Compactness Research Articles

Quasi-uniform entropy vs topological entropy

In 2023 Haihambo and Olela Otafudu introduced and studied the notion of quasi-uniform entropy hQU(ψ) for a uniformly continuous self-map ψ of a quasi-metric or a quasi-uniform space X. In this paper, we discuss the connection between the topological entropy functions h,hf and the quasi-uniform entropy function hQU on a quasi-uniform space X, where h and hf are the topological entropy functions defined using compact sets and finite open covers, respectively. In particular, we have shown that for a uniformly continuous self-map ψ of a T0-quasi-uniform space (X,U) we have h(ψ)≤hQU(ψ) when X is compact and hQU(ψ)≤hf(ψ) with equality if X is a compact T2 space.

Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the p-Laplacian

For a given constant λ>0 and a bounded Lipschitz domain D⊂Rn (n≥2), we establish that almost-minimizers of the functionalJ(v;D)=∫D∑i=1m|∇vi(x)|p+λχ{|v|>0}(x)dx,1<p<∞, where v=(v1,⋯,vm), and m∈N, exhibit optimal Lipschitz continuity in compact sets of D. Furthermore, assuming p≥2 and employing a distinctly different methodology, we tackle the issue of boundary Lipschitz regularity for v. This approach simultaneously yields an alternative proof for the optimal local Lipschitz regularity for the interior case.

Neural networks-based composite learning control for robotic systems with predefined time error constraints

This paper investigates the neural networks-based control problem for robotic systems with error constraints. By embedding a new predefined time performance function into an asymmetric barrier function, a predefined time constraints method is proposed with the following features: (1) it is a general approach that can handle asymmetric error constraints and can be directly applied to the predefined performance control for other high-order nonlinear systems without any changing; (2) both settling time and convergent compact set can be preassigned by the user; (3) the results are global i.e., the robot can start arbitrarily within the physical domain. By extending the regression operators with online data memory, a prediction error is constructed, by using which, a new neural networks based composite learning law is constructed to handle unknown dynamics. Compared with the traditional method such as gradient descent algorithm, σ-modification and ϵ-modification which cannot guarantee the convergence of the neural networks weights or can only force the weights to stay around preselected values, the neural networks guarantees that the weights converge to the region around the true values, in particular under a weak interval excitation (IE) condition rather than the typically stringent persistent excitation (PE) condition. The benefits and effectiveness of the proposed control are theoretically authenticated and experimentally validated.

Universal approximation theorem for vector- and hypercomplex-valued neural networks

The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications, including regression and classification tasks. Furthermore, it is valid for real-valued neural networks and some hypercomplex-valued neural networks such as complex-, quaternion-, tessarine-, and Clifford-valued neural networks. However, hypercomplex-valued neural networks are a type of vector-valued neural network defined on an algebra with additional algebraic or geometric properties. This paper extends the universal approximation theorem for a wide range of vector-valued neural networks, including hypercomplex-valued models as particular instances. Precisely, we introduce the concept of non-degenerate algebra and state the universal approximation theorem for neural networks defined on such algebras.

An index theorem for loop spaces

We formulate and prove an index theorem for loop spaces of compact manifolds in the framework of KK-theory. It is a strong candidate for the noncommutative geometrical definition (or the analytic counterpart) of the Witten genus. In order to specify an “appropriate form” of the index theorem to formulate a loop space version, we formulate and prove an equivariant index theorem for non-compact S1-manifolds with a compact fixed-point set. In order to formulate it, we use a ring of formal power series.

Inverse compensation mechanism-based adaptive fuzzy-neural fault-tolerant control for an uncertain quadrotor UAV

Actuator faults, system uncertainties, and external disturbances can affect tracking control performance of the quadrotor UAV, and even cause the instability of the system. To address this issue, an adaptive fuzzy-neural fault-tolerant control (AFNFTC) scheme is proposed for the quadrotor UAV integrated with inverse compensation strategy and fuzzy neural networks (FNNs) technology. Firstly, we design an inverse compensation mechanism to address actuator faults and utilize the approximation capabilities of the FNNs to compensate for system uncertainties. Then, based on the Lyapunov direct method, the AFNFTC scheme is designed to ensure the stability of the quadrotor UAV. Under the proposed scheme, the closed-loop system for the quadrotor UAV is proven to be uniformly bounded, and the tracking error of the quadrotor UAV is demonstrated to converge to a small compact set ultimately by appropriately selecting design parameters. Finally, the feasibility and effectiveness of the proposed scheme is validated through simulation results.

Shape optimization for a nonlinear elliptic problem related to thermal insulation

In this paper, we consider a minimization problem of a nonlinear functional I_{\beta,p}(D,\Omega) related to a thermal insulation problem with a convection term, where \Omega is a bounded connected open set in \R^{n} and D\subset\overline{\Omega} is a compact set. The Euler–Lagrange equation relative to I_{\beta,p} is a p -Laplace equation, 1&lt;p&lt;\infty , with a Robin boundary condition with parameter \beta&gt;0 . The main aim is to study extremum problems for I_{\beta,p}(D,\Omega) , among domains D with given geometrical constraints and \Omega\setminus D of fixed thickness. In the planar case, we show that under perimeter constraint the disk maximizes I_{\beta,p} . In the n -dimensional case we restrict our analysis to convex sets showing that the same is true for the ball but under different geometrical constraints.

Solutions to the nonlinear obstacle problem with compact contact sets

For the obstacle problem with a nonlinear operator, we characterize the space of global solutions with compact contact sets. This is achieved by constructing a bijection onto a class of quadratic polynomials describing the asymptotic behavior of solutions.

Sampled‐data cooperative semi‐global robust practical output regulation for nonlinear multi‐agent systems with an uncertain leader

AbstractThis article explores the problem of sampled‐data cooperative robust practical output regulation for nonlinear multi‐agent systems whose leader system is unknown. In this study, the nonlinearities of the follower systems are not constrained by the linear growth condition. Based on the robust control technology and the adaptive control technology, a distributed output‐based sampled‐data control protocol is developed to deal with the uncertainty of the exosystem which can belong to any large known compact set. We first perform the coordinate transformation of closed‐loop system to obtain the so‐called augmented system, and then perform the stability analysis for this system. As a result, it is proven that the steady‐state tracking errors are in any small given neighborhood of the origin. Finally, the performance of the proposed controller is validated by a numerical example.

Sequential Patching Lattice for Image Classification and Enquiry: Streamlining Digital Pathology Image Processing

Digital pathology and the integration of artificial intelligence (AI) models have revolutionized histopathology, opening new opportunities. With the increasing availability of Whole Slide Images (WSIs), there’s a growing demand for efficient retrieval, processing, and analysis of relevant images from vast biomedical archives. However, processing WSIs presents challenges due to their large size and content complexity. Full computer digestion of WSIs is impractical, and processing all patches individually is prohibitively expensive. In this paper, we propose an unsupervised patching algorithm, Sequential Patching Lattice for Image Classification and Enquiry (SPLICE). This novel approach condenses a histopathology WSI into a compact set of representative patches, forming a “collage” of WSI while minimizing redundancy. SPLICE prioritizes patch quality and uniqueness by sequentially analyzing a WSI and selecting nonredundant representative features. We evaluated SPLICE for search and match applications, demonstrating improved accuracy, reduced computation time, and storage requirements compared to existing state-of-the-art methods. As an unsupervised method, SPLICE effectively reduces storage requirements for representing tissue images by 50%. This reduction enables numerous algorithms in computational pathology to operate much more efficiently, paving the way for accelerated adoption of digital pathology.

Two-dimensional Jacobians $$\textrm{det}\hspace{0.56905pt}$$ and $$\textrm{Det}\hspace{0.56905pt}$$ for bounded variation functions and applications

AbstractThe paper deals with the comparison in dimension two between the strong Jacobian determinant $$\textrm{det}\hspace{0.56905pt}$$ det and the weak (or distributional) Jacobian determinant $$\textrm{Det}\hspace{0.56905pt}$$ Det . Restricting ourselves to dimension two, we extend the classical results of Ball and Müller as well as more recent ones to bounded variation vector-valued functions, providing a sufficient condition on a vector-valued U in $$BV(\Omega )^2$$ B V ( Ω ) 2 such that the equality $$\textrm{det}\hspace{0.56905pt}(\nabla U)=\textrm{Det}\hspace{0.56905pt}(\nabla U)$$ det ( ∇ U ) = Det ( ∇ U ) holds either in the distributional sense on $$\Omega $$ Ω , or almost-everywhere in $$\Omega $$ Ω when U is in $$W^{1,1}(\Omega )^2$$ W 1 , 1 ( Ω ) 2 . The key-assumption of the result is the regularity of the Jacobian matrix-valued $$\nabla U$$ ∇ U along the direction of a given non vanishing vector field $$b\in C^1(\Omega )^2$$ b ∈ C 1 ( Ω ) 2 , i.e.$$\nabla U\, b$$ ∇ U b is assumed either to belong to $$C^0(\Omega )^2$$ C 0 ( Ω ) 2 with one of its coordinates in $$C^1(\Omega )$$ C 1 ( Ω ) , or to belong to $$C^1(\Omega )^2$$ C 1 ( Ω ) 2 . Two examples illustrate this new notion of two-dimensional distributional determinant. Finally, we prove the lower semicontinuity of a polyconvex energy defined for vector-valued functions U in $$BV(\Omega )^2$$ B V ( Ω ) 2 , assuming that the vector field b and one of the coordinates of $$\nabla U\, b$$ ∇ U b lie in a compact set of regular vector-valued functions.

Parsimonious airfoil Parameterisation: A deep learning framework with Bidirectional LSTM and Gaussian Mixture models

The choice of airfoil parameterisation method significantly influences the overall wing optimisation performance by affecting the flexibility and computational efficiency of the process. Ideally, one should be able to intuitively constrain airfoil shape and structural characteristics as input to the optimisation process. Current parameterisation techniques lack the flexibility to generate airfoils efficiently by specifying parsimonious shape and structural features. To address this limitation, a deep learning framework is proposed, enabling conditional airfoil generation from an airfoil’s shape and structural feature definition.Specifically, we demonstrate the application of Bidirectional Long Short Term Memory models and Bayesian Gaussian Mixture models to derive airfoil coordinates from a compact set of shape and structural characteristics that we define. The proposed framework is shown to achieve favorable airfoil performance optimisation due to improved exploration and exploitation of the design space, compared to traditional approaches. Overall, the proposed optimisation framework is able to realise a 9.04% performance improvement over an airfoil design optimised with traditional parameterisation techniques.

Analytical Solution of Concentrated Mixtures of Hydrogen Sulfide and Methanol in Steady State in Biofilm Model

The mathematical model of steady-state biofiltration of hydrogen sulfide and methanol mixtures is explored in this work. The non-linear term associated with monoid kinetics is a part of the system of non-linear reaction-diffusion equations on which this model is built. By applying the Akbari Ganji Method and the Homotopy Perturbation Method to resolve the non-linear equations, an approximate analytical solution of the concentrate mixtures in the biofilm model is obtained. Furthermore, this paper reports numerically by exploiting the Matlab software. By comparing the analytical solution with numerical findings, the accuracy of the method is presented. The new analytical results contribute to optimizing the consistency of this model. These novel approaches produce a compact set of analytical approximations that are straightforward to compute and verify as well.

Caratheodory periodic perturbations of degenerate systems

We study the structure of the set of harmonic solutions to T-periodically perturbed coupled differential equations on differentiable manifolds, where the perturbation is allowed to be of Caratheodory-type regularity. Employing degree-theoretic methods, we prove the existence of a noncompact connected set of nontrivial T-periodic solutions that, in a sense, emanates from the set of zeros of the unperturbed vector field. The latter is assumed to be ''degenerate'': Meaning that, contrary to the usual assumptions on the leading vector field, it is not required to be either trivial nor to have a compact set of zeros. In fact, known results in the ``nondegenerate case can be recovered from our ones. We also provide some illustrating examples of Lienard- and \(\phi\)-Laplacian-type perturbed equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/39/abstr.html

Randomized low‐rank approximation of parameter‐dependent matrices

AbstractThis work considers the low‐rank approximation of a matrix depending on a parameter in a compact set . Application areas that give rise to such problems include computational statistics and dynamical systems. Randomized algorithms are an increasingly popular approach for performing low‐rank approximation and they usually proceed by multiplying the matrix with random dimension reduction matrices (DRMs). Applying such algorithms directly to would involve different, independent DRMs for every , which is not only expensive but also leads to inherently non‐smooth approximations. In this work, we propose to use constant DRMs, that is, is multiplied with the same DRM for every . The resulting parameter‐dependent extensions of two popular randomized algorithms, the randomized singular value decomposition and the generalized Nyström method, are computationally attractive, especially when admits an affine linear decomposition with respect to . We perform a probabilistic analysis for both algorithms, deriving bounds on the expected value as well as failure probabilities for the approximation error when using Gaussian random DRMs. Both, the theoretical results and numerical experiments, show that the use of constant DRMs does not impair their effectiveness; our methods reliably return quasi‐best low‐rank approximations.

EVALUATION OF INTERACTIVE SPACES FOR ENHANCED LEARNING IN ARCHITECTURE DEPARTMENT BUILDINGS IN NIGER STATE, NIGERIA

The design studio is still the focal point of curricular programmes in architectural faculties around the world, and it is regarded as the gold standard for architectural design education. However, the architecture studio's primary role in the existing academic framework of architectural education needs to be reconsidered. As a result, a shift to architecture studio design is required to create an environment that delivers an immersive and collaborative feeling of setting for studio users. The job of architecture is to build and develop places that fulfil the demands of architecture users and the usage of spaces by different individuals and groups of people, just as an architecture design studio is considered as a learning environment and usually a location where interactions occur. This study investigates the methodologies and goals of architectural design for open space facilities that relieve stress in learning environments such as those found on university campuses in a compact urban setting. The literature reviews, along with expert input, indicate strategies for integrating sustainability as the foundation for achieving a functional institutional environment structure for Nigerian schools of architecture; thus, this serves as the foundation for the development of a questionnaire to collect relevant data. Improving quality and service delivery through better interactive space organisation. The findings suggest that architects should incorporate sustainable building areas into their theoretical preparation. According to the study, it is critical to set standards for an alternative design approach that is user-centered while also strengthening the training of student architects by creating more interactive spaces.

Algebras of polynomials generated by linear operators

Let $E$ be a Banach space and $A$ be a commutative Banach algebra with identity. Let $\mathbb{P}(E, A)$ be the space of $A$-valued polynomials on $E$ generated by bounded linear operators (an $n$-homogenous polynomial in $\mathbb{P}(E,A)$ is of the form $P=\sum_{i=1}^\infty T^n_i$, where $T_i:E\to A$, $1\leq i &lt;\infty$, are bounded linear operators and $\sum_{i=1}^\infty \|T_i\|^n &lt; \infty$). For a compact set $K$ in $E$, we let $\mathbb{P}(K, A)$ be the closure in $\mathscr{C}(K,A)$ of the restrictions $P|_K$ of polynomials $P$ in $\mathbb{P}(E,A)$. It is proved that $\mathbb{P}(K, A)$ is an $A$-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product $\mathcal{P}_N(K){\widehat\otimes}_\epsilon A$, where $\mathcal{P}_N(K)$ is the uniform algebra on $K$ generated by nuclear scalar-valued polynomials. The character space of $\mathbb{P}(K, A)$ is then identified with $\hat{K}_N\times \mathfrak{M}(A),$ where $\hat K_N$ is the nuclear polynomially convex hull of $K$ in $E$, and $\mathfrak{M}(A)$ is the character space of $A$.

Resilient and robust management policy for multi‐stage supply chains with perishable goods and inaccurate forecast information: A distributed model predictive control approach

AbstractAn efficient supply chain (SC) management requires that decisions are taken to minimize the effects of parametric uncertainties and unpredictable external disturbances. In this article, we consider this problem with reference to a multi‐stage SC (MSSC) whose dynamics is characterized by the following elements of complexity: perishable goods with uncertain perishability rate, an uncertain future customer demand that is only known to fluctuate inside a given compact set. The problem we face is to define a resilient and robust Replenishment Policy (RP) such that at any stage the following requirements are satisfied: the fulfilled demand is maximized, overstocking is avoided, the bullwhip effect (BE) is mitigated. These objectives should be pursued despite the mentioned uncertainties and unexpected customer demand behaviors violating the bounds of the compact set. Robustness is here intended with respect to uncertainty on the perishability rate, and resiliency as the ability to quickly react to the mentioned unforeseen customer demands. We propose a method based on a distributed resilient robust model predictive control (DRRMPC) approach. Each local robust MPC (RMPC) involves solving a Min‐Max constrained optimization problem (MMCOP). To drastically reduce the numerical complexity of each MMCOP, we parametrize its solution by means of B‐spline functions.

Robust non-computability of dynamical systems and computability of robust dynamical systems

In this paper, we examine the relationship between the stability of the dynamical system $x^{\prime}=f(x)$ and the computability of its basins of attraction. We present a computable $C^{\infty}$ system $x^{\prime}=f(x)$ that possesses a computable and stable equilibrium point, yet whose basin of attraction is robustly non-computable in a neighborhood of $f$ in the sense that both the equilibrium point and the non-computability of its associated basin of attraction persist when $f$ is slightly perturbed. This indicates that local stability near a stable equilibrium point alone is insufficient to guarantee the computability of its basin of attraction. However, we also demonstrate that the basins of attraction associated with a structurally stable - globally stable (robust) - planar system defined on a compact set are computable. Our findings suggest that the global stability of a system and the compactness of the domain play a pivotal role in determining the computability of its basins of attraction.

Conditions for ultimate boundedness of solutions and permanence for a hybrid Lotka–Volterra system

In the paper, a generalized Lotka–Volterra – type system with switching is considered. The conditions for the ultimate boundedness of solutions and the permanence of the system are studied. With the aid of the direct Lyapunov method, the requirements for the switching law are established to guarantee the necessary dynamics of the system. An attractive compact invariant set is constructed in the phase space of the system, and a given region of attraction for this set is provided. A distinctive feature of the work is the use of a combination of two different Lyapunov functions, each of which plays its own special role in solving the problem.