Mapping Properties Research Articles

Approximation of Time-Frequency Shift Equivariant Maps by Neural Networks

Based on finite-dimensional time-frequency analysis, we study the properties of time-frequency shift equivariant maps that are generally nonlinear. We first establish a one-to-one correspondence between Λ-equivariant maps and certain phase-homogeneous functions and also provide a reconstruction formula that expresses Λ-equivariant maps in terms of these phase-homogeneous functions, leading to a deeper understanding of the class of Λ-equivariant maps. Next, we consider the approximation of Λ-equivariant maps by neural networks. In the case where Λ is a cyclic subgroup of order N in ZN×ZN, we prove that every Λ-equivariant map can be approximated by a shallow neural network whose affine linear maps are simply linear combinations of time-frequency shifts by Λ. This aligns well with the proven suitability of convolutional neural networks (CNNs) in tasks requiring translation equivariance, particularly in image and signal processing applications.

Mapping Properties of Associate Laguerre Polynomial in Symmetric Domains

The significant characteristics of Associate Laguerre polynomials (ALPs) have noteworthy applications in the fields of complex analysis and mathematical physics. The present article mainly focuses on the inclusion relationships of ALPs and various analytic domains. Starting with the investigation of admissibility conditions of the analytic functions belonging to these domains, we obtained the conditions on the parameters of ALPs under which an ALP maps an open unit disc inside such analytical domains. The graphical demonstration enhances the outcomes and also proves the validity of our obtained results.

A Squared Smoothing Newton Method for Semidefinite Programming

This paper proposes a squared smoothing Newton method via the Huber smoothing function for solving semidefinite programming problems (SDPs). We first study the fundamental properties of the matrix-valued mapping defined upon the Huber function. Using these results and existing ones in the literature, we then conduct rigorous convergence analysis and establish convergence properties for the proposed algorithm. In particular, we show that the proposed method is well-defined and admits global convergence. Moreover, under suitable regularity conditions, that is, the primal and dual constraint nondegenerate conditions, the proposed method is shown to have a super-linear convergence rate. To evaluate the practical performance of the algorithm, we conduct extensive numerical experiments for solving various classes of SDPs. Comparison with the state-of-the-art SDP solvers demonstrates that our method is also efficient for computing accurate solutions of SDPs. Funding: The research of D. Sun was supported in part by the Hong Kong Research Grants Council under Grant 15307523, and the research of K.-C. Toh was supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant call [MOE-2019-T3-1-010].

Uniqueness and Liouville Properties of Subelliptic Harmonic Maps with Potential

Uniqueness and Liouville Properties of Subelliptic Harmonic Maps with Potential

A Symmetric View of Fixed-Point Results in Non-Archimedean Generalized Neutrosophic Metric Spaces

In this paper, we extend prominent fixed-point theorems within the framework of symmetry, a structure increasingly relevant in decision-making, optimization, and uncertainty modeling. While previous studies have explored fixed-point theorems in non-Archimedean spaces, the influence of symmetry on the properties of mappings remains underexamined. To address this gap, we introduce and analyze the concepts of χ-contractions and χ-weak contractions, demonstrating how symmetry impacts the conditions for the existence of fixed points. Our methodology integrates these concepts in generalized neutrosophic metric spaces, providing a novel perspective on fixed-point theory. We perform a rigorous analysis, revealing new insights into their practical applications. However, our proposed system may face limitations in complex or dynamic environments, where additional conditions may be necessary to ensure the existence or uniqueness of fixed points.

FFPNet: Fine-Grained Feature Perception Network for Semantic Change Detection on Bi-Temporal Remote Sensing Images

Semantic change detection (SCD) is a newly important topic in the field of remote sensing (RS) image interpretation since it provides semantic comprehension for bi-temporal RS images via predicting change regions and change types and has great significance for urban planning and ecological monitoring. With the availability of large scale bi-temporal RS datasets, various models based on deep learning (DL) have been widely applied in SCD. Since convolution operators in DL extracts two-dimensional feature matrices in the spatial dimension of images and stack feature matrices in the dimension termed the channel, feature maps of images are tri-dimensional. However, recent SCD models usually overlook the stereoscopic property of feature maps. Firstly, recent SCD models are usually limited in capturing spatial global features in the process of bi-temporal global feature extraction and overlook the global channel features. Meanwhile, recent SCD models only focus on spatial cross-temporal interaction in the process of change feature perception and ignore the channel interaction. Thus, to address above two challenges, a novel fine-grained feature perception network (FFPNet) is proposed in this paper, which employs the Omni Transformer (OiT) module to capture bi-temporal channel–spatial global features before utilizing the Omni Cross-Perception (OCP) module to achieve channel–spatial interaction between cross-temporal features. According to the experiments on the SECOND dataset and the LandsatSCD dataset, our FFPNet reaches competitive performance on both countryside and urban scenes compared with recent typical SCD models.

Isolated Calmness of Perturbation Mappings and Superlinear Convergence of Newton-Type Methods

In this paper, we characterize Lipschitzian properties of different multiplier-free and multiplier-dependent perturbation mappings associated with the stationarity system of a so-called generalized nonlinear program popularized by Rockafellar. Special emphasis is put on the investigation of the isolated calmness property at and around a point. The latter is decisive for the locally fast convergence of the so-called semismooth* Newton-type method by Gfrerer and Outrata. Our central result is the characterization of the isolated calmness at a point of a multiplier-free perturbation mapping via a combination of an explicit condition and a rather mild assumption, automatically satisfied e.g. for standard nonlinear programs. Isolated calmness around a point is characterized analogously by a combination of two stronger conditions. These findings are then related to so-called criticality of Lagrange multipliers, as introduced by Izmailov and extended to generalized nonlinear programming by Mordukhovich and Sarabi. We derive a new sufficient condition (a characterization for some problem classes) of nonexistence of critical multipliers, which has been also used in the literature as an assumption to guarantee local fast convergence of Newton-, SQP-, or multiplier-penalty-type methods. The obtained insights about critical multipliers seem to complement the vast literature on the topic.

On the role of the surface geometry in convex billiards

Abstract This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some properties that have long been known for billiards on the plane are established. We prove the twist property of the billiard maps and establish some conditions for the existence and non-existence of rotational invariant curves. Although we prove that Lazutkin’s and Hubacher’s theorems are valid for general surfaces, we also find that Mather’s theorem does not apply to surfaces of non-negative curvature.

Mild Solutions of the (ρ1,ρ2,k1,k2,φ)-Proportional Hilfer–Cauchy Problem

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the (ρ1,ρ2,k1,k2,φ)-proportional integral and the (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the (ρ1,ρ2,k1,k2,φ)-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the (ρ1,ρ2,k1,k2,φ)-proportional integral and (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional derivative. The results obtained suggest that the most comprehensive formulation of this fractional calculus has been achieved. Under the guidance of the findings from earlier sections, we investigate the existence of mild solutions for the (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results.

Exploring Soliton Solutions and Chaotic Dynamics in the (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation: A Generalized Rational Exponential Function Approach

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations.

Estimates for Certain Rough Multiple Singular Integrals on Triebel–Lizorkin Space

This paper focuses on studying the mapping properties of singular integral operators over product symmetric spaces. The boundedness of such operators is established on Triebel–Lizorkin spaces whenever their rough kernel functions belong to the Grafakos and Stefanov class. Our findings generalize, extend and improve some previously known results on singular integral operators.

Invasive management of atrial tachycardias using a novel lattice-tip catheter combining high-density mapping and dual ablation properties: initial real-world experience.

Invasive management of atrial tachycardias(ATs) requires proper diagnosis of the mechanism followed by elimination of the responsible substrate. A novel lattice-tip catheter with both high-density mapping and dual ablation properties(radiofrequency-RF/pulsed field ablation-PFA) has been recently introduced for catheter ablation of atrial fibrillation. We present the first study to assess its performance in the management of ATs (diagnostic and therapeutic). Patients with documented ATs were selected. Activation mapping was used for the establishment of the AT mechanism. Confirmation with entrainment was performed, whenever appropriate. Accuracy of the activation mapping in diagnosis, acute ablation efficacy, and procedural characteristics were the study endpoints. Twenty patients were included (12 cavotricuspid isthmus-dependent atrial flutters, 5 mitral flutters, 2 roof flutters, and 2 focal ATs). Proper diagnosis was established by activation mapping in all cases. The mean mapping time was 7.85 ± 3.06 min with 296.82 ± 150.9 mean mapping points/minute. The mean ablation time was 54.25 ± 42.97 s. Conversion to sinus rhythm during ablation was achieved in all cases with the exception of a roof flutter that converted to mitral flutter and a case of a parahisian AT in which ablation was not attempted. Patients that received ablation did not experience any arrhythmia recurrence in a mean follow up of 4.14 ± 0.91months. No major or minor complications occurred. The lattice-tip catheter and its dedicated electroanatomical mapping system provided sufficiently detailed activation mapping for the diagnosis of the AT mechanism. The delivered lesions were highly effective acutely, with no adverse events. However, limitations exist and should be acknowledged.

On sharp heat kernel estimates in the context of Fourier–Dini expansions

We prove sharp estimates of the heat kernel associated with Fourier–Dini expansions on (0,1) equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier–Dini semigroup.

Hardy–Littlewood maximal operator in a new vanishing Orlicz–Morrey space

We study mapping properties of maximal operator in a new vanishing subspace of Orlicz–Morrey spaces. We show that the vanishing property defining that subspace is preserved under the action of maximal operator. As an application, we also investigate the behavior of fractional operators.

Electrographic flow mapping of atrial fibrillation.

A low ceiling of efficacy exists for the treatment of persistent atrial fibrillation via pulmonary vein isolation without adjunctive ablations, which is likely because they do not target an individual patient's specific underlying disease mechanisms. Electrographic flow (EGF) mapping is the first system that reliably displays wavefront propagation through the atria. It is a promising tool for localizing sources of atrial fibrillation, guiding targeted ablation, and visualizing conduction through the atrial substrate. We describe EGF mapping with emphasis on contemporary studies examining map reproducibility and use cases in the preclinical and clinical environment. Animal experiments demonstrated that maps were interpretable across increasingly complex rhythms with pacing during spontaneously persistent atrial fibrillation reliably simulating EGF sources. The FLOW-AF randomized controlled trial showed that source ablation improved outcomes and that EGF map properties may be used to phenotype patients based on their atrial fibrillation mechanisms and recurrence likelihoods. Targeted ablation strategies balance the risks of insufficiently ablating atrial fibrillation triggers with exacerbating disease through additional scar formation. EGF mapping leverages spatiotemporal relationships in voltage to localize sources and quantify substrate health. Further research is needed to optimize phenotyping and treatment efforts.

The Teodorescu and the Π-operator in octonionic analysis and some applications

In the development of function theory in octonions, the non-associativity property produces an additional associator term when applying the Stokes formula. To take the non-associativity into account, particular intrinsic weight factors are implemented in the definition of octonion-valued inner products to ensure the existence of a reproducing Bergman kernel. This Bergman projection plays a pivotal role in the L2-space decomposition demonstrated in this paper for octonion-valued functions. In the unit ball, we explicitly show that the intrinsic weight factor is crucial to obtain the reproduction property and that the latter precisely compensates an additional associator term that otherwise appears when leaving out the weight factor.Furthermore, we study an octonionic Teodorescu transform and show how it is related to the unweighted version of the Bergman transform and establish some operator relations between these transformations. We apply two different versions of the Borel-Pompeiu formulae that naturally arise in the context of the non-associativity. Next, we use the octonionic Teodorescu transform to establish a suitable octonionic generalization of the Ahlfors-Beurling operator, also known as the Π-operator. We prove an integral representation formula that presents a unified representation for the Π-operator arising in all prominent hypercomplex function theories. Then we describe some basic mapping properties arising in context with the L2-space decomposition discussed before.Finally, we explore several applications of the octonionic Π-operator. Initially, we demonstrate its utility in solving the octonionic Beltrami equation, which characterizes generalized quasi-conformal maps from R8 to R8 in a specific analytical sense. Subsequently, analogous results are presented for the hyperbolic octonionic Dirac operator acting on the right half-space of R8. Lastly, we discuss how the octonionic Teodorescu transform and the Bergman projection can be employed to solve an eight-dimensional Stokes problem in the non-associative octonionic setting.

Hardy–Littlewood maximal function on Lorentz–Herz spaces

AbstractThis paper extends the study of the generalized Lorentz spaces to the Lorentz–Herz spaces. The Lorentz–Herz spaces consist of all Lebesgue measurable functions such that theirs non-increasing rearrangements belong to the weighted Herz space. The main result of this paper establishes the mapping properties of the Hardy–Littlewood maximal function on the Lorentz–Herz spaces.

On the existence and Lipschitz-like property of the solution map to parametric equilibrium problems governed by trifunction with applications

In this paper, we study a class of parametric equilibrium problems governed by a trifunction. We apply the Fan-KKM theorem under new type of monotonicity and coercivity conditions for trifunctions to obtain the existence result of a solution for the equilibrium problem. For the solution map of parametric equilibrium problems, we obtain the Hölder-type Lipschitz-like property relative to a closed and convex set, which generalizes some existing results in the literature. We illustrate the application of our abstract results in the study of a new model of a fluid flow problem with slip and leak boundary conditions of frictional type.

I-Symmetric Spaces and the π-Images of Metric Spaces

Symmetric spaces and sn-symmetric spaces, as a generalization of metric spaces, have many important properties and have been widely discussed. We consider characterizations and mapping properties of sn-symmetric spaces under ideal convergence. I-symmetric spaces and I-sn-symmetric spaces are defined and studied. These not only generalize some classical results on symmetric spaces but also provide new directions to study generalized metric spaces using the notion of ideal convergence. As an application of I-sn-symmetric spaces, some relevant properties of statistical convergence are obtained. Some unanswered questions in this field are raised.

Adapting high-speed indentation mapping for investigating microstructure-property correlations in chromium carbide-nickel alloy coatings: Challenges and solutions

This study investigates the microstructural and mechanical characteristics of chromium carbide‑nickel rich alloy coatings produced through laser cladding, detonation spraying, and plasma spraying techniques. While all three processing techniques used the same feedstock powder, each method yields coatings with unique microstructures encompassing varying compositions and length scales. Employing Field Emission Scanning Electron Microscopy (FE-SEM) in combination with Energy Dispersive Spectroscopy (EDS) and nanoindentation mapping, local microstructure and mechanical properties were assessed at the micrometre length scale. Laser-clad coatings exhibit a typical metal matrix composite microstructure with high carbide content, distinct (MoNb)C2 phases, and Fe in the metallic matrix, while the thermal sprayed coatings showed carbides of varying sizes and metallic matrix with varying degrees of chromium dissolved in it. The instrumented indentation technique helped precisely record the microstructural features in all the coatings. Chromium carbide consistently emerges as the hardest phase across all coatings, with variations in metallic matrix hardness. Higher matrix hardness in thermal sprayed coatings correlates with increased Cr content, attributed to extended solid solubility of Cr and the presence of Mo and Nb. Energy dispersive spectroscopy and transmission electron microscopy provided clearer insight into the microstructure. This study highlights a direct one-to-one correlation between microstructure and mechanical properties mapped using instrumented indentation techniques in chromium carbide‑nickel rich coatings across different deposition methods.