Uniform Convergence Research Articles

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.

On uniform polynomial approximation

On uniform polynomial approximation

Uniform Asymptotic Approximations for the Phase Plane Trajectories of the SIR Model with Vital Dynamics

Uniform Asymptotic Approximations for the Phase Plane Trajectories of the SIR Model with Vital Dynamics

A new method for solving fractional optimal control problems

In this article, we present a novel approach that utilizes the operational matrix method to solve fractional optimal control problems. We address the challenge of missing initial conditions for the co-state function by incorporating the linear shooting method. Our objective is to provide a method that not only enhances computational efficiency and reduces cost but also improves accuracy and usability. With our new approach, we can calculate the coefficients of the expansion solution without the need to solve an algebraic system. These coefficients can be generated explicitly or through an iterative process. We provide a proof of the uniform convergence of the approximate series of functions to the unique solution of the optimal control problem. To demonstrate the effectiveness of our proposed method, we solve several numerical examples and compare the results with those obtained by other researchers. Additionally, we extend the application of this new method to solve problems with multiple states, thereby expanding its scope to a wider range of problem domains. This allows us to address diverse scenarios using the same efficient and accurate approach. Through our research, we contribute to the development of a powerful and versatile method for solving fractional optimal control problems, offering significant advantages over existing techniques.

Torsional buckling of a tape-spring: Review and renew

Tape-springs – straight strips with transversely curved cross-sections – undergo a flattening of the cross-section upon bending, and accompanying torsional buckling in equal-sense bending which then “recedes” at greater deformation. The exact analytical solution for tape-spring bending was solved by Mansfield; however, the resulting expression for the twisting curvature at the onset and recession of torsional buckling cannot be solved in closed-form. Approximate closed-form expressions are therefore sought: firstly, by utilising Taylor series approximations of Mansfield’s extensive functions; and secondly, by assuming a uniform transverse curvature, thereby rendering the problem algebraic in nature, and deriving new governing expressions. Further approximations assume the cross-section does not deform upon bending, and that the tape-spring remains a developable surface in the inextensible limit of zero thickness. All approximate solutions are shown to agree closely with Mansfield’s solution at small curvatures, with the uniform curvature approximation also capturing the recession of torsional buckling to a reasonable degree of accuracy. The accuracy of the uniform curvature approximation is related to a pertinent dimensionless parameter, which governs the size of a boundary layer in Mansfield’s exact solution: as this parameter tends to infinity, the uniform curvature approximation and Mansfield’s exact solution are shown to converge; as it tends to zero, all solutions are shown to converge on the inextensible limit.

Integrating single-cell RNA-seq to identify fibroblast-based molecular subtypes for predicting prognosis and therapeutic response in bladder cancer.

Bladder cancer (BLCA) is a highly aggressive and heterogeneous disease, posing challenges for diagnosis and treatment. Cancer immunotherapy has recently emerged as a promising option for patients with advanced and drug-resistant cancers. Fibroblasts, a significant component of the tumor microenvironment, play a crucial role in tumor progression, but their precise function in BLCA remains uncertain. Single-cell RNA sequencing (scRNA-seq) data for BLCA were obtained from the Gene Expression Omnibus database. The R package "Seurat" was used for processing scRNA-seq data, with uniform manifold approximation and projection (UMAP) for downscaling and cluster identification. The FindAllMarkers function identified marker genes for each cluster. Differentially expressed genes influencing overall survival (OS) of BLCA patients were identified using the limma package. Differences in clinicopathological characteristics, immune microenvironment, immune checkpoints, and chemotherapeutic drug sensitivity between high- and low-risk groups were investigated. RT-qPCR and immunohistochemistry validated the expression of prognostic genes. Fibroblast marker genes identified three molecular subtypes in the testing set. A prognostic signature comprising ten genes stratified BLCA patients into high- and low-score groups. This signature was validated in one internal and two external validation sets. High-score patients exhibited increased immune cell infiltration, elevated chemokine expression, and enhanced immune checkpoint expression but had poorer OS and a reduced response to immunotherapy. Six sensitive anti-tumor drugs were identified for the high-score group. RT-qPCR and immunohistochemistry showed that CERCAM, TM4SF1, FN1, ANXA1, and LOX were highly expressed, while EMP1, HEYL, FBN1, and SLC2A3 were downregulated in BLCA. A novel fibroblast marker gene-based signature was established, providing robust predictions of survival and immunotherapeutic response in BLCA patients.

Connecting palettes — An integrated spectroscopic analysis and UMAP visualisation of Wu Guanzhong's paint palette and his painting

A paint palette of the 20th-century Chinese modern artist Wu Guanzhong (1919–2010) was studied with an integrated spectroscopic approach, macro-scale non-invasive techniques hyperspectral imaging (HSI) and macro X-ray fluorescence (MA-XRF), complemented with Raman spectroscopy. The palette is revealed to be comprised of synthetic inorganic and organic pigments, typical of the 20th century modern palette that is dominated by azo and phthalocyanine pigments. HSI datasets of the paint palette and those of Wu's oil painting ‘Xidi Village’ (2001) were then processed within one unsupervised Uniform Manifold Approximation and Projection (UMAP) model, in an attempt to identify connections between the artefact and the artwork, so that known pigment results of the palette may provide preliminary pigment assignment for the painting. This study describes the steps taken to establish this automated assisted pigment analysis workflow, which involves processing multiple HSI datasets within one UMAP model, density clustering, non-negative least square (NNLS) fitting to extract endmembers (EM) spectra and to generate distribution maps. The approach to perform micro-sampling pigment analysis on the artist's palette (or artist's materials) is considered less invasive to the integrity than that from artworks, the results of which have demonstrated to serve as a critical resource to support further pigment analysis studies of the artist's paintings via non-invasive analytical techniques.

Approximation of the Hilbert transform on the half–line

The paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule.

Uniform Approximation of Continuous Couplings

AbstractWe study the approximation of non-negative multi-variate couplings in the uniform norm while matching given single-variable marginal constraints.

Unsupervised Machine Learning‐Derived Anion‐Exchange Membrane Polymers Map: A Guideline for Polymers Exploration and Design

AbstractAlthough anion‐exchange membranes (AEMs) are commonly used in fuel cells and water electrolyzers, their widespread commercialization is hindered by problems such as low anion conductivity and durability. Moreover, the development of high‐performance AEMs remains complex and time consuming. Here, we address these challenges by proposing an innovative approach for the efficient design and screening of AEM polymers using unsupervised machine learning. Our model, which combines principal component analysis with uniform manifold approximation and projection, generates an intuitive map that clusters AEM polymers based on structural similarities without any predefined knowledge regarding anion conductivity or other experimentally derived variables. As a powerful navigation tool, this map provides insights into promising main‐chain structures, such as poly(arylene alkylene)s with consistently high conductivity and polyolefins with exceptional performance depending on the substituent. Furthermore, assisted by key molecular descriptors, inverse analysis with this model allows targeted design and property prediction before synthesis, which will significantly accelerate the discovery of novel AEM polymers. This work represents a paradigm shift not only in AEM research but also generally in materials research, moving from black‐box predictions toward interpretable guidelines that foster collaboration between researchers and machine learning for efficient and informed material development.

Failure of almost uniform convergence for noncommutative martingales

Failure of almost uniform convergence for noncommutative martingales

Superconvergence analysis of the conforming discontinuous Galerkin method on a Bakhvalov-type mesh for singularly perturbed reaction–diffusion equation

The conforming discontinuous Galerkin (CDG) method maximizes the utilization of all degrees of freedom of the discontinuous Pk polynomial to achieve a convergence rate two orders higher than its counterpart conforming finite element method employing continuous Pk element. Despite this superiority, there is little theory of the CDG methods for singular perturbation problems. In this paper, superconvergence of the CDG method is studied on a Bakhvalov-type mesh for a singularly perturbed reaction–diffusion problem. For this goal, a pre-existing least squares method has been utilized to ensure better approximation properties of the projection. On the basis of that, we derive superconvergence results for the CDG finite element solution in the energy norm and L2-norm and obtain uniform convergence of the CDG method for the first time.

Phenotypic Landscape of Immune Cells in Sepsis: Insights from High-Dimensional Mass Cytometry.

Understanding the sepsis-induced immunological response can be facilitated by identifying phenotypic changes in immune cells at the single-cell level. Mass cytometry, a novel multiparametric single-cell analysis technique, offers considerable benefits in characterizing sepsis-induced phenotypic changes in peripheral blood mononuclear cells. Here, we analyzed peripheral blood mononuclear cells from 20 sepsis patients and 10 healthy donors using mass cytometry and employing 23 markers. Both manual gating and automated clustering approaches (PhenoGraph) were used for cell identification, complemented by uniform manifold approximation and projection (UMAP) for dimensionality reduction and visualization. Our study revealed that patients with sepsis exhibited a unique immune cell profile, marked by an increased presence of monocytes, B cells, and dendritic cells, alongside a reduction in natural killer (NK) cells and CD4/CD8 T cells. Notably, significant changes in the distributions of monocytes and B and CD4 T cells were observed. Clustering with PhenoGraph unveiled the subsets of each cell type and identified elevated CCR6 expression in sepsis patients' monocyte subset (PG#5), while further PhenoGraph clustering on manually gated T and B cells discovered sepsis-specific CD4 T cell subsets (CCR4low CD20low CD38low) and B cell subsets (HLA-DRlow CCR7low CCR6high), which could potentially serve as novel diagnostic markers for sepsis.

Asymptotic matching the self-consistent expansion to approximate the modified Bessel functions of the second kind

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem—too much freedom! The SCE can be used to generate an enormous number of approximations but distinguishing the superb approximations from the deficient ones can only be achieved after the fact by comparison to experimental or numerical results. Here, we propose a method of using the SCE to a priori obtain uniform approximations, namely asymptotic matching. If the asymptotic behaviour of a problem can be identified, then the approximations generated by the SCE can be tuned to asymptotically match the desired behaviour and this can be used to obtain uniform approximations over the entire domain of consideration, without needing to resort to empirical comparisons. We demonstrate this method by applying it to the task of obtaining uniform approximations of the modified Bessel functions of the second kind, Kα(x) .

Smooth Logistic Real and Complex, Ordinary and Fractional Neural Network Approximations over Infinite Domains

In this work, we study the univariate quantitative smooth approximations, including both real and complex and ordinary and fractional approximations, under different functions. The approximators presented here are neural network operators activated by Richard’s curve, a parametrized form of logistic sigmoid function. All domains used are obtained from the whole real line. The neural network operators used here are of the quasi-interpolation type: basic ones, Kantorovich-type ones, and those of the quadrature type. We provide pointwise and uniform approximations with rates. We finish with their applications.

Quantitative Uniform Approximation by Activated Singular Operators

In this article, we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp, and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness.