Maximum Norm Research Articles

Coexistence and locally exponential stability of multiple equilibrium points for fractional-order impulsive control Cohen–Grossberg neural networks

Different from the existing multiple Mittag-Leffler stability or multiple asymptotic stability, the multiple exponential stability, which has explicit and faster convergence rate, is investigated in this paper for fractional-order impulsive control Cohen–Grossberg neural networks. First, by using the definition of Dirac delta function, the fractional-order control Cohen–Grossberg neural networks are translated into fractional-order impulsive neural networks, in which pulse effect relies on the fractional order of the addressed system. Then, based on maximum norm, 1−norm and general q−norm (q=2n), a series of novel criteria are obtained respectively to ensure that such n−neuron neural networks can have ∏i=1n(2Li+1) total equilibrium points and ∏i=1n(Li+1) locally exponentially stable equilibrium points, by utilizing the known fixed point theorem, the method of average impulsive interval, the theory of fractional-order differential equations, and the method of Lyapunov function. This paper’s investigation reveals the effects of impulsive function, impulsive interval, and fractional order on the dynamic behaviors. Finally, theoretical results are shown to be effective by four illustrative examples.

Bayesian Non-Parametric Inference for Multivariate Peaks-over-Threshold Models.

We consider a constructive definition of the multivariate Pareto that factorizes the random vector into a radial component and an independent angular component. The former follows a univariate Pareto distribution, and the latter is defined on the surface of the positive orthant of the infinity norm unit hypercube. We propose a method for inferring the distribution of the angular component by identifying its support as the limit of the positive orthant of the unit p-norm spheres and introduce a projected gamma family of distributions defined through the normalization of a vector of independent random gammas to the space. This serves to construct a flexible family of distributions obtained as a Dirichlet process mixture of projected gammas. For model assessment, we discuss scoring methods appropriate to distributions on the unit hypercube. In particular, working with the energy score criterion, we develop a kernel metric that produces a proper scoring rule and presents a simulation study to compare different modeling choices using the proposed metric. Using our approach, we describe the dependence structure of extreme values in the integrated vapor transport (IVT), data describing the flow of atmospheric moisture along the coast of California. We find clear but heterogeneous geographical dependence.

Sampling discretization of the uniform norm and applications

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any N-dimensional subspace of the space of continuous functions it is sufficient to use eCN sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in N number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on N. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

Adaptive backstepping controller based on a novel framework for dynamic solution of an ankle rehabilitation spherical parallel robot

AbstractThis research offers an adaptive model-based methodology for autonomous control of 3-RRR spherical parallel manipulator (RSPM) based on a novel modeling framework. RSPM is an overconstrained parallel mechanism that has a variety of applications in medical procedures such as ankle rehabilitation because of its precision and accuracy. However, obtaining a complete explicit dynamic model of these mechanisms for tracking purposes has been a problematic challenge due to their inherent singularities, coupling effects of the limbs, and redundant constraints imposed by the intermediate joints. This paper presents a novel algorithm to obtain the analytical kinematic solutions of RSPMs based on the closed-loop vector method, which includes constraint analysis. By incorporating constrained kinematics into the dynamic model, a comprehensive explicit dynamic solution of the non-overconstrained version 3-RCC of RSPM is developed in task space, based on screw theory and the linear homogeneous property of algebraic equations on the manipulator twist. Based on the proposed computational framework, a robust self-tuning backstepping control (STBC) strategy is applied to the robot to overcome the effect of external disturbances and time-varying uncertainties. Furthermore, an observer-based compensation (OBC) method is presented for dealing with the nonlinear hysteresis loops of the ankle during trajectory tracking purposes. The closed-loop stability of the whole system including STBC and OBC is theoretically performed by Lyapunov methods. The proposed methodologies are validated by realistic co-simulations in different scenarios. For instant, in the presence of external disturbances, the maximum tracking error norm of STBC is 37.5% less than the sliding mode approach.

The Metaverse is the future research space for jurisprudence

The article is devoted to the study of the concept of the Metaverse as a future research space for jurisprudence. It has been established that the Metaverse is known as a special virtual space created by computer technologies, where users can interact with each other, between various digital objects and the digital environment thanks to the created avatars – virtual digital images. It has been determined that the Metaverse can be called a copy of the real world, but in a virtual space that combines modern technologies, virtual and augmented reality, three-dimensional graphics, artificial intelligence, social networks, as well as other advanced innovative technologies. The focus is on the fact that the hypothetical transfer of real life to the virtual world raises a large number of legal issues and creates a huge field for scientific research, in particular, in jurisprudence. It was found that legal problems that may arise in the future in the Metaverse should be solved by lawyers together with philosophers, technologists, experts, and scientists from other fields of science based on developed uniform standards and norms. The opinion of scientists regarding the creation of a single international legal framework for the regulation of the Metaverse due to the absence of borders in the Metaverse and non-belonging to a specific state is supported. It is stated that the process of developing a unified international legal framework for the regulation of the Metaverse is a very large, complex, and long-lasting work that requires taking into account the cultural, national, legal, and moral differences of different countries. Such a difficult task requires the approval and consent of states, international organizations, and societies. However, given the potential for active development of the Metaverse and its impact on society, these efforts can contribute to its effective and consistent regulation.

Orthogonal subspace exploration for matrix completion

Matrix completion, aiming at restoring a low-rank matrix from observed entries, indicates the connection with the subspace clustering due to the low-rank property. However, it is expensive to incorporate subspace learning into the pervasive surrogate of matrix completion, the nuclear norm. In this paper, we design an orthogonal subspace exploration model for matrix completion, which can be easily integrated due to the succinct formulation. Then, we propose a non-convex surrogate with tractable solutions for low-rank matrix completion, so that the subspace exploration can be performed simultaneously. Compared with the existing surrogates (e.g., nuclear norm, Schatten-p norm, max norm, etc.), the proposed surrogate is differential such that the optimization is still simple even after the subspace exploration is incorporated. Although the surrogate is non-convex, a parameter-free algorithm that is proved to converge into the global optimum is developed. The optimization consists of closed-form solutions so that the orthogonal subspace exploration will not distinctly bring additional costs and the algorithm empirically converges within dozens of iterations. Experiments illustrate the efficiency and superiority of our model.

Exact Estimates of Functions in Sobolev Spaces with Uniform Norm

Exact Estimates of Functions in Sobolev Spaces with Uniform Norm

Tracially amenable actions and purely infinite crossed products

AbstractWe introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C$$^*$$ ∗ -algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group $$F_2$$ F 2 , we show that outer, tracially amenable actions on simple, unital, $$\mathcal {Z}$$ Z -stable C$$^*$$ ∗ -algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.

Gradient-Based Empirical Risk Minimization Using Local Polynomial Regression

In this paper, we consider the widely studied problem of empirical risk minimization (ERM) of strongly convex and smooth loss functions using iterative gradient-based methods. A major goal of the existing literature has been to compare different prototypical algorithms, such as batch gradient descent (GD) or stochastic gradient descent (SGD), by analyzing their rates of convergence to ϵ-approximate solutions with respect to the number of gradient computations, which is also known as the oracle complexity. For example, the oracle complexity of GD is [Formula: see text], where n is the number of training samples and p is the parameter space dimension. When n is large, this can be prohibitively expensive in practice, and SGD is preferred due to its oracle complexity of [Formula: see text]. Such standard analyses only utilize the smoothness of the loss function in the parameter being optimized. In contrast, we demonstrate that when the loss function is smooth in the data, we can learn the oracle at every iteration and beat the oracle complexities of GD, SGD, and their variants in important regimes. Specifically, at every iteration, our proposed algorithm, Local Polynomial Interpolation-based Gradient Descent (LPI-GD), first performs local polynomial regression with a virtual batch of data points to learn the gradient of the loss function and then estimates the true gradient of the ERM objective function. We establish that the oracle complexity of LPI-GD is [Formula: see text], where d is the data space dimension, and the gradient of the loss function is assumed to belong to an η-Hölder class with respect to the data. Our proof extends the analysis of local polynomial regression in nonparametric statistics to provide supremum norm guarantees for interpolation in multivariate settings and also exploits tools from the inexact GD literature. Unlike the complexities of GD and SGD, the complexity of our method depends on d. However, our algorithm outperforms GD, SGD, and their variants in oracle complexity for a broad range of settings where d is small relative to n. For example, with typical loss functions (such as squared or cross-entropy loss), when [Formula: see text] for any [Formula: see text] and [Formula: see text] is at the statistical limit, our method can be made to require [Formula: see text] oracle calls for any [Formula: see text], while SGD and GD require [Formula: see text] and [Formula: see text] oracle calls, respectively. Funding: This work was supported in part by the Office of Naval Research [Grant N000142012394], in part by the Army Research Office [Multidisciplinary University Research Initiative Grant W911NF-19-1-0217], and in part by the National Science Foundation [Transdisciplinary Research In Principles Of Data Science, Foundations of Data Science].

Learning Ultrametric Trees for Optimal Transport Regression

Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the space. However, measures supported on trees admit a closed-form optimal transport that can be computed in linear time. In this paper, we aim to find an optimal tree structure for a given discrete metric space so that the tree-Wasserstein distance approximates the optimal transport distance in the original space. One of our key ideas is to cast the problem in ultrametric spaces. This helps us optimize over the space of ultrametric trees --- a mixed-discrete and continuous optimization problem --- via projected gradient decent over the space of ultrametric matrices. During optimization, we project the parameters to the ultrametric space via a hierarchical minimum spanning tree algorithm, equivalent to the closest projection to ultrametrics under the supremum norm. Experimental results on real datasets show that our approach outperforms previous approaches (e.g. Flowtree, Quadtree) in approximating optimal transport distances. Finally, experiments on synthetic data generated on ground truth trees show that our algorithm can accurately uncover the underlying trees.

Stronger and Transferable Node Injection Attacks

Despite the increasing popularity of graph neural networks (GNNs), the security risks associated with their deployment have not been well explored. Existing works follow the standard adversarial attacks to maximize cross-entropy loss within an L-infinity norm bound. We analyze the robustness of GNNs against node injection attacks (NIAs) in black-box settings by allowing new nodes to be injected and attacked. In this work, we propose to design stronger and transferable NIAs. First, we propose margin aware attack (MAA) that uses a maximum margin loss to generate NIAs. We then propose a novel margin and direction aware attack (MDA) that diversifies the initial directions of MAA attack by minimizing the cosine similarity of the injected nodes with respect to their respective random initialization in addition to the maximization of max-margin loss. This makes the NIAs stronger. We further observe that using L2 norm of gradients in the attack step leads to an enhanced diversity amongst the node features, thereby further enhancing the strength of the attack. We incorporate transferability in NIAs by perturbing the surrogate model before generating the attack. An analysis of eigen spectrum density of the hessian of the loss emphasizes that perturbing the weights of the surrogate model improves the transferability. Our experimental results demonstrate that the proposed resilient node injection attack (R-NIA) consistently outperform PGD by margins about 7-15% on both large and small graph datasets. R-NIA is significantly stronger and transferable than existing NIAs on graph robustness benchmarks.

Efficient Learning in Polyhedral Games via Best-Response Oracles

We study online learning and equilibrium computation in games with polyhedral decision sets, a property shared by normal-form games (NFGs) and extensive-form games (EFGs), when the learning agent is restricted to utilizing a best-response oracle. We show how to achieve constant regret in zero-sum games and O(T^0.25) regret in general-sum games while using only O(log t) best-response queries at a given iteration t, thus improving over the best prior result, which required O(T) queries per iteration. Moreover, our framework yields the first last-iterate convergence guarantees for self-play with best-response oracles in zero-sum games. This convergence occurs at a linear rate, though with a condition-number dependence. We go on to show a O(T^(-0.5)) best-iterate convergence rate without such a dependence. Our results build on linear-rate convergence results for variants of the Frank-Wolfe (FW) algorithm for strongly convex and smooth minimization problems over polyhedral domains. These FW results depend on a condition number of the polytope, known as facial distance. In order to enable application to settings such as EFGs, we show two broad new results: 1) the facial distance for polytopes in standard form is at least γ/k where γ is the minimum value of a nonzero coordinate of a vertex of the polytope and k≤n is the number of tight inequality constraints in the optimal face, and 2) the facial distance for polytopes of the form Ax=b, Cx≤d, x≥0 where x∈R^n, C≥0 is a nonzero integral matrix, and d≥0, is at least 1/(c√n), where c is the infinity norm of C. This yields the first such results for several problems such as sequence-form polytopes, flow polytopes, and matching polytopes.

Explicit P1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell’s equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell’s equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space.

Abstract 318: Demystifying tumor heterogeneity with a fully automated, high-throughput single-cell DNA-Seq (scDNA-seq) workflow

Abstract Objective: Genetic heterogeneity is a key factor underlying tumor drug resistance and metastatic potential. Understanding this heterogeneity is therefore vital to improving both prognosis and treatment. Next-generation sequencing is a valuable tool for analyzing the genetic makeup of tumors. However, bulk sequencing methods lack the sensitivity to fully resolve tumor heterogeneity. While single cell methods provide a powerful approach to dissect such heterogeneity, to date, these methods have been limited in their throughput. To address this bottleneck, we have developed a fully automated workflow for generating scDNA-seq libraries based on PicoPLEX® whole genome amplification (WGA) technology. This high-throughput method, which has been optimized on our ICELL8® cx Single-Cell System, enables the generation of WGA libraries for >1,000 single cells within one day. Methods: We first sought to demonstrate the ability of this new high-throughput method to generate WGA libraries of comparable quality to the standard PicoPLEX workflow in terms of genome coverage, GC bias, and other typical quality metrics. We additionally assessed copy number variant (CNV) detection sensitivity using two cell lines with well-characterized small segmental CNVs: GM22601 (~25Mb deletion on chromosome 4) and GM05067 (~45Mb gain on chromosome 9). We also analyzed a lymphoblastoid line (K562) that carries a range of chromosomal aneuploidies. A total of 1,288 single-cell WGA libraries were generated and sequenced to a depth of 250K paired-end reads per cell. Data analysis was done using the Ginkgo CNV pipeline with an average bin size of 500 kb. As a final proof of principal, we also generated single cell data using tumor and adjacent normal tissue from two clear cell renal cell carcinoma (ccRCC) samples. Results: Libraries generated using our high-throughput workflow had a high mapping rate, with 94.1% of the reads being uniquely mapped. Additionally, the libraries were comparable to those generated with the standard PicoPLEX workflow in terms of coverage uniformity, GC bias and other metrics. Moreover, the segmental aneuploidies in both GM22601 and GM05067 were reliably detected in >90% of cells at a read depth as low as 250,000 reads per cell. Analysis of the ccRCC samples revealed subclonal heterogeneity with various CNVs common to ccRCC, including deletion of chr. 3p, amplification of chr. 5q, and duplication of chr. 2. Conclusion: By adapting PicoPLEX technology to high throughput using the ICELL8 cx Single-Cell System, we have obtained single-cell WGA libraries from up to 1,200 cells at once and enabled the reliable detection of CNVs and tumor subclones at a shallow sequencing depth. In addition, by leveraging the automated nanoliter-dispensing capabilities of ICELL8 cx system this method provides a significant reduction in reagent use and labor compared to plate-based methods. Citation Format: Xuan Li, Raymond Mendoza, Samantha Leong, Hima Anbunathan, Mike Covington, Mohammad Fallahi, Bryan Bell, Shuwen Chen, Yue Yun, Andrew Farmer. Demystifying tumor heterogeneity with a fully automated, high-throughput single-cell DNA-Seq (scDNA-seq) workflow [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2024; Part 1 (Regular Abstracts); 2024 Apr 5-10; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2024;84(6_Suppl):Abstract nr 318.

Higher order Galerkin finite element method for [formula omitted]-dimensional generalized Benjamin–Bona–Mahony–Burgers equation: A numerical investigation

In this article, we study solitary wave solutions of the generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation using higher-order shape elements in the Galerkin finite element method (FEM). Higher-order elements in FEMs are known to produce better results in solution approximations; however, these elements have received fewer studies in the literature. As a result, for the finite element analysis of the gBBMB equation, we consider Lagrange quadratic shape functions. We employ the Galerkin finite element approximation to derive a priori error estimates for semi-discrete solutions. For fully discrete solutions, we adopt the Crank–Nicolson approach, and to handle nonlinearity, we utilize a predictor–corrector scheme with Crank–Nicolson extrapolation. Additionally, we perform a stability analysis for time using the energy method. In the space, O(h3) convergence in L2(Ω) norm and O(h2) convergence in H1(Ω) norm are observed. Furthermore, an optimal O(Δt2) convergence in the maximum norm for the temporal direction is also obtained. We test the theoretical results on a few numerical examples in one- and two-dimensional spaces, including the dispersion of a single solitary wave and the interaction of double and triple solitary waves. To demonstrate the efficiency and effectiveness of the present scheme, we compute L2 and L∞ normed errors, along with the mass, momentum, and energy invariants. The obtained results are compared with the existing literature findings both numerically and graphically. We find quadratic shape functions improve accuracy in mass, momentum, energy invariants and also give rise to a higher order of convergence for Galerkin approximations for the considered nonlinear problems.

The Banach–Mazur distance between isomorphic spaces of continuous functions is not always an integer

For over half a century, Aleksander Pełczyński's question, whether the Banach–Mazur distance between any two isomorphic C(K) spaces is an integer, has remained open; as usual, K is a compact Hausdorff space and C(K) denotes the Banach space of all continuous real-valued functions on K, provided with the maximum norm. We answer this question in the negative. Moreover, we prove that the Banach–Mazur distance between isomorphic spaces C(K) and C0(L) is not always an integer; here L is a locally compact but noncompact Hausdorff space and C0(L) denotes the Banach space of all continuous real-valued functions vanishing at infinity on L, furnished with the maximum norm.

New completeness theorems on the boundary in elasticity

The completeness on the boundary (in the sense of Picone) of certain systems related to the III and IV BVPs for the elasticity system is proved. The completeness is obtained in both Lp (1⩽1<∞) and uniform norms.

IMPROVING WORKING CONDITIONS FOR WORKERS INSTALLING MONUMENTS IN RITUAL ECONOMY

The ritual industry in everyday life is emotionally associated with mourning. Оccupational safety issues in this industry are not discussed as widely in society as in other industries. The ritual industry does not have its own sectoral labor protection rules, therefore, labor protection issues are regulated only by intersectoral labor protection rules.&#x0D; Memorial structures in terms of weight, as a rule, exceed the maximum permissible norms for single lifting and moving weights for men. Therefore, when lifting and moving memorial structures, means of small mechanization are required.&#x0D; In large cities, burial in cemeteries is very dense and the passages between the graves are getting narrower and narrower. Therefore, at present, an essential requirement for the means of mechanization for the transportation&#x0D; of monuments is the requirement that the dimensions of transport mechanisms correspond to the dimensions of narrow passages. Various types of mechanization equipment (trolleys) for transporting monuments are presented on the Russian market, but most of them are wider in size than necessary.&#x0D; This article examines the working conditions and risks of a worker installing monuments in a ritual household. The article presents technical solutions for moving memorial structures around the cemetery using small-scale mechanization tools to reduce the severity of workers» labor. Reducing the severity of workers» work will reduce the number of.

On solutions vanishing at infinity of infinite systems of quadratic Urysohn integral equations

The paper is devoted to present a result on the existence of solutions of an infinite system of quadratic integral equations of the Urysohn type considered on the real half-axis. Our investigations are conducted in the Banach space consisting of bounded and continuous functions defined on the real half-axis with values in the space of real sequences converging to zero. That space is equipped with the standard supremum norm. The main tools used in our study is the technique of measures of noncompactness and the Schauder fixed point principle. We illustrate our result by a suitable example.

Toward coordinated planning and hierarchical optimization control for highly redundant mobile manipulator

This paper represents a constraint planning and optimization control scheme for a highly redundant mobile manipulator considering a complex indoor environment. Compared with the traditional optimization solution of a redundant manipulator, infinity norm and slack variable are additionally introduced and leveraged by the optimization algorithm. The former takes into account the joint limits effectively by considering individual joint velocities and the latter relaxes the equality constraint by decreasing the infeasible solution area. By using derived kinematic equations, the tracking control problem is expressed as an optimization problem and converted into a new quadratic programming (QP) problem. To address the optimization problem, the two-timescale recurrent neural networks optimization scheme is proposed and tested with a 9 DOFs nonholonomic mobile-based manipulator. Additionally, the BI2RRT∗ path-planning algorithm incorporates path planning in the complex environment where different obstacles are positioned. To test and evaluate the proposed optimization scheme, both predefined and generated paths are tested in the Neurorobotics Platform (NRP) 22https://neurorobotics.net.which is open access and open source integrative simulation framework powered by Gazebo and developed by our team.