Uniform Convergence Research Articles

Approximation in Weighted Space with Generalized Max-Product Type Favard-Szász-Mirakyan-Durrmeyer Operators

In this paper, we explore the uniform approximation of functions using Generalized Favard- Szász-Mirakyan -Durrmeyer operators of the max-product type with specific exponential weighted spaces. We analyze the approximation rate with an appropriate continuity modulus.

Numerical solution of time-dependent two-parameter singularly perturbed problems via trigonometric quintic B-spline collocation technique

Abstract This paper introduces a novel algorithm for solving time-dependent two-parameter singularly perturbed parabolic convection-diffusion-reaction equations with Dirichlet boundary conditions. The algorithm is formulated using the Crank-Nicolson (CN) scheme for the temporal derivative discretization. Then, the Trigonometric Quintic B-spline (TQBS) is applied to approximate the state variable and its spatial derivatives on nonuniform collocation points. We conducted a comprehensive convergence analysis and stability of the proposed method and proved that the scheme achieved a parameter uniform convergence of approximately fourth order in space and second order in time. To make additional evidence to support the theoretical findings and further assess the proposed method, we implemented the numerical algorithm to solve three test examples. Furthermore, using these test examples, we demonstrated the parameter-uniform convergence of the proposed numerical scheme.

A note on the space of all Toeplitz operators

Abstract We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.

IMMU-03. EXPLORING CANCER-SPECIFIC T CELLS AND ANTIGENS IN GLIOBLASTOMA BASED ON SINGLE-CELL SEQUENCING OF CD8+ TILS

Abstract BACKGROUND The effectiveness of cancer immunotherapy against glioblastoma (GBM) remains limited. This study aims to identify cancer-specific antigens to develop antigen-based cancer immunotherapies for GBM. We investigated candidate tumor antigen-specific T cells in GBM by single-cell RNA sequencing (scRNA-seq) and single-cell TCR sequencing (scTCR-seq), and we explored candidate antigens through genetic analysis of tumor tissues. METHODS Flow cytometry analysis was conducted on fresh tumor digest samples to evaluate tumor-infiltrating lymphocytes (TILs) of GBM. Single-cell RNA and TCR sequencing were performed on CD8+ T cells in TILs. Both data generated by Cell Ranger software were loaded into Seurat at R studio. The dimensional reduction for clustering was performed with uniform manifold approximation and projection (UMAP). Additionally, we performed bulk RNA sequencing (RNA-seq) and whole exome sequencing (WES) on tumor tissues. RESULTS Two out of 20 patients (10%) exhibited abundant TILs in GBM. From these two patients, approximately 20,000 CD8+ T cells in total were analyzed in single-cell analysis. Clustering based on UMAP revealed a predominance of clusters expressing exhaustion markers, with high TCR clonality centered on exhausted T cell (TEX) clusters, like our previous date of lung cancer TILs recognizing tumor antigens. Moreover, within the TEX clusters, the expression of CXCL13, often reported as an antigen-specific marker in recent years, was significantly higher in our data compared to two publicly available datasets. Subsequently, based on RNA-seq and WES, 61 candidate neoantigens were estimated using machine-learning prediction algorithms from NEC Corporation. Furthermore, 5 overexpressed cancer/testis antigens, and 3 bacterial species-derived microbial peptides using Kraken2 were selected as antigen candidates. CONCLUSIONS We identified prospective tumor antigen-specific T cell subsets with high CXCL13 expression from two GBM patients. We plan to identify cancer-specific T cells and antigens from these candidates by T cell activation assay.

BIOM-04. SINGLE-CELL RNA SEQUENCING PROTOCOL OF THE HUMAN METASTATIC PROSTATE SPINE TUMOR MICROENVIRONMENT

Abstract INTRODUCTION 90% of spine tumor patients can develop metastases with debilitating complications, such as sensorimotor dysfunction, paralysis, and spinal instability. As spine tumor patients have limited eligibility to participate in clinical trials and have heterogenous tumor pathology, there is a distinct lack of therapeutic targets and prognostication markers of disease. Thus, there is a distinct need for molecular profiling of spine tumors. OBJECTIVE We describe an OR-to-bench-top processing of spine tumors for single-cell omic studies, with samples from a metastatic prostate cancer patient to demonstrate the ability to compare molecular markers and microenvironments of difficult-to-process spine tumors and adjacent normal tissue. METHODS Under IRB protocol #Pro00101198, patients undergoing spine surgery with the Department of Neurosurgery at Duke University were consented for tissue collection. Vertebral osseous metastases and adjacent normal tissue were confirmed histologically (H&E, immunohistochemistry) and radiologically (MRI, PET, CT, and x-ray). Tissues were manually and enzymatically homogenized in serum-free media with collagenase A and DNAse I to obtain a single cell suspension. Cells were then filter sterilized, erythrolyzed, counted to determine live-death ratio, resuspended to 10-20 million cells per mL, then cryopreserved for downstream standard library preparation for single-cell sequencing using the 10X genomics platform. Downstream analysis was performed using Seurat pipeline, InferCNV analysis, and SingleR. RESULTS P404S and PSA expression confirmed tumor was prostate in origin. Radiological imaging and H&E staining confirmed the presence of T7 osseous metastasis. Uniform manifold approximation and project (UMAP) plots identified 12 unique clusters of immune cell subpopulations between normal and tumorous vertebrae, with their own subsets of differentially expressed genes. CONCLUSION We have developed a protocol to generate granular single cell molecular profiling for spine tumors and metastases. Applications of this protocol can aid in identifying therapeutic targets, molecular predictors of disease, and potentiate increased eligibility of patients for clinical trials.

BIOM-67. DIFFERENTIAL DNA METHYLATION PATTERNS IN THE CSF OF PATIENTS WITH DIFFUSE GLIOMAS

Abstract Studies have shown that different central nervous system (CNS) tumor types exhibit unique DNA methylation profiles. DNA methylation-based machine learning classification of CNS tumor tissue has been demonstrated to be highly-accurate in differentiating between tumor types, including between diffuse glioma subtypes. While similar analyses have been performed using cell-free DNA (cfDNA) from blood, few studies have looked at the potential of DNA methylation as a biomarker for diffuse gliomas using CSF-derived cfDNA. In this study, we analyzed CSF cfDNA methylation patterns in diffuse glioma patients. DNA methylation profiling of CSF of patients with glioblastoma, IDH-wildtype (GBM) (n=37), IDH-mutant diffuse gliomas (n=12), and non-neoplastic conditions (n=5) was performed using cell-free methylated DNA immunoprecipitation and high-throughput sequencing (cfMeDIP-seq). Methylation quantification was estimated for each genomic region. Significance testing was performed between diagnostic groups to discover differentially-methylated regions (DMRs). The 300 most differentially-methylated regions (DMRs) for each comparison (IDH-wildtype vs. IDH-mutant, IDH-mutant vs. control, IDH-wildtype vs. control), for a total of 900 DMRs, were utilized as features in dimensionality reduction analysis and machine learning. Out of ~1×107 defined genomic regions, there were ~7×105 DMRs between GBM and control samples and ~1.5×106 DMRs between GBM and IDH-mutant samples, accounting for ~7% and ~15% of the genome, respectively. Uniform manifold approximation and projection (UMAP) and hierarchical clustering analysis revealed that samples were mostly clustered by diagnosis. Random forest analysis of the 900 DMRs with an 80/20 train/test split resulted in an overall accuracy of 0.815. In conclusion, CSF cfDNA methylation profiles of diffuse glioma patients exhibit differential methylation patterns between IDH-mutant and IDH-wildtype tumors, as well as between tumor and control, and could be informative biomarkers for the diagnosis of patients with diffuse gliomas.

CNSC-52. OLIGODENDROGLIOMA SPATIAL TRANSCRIPTOMIC SEQUENCING REVEALS THERAPY-DEPENDENT EVOLUTION OVER TIME

Abstract Oligodendroglioma evolution is incompletely understood. Here we analyze serial oligodendroglioma samples to define genomic and cellular mechanisms underlying treatment resistance. RNA sequencing was performed on 53,406 spatial transcriptomes from 31 oligodendrogliomas (IDH mutant, 1p/19q-codeleted) that were resected from 16 patients. There were 15 CNS WHO grade 2 and 16 grade 3 oligodendrogliomas, which were stratified for temporal progression across grades (n=5 grade 2 to 2, n=5 grade 2 to 3, n=6 grade 3 to 3) and treatments between serial resections (n=2 TMZ/RT, n=2 TMZ, n=1 mTOR inhibition, n=1 IDH inhibition, n=10 surgical monotherapy). Spatial transcriptomic clusters were defined using cell signature gene sets, cell cycle analysis, differentially expressed marker genes, immunohistochemistry, spatial copy number alterations (CNAs), and deconvolution of 18,628 oligodendroglioma single-cell transcriptomes from 16 independent patients. Specificity was tested using immunofluorescence on independent oligodendrogliomas that did not recur after surgical monotherapy (n=8 samples, median follow-up 8.4 years) and spatial transcriptomic sequencing on 8 matched-paired astrocytomas from 4 patients. Integrated uniform manifold approximation and projection revealed a proneural/NPC-like cluster that was enriched in grade 3 oligodendroglioma samples and represented the starting point in oligodendroglioma lineage differentiation using RNA velocity, Monocle, and subclonal spatiotemporal CNA evolution. Comparison across treatment regimens showed (1) enrichment of the proneural/NPC-like cluster in recurrent versus primary tumors that were treated with surgical monotherapy, (2) enrichment of a proneural/OPC-like cluster and depletion of an oligodendrocyte-like cluster after TMZ/RT, (3) enrichment of proneural/NPC-like and immediate-early response clusters after TMZ/RT, (4) enrichment of the oligodendrocyte-like cluster following mTOR inhibition, and (5) enrichment of cancer associated fibroblasts (CAFs) following IDH inhibition. These data show the spatiotemporal landscape of oligodendroglioma is associated with branched lineage differentiation that is modified by postoperative therapy, and suggest that therapy-dependent druggable dependencies may exist to treat oligodendrogliomas that are resistant to standard interventions.

Signature analysis of high-throughput transcriptomics screening data for mechanistic inference and chemical grouping.

High-throughput transcriptomics (HTTr) uses gene expression profiling to characterize the biological activity of chemicals in in vitro cell-based test systems. As an extension of a previous study testing 44 chemicals, HTTr was used to screen an additional 1,751 unique chemicals from the EPA's ToxCast collection in MCF7 cells using 8 concentrations and an exposure duration of 6 h. We hypothesized that concentration-response modeling of signature scores could be used to identify putative molecular targets and cluster chemicals with similar bioactivity. Clustering and enrichment analyses were conducted based on signature catalog annotations and ToxPrint chemotypes to facilitate molecular target prediction and grouping of chemicals with similar bioactivity profiles. Enrichment analysis based on signature catalog annotation identified known mechanisms of action (MeOAs) associated with well-studied chemicals and generated putative MeOAs for other active chemicals. Chemicals with predicted MeOAs included those targeting estrogen receptor (ER), glucocorticoid receptor (GR), retinoic acid receptor (RAR), the NRF2/KEAP/ARE pathway, AP-1 activation, and others. Using reference chemicals for ER modulation, the study demonstrated that HTTr in MCF7 cells was able to stratify chemicals in terms of agonist potency, distinguish ER agonists from antagonists, and cluster chemicals with similar activities as predicted by the ToxCast ER Pathway model. Uniform manifold approximation and projection (UMAP) embedding of signature-level results identified novel ER modulators with no ToxCast ER Pathway model predictions. Finally, UMAP combined with ToxPrint chemotype enrichment was used to explore the biological activity of structurally related chemicals. The study demonstrates that HTTr can be used to inform chemical risk assessment by determining in vitro points of departure, predicting chemicals' MeOA and grouping chemicals with similar bioactivity profiles.

The uniform convergence topology on separable subsets

The uniform convergence topology on separable subsets

Refining uniform approximation algorithm for low-rank Chebyshev embeddings

Abstract Nowadays, low-rank approximations are a critical component of many numerical procedures. Traditionally the problem of low-rank approximation of matrices is solved in unitary invariant norms such as Frobenius or spectral norm due to the existence of efficient methods for constructing approximations. However, recent results discover the potential of low-rank approximations in the Chebyshev norm, which naturally arises in many applications. In this paper, we investigate the problem of uniform approximation of vectors, which is the main component in the low-rank approximations of matrices in the Chebyshev norm. The principal novelty of this paper is the accelerated algorithm for solving uniform approximation problems. We also analyze the iterative procedure of the proposed algorithm and demonstrate that it has a geometric convergence rate. Finally, we provide an extensive numerical evaluation, which demonstrates the effectiveness of the proposed procedures.

An improved uniform convergence bound with fat-shattering dimension

The fat-shattering dimension characterizes the uniform convergence property of real-valued function classes. The state-of-the-art upper bounds in [6] feature a multiplicative squared logarithmic factor on the sample complexity, leaving an open gap with the existing lower bound. By relying on a refined packing number bound given in [20], we provide an improved uniform convergence bound that closes this gap.

Enhancing Streamflow Prediction in Ungauged Basins Using a Nonlinear Knowledge‐Based Framework and Deep Learning

AbstractIn hydrology, a fundamental task involves enhancing the predictive power of a model in ungagged basins by transferring information on physical attributes and hydroclimate dynamics from gauged basins. Introducing an integrated nonlinear clustering framework, this study aims to develop a comprehensive framework that augments predictive performance in basins where direct measurements are sparse or absent. In this framework, uniform manifold approximation and projection (UMAP) is used as a nonlinear method to extract the essential features embedded in hydro‐climatological attributes and physical properties. Then, the Growing Neural Gas (GNG) clustering model is used to find the basins that potentially share similar hydro‐climatological behaviors. Besides UMAP‐GNG, the integration of Principal Component Analysis (PCA) as a linear method to reduce dimensionality with common clustering methods are also assessed to serve as benchmarks. The results reveal that the combination of clustering algorithms with the PCA method may lead to loss of information while the nonlinear method (UMAP) can extract more informative features. The efficacy of the proposed framework is assessed across the Contiguous United States (CONUS) by training a single Base Model using long short‐term memory (LSTM) for the centroids of all clusters and then, fine‐tuning the model on the centroids of each cluster separately to create a regional model. The results indicate that using the information extracted by the UMAP‐GNG method to guide a Base Model can significantly improve the accuracy in most of the clusters and enhance the median prediction accuracy within different clusters from 0.04 to 0.37 of KGE in ungauged basins.

All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function

AbstractWe construct a differentiable locally Lipschitz function in with the property that for every convex body there exists such that coincides with the set of limits of derivatives of sequences converging to . The technique can be further refined to recover all compact connected subsets with nonempty interior, disclosing an important difference between differentiable and continuously differentiable functions. It stems out from our approach that the class of these pathological functions contains an infinite‐dimensional vector space and is dense in the space of all locally Lipschitz functions for the uniform convergence.

Large magnetocrystalline anisotropic energy and its impact on magnetostriction of L10-FePt

Abstract The origin of magnetocrystalline anisotropic energy (MAE) guided by spin–orbit coupling in the L10-FePt alloy was analyzed. Correlation of MAE with the calculated magnetoelastic constants in the single and polycrystal model was determined by means of first-principles calculations using density functional theory (DFT). More specifically, a systematic analysis of the large MAE and magnetostriction coefficients (λ’s) were done by means of post-processing of eigenvalues (orbital energies) and eigenfunctions (orbital occupancies). Our numerical analysis includes the convolution of the projected wave-function (density of states) of each orbital of the Fe and Pt sub-lattices into their orbital energies, which in principle should be valid for any solid regardless of the strength of the MAE. The determined anisotropic magnetostriction coefficients are of the λ ∼ 10 − 4 –10–3 order for several crystallographic directions; which is in accord to the known experimental data. However, the polycrystalline model based on the uniform stress approximation leads to a decrease of the λ by two orders in the overall magnetostrictive behavior ( λ ∼ 10 − 5 ). Such a large difference is not unexpected as the magnetostriction for polycrystalline FePt thin films were recorded as very low, thus validating the robustness of the current theoretical proposition of analyzing MAE.

Convergence analysis of non-matching finite elements for a linear monotone additive Schwarz scheme for semi-linear elliptic problems

Abstract In this article, we are interested in the standard finite element approximation method of linear additive Schwarz iterations for a class of semi-linear elliptic problems, for two subdomains, in the context of non-matching grids. More precisely, by means of a uniform convergence result from the study by Lui and a fundamental lemma consisting of estimating, at each iteration, the gap between the continuous and the finite element Schwarz iterates, we prove that the discrete Schwarz sequences converge, in the maximum norm, to the true solution. Moreover, we also give numerical results to support the theoretical findings.

Exploring tumor endothelial cells heterogeneity in hepatocellular carcinoma: insights from single-cell sequencing and pseudotime analysis.

This study aimed to explore the heterogeneity of tumor endothelial cells (TECs) in hepatocellular carcinoma (HCC) and their role in tumor progression, with the goal of identifying new therapeutic targets and strategies to improve patient prognosis. Single-cell RNA sequencing data from nine primary liver cancer samples were analyzed, obtained from the Gene Expression Omnibus (GEO) database. Data preprocessing, normalization, dimensionality reduction, and batch effect correction were performed based on the Seurat package. HCC cell types were identified using uniform manifold approximation and projection (UMAP) and cluster analysis, and the different cell types were annotated using the CellMarker database. Pseudotime trajectory analysis was conducted with Monocle to explore the differentiation trajectory of TECs. MAPK signaling pathway activity and copy number variations (CNV) in TECs were analyzed in conjunction with data from The Cancer Genome Atlas (TCGA), the trans-well and wound healing assay was used for cell invasion and migration activity assessment. Two subgroups of TECs (TECs 1 and TECs 2) were identified, exhibiting distinct functional activities and signaling pathways. Specifically, TECs 1 may be involved in tumor cell proliferation and inflammatory responses, whereas TECs 2 is not only involved in cell proliferation pathways, but also enriched in pathways such as metabolic synthesis. Pseudotime analysis revealed dynamic changes in TECs subgroups during HCC progression, correlating specific gene expressions (such as PDGFRB, PGF, JUN, and NR4A1). Subsequently, the JUN gene was predicted by performing binding sites and was shown to act as a transcription factor that may regulate the expression of the PGF gene. CNV analysis highlighted key genes and pathways in TECs that might influence HCC progression, and the PGF as key regulatory factor mediated cell proliferation and migration. The study revealed the heterogeneity of TECs in HCC and their potential roles in tumor progression, offering new perspectives and potential therapeutic targets for HCC molecular mechanisms. The findings emphasize the importance of further exploring TECs heterogeneity for understanding HCC pathogenesis and developing personalized treatment strategies.

Two remarks on properties of functions of bounded variation

In terms of variations, a sufficient condition for the uniform convergence of sequences of continuous functions is proved. Using this result, we obtain an addition to the classical Helly theorem on the selection of convergent sequences of functions with uniformly bounded variations in the case when the limit function is continuous. Also, by using an example we show that the condition of continuous differentiability of a function, ensuring the differentiability of its variation with the variable upper limit, is in a certain sense sharp.

Approximation in quantum calculus of the Phillips operators by using the sequences of q-Appell polynomials

In this paper, we attempt to use the Dunkl analog to study the convergence properties of q-Phillips operators by using the q-Appell polynomials. By applying the new sequences of continuous functions νs,q(z)=(z−12[s]q)ϱ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\ u _{s,q}(z)=\\Bigl(z- \\frac{1}{2[s]_{q}}\\Bigr)_{\\varrho}$\\end{document} on [0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[0, \\infty )$\\end{document}, we construct an improved version of the q-Phillips operators. We calculate the qualitative outcomes in weighted Korovkin spaces to better understand the Phillips operators’ uniform convergence results. We obtain the approximation properties by use of the modulus of continuity and functions belonging to the Lipschitz class. Moreover, we give some direct theorems for the function belonging to Peetre’s K-functional.

Approximation results of Phillips type operators including exponential function

<p>The current article deals with a study on some moderation of the Phillips operators, including constant and exponential functions. Here, we derive the moments applying the notion of moment-generating function for the well-known Phillips operators. The authors also establish uniform convergence estimates for the improved form of these operators. Additionally, some direct estimates involving the asymptotic-type results are discussed.</p>

Kantorovich Variant of the Blending Type Bernstein Operators

In this paper, we introduce a novel class of blending-type Bernstein–Kantorovich operators. These operators depend on three parameters: α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}, and s. We establish results on the uniform convergence and rate of convergence of these operators in terms of the first and second order modulus of continuity. We also investigate the shape-preserving properties of the operators, such as monotonicity and convexity, for each choice of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}, and s. Finally, we provide graphical and numerical results to illustrate the accuracy of the operators and to demonstrate how they approach certain functions.