisfies the one point large deviations principle (LDP) with good rate function. Further, the local uniform convergence orders of the one point large deviations rate function are derived by means of the equivalent characterizations for the original and numerical rate functions. These theoretical results are finally supported by a series of numerical experiments.

Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay

A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically.

Stability analysis and numerical evaluations of a COVID-19 model with vaccination

A novel (nonlinear) mathematical model for the transmission of Coronavirus 19 (COVID-19) with eight compartments and considering the impact of vaccination is examined in this manuscript. The qualitative behavior of the system such as the boundedness of solutions, the basic reproduction number, and the stability of the equilibrium points is investigated in detail. Some domestic real data collected from the Kerman University of Medical Science (KUMC) is used to estimate the parameters of the proposed model. We predict the dynamical behavior of the system through numerical simulations based on a combined spectral matrix collocation methodology. In this respect, we first linearize the nonlinear system of equations by the method of quasilinearization (QLM). Hence, the shifted version of Chebyshev polynomials of the second kind (SCPSK) is utilized along with the domain-splitting strategy to acquire the solutions of the system over a long time interval. The uniform convergence and upper bound estimation of the SCPSK bases are proved in a rigorous manner. Moreover, the technique of residual error functions is used to testify the accuracy of the QLM-SCPSK method. The presented numerical results justify the robustness and good accuracy of the QLM-SCPSK technique. The achieved numerical orders of convergence indicate that the QLM-SCSK algorithm has exponential rate of convergence. Using the linearization technique in one hand and the domain-splitting strategy on the other hand, enable us to predict the behaviour of similar disease problems with high accuracy and maximum efficiency on an arbitrary domain of interest.

K-nearest-neighbors induced topological PCA for single cell RNA-sequence data analysis

Single-cell RNA sequencing (scRNA-seq) is widely used to reveal heterogeneity in cells, which has given us insights into cell–cell communication, cell differentiation, and differential gene expression. However, analyzing scRNA-seq data is a challenge due to sparsity and the large number of genes involved. Therefore, dimensionality reduction and feature selection are important for removing spurious signals and enhancing downstream analysis. Traditional PCA, a main workhorse in dimensionality reduction, lacks the ability to capture geometrical structure information embedded in the data, and previous graph Laplacian regularizations are limited by the analysis of only a single scale. We propose a topological Principal Components Analysis (tPCA) method by the combination of persistent Laplacian (PL) technique and L2,1 norm regularization to address multiscale and multiclass heterogeneity issues in data. We further introduce a k-Nearest-Neighbor (kNN) persistent Laplacian technique to improve the robustness of our persistent Laplacian method. The proposed kNN-PL is a new algebraic topology technique which addresses the many limitations of the traditional persistent homology. Rather than inducing filtration via the varying of a distance threshold, we introduced kNN-tPCA, where filtrations are achieved by varying the number of neighbors in a kNN network at each step, and find that this framework has significant implications for hyper-parameter tuning. We validate the efficacy of our proposed tPCA and kNN-tPCA methods on 11 diverse benchmark scRNA-seq datasets, and showcase that our methods outperform other unsupervised PCA enhancements from the literature, as well as popular Uniform Manifold Approximation (UMAP), t-Distributed Stochastic Neighbor Embedding (tSNE), and Projection Non-Negative Matrix Factorization (NMF) by significant margins. For example, tPCA provides up to 628%, 78%, and 149% improvements to UMAP, tSNE, and NMF, respectively on classification in the F1 metric, and kNN-tPCA offers 53%, 63%, and 32% improvements to UMAP, tSNE, and NMF, respectively on clustering in the ARI metric.

Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation

Abstract This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation. Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the L 2 L^{2} -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations 46 (2021), 9, 1601–1627]. Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris 350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation. Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936]. Finally, we study the nonlinear smoothing and uniform convergence of 𝑛-dimensional generalized Zakharov–Kuznetsov equation with n ≥ 3 n\geq 3 .