Uniform Convergence Research Articles

Influence of pore length non-uniformity on transport properties of random nanopore networks

The modeling of fluid transport in nanoporous materials has long attracted attention because of the numerous industrial applications of such materials, particularly in membrane-based separations, and others such as adsorptive gas separation and storage, catalysis, and the emerging area of nanofluidics. While effective medium theory has been widely used in modelling this transport, a critical approximation that all pores have the same length is commonly made, whose adequacy is unknown. Here we apply effective medium theory to a randomly overlapping capillary model having an inherent pore length distribution related to the pore radius distribution, and compare the network transport properties with those of the commonly used uniform pore length random network. We find that the consideration of nonuniform pore length distribution leads to significantly higher effective diffusivity and lower associated apparent tortuosity, mediated by the lower mean pore length of large pores due to their higher probability of overlap. The decrease in pore length with an increase in pore radius and the competing effects of adsorption and diffusion, lead to more complex behaviour of the transport properties with variation in temperature and relative standard deviation of the pore radius distribution. Our results clearly demonstrate the importance of considering nonuniformity of pore length, as the conventional uniform pore length approximation leads to significantly different and therefore misleading results.

Uniform Convergence of Fourier Series for Hölder Continuous Functions and Functions of Bounded Variation

This article discusses two significant results about the uniform convergence of Fourier series. One is the uniform convergence of the Fourier series for Hölder-α(α>0) continuous functions, while the other is the uniform convergence of the Fourier series for functions that can be expressed as the sum of finitely many continuous monotonic functions. The two results are particularly useful. Because Hölder continuity condition ensures a specific regularity that facilitates the analysis and provides a robust framework for proving uniform convergence and many practical functions can be decomposed into monotonic components, making the theorem broadly applicable. By providing detailed proofs and applications, this study demonstrates the significant implications of uniform convergence in both theoretical and practical contexts. Moreover, this work paves the way for future research in higher-dimensional Fourier analysis, non-periodic functions, and adaptive Fourier techniques, underscoring the ongoing relevance and versatility of Fourier series in modern mathematics. Several well-known results emerge as corollaries of these findings, highlighting the robustness and applicability of these theorems in Fourier analysis.

Singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme

In this paper, we proposed an accurate ϵ-uniformly convergent numerical method to solve singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme. The time-fractional derivative is defined in the sense of Caputo with order η∈(0,1). The time-fractional derivative is discretized by employing the Crank–Nicolson method on a uniform mesh, and an exponential fitted operator scheme along with the standard upwind method is used to mesh-grid the space domain. The truncation error and uniform stability of the discretized problems are examined in order to prove the parameter uniform convergence of the proposed scheme. It is demonstrated that the scheme is ϵ-uniformly convergent of order O((Δt)2−η+Δx), where Δt and Δx represent the step sizes of the time and space domains, respectively. Two numerical examples are provided in order to assess the accuracy of the suggested scheme and validate the theoretical concepts discussed. To demonstrate the efficiency of the numerical scheme presented, comparisons have been made with the numerical solution obtained by the finite difference method that exists in the literature. Consequently, it is observed that the results obtained by the present scheme are more accurate and have a better convergence rate.

Weighted approximations by sampling type operators: recent and new results

In this paper, we collect some recent results on the approximation properties of generalized sampling operators and Kantorovich operators, focusing on pointwise and uniform convergence, rate of convergence, and Voronovskaya-type theorems in weighted spaces of functions. In the second part of the paper, we introduce a new generalization of sampling Durrmeyer operators including a special function $\rho$ which satisfies certain assumptions. For the family of newly constructed operators, we obtain pointwise convergence, uniform convergence and rate of convergence for functions belonging to weighted spaces of functions.

Single-cell transcriptomic analysis reveals a decrease in the frequency of macrophage-RGS1high subsets in patients with osteoarticular tuberculosis

BackgroundCell subsets differentially modulate host immune responses to Mycobacterium tuberculosis (MTB) infection. However, the nature and functions of these subsets against osteoarticular tuberculosis (OTB) are unclear. Here, we aimed to understand the phenotypes and functions of immune cell subsets in patients with OTB using single-cell RNA sequencing (scRNA-Seq).MethodsPathological and healthy adjacent tissues were isolated from patients with OTB and subjected to scRNA-Seq. Unsupervised clustering of cells was performed based on gene expression profiles, and uniform manifold approximation and projection was used for clustering visualization.ResultsThirteen cell subsets were identified in OTB tissues. scRNA-seq datasets of patients and healthy controls (HCs) showed that infection changed the frequency of immune cell subsets in OTB tissues. Myeloid cell examination revealed nine subsets. The frequency of macrophage-RGS1high subsets decreased in OTB tissues; this increased MTB susceptibility in an SLC7A11/ferroptosis-dependent manner. Immunohistochemistry assays and flow cytometry for patients with OTB and osteoarticular bacterial infection (OBI) and HCs verified that the frequency of macrophage-RGS1high subset decreased in OTB tissues and blood samples, thereby distinguishing patients with OTB from HCs and patients with OBI.ConclusionThe macrophage-RGS1high subset levels were decreased in patients with OTB, and would be up-regulated after effective treatment. Therefore, the clinical significance of this study is to discover that macrophage-RGS1high subset may serve as a potential biomarker for OTB diagnosis and treatment efficacy monitoring.

Why and how to construct an epistemic justification of machine learning?

Consider a set of shuffled observations drawn from a fixed probability distribution over some instance domain. What enables learning of inductive generalizations which proceed from such a set of observations? The scenario is worthwhile because it epistemically characterizes most of machine learning. This kind of learning from observations is also inverse and ill-posed. What reduces the non-uniqueness of its result and, thus, its problematic epistemic justification, which stems from a one-to-many relation between the observations and many learnable generalizations? The paper argues that this role belongs to any complexity regularization which satisfies Norton’s Material Theory of Induction (MTI) by localizing the inductive risk to facts in the given domain. A prime example of the localization is the Lottery Ticket Hypothesis (LTH) about overparameterized neural networks. The explanation of MTI’s role in complexity regularization of neural networks is provided by analyzing the stability of Empirical Risk Minimization (ERM), an inductive rule that controls the learning process and leads to an inductive generalization on the given set of observations. In cases where ERM might become asymptotically unstable, making the justification of the generalization by uniform convergence unavailable, LTH and MTI can be used to define a local stability. A priori, overparameterized neural networks are such cases and the combination of LTH and MTI can block ERM’s trivialization caused by equalizing the strengths of its inductive support for risk minimization. We bring closer the investigation of generalization in artificial neural networks and the study of inductive inference and show the division of labor between MTI and the optimality justifications (developed by Gerhard Schurz) in machine learning.

Benchmarking clustering, alignment, and integration methods for spatial transcriptomics

BackgroundSpatial transcriptomics (ST) is advancing our understanding of complex tissues and organisms. However, building a robust clustering algorithm to define spatially coherent regions in a single tissue slice and aligning or integrating multiple tissue slices originating from diverse sources for essential downstream analyses remains challenging. Numerous clustering, alignment, and integration methods have been specifically designed for ST data by leveraging its spatial information. The absence of comprehensive benchmark studies complicates the selection of methods and future method development.ResultsIn this study, we systematically benchmark a variety of state-of-the-art algorithms with a wide range of real and simulated datasets of varying sizes, technologies, species, and complexity. We analyze the strengths and weaknesses of each method using diverse quantitative and qualitative metrics and analyses, including eight metrics for spatial clustering accuracy and contiguity, uniform manifold approximation and projection visualization, layer-wise and spot-to-spot alignment accuracy, and 3D reconstruction, which are designed to assess method performance as well as data quality. The code used for evaluation is available on our GitHub. Additionally, we provide online notebook tutorials and documentation to facilitate the reproduction of all benchmarking results and to support the study of new methods and new datasets.ConclusionsOur analyses lead to comprehensive recommendations that cover multiple aspects, helping users to select optimal tools for their specific needs and guide future method development.

A universal optoelectronic imaging platform with wafer-scale integration of two-dimensional semiconductors

Photodetectors (PDs) are crucial in modern society, as they enable the detection of a diverse range of light-based signals. With the exponential increase in their development, materials are being used to create a wide range of PDs that play critical roles in enabling various applications and technologies. Image sensor technology has been hindered due to the lack of a universal system that can integrate all types of PDs with silicon-based readout integrated circuits (ROICs). To address this issue, we conducted experiments using two-dimensional materials as an example. We fabricated high-performance MoS2/MoTe2-based photodetectors and identified the most suitable ROICs to pair with them. This established a solid foundation for further research in the field of image sensors. We developed and implemented a versatile testing system that uses a printed circuit board to connect the PD and ROIC. After the ROIC generates the sampled signal, it is collected and processed by algorithms, which overcome device uniformity limitations and produce a high-quality image that is visible to the naked eye. This universal system can be used with a wide range of PD and ROIC types made from different materials, making it highly convenient for diverse testing applications and the development of diverse image sensor types. This robust new platform is expected to spur further innovation and advancements in this rapidly developing field.

Crop Water Status Analysis from Complex Agricultural Data Using UMAP-Based Local Biplot

To optimize growth and management, precision agriculture relies on a deep understanding of agricultural dynamics, particularly crop water status analysis. Leveraging unmanned aerial vehicles, we can efficiently acquire high-resolution spatiotemporal samples by utilizing remote sensors. However, non-linear relationships among data features, localized within specific subgroups, frequently emerge in agricultural data. Interpreting these complex patterns requires sophisticated analysis due to the presence of noise, high variability, and non-stationarity behavior in the collected samples. Here, we introduce Local Biplot, a methodological framework tailored for discerning meaningful data patterns in non-stationary contexts for precision agriculture. Local Biplot relies on the well-known uniform manifold approximation and projection method, such as UMAP, and local affine transformations to codify non-stationary and non-linear data patterns while maintaining interpretability. This lets us find important clusters for transformation and projection within a single global axis pair. Hence, our framework encompasses variable and observational contributions within individual clusters. At the same time, we provide a relevance analysis strategy to help explain why those clusters exist, facilitating the understanding of data dynamics while favoring interpretability. We demonstrated our method’s capabilities through experiments on both synthetic and real-world datasets, covering scenarios involving grass and rice crops. Moreover, we use random forest and linear regression models to predict water status variables from our Local Biplot-based feature ranking and clusters. Our findings revealed enhanced clustering and prediction capability while emphasizing the importance of input features in precision agriculture. As a result, Local Biplot is a useful tool to visualize, analyze, and compare the intricate underlying patterns and internal structures of complex agricultural datasets.

Repetitive Impedance Learning-Based Physically Human-Robot Interactive Control.

Model-based impedance learning control can provide variable impedance regulation for robots through online impedance learning without interaction force sensing. However, the existing related results only guarantee the closed-loop control systems to be uniformly ultimately bounded (UUB) and require the human impedance profiles being periodic, iteration-dependent, or slowly varying. In this article, a repetitive impedance learning control approach is proposed for physical human-robot interaction (PHRI) in repetitive tasks. The proposed control is composed of a proportional-differential (PD) control term, an adaptive control term, and a repetitive impedance learning term. Differential adaptation with projection modification is designed for estimating robotic parameters uncertainties in the time domain, while fully saturated repetitive learning is proposed for estimating time-varying human impedance uncertainties in the iterative domain. Uniform convergence of tracking errors is guaranteed by the PD control and the use of projection and full saturation in the uncertainties estimation and is theoretically proved based on a Lyapunov-like analysis. In impedance profiles, the stiffness and damping are composed of an iteration-independent term and an iteration- dependent disturbance, which are estimated by repetitive learning and compressed by the PD control, respectively. Therefore, the developed approach can be applied to the PHRI where iteration-dependent disturbances exist in the stiffness and damping. The control effectiveness and advantages are validated by simulations on a parallel robot in a repetitive following task.

Skew-product attractors of non-autonomous Caputo fractional differential equations.

A non-autonomous Caputo fractional differential equation (FDE) of order α∈(0,1) in Rd with a driving system on a compact base space P is shown to generate a skew-product semi-flow on Cα×P, where Cα is the space of continuous functions f:R+→Rd with a weighted norm giving uniform convergence on compact time subsets. This skew-product semi-flow is then shown to have a bounded and closed attractor when the vector field of the Caputo FDE satisfies a uniform dissipativity condition. It attracts bounded sets of constant initial functions f in here Cα. The properties and structure of this attractor in Cα×P are also discussed.

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ψ$$ \psi $$‐Bernstein–Kantorovich operators

In this article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function and investigate their approximation properties. By choosing an appropriate function , the order of approximation of our operators to a function is at least as good as the classical Bernstein–Kantorovich operators on the interval . We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided.

The stabilized nonconforming virtual element method for the Darcy–Stokes problem

A stabilized nonconforming virtual element method is designed in order to solve the Darcy–Stokes problem, which preserves a divergence-free approximation to the velocity. The same degrees of freedom as the usual Crouzeix–Raviart-type virtual element is used, but a different virtual element space is obtained by modifying the conforming Stokes virtual element with the H1-projection operator. The proposed stabilized scheme contains two jump penalty terms over edges. One is the penalty for jumps of velocity approximation and the other one is the penalty for jumps of its normal component. We analyze this method’s well-posedness and prove its uniform convergence in a discrete energy norm. Finally, we verify the validity of this stabilized scheme by some numerical experiments.

A Numerical Method for Investigating Fractionali Volterra-Fredholm Integro-Differential Model

In this article, we investigate the fractional Volterra-Fredholm integro-differential equations. These equations appear in several applications such as control theory, biology, and particle dynamics in physics. We derive a numerical method based on the operational matrix method to solve this class of integro-differential equations. We prove the existence and uniqueness of the ex-act solution. Additionally, we demonstrate the uniform convergence of the numerical solutions to the exact solution. We present several numerical examples to show the numerical efficiency of the proposed method. In the first example, we choose a linear problem and find that the approximate solution converges to the exact solution when the number of block pulse functions is very large. In the next two examples, we consider the nonlinear case and compute the L2-local truncation error since exact solutions are not available. The error was of order 10−12. Furthermore, we sketch the graph of the approximate solutions for different values of the fractional derivative to observe the influence of the fractional derivative on the profile of the solutions. Theoretical and numerical results show that the proposed method is accurate and can be applied to other nonlinear problems in science.

Enhanced classification of pyrite generations based on mineral chemistry using uniform manifold approximation and projection (UMAP)

Trace element signatures are tracked to unravel the genetic history of ore deposits in several mineral systems. This is particularly true for pyrite, a ubiquitous component of many ore-forming systems, including orogenic gold deposits. Here, we critically compare the efficacy of utilizing Uniform Manifold Approximation and Projection (UMAP) versus Principal Component Analysis (PCA) as dimensionality reduction tools applied to a 31 element dataset collected using Laser Ablation Inductively Coupled Mass Spectrometry of pyrite grains from the Kibali gold district (Democratic Republic of Congo). Because of the non-linearity inherent to mineral chemistry and because of its superior preservation of local (distances within clusters) and global (separation between clusters) data relationships, the UMAP approach outperforms dimensionality reduction by PCA. We further present a workflow in which UMAP dimensionality reduction is followed by k-means clustering to guide the classification of pyrite generations in the Kibali case study. Validating this approach with trace elements and UMAP + k-means on a seed-by-seed petrography basis shows that the workflow significantly enhances the original pyrite classification, previously based on texture. This study thus emphasizes the utility of employing advanced statistical analysis methods to capture the intricate nature of pyrite formation. These findings will shape best practices for handling large multi-element datasets in pyrite mineral chemistry studies and are extrapolatable to other mineral systems in which trace element signatures are used to infer the conditions of ore deposit genesis.

Enhancing the rationale of convolutional neural networks for glitch classification in gravitational wave detectors: a visual explanation

In the pursuit of detecting gravitational waves, ground-based interferometers (e.g. LIGO, Virgo, and KAGRA) face a significant challenge: achieving the extremely high sensitivity required to detect fluctuations at distances significantly smaller than the diameter of an atomic nucleus. Cutting-edge materials and innovative engineering techniques have been employed to enhance the stability and precision of the interferometer apparatus over the years. These efforts are crucial for reducing the noise that masks the subtle gravitational wave signals. Various sources of interference, such as seismic activity, thermal fluctuations, and other environmental factors, contribute to the total noise spectra characteristic of the detector. Therefore, addressing these sources is essential to enhance the interferometer apparatus’s stability and precision. Recent research has emphasised the importance of classifying non-stationary and non-Gaussian glitches, employing sophisticated algorithms and machine learning methods to distinguish genuine gravitational wave signals from instrumental artefacts. The time-frequency-amplitude representation of these transient disturbances exhibits a wide range of new shapes, variability, and features, reflecting the evolution of interferometer technology. In this study, we developed a convolutional neural network model to classify glitches using spectrogram images from the Gravity Spy O1 dataset. We employed score-class activation mapping and the uniform manifold approximation and projection algorithm to visualise and understand the classification decisions made by our model. We assessed the model’s validity and investigated the causes of misclassification from these results.

A 2-dimensional real Banach space with constant of analyticity less than one

We show that on the real 2-dimensional Banach space ℓ12 there is an analytic function f:Bℓ12→R such that its power series at origin has radius of uniform convergence one, but for some a∈Bℓ12 the power series centred at that point has radius of uniform convergence strictly less than 1−‖a‖. This result highlights a fundamental distinction in real analytic functions (compared to complex analytic functions), where the radius of analyticity can differ from the radius of uniform convergence. Moreover, this example provides the first non-trivial upper bound for the constant of analyticity.

A parameter uniform numerical method on a Bakhvalov type mesh for singularly perturbed degenerate parabolic convection–diffusion problems

AbstractWe are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second step, we discretize in space using an upwind scheme on a Bakhvalov-type mesh and prove its uniform convergence using the truncation error and barrier function approach, wherein several bounds derived for the mesh step sizes are used. Numerical results for a couple of examples are presented to support the theoretical bounds derived in the paper.

Compressed representation of brain genetic transcription.

The architecture of the brain is too complex to be intuitively surveyable without the use of compressed representations that project its variation into a compact, navigable space. The task is especially challenging with high-dimensional data, such as gene expression, where the joint complexity of anatomical and transcriptional patterns demands maximum compression. The established practice is to use standard principal component analysis (PCA), whose computational felicity is offset by limited expressivity, especially at great compression ratios. Employing whole-brain, voxel-wise Allen Brain Atlas transcription data, here we systematically compare compressed representations based on the most widely supported linear and non-linear methods-PCA, kernel PCA, non-negative matrix factorisation (NMF), t-stochastic neighbour embedding (t-SNE), uniform manifold approximation and projection (UMAP), and deep auto-encoding-quantifying reconstruction fidelity, anatomical coherence, and predictive utility across signalling, microstructural, and metabolic targets, drawn from large-scale open-source MRI and PET data. We show that deep auto-encoders yield superior representations across all metrics of performance and target domains, supporting their use as the reference standard for representing transcription patterns in the human brain.