Dual Space Research Articles

Dual magnetic curves and flux ruled surfaces

In this paper, we introduce the dual magnetic trajectories traced by the dual Frenet vectors of any dual curves in dual space via the dual Lorentz force, and then we give the characterizations of dual magnetic curves. Moreover, we define the dual flux ruled surfaces corresponding to the Killing vectors on the dual unit sphere and identify the conditions for these surfaces to be minimal or developable. Additionally, we present certain characterizations related to these surfaces. We demonstrate the visual representations of some dual flux ruled surfaces in some examples.

On Semidefinite Programming Characterizations of the Numerical Radius and Its Dual Norm

On Semidefinite Programming Characterizations of the Numerical Radius and Its Dual Norm

Constitutive behavior of asymmetric multi-material honeycombs with bi-level variably-thickened composite architecture

Bi-level tailoring of cellular metamaterials involving a dual design space of unit cell and elementary beam level architectures has recently gained traction for the ability to achieve extreme elastic constitutive properties along with modulating multi-functional mechanical behavior in an unprecedented way. This article proposes an efficient analytical approach for the accurate evaluation of all constitutive elastic constants of asymmetric multi-material variably-thickened hexagonal lattices by considering the combined effect of bending, stretching, and shearing deformations of cell walls along with their rigid rotation. A tri-member unit cell is conceptualized, wherein all nine constitutive constants are obtained through the mechanics under one cell wall direction and subsequent repetitive coordinate transformations. We enhance the design space of lattice metamaterials substantially here by introducing multiple exploitable dimensions such as asymmetric geometry, multi-material unit cells and variably-thickened cell walls, wherein the conventional monomaterial auxetic and non-auxetic hexagonal configurations can be analyzed as special cases along with other symmetric and asymmetric lattices such as a range of rectangular and rhombic geometries. The generic analytical approach along with extensive numerical results presented in this paper opens up new avenues for efficient optimized design of the next-generation multi-functional lattices and cellular metamaterials with highly tailored effective elastic properties.

Towards Unified Defense for Face Forgery and Spoofing Attacks via Dual Space Reconstruction Learning

Towards Unified Defense for Face Forgery and Spoofing Attacks via Dual Space Reconstruction Learning

An Introduction to Extended Gevrey Regularity

Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example, when initial value problems are ill-posed in Gevrey settings. In this paper, we consider a convenient framework for studying smooth functions that possess weaker regularity than any Gevrey function. Since the available literature on this topic is scattered, our aim is to provide an overview of extended Gevrey regularity, highlighting its most important features. Additionally, we consider related dual spaces of ultra distributions and review some results on micro-local analysis in the context of extended Gevrey regularity. We conclude the paper with a few selected applications that may motivate further study of the topic.

Fermion quadrature bases for Wigner functionals

A Grassmann functional phase space is formulated for the definition of fermionic Wigner functionals by identifying suitable fermionic operators that are analogues to boson quadrature operators. Instead of the Majorana operators, we use operators that are defined with relative spin transformations between the ladder operators. The eigenstates of these operators are shown to provide orthogonal bases, provided that the dual space is defined with the incorporation of a spin transformation. These bases then serve as quadrature bases in terms of which Wigner functionals are defined in a way equivalent to the bosonic case. As an application, we consider a two-level fermion system.

Next-Generation Dual Transceiver FSO Communication System for High-Speed Trains in Neom Smart City

Smart cities like Neom require efficient and reliable transportation systems to support their vision of sustainable and interconnected urban environments. High-speed trains (HSTs) play a crucial role in connecting different areas of the city and facilitating seamless mobility. However, to ensure uninterrupted communication along the rail lines, advanced communication systems are essential to expand the coverage range of each base station (BS) while reducing the handover frequency. This paper presents the dual transceiver free space optical (FSO) communication system as a solution to achieve these objectives in the operational environment of HSTs in Neom city. Our channel model incorporates log-normal (LN) and gamma–gamma (GG) distributions to represent channel impairments and atmospheric turbulence in the city. Furthermore, we integrated the siding loop model, providing valuable insights into the system in real-world scenarios. To assess the system’s performance, we formulated the received signal-to-noise ratio (SNR) of the network under assumed fading conditions. Additionally, we analyzed the system’s bit error rate (BER) analytically and through Monte Carlo simulation. A comparative analysis with reconfigurable intelligent surfaces (RIS) and relay-assisted FSO communications shows the superior coverage area and efficiency of the dual transceiver model. A significant reduction of up to 76% and 99% in the number of required BSs compared to RIS and relay, respectively, is observed. This reduction leads to fewer handovers and lower capital expenditure (CAPEX) costs.

Discrete-time Euler-smoothing methods for time-varying convex constrained optimization

In this paper, we propose two kinds of prediction–correction methods, discrete-time Euler-smoothing methods, for solving the general time-varying convex constrained optimization problem. By using the complementarity function and smoothing technique, the Karush–Kuhn–Tucker (KKT) trajectory of the problem can be reformulated as a system of smooth equations and considering the dynamics of the system of smooth equations in a discrete constant time-sampling periods scheme. The proposed methods mainly consist of the prediction step derived by Euler approximation and the correction step generated by one or multiple Newton iterations. Under some reasonable conditions, the approximate solution trajectories produced by the methods achieve the asymptotic error bound behaving as O(h4) with respect to optimal trajectory both in the primal and dual space as the smoothing parameter approaches to zero, where h is the sampling period. Finally, the experimental results show that our methods are efficient, and the steady-state tracking errors are much smaller than the other prediction–correction methods under the same conditions.

Mirror symmetry and Hitchin system on Deligne–Mumford curves: Strominger–Yau–Zaslow duality

Abstract We systematically study the moduli stacks of Higgs bundles, spectral curves, and Norm maps on Deligne–Mumford curves. As an application, under some mild conditions, we prove the Strominger–Yau–Zaslow duality for the moduli spaces of Higgs bundles over a hyperbolic stacky curve.

Linearization of holomorphic Lipschitz functions

AbstractLet and be complex Banach spaces with denoting the open unit ball of . This paper studies various aspects of the holomorphic Lipschitz space , endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets of Lipschitz mappings and of bounded holomorphic mappings, from to . Thanks to the Dixmier–Ng theorem, is indeed a dual space, whose predual shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of . We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that contains a 1‐complemented subspace isometric to and that has the (metric) approximation property whenever has it. We also analyze when is a subspace of , and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.

Delta‐points and their implications for the geometry of Banach spaces

AbstractWe show that the Lipschitz‐free space with the Radon–Nikodým property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to . Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of , with a ‐point. Building on these two results, we are able to renorm every infinite‐dimensional Banach space to have a ‐point. Next, we establish powerful relations between existence of ‐points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of ‐points for the asymptotic geometry of Banach spaces. In addition, we prove that if is a Banach space with a shrinking ‐unconditional basis with , or if is a Hahn–Banach smooth space with a dual satisfying the Kadets–Klee property, then and its dual fail to contain ‐points. In particular, we get that no Lipschitz‐free space with a Hahn–Banach smooth predual contains ‐points. Finally, we present a purely metric characterization of the molecules in Lipschitz‐free spaces that are ‐points, and we solve an open problem about representation of finitely supported ‐points in Lipschitz‐free spaces.

Adaptive Deep Fourier Residual method via overlapping domain decomposition

The Deep Fourier Residual (DFR) method is a specific type of variational physics-informed neural network (VPINN). It provides a robust neural network-based solution to partial differential equations (PDEs). The DFR strategy is based on minimizing the dual norm of the weak residual of a PDE, which is equivalent to minimizing the energy norm of the error. To compute the dual norm of the weak residual, the DFR method employs an orthonormal spectral basis of the test space, known for rectangles or cuboids for multiple function spaces.In this work, we extend the DFR method with ideas of traditional domain decomposition (DD). This enables two improvements: (a) to solve problems in more general polygonal domains and (b) to develop an adaptive refinement technique in the test space using a Döfler marking algorithm. In the former case, we retain the desirable equivalence between the employed loss function and the H1 error under non-restrictive assumptions, numerically demonstrating adherence to explicit bounds in the case of the L-shaped domain problem. In the latter, we show how refinement strategies lead to potentially significant improvements against a reference, classical DFR implementation with a test function space of significantly lower dimensionality, allowing us to better approximate singular solutions at a more reasonable computational cost.

Some aspects of the Bergman and Hardy spaces associated with a class of generalized analytic functions

For λ≥0, a C2 function f defined on the unit disk D is said to be λ-analytic if Dz̄f=0, where Dz̄ is the (complex) Dunkl operator given by Dz̄f=∂z̄f−λ(f(z)−f(z̄))/(z−z̄). The aim of the paper is to study several problems on the associated Bergman spaces Aλp(D) and Hardy spaces Hλp(D) for p≥2λ/(2λ+1), such as boundedness of the Bergman projection, growth of functions, density, completeness, and the dual spaces of Aλp(D) and Hλp(D), and characterization and interpolation of Aλp(D).

Counterexamples in rotundity of norms in Banach spaces

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense.In the last part of this note, we solve a fourth question posed by the same three authors by constructing a C∞-smooth norm in c0 whose dual norm is not strictly convex.

Weighted estimates for product singular integral operators in Journé’s class on RD-spaces

Abstract An RD-space 𝑀 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝑀. In this paper, firstly, the authors give the Plancherel–Pôlya characterization of product weighted Triebel–Lizorkin spaces and product weighted Besov spaces on RD-spaces and make some estimates for the product singular integral operators in Journé’s class on these function spaces. As a result of these conclusions, they present some sufficient conditions for the boundedness of product singular integral operators on the product Lipschitz spaces and product weighted Hardy spaces. Secondly, by the boundedness of lifting and projection operators, they also obtain that the dual spaces of the product weighted Hardy spaces are product weighted Carleson measure spaces. Using the idea of dual, the authors obtain the weighted boundedness of singular integral operators on the product weighted Carleson measure spaces.

Multiplication between elements in martingale Hardy spaces and their dual spaces

In the first part of the paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces Hp(0<p⩽1) and functions in their dual spaces. Our decompositions are based on martingale paraproducts. The second part of the paper concerns applications of the method developed for the classical martingales in the first part. In particular, we build a connection between Hardy spaces on spaces of homogenous type equipped with a doubling measure and Hardy spaces with respect to the corresponding dyadic martingales. Using the method introduced in the first part, we obtain analogous results for dyadic martingales on spaces of homogenous type, which, thanks to the aforementioned connection, yield conclusions for products between elements in Hardy spaces and those in their duals on spaces of homogenous type. The key property of martingale Hardy spaces in the study is that they admit the atomic decomposition, for which we provide an interpretation via duality.

Generating operators between Banach spaces

We introduce and study the notion of generating operators as those norm-one operators G:X⟶Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G:X\\longrightarrow Y$$\\end{document} such that for every 0<δ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\delta <1$$\\end{document}, the set {x∈X:‖x‖⩽1,‖Gx‖>1-δ}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{x\\in X:\\Vert x\\Vert \\leqslant 1,\\ \\Vert Gx\\Vert >1-\\delta \\}$$\\end{document} generates the unit ball of X by closed convex hull. This class of operators includes isometric embeddings, spear operators (actually, operators with the alternative Daugavet property), and other examples like the natural inclusions of ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document} into c0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_0$$\\end{document} and of L∞[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty [0,1]$$\\end{document} into L1[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1[0,1]$$\\end{document}. We first present a characterization in terms of the adjoint operator, make a discussion on the behaviour of diagonal generating operators on c0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_0$$\\end{document}-, ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-, and ℓ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _\\infty $$\\end{document}-sums, and present examples in some classical Banach spaces. Even though rank-one generating operators always attain their norm, there are generating operators, even of rank-two, which do not attain their norm. We discuss when a Banach space can be the domain of a generating operator which does not attain its norm in terms of the behaviour of some spear sets of the dual space. Finally, we study when the set of all generating operators between two Banach spaces X and Y generates all non-expansive operators by closed convex hull. We show that this is the case when X=L1(μ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=L_1(\\mu )$$\\end{document} and Y has the Radon-Nikodým property with respect to μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}. Therefore, when X=ℓ1(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=\\ell _1(\\Gamma )$$\\end{document}, this is the case for every target space Y. Conversely, we also show that a real finite-dimensional space X satisfies that generating operators from X to Y generate all non-expansive operators by closed convex hull only in the case that X is an ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-space.

Risk Measures’ Duality on Ordered Linear Spaces

The aim of this paper is to provide a dual representation of convex and coherent risk measures in partially ordered linear spaces with respect to the algebraic dual space. An algebraic robust representation is deduced by weak separation of convex sets by functionals, which are assumed to be only linear; thus, our framework does not require any topological structure of the underlying spaces, and our robust representations are found without any continuity requirement for the risk measures. We also use such extensions to the representation of acceptability indices.

Advancements in the Analysis of Sobolev Spaces and Function Spaces on Manifolds: Theoretical Framework and Applications

This research paper delves into Sobolev spaces and function spaces on smooth manifolds, revealing fundamental theorems such as existence, embeddings, and compactness properties. Noteworthy results include the Poincare inequality elucidating function behavior on compact manifolds and compactness properties of Sobolev spaces on Riemannian manifolds. The study establishes trace theorems for functions on the boundary and interpolation results between Sobolev spaces. Isoperimetric inequalities and stability under weak convergence contribute to a holistic understanding of geometric and analytical aspects of Sobolev spaces. The research concludes by exploring invariance under diffeomorphisms and compactness in dual spaces, providing a unified framework for analyzing function spaces on manifolds.

Dual-space high-frequency learning for transformer-based MRI super-resolution

Background and ObjectiveMagnetic resonance imaging (MRI) can provide rich and detailed high-contrast information of soft tissues, while the scanning of MRI is time-consuming. To accelerate MR imaging, a variety of Transformer-based single image super-resolution methods are proposed in recent years, achieving promising results thanks to their superior capability of capturing long-range dependencies. Nevertheless, most existing works prioritize the design of transformer attention blocks to capture global information. The local high-frequency details, which are pivotal to faithful MRI restoration, are unfortunately neglected. MethodsIn this work, we propose a high-frequency enhanced learning scheme to effectively improve the awareness of high frequency information in current Transformer-based MRI single image super-resolution methods. Specifically, we present two entirely plug-and-play modules designed to equip Transformer-based networks with the ability to recover high-frequency details from dual spaces: 1) in the feature space, we design a high-frequency block (Hi-Fe block) paralleled with Transformer-based attention layers to extract rich high-frequency features; while 2) in the image intensity space, we tailor a high-frequency amplification module (HFA) to further refine the high-frequency details. By fully exploiting the merits of the two modules, our framework can recover abundant and diverse high-frequency information, rendering faithful MRI super-resolved results with fine details. ResultsWe integrated our modules with six Transformer-based models and conducted experiments across three datasets. The results indicate that our plug-and-play modules can enhance the super-resolution performance of all foundational models to varying degrees, surpassing the capabilities of existing state-of-the-art single image super-resolution networks. ConclusionComprehensive comparison of super-resolution images and high-frequency maps from various methods, clearly demonstrating that our module possesses the capability to restore high-frequency information, showing huge potential in clinical practice for accelerated MRI reconstruction.