Regular Space Research Articles

Dominated and absolutely summing operators on the space Crc(X,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\,C_{rc}(X,E)$$\\end{document} of vector-valued continuous functions

Let X be a completely regular Hausdorff space and E and F be Banach spaces. Let Crc(X,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{rc}(X,E)$$\\end{document} denote the Banach space of all continuous functions f:X→E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:X\\rightarrow E$$\\end{document} such that f(X) is a relatively compact set in E, and βσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _\\sigma $$\\end{document} be the strict topology on Crc(X,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{rc}(X,E)$$\\end{document}. We characterize dominated and absolutely summing operators T:Crc(X,E)→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T:C_{rc}(X,E)\\rightarrow F$$\\end{document} in terms of their representing operator-valued Baire measures. It is shown that every absolutely summing (βσ,‖·‖F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\beta _\\sigma ,\\Vert \\cdot \\Vert _F)$$\\end{document}-continuous operator T:Crc(X,E)→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T:C_{rc}(X,E)\\rightarrow F$$\\end{document} is dominated. Moreover, we obtain that every dominated operator T:Crc(X,E)→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T:C_{rc}(X,E)\\rightarrow F$$\\end{document} is absolutely summing if and only if every bounded linear operator U:E→F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U:E\\rightarrow F$$\\end{document} is absolutely summing.

Effect of laser remelting on the microstructure and properties of an AlCoFeNi eutectic high-entropy alloy fabricated through double-wire arc additive manufacturing

Additive manufacturing (AM) technology is a new method for preparing eutectic high-entropy alloys (EHEAs). The resulting EHEAs have excellent mechanical properties. However, in engineering, material failure, such as wear or fatigue fracture, often occurs on the surfaces of components. There is no report on surface modification and strengthening of additively manufactured EHEAs. Therefore, this study successfully utilized laser remelting (LR) technology to enhance the surface properties of an AlCoFeNi EHEA fabricated by using double-wire arc AM technology. After LR, the distance between adjacent regular eutectic regions and lamellar spacing of the EHEAs significantly decreased, and the proportion of strictly semicoherent face-centered cubic/body-centered cubic interfaces increased. When microhardness increased from 277.7 HV to 302.3 HV, the average friction coefficient and wear rate of the alloys decreased by 32.5% and 51.1%, respectively, and hardness and wear resistance significantly improved. This study has laid a solid foundation for the engineering application of the combination of LR and AM and also provided new ideas for improving the surface performance of additively manufactured EHEAs.

Study on the Method of Vineyard Information Extraction Based on Spectral and Texture Features of GF-6 Satellite Imagery

Accurately identifying the distribution of vineyard cultivation is of great significance for the development of the grape industry and the optimization of planting structures. Traditional remote sensing techniques for vineyard identification primarily depend on machine learning algorithms based on spectral features. However, the spectral reflectance similarities between grapevines and other orchard vegetation lead to persistent misclassification and omission errors across various machine learning algorithms. As a perennial vine plant, grapes are cultivated using trellis systems, displaying regular row spacing and distinctive strip-like texture patterns in high-resolution satellite imagery. This study selected the main oasis area of Turpan City in Xinjiang, China, as the research area. First, this study extracted both spectral and texture features based on GF-6 satellite imagery, subsequently employing the Boruta algorithm to discern the relative significance of these remote sensing features. Then, this study constructed vineyard information extraction models by integrating spectral and texture features, using machine learning algorithms including Naive Bayes (NB), Support Vector Machines (SVMs), and Random Forests (RFs). The efficacy of various machine learning algorithms and remote sensing features in extracting vineyard information was subsequently evaluated and compared. The results indicate that three spectral features and five texture features under a 7 × 7 window have significant sensitivity to vineyard recognition. These spectral features include the Normalized Difference Vegetation Index (NDVI), Enhanced Vegetation Index (EVI), and Normalized Difference Water Index (NDWI), while texture features include contrast statistics in the near-infrared band (B4_CO) and the variance statistic, contrast statistic, heterogeneity statistic, and correlation statistic derived from NDVI images (NDVI_VA, NDVI_CO, NDVI_DI, and NDVI_COR). The RF algorithm significantly outperforms both the NB and SVM models in extracting vineyard information, boasting an impressive accuracy of 93.89% and a Kappa coefficient of 0.89. This marks a 12.25% increase in accuracy and a 0.11 increment in the Kappa coefficient over the NB model, as well as an 8.02% enhancement in accuracy and a 0.06 rise in the Kappa coefficient compared to the SVM model. Moreover, the RF model, which amalgamates spectral and texture features, exhibits a notable 13.59% increase in accuracy versus the spectral-only model and a 14.92% improvement over the texture-only model. This underscores the efficacy of the RF model in harnessing the spectral and textural attributes of GF-6 imagery for the precise extraction of vineyard data, offering valuable theoretical and methodological insights for future vineyard identification and information retrieval efforts.

Existence of a Global Attractor for the Reaction–Diffusion Equation with Memory and Lower Regularity Terms

This paper investigates the large time behavior for the reaction–diffusion equation with memory and the forcing term g∈H−1(Ω). We prove the existence of a global attractor in L2(Ω)×Lμ2(R;H01(Ω)). Due to the lower regularity of g, one can hardly use the traditional energy estimates to derive the existence of a bounded absorbing set in the higher regularity space and then the compactness of the semigroup. Here, we utilize the contractive function method to establish the asymptotic smoothness of the semigroup.

Optimization of Fuzzy Mathematical Model of Regular Octagon-Shaped Parking Space

In the vigorous development of the city population, there is a need to set up clear dimensions for the parking space. Car parking plays a significant part in the residential apartments and commercial buildings. Roadside parking spaces will create a huge traffic problem if the parking lots are not properly designed. Sometimes, it leads to the accident. Hence, keeping all these causes in mind, the flexible parking space is the necessary of the time. In the proposed research work, the parking space considered in the problem is of a regular octagon shape, and the mathematical model is developed under a fuzzy environment. LINGO Optimization Software is used to solve the mathematical model and MATLAB(Matrix Laboratory) is used to define the fuzzy variable. At the outset, the results will reveal the importance of making the length of the parking lot under fuzzy environment.

Gravity-driven flow of liquid bridges between vertical fibres

Liquid bridges are formed when a flowing liquid interacts with multiple parallel fibres, as relevant to heat and mass transfer applications that utilize flow down fibre arrays. We perform a comprehensive experimental study of flowing liquid bridges between two vertical fibres whose spacing is controlled dynamically in our experimental apparatus. The bridge patterns exhibit a regular periodic spacing typical of absolute instability for low flow rates, but become spatially inhomogeneous above a critical flow rate where the base flow is convectively unstable. The shapes of individual bridges and their associated dynamics are measured, as they depend upon the liquid properties, and fibre geometry/spacing. The bridge length scales similarly to static bridges between parallel fibres. The bridge dynamics exhibits a dependence on viscosity and scale with the impedance. A simple energy balance is used to derive a scaling relationship for the bridge velocity that captures the general trend of our experimental data. Finally, we demonstrate that these scalings similarly apply when the fibres are dynamically separated or brought together.

Mechanical Microdeformation and Kink–band Formation in Mica from the Colônia Impact Crater, São Paulo, Brazil

A comprehensive examination using a transmitted light optical microscope was conducted to analyze the morphology and geometric parameters of kink band development in mica from the Colônia impact crater's crystalline basement rocks. Significant differences were observed between kink bands formed perpendicular and parallel to the [001] plane. Kink bands perpendicular to the [001] are narrow and elongated, exhibiting two distinct coplanar orientations with regular spacing and parallel planar fractures, which suggest a slip-dislocation strain mechanism. In contrast, kink bands parallel to the [001] plane exhibit a greater variety of shapes and a more complex internal structure. These kink bands can be grouped into six types: sigmoidal-shape, S-shape, lenticular-shape, Z-shape, Z-shape with internal dislocation, and Z-shape with internal rotation. The deformation patterns in these kink bands indicate two primary processes: flexural and shear strain. The most notable deformations induced by these strain mechanisms include the curvature of cleavage lamellae, delamination cracks, and fractures along the boundaries of kink bands. More severely deformed kink bands exhibit dislocation, rotation, and partial obliteration of internal cleavage lamellae. These characteristics differ from those of typical kink bands in tectonically deformed rocks, thereby reviving the longstanding debate regarding their potential use as alternative evidence in impact cratering investigations.

Error analysis of kernel/GP methods for nonlinear and parametric PDEs

We introduce a priori Sobolev-space error estimates for the solution of arbitrary nonlinear, and possibly parametric, PDEs that are defined in the strong sense, using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

Order-boundary characterization of the linear lattice of Riemann 𝜇-integrable functions as a certain completion of the linear lattice of bounded continuous functions

The linear lattice of Riemann μ \mu -integrable functions on a completely regular space with a bounded positive Radon measure μ \mu is viewed as an extension of the linear lattice of bounded continuous functions. To characterize this Riemann extension, the new functional analysis category of c c -latlineals with refinements ( ≡ c r \equiv cr -latlineals) is introduced. On this basis, the procedure of c r cr -completion of certain order-boundary type is defined. The μ \mu -Riemann extension is shown to be the result of applying this procedure to the c r μ cr_{\mu } -latlineal of bounded continuous functions.

The class C(ω1) and countable net weight

Hart and Kunen in [5] and, independently, Ríos-Herrejón in [14] defined and studied the class C(ω1) of topological spaces X having the property that for every neighborhood assignment {U(y):y∈Y} with Y∈[X]ω1 there is Z∈[Y]ω1 such thatZ⊂⋂{U(z):z∈Z}. It is obvious that spaces of countable net weight, i.e. having a countable network, belong to this class. In this paper we present several independence results concerning the relationships of these two and several other natural classes that are sandwiched between them, thus clarifying some of the main problems that were raised in [14].In particular, we prove that the continuum hypothesis, in fact a weaker combinatorial principle called super stick, implies that every regular space in C(ω1) has countable net weight, answering a question that was raised by Hart and Kunen in both [5] and [6].

Verification ofHarmonic Functions on Unbounded Domains in Ahlfors Regular Spaces using Sphericalization

Verification ofHarmonic Functions on Unbounded Domains in Ahlfors Regular Spaces using Sphericalization

The logarithmic Dirichlet Laplacian on Ahlfors regular spaces

We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on Riemannian manifolds. Specifically, its heat semigroup consists of compact operators which are trace-class after some critical point in time. Moreover, its domain is a Banach module over the Dini continuous functions and every Hölder continuous function is a smooth vector. Finally, the operator is compatible, in the sense of noncommutative geometry, with the action of a large class of non-isometric homeomorphisms.

One-point connectifications of regular spaces

It is well known that, a locally compact Hausdorff space has a Hausdorff one-point compactification (known as the Alexandroff compactification) if and only if it is non-compact. There is also, an old question of Alexandroff of characterizing spaces which have a one-point connectification. Here, we study one-point connectifications in the realm of regular spaces and prove that a locally connected space has a regular one-point connectification if and only if the space has no regular-closed component. This, also gives an answer to the conjecture raised by M. R. Koushesh. Then, we consider the set of all one-point connectifications of a locally connected regular space and show that, this set (naturally partially ordered) is a compact conditionally complete lattice. Further, we extend our theorem for locally connected regular spaces with a topological property P and give conditions on P which guarantee the space to have a regular one-point connectification with P.

Regular and Complete Spaces over Topological Quasigroups

Regular and Complete Spaces over Topological Quasigroups

A new class of efficient high order semi-Lagrangian IMEX discontinuous Galerkin methods on staggered unstructured meshes

In this paper we present a new high order semi-implicit discontinuous-Galerkin (DG) scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. This allows for a fast computation of the DG operators and matrices without any need to store them for each mesh element. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations because the semi-Lagrangian approach is the only explicit discretization for convection terms that does not need any stability restriction on the maximum admissible time step. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time.

Long‐time asymptotics of solution for the fifth‐order modified KdV equation in the presence of discrete spectrum

AbstractWe investigate the Cauchy problem of an integrable focusing fifth‐order modified Korteweg–de Vries (KdV) equation, which contains the fifth‐order dispersion and relevant higher order nonlinear terms. The long‐time asymptotics of solution is established in the case of initial conditions that lie in some low regularity weighted Sobolev spaces and allow for the presence of discrete spectrum. Our method is based on a generalization of the nonlinear steepest descent method of Deift and Zhou. We show that the solution decomposes in the long time into three main regions: (i) an expanding oscillatory region where solitons and breathers travel with positive velocities, the leading order term has the form of a multisoliton/breather and soliton/breather–radiation interactions; (ii) a Painlevé region, which does not have traveling solitons and breathers, the asymptotics can be characterized with the solution of a fourth‐order Painlevé II equation; (iii) a region of breathers traveling with negative velocities. Employing a global approximation via PDE techniques, the asymptotic behavior of solution is extended to lower regularity spaces with weights.

Global attractor for the damped BBM equation in the sharp low regularity space

Global attractor for the damped BBM equation in the sharp low regularity space

Maximizing the symmetry of Maxwell’s equations

Maxwell’s equations can be successfully extended to electromagnetic fields having three complex-valued components rather than their usual three real-valued components. Here the implications of interpreting the imaginary-valued components as extending into time rather than space are explored. The complex-valued Maxwell equations remain consistent with the original Maxwell equations and the experimental results that they predict. Further, the extended equations predict novel phenomena such as the existence of electromagnetic waves that propagate not only through regular space but also through a separate temporal space (time) that is implied by the three imaginary components of the fields. In a vacuum, part of these imaginary valued waves propagates through time at the same rate as an observer stationary in space. While the imaginary valued field components are not directly observable, analysis indicates that they should be indirectly detectable experimentally based on secondary effects that occur under special circumstances. Experimental investigation attempting to falsify or support the existence of complex valued electromagnetic fields extending into time is merited due to the substantial theoretical and practical implications involved.

Transition from dynamic equilibrium to unconstrained growth in cooperative systems

We introduce a model of cooperative evolution of agents (CE model) on any lattice or graph with agents that have only one opportunity to evolve or grow during their lifespan, then shrink or shrivel to die. The CE model cooperative rule: growing agents meet in the model space, and each agent involved is granted a bonus lifespan. Such synergy is easy to find in studies of social media, economics, epidemiology, etc.In the current work, the CE model uses the 2D square regular lattice cellular space and the simplest evolution rules. Numerical simulation shows that the bonus lifespan introduces a threshold into the CE model and the agent-occupied space shows chaotic behavior depending on the CE model parameters.The CE model of cooperative evolution is rather simple but opens up many possibilities for further research to understand in depth the effects of cooperation and synergy, and thus on real-life evolution.

Spatiotemporal Fusion Prediction of Sea Surface Temperatures Based on the Graph Convolutional Neural and Long Short-Term Memory Networks

Sea surface temperature (SST) prediction plays an important role in scientific research, environmental protection, and other marine-related fields. However, most of the current prediction methods are not effective enough to utilize the spatial correlation of SSTs, which limits the improvement of SST prediction accuracy. Therefore, this paper first explores spatial correlation mining methods, including regular boundary division, convolutional sliding translation, and clustering neural networks. Then, spatial correlation mining through a graph convolutional neural network (GCN) is proposed, which solves the problem of the dependency on regular Euclidian space and the lack of spatial correlation around the boundary of groups for the above three methods. Based on that, this paper combines the spatial advantages of the GCN and the temporal advantages of the long short-term memory network (LSTM) and proposes a spatiotemporal fusion model (GCN-LSTM) for SST prediction. The proposed model can capture SST features in both the spatial and temporal dimensions more effectively and complete the SST prediction by spatiotemporal fusion. The experiments prove that the proposed model greatly improves the prediction accuracy and is an effective model for SST prediction.