Smoothness Properties Research Articles

Smoothing properties for a two-dimensional Kawahara equation

In this paper we study smoothness properties of solutions to a two-dimensional Kawahara equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data ϕ possesses certain regularity and sufficient decay as x→∞, then the solution u(t) will be smoother than ϕ for 0<t≤T where T is the existence time of the solution.

Adaptive estimation for functional data: Using a framelet block‐thresholding method

This article considers a framelet block‐thresholding method for estimating mean and covariance functions from discretely sampled noisy observations. Estimated convergence rates are established for all types of sampling schemes. In particular, the results reveal a phase transition phenomenon related to the number of observations on each curve. It is shown that the proposed procedures are adaptive in automatically adjusting the smoothness properties of the underlying mean and covariance functions. In contrast, theoretical results for other smoothing methods hold in the setting where smoothness parameters are assumed to be known, since the regularization parameters of estimators that depend on smoothness properties need to be chosen carefully. Simulation studies are provided to offer empirical support for the theoretical results. A comparison with other methods demonstrates that the proposed method outperforms in adaptivity. An application to a real dataset is also provided to illustrate the proposed estimation procedure.

WELL-POSEDNESS AND REGULARITY OF CAPUTO–HADAMARD TIME-FRACTIONAL DIFFUSION EQUATIONS

Ultraslow diffusion describes the long-time diffusion of particles whose mean square displacement (MSD) grows logarithmically in time. We prove the well-posedness of a Caputo–Hadamard time-fractional diffusion model in multiple space dimensions, in which the MSD in time grows logarithmically and thus provides adequate descriptions for the ultraslow diffusion processes, as well as the smoothing properties of the solutions.

A nonhomogeneous boundary‐valued problem for the coupled Korteweg–de Vries system

In this paper, we study the initial‐boundary‐value problem (IBVP) for coupled Korteweg–de Vries equations posed on a finite interval with nonhomogeneous boundary conditions. We overcome the requirement for stronger smooth boundary conditions in the traditional method via the Laplace transform. Our approach uses the strong Kato smoothing property and the contraction mapping principle.

TSE_Fuse: Two stage enhancement method using attention mechanism and feature-linking model for infrared and visible image fusion

Infrared and visible image fusion is a common solution that joints multispectral modals to address weaknesses of observation in a complex environment to achieve all-weather and full-time perception. The core concept of pixel-level fusion is to obtain complementary features and remove redundant factors. The existing works based on enhancement aim to improve the details of raw information, but most of them easily suffer from noise while in the enhancement process since they ignore well-designed feature selection. To further suppress noise and bridge infrared and visible latent cues, we designed a two-stage enhancement (TSE) framework by using an attention mechanism and feature-linking model (FLM). First of all, we construct a novel decomposition scheme that combines structure adaptive total-variational (SATV) and l1 sparsity term, which is called SATV-l1, to extract two scale detail layers and base layer. Furthermore, the l1 sparsity term is exploited on the base layer to compute its piecewise smoothness property owing to the initial base layer containing texture features. In the final stage, we introduce the TSE to fulfill the detail layer and reconstruct the fused image. Extensive experimental validations on public datasets are performed, demonstrating the robustness and effectiveness of our approach.

Extreme-ultraviolet vector-vortex beams from high harmonic generation

Structured light in the short-wavelength regime opens exciting avenues for the study of ultrafast spin and electronic dynamics. Here, we demonstrate theoretically and experimentally the generation of vector-vortex beams (VVB) in the extreme ultraviolet through high-order harmonic generation (HHG). The up-conversion of VVB, which are spatially tailored in their spin and orbital angular momentum, is ruled by the conservation of the topological Pancharatnam charge in HHG. Despite the complex propagation of the driving beam, high-harmonic VVB are robustly generated with smooth propagation properties. Remarkably, we find out that the conversion efficiency of high-harmonic VVB increases with the driving topological charge. Our work opens the possibility to synthesize attosecond helical structures with spatially varying polarization, a unique tool to probe spatiotemporal dynamics in inhomogeneous media or polarization-dependent systems.

Analysis of Soil Quality Index of mixed garden land use type on dry land in Blang Bintang sub-district, Aceh Besar district

This research was conducted by using a descriptive method based on the results of surveys and field observations and laboratory analysis. Field survey activities were carried out to obtain primary data in the form of general biophysical conditions of the area and physical and chemical characteristics of the soil obtained from observations and indicators of soil quality through soil analysis in the laboratory. Soil sampling points were determined using the purposive sampling method, namely points that have been determined in selected dry land areas in Blang Bintang District, Aceh Besar District. Soil sampling for analysis of chemical properties was carried out by drilling. Soil drilling was carried out to determine the thickness of the soil solum. Sampling was focused only on the top soil layer with a thickness of 0 - 20 cm. In mixed garden land use type (LUT) 5 - 6 sample points were taken which were then analyzed in the laboratory. From the data from the soil analysis, the fertility status of each type of land use will be determined. Field observations and sampling were carried out at each observation point (LUT) in the Blang Bintang District, Aceh Besar District.The soil characteristics and a large percentage of sand compared to the percentage of silt and clay. The low content of clay fraction in both lands affected the formation of soil aggregates. The position and composition of organic matter greatly determine the process of forming stability and distribution of aggregates. Sandy soil in mixed garden vegetation is difficult to absorb water and nutrients due to large grains and small surface area per unit weight. The soil which is dominated by the sand fraction is porous with high aeration pores. Smooth aeration properties can increase the oxidationof organic matter.

Roles of Epithelial and Mesenchymal TRP Channels in Mediating Inflammatory Fibrosis.

The maintenance of normal vision is dependent on preserving corneal transparency. For this to occur, this tissue must remain avascular and its stromal architecture needs to be retained. Epithelial transparency is maintained provided the uppermost stratified layers of this tissue are composed of terminally differentiated non-keratinizing cells. In addition, it is essential that the underlying stromal connective tissue remains avascular and scar-free. Keratocytes are the source of fibroblasts that are interspersed within the collagenous framework and the extracellular matrix. In addition, there are sensory nerve fibers whose lineage is possibly either neural crest or mesenchymal. Corneal wound healing studies have been undertaken to delineate the underlying pathogenic responses that result in the development of opacification following chemical injury. An alkali burn is one type of injury that can result in severe and long- lasting losses in ocular transparency. During the subsequent wound healing process, numerous different proinflammatory cytokines and proteolytic enzymes undergo upregulation. Such increases in their expression levels induce maladaptive expression of sustained stromal inflammatory fibrosis, neovascularization, and losses in the smooth optical properties of the corneal outer surface. It is becoming apparent that different transient receptor potential channel (TRP) isoforms are important players in mediating these different events underlying the wound healing process since injury upregulates both their expression levels and functional involvement. In this review, we focus on the involvement of TRPV1, TRPA1 and TRPV4 in mediating some of the responses that underlie the control of anterior ocular tissue homeostasis under normal and pathological conditions. They are expressed on both different cell types throughout this tissue and also on corneal sensory nerve endings. Their roles have been extensively studied as sensors and transducers of environmental stimuli resulting from exposure to intrinsic modulators and extrinsic ligands. These triggers include alteration of the ambient temperature and mechanical stress, etc., that can induce pathophysiological responses underlying losses in tissue transparency activated by wound healing in mice losses in tissue transparency. In this article, experimental findings are reviewed about the role of injury-induced TRP channel activation in mediating inflammatory fibrotic responses during wound healing in mice.

Modeling spatial tail dependence with Cauchy convolution processes

We study the class of dependence models for spatial data obtained from Cauchy convolution processes based on different types of kernel functions. We show that the resulting spatial processes have appealing tail dependence properties, such as tail dependence at short distances and independence at long distances with suitable kernel functions. We derive the extreme-value limits of these processes, study their smoothness properties, and detail some interesting special cases. To get higher flexibility at sub-asymptotic levels and separately control the bulk and the tail dependence properties, we further propose spatial models constructed by mixing a Cauchy convolution process with a Gaussian process. We demonstrate that this framework indeed provides a rich class of models for the joint modeling of the bulk and the tail behaviors. Our proposed inference approach relies on matching model-based and empirical summary statistics, and an extensive simulation study shows that it yields accurate estimates. We demonstrate our new methodology by application to a temperature dataset measured at 97 monitoring stations in the state of Oklahoma, US. Our results indicate that our proposed model provides a very good fit to the data, and that it captures both the bulk and the tail dependence structures accurately.

ОБ ИГРАХ, ПОДОБНЫХ ИГРЕ БАНАХА-МАЗУРА И ИСПОЛЬЗУЕМЫХ В ЭКОНОМИКЕ

This article introduces a new class of games, which we will call G-games. One of the representatives of this class is the famous Banach–Mazur game. Some properties of convexity and smoothness of Banach spaces in connection with G-games are analyzed. An overview of mainly dynamic games developed by a number of authors in recent years is presented, the application of which can be expanded beyond the scope of the situations under consideration and used to analyze other processes.

Partial Differential Equation (PDE) based Hybrid Diffusion Filters for Enhancing Noise Performance of Point of Care Ultrasound (POCUS) Images

A hybrid filter is developed by combining smoothing and edge preservation properties of anisotropic diffusion (AD) filters and noise reduction features of median filtering. Mixed Gaussian Impulse noise and speckle noise are considered for analysis. The performance of this hybrid filter is verified using ultrasound images. The effectiveness of this filter is assessed with Point of Care Ultrasound (POCUS) images to verify whether the algorithm developed is applicable to them. POCUS refers to a handheld portable ultrasound instrument that can be used at patient bedside. Quantitative analysis with COVID-19 POCUS images, in terms of SNR, SSIM and MSE is performed. Results demonstrate that for all test images, the proposed filter has the best SNR, least MSE, and highest SSIM. Significant improvement in image quality is thus observed both qualitatively and quantitatively. The novelty of suggested technique is its effectiveness in reducing both mixed Gaussian impulse noise and speckle noise in ultrasound as well as POCUS images without the need for separate filters. POCUS has played a significant role in the diagnosis and management of pulmonary, cardiac and vascular pathologies associated with COVID-19. Automatic segmentation of these images and subsequent automatic detection and diagnosis are becoming increasingly popular due to the rapid development of artificial intelligence technologies. These results are useful in implementing better pre-processing prior to segmentation of ultrasound images to facilitate improved patient care.

Modified generalized neo-fuzzy system with combined online fast learning in medical diagnostic task for situations of information deficit

&lt;abstract&gt; &lt;p&gt;In the paper, we propose the modified generalized neo-fuzzy system. It is designed to solve the pattern-image recognition task by working with data that are fed to the system in the image form. The neo-fuzzy system can work with small training datasets, where classes can overlap in a features space. The core of the system under consideration is a modification of multidimensional generalized neuro-fuzzy neuron with an additional softmax activation function in the output layer instead of the defuzzification layer and quartic-kernel functions as membership ones. The learning procedure of the system combined cross-entropy criterion optimization using a matrix version of the optimal by speed Kaczmarz-Widrow-Hoff algorithm with the additional filtering (smoothing) properties. In comparison to the well-known systems, the modified neo-fuzzy one provides both numerical and computational implementation simplicity. The computational experiments have proved the effectiveness of the modified generalized neo-fuzzy-neuron, including the situation with shot training datasets.&lt;/p&gt; &lt;/abstract&gt;

Hyperspectral Image Denoising Using Adaptive Weight Graph Total Variation Regularization and Low-Rank Matrix Recovery

Hyperspectral image (HSI) is often corrupted by various kinds of noises. This letter proposes an innovative HSI denoising approach by leveraging the graph signal processing (GSP) theory and the low-rank (LR) matrix recovery model. With GSP, the piecewise smoothness (PWS) property of the HSI can be efficiently characterized, leading to a new regularization for HSI denoising, termed the adaptive weight graph total variation (AWGTV) regularization. Then, the denoising problem is formulated into a constrained optimization problem that incorporates the AWGTV and the LR property of the HSI. An augmented Lagrange multiplier method is adopted to solve the problem. Numerical experiments conducted on synthetic and real-world datasets and comparisons with existing methods demonstrate the effectiveness of the proposed denoising algorithm.

Coupled Splines for Sparse Curve Fitting.

We formulate as an inverse problem the construction of sparse parametric continuous curve models that fit a sequence of contour points. Our prior is incorporated as a regularization term that encourages rotation invariance and sparsity. We prove that an optimal solution to the inverse problem is a closed curve with spline components. We then show how to efficiently solve the task using B-splines as basis functions. We extend our problem formulation to curves made of two distinct components with complementary smoothness properties and solve it using hybrid splines. We illustrate the performance of our model on contours of different smoothness. Our experimental results show that we can faithfully reconstruct any general contour using few parameters, even in the presence of imprecisions in the measurements.

On the relationship and the norm discretization for Lizorkin-Triebel spaces with generalized smoothness

Homogeneous and non-homogeneous Lizorkin-Triebel spaces with generalized smoothness ?F ?(.)pq (Rn) and F?(.)pq (Rn) have been considered. In particular, under some assumptions of the function ?(t): R+ ? R+,?(1) = 1 determining the generalized smoothness properties, the discretization procedure is realized and the relationship is established between these spaces on Rn and their discrete analogues.

Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation

&lt;p style='text-indent:20px;'&gt;The main purpose of this paper is to study local regularity properties of the fourth-order nonlinear Schrödinger equations on the half line. We prove the local existence, uniqueness, and continuous dependence on initial data in low regularity Sobolev spaces. We also obtain the nonlinear smoothing property: the nonlinear part of the solution on the half line is smoother than the initial data.&lt;/p&gt;

Hyperspectral Image Classification Based on Kernel-Guided Deformable Convolution and Double-Window Joint Bilateral Filter

Convolutional neural networks (CNNs) have been widely used in hyperspectral image (HSI) classification. However, a shape-fixed convolution kernel cannot extract appropriate spatial-spectral features. Thus, we propose a novel two-stage classification method based on kernel-guided deformable convolution networks and double-window joint bilateral filter (KDCDWBF) for HSIs. First, according to the calculated similarity map, the shape of the kernel-guided deformable convolution (KDC) is more consistent with the real shape of land covers, so the KDC can extract more pure neighborhood spatial-spectral information. Then, using the piecewise smoothness property of the HSI, a double-window joint bilateral filter (DWJBF) is designed to complete the coarse-to-fine classification stage, which can solve the misclassification problem of single pixels and small regions. Experiments on two HSI datasets demonstrate that the proposed network can achieve better classification performance when compared with other state-of-the-art methods.

Sharp Kato smoothing properties of weakly dissipated KdV equations with variable coefficients on a periodic domain

&lt;p style='text-indent:20px;'&gt;It is well known that the solutions of the Cauchy problem of the Korteweg-de Vries (KdV) equation on a periodic domain &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ {\mathbb{T}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ u_t +uu_x +u_{xxx} = 0, \quad u(x,0) = \phi (x), \quad x\in {\mathbb{T}}, \ t\in {\mathbb{R}}, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;possess neither the sharp Kato smoothing property,&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE2"&gt; \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies \partial ^{s+1}_xu \in L^{\infty}_x ({\mathbb{T}}, L^2 (0,T)), $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;nor the Kato smoothing property,&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE3"&gt; \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies u\in L^2 (0,T; H^{s+1} ({\mathbb{T}})). $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;This paper shows that the solutions of the Cauchy problem of following weakly dissipated KdV equation with variable coefficients posed on a periodic domain &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ {\mathbb{T}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;,&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE4"&gt; \begin{document}$ u_t +uu_x +a(x,t) u_{xxx} - (g(x,t)u_{x})_x = 0, \qquad u(x,0) = \phi (x), $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;where &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ a $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ g $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are given real-valued smooth functions periodic in &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ x $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; satisfying&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE5"&gt; \begin{document}$ a(x,t)\ne 0, \quad x\in {\mathbb{T}}, \ t\geq 0 \,\quad \mbox{and} \quad \int_{\mathbb{T}}\frac{g(x,t)}{|a(x,t)|} dx &amp;gt; 0 \quad \forall t\geq 0, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;possess the sharp Kato smoothing property,&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE6"&gt; \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies \partial ^{s+1}_xu \in L^{\infty}_x ({\mathbb{T}}, L^2 (0,T)), \quad \forall \, s\geq 0, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;and the nonlinear part of its solution &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; possesses &lt;i&gt;the strong Kato smoothing property&lt;/i&gt;,&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE7"&gt; \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies (u -v)\in C([0,T]; H^{s+1} ({\mathbb{T}})), \quad \forall \, s&amp;gt;\frac12, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;and the &lt;i&gt;sharp double Kato smoothing property&lt;/i&gt;,&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE8"&gt; \begin{document}$ \phi \in H^s ({\mathbb{T}}) \implies \partial ^{s+2}_x(u -v)\in L^{\infty}_x ({\mathbb{T}}, L^2 (0,T)), \quad \forall \, s&amp;gt;\frac12, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;with &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ v $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; being the solution of the linear problem&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE9"&gt; \begin{document}$ v_t+ a(x,t)v_{xxx} -(g(x,t)v_{x} )_x = 0, \quad v(x,0) = \phi (x), \quad x\in {\mathbb{T}}, \ t&amp;gt;0. $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;

On a new family of generalized Bernstein operators

"In this paper we remark that alfa-Bernstein operators, introduced by X. Y. Chen et al., are combinations of two known operators (Stancu and Bernstein operators) and we establish the preservation of global smoothness properties by these linear operators, the global smoothness being expressed by a Lipschitz condition with a certain second order modulus of continuity."

Stabilization of higher order Schrödinger equations on a finite interval: Part II

&lt;p style='text-indent:20px;'&gt;Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [&lt;xref ref-type="bibr" rid="b3"&gt;3&lt;/xref&gt;] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ L^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.&lt;/p&gt;