Local Regularity Research Articles

Coupling local and nonlocal fourth-order evolution equations for image denoising

Local and nonlocal regularization methods have attracted an extensive interest in image denoising. In this paper, we propose two hybrid energy functionals that benefit from the local Laplacian regularization in homogeneous regions and the nonlocal Laplacian regularization in texture regions. The existence and uniqueness of solutions for the gradient flows, which are local and nonlocal coupled fourth-order evolution equations, are proven via the fixed point method. We consider nonlinear generalizations of the coupled equations for image denoising by proposing a new fourth-order operator that reduces the contrast losing effect of the second-order operator. The operator splitting method is adopted for the numerical implementation such that the local and nonlocal parts are handled separately. By numerical experiments, we illustrate that the proposed equations inherit advantages of the local and nonlocal methods for image denoising.

Local well-posedness and regularity criterion for nonhomogeneous magneto-micropolar fluid equations without angular viscosity

Local well-posedness and regularity criterion for nonhomogeneous magneto-micropolar fluid equations without angular viscosity

Stochastic homogenization on perforated domains Ⅰ – Extension Operators

<abstract><p>In this first part of a series of 3 papers, we set up a framework to study the existence of uniformly bounded extension and trace operators for $ W^{1, p} $-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimally smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from $ W^{1, p} $ to $ W^{1, r} $ with the strict inequality $ r < p $. In particular, we estimate the $ L^{r} $-norm of the extended gradient in terms of the $ L^{p} $-norm of the original gradient. Similar relations hold for the symmetric gradients (for $ {\mathbb{R}^{d}} $-valued functions) and for traces on the boundary. As a byproduct we obtain some Poincaré and Korn inequalities of the same spirit.</p> <p>Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions: local $ (\delta, M) $-regularity to quantify statistically the local Lipschitz regularity and isotropic cone mixing to quantify the density of the geometry and the mesoscopic properties. These two properties are sufficient to reduce the problem of extension operators to the connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process, for which we can explicitly estimate the connectivity terms.</p></abstract>

Estimates for evolutionary partial differential equations in classical function spaces

Abstract We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.

Пластическое течение в твердых растворах Cu-Ni как автоволновой процесс

The influence of Ni content in Cu-Ni alloys on the localization of plastic flow under uniaxial tension has been studied. The regularities of localization of plastic deformation during linear and parabolic deformation hardening, as well as at the yield point, depending on the Ni content in the solid solution, have been studied by the speckle photography method. The influence of the Ni content on the duration of the stages of parabolic deformation hardening in alloys of different composition was found. The dependence of the spatial period of localization of plastic deformation (the length of the localization autowave) on the Ni content in the studied alloys is established. Keywords: localization of plastic deformation, autowaves, copper-nickel alloys, structure, mechanical properties, deformation and impurity hardening.

Plastic flow in Cu-Ni solid solutions as an autowave process

The influence of Ni content in Cu-Ni alloys on the localization of plastic flow under uniaxial tension has been studied. The regularities of localization of plastic deformation during linear and parabolic deformation hardening, as well as at the yield point, depending on the Ni content in the solid solution, have been studied by the speckle photography method. The influence of the Ni content on the duration of the stages of parabolic deformation hardening in alloys of different composition was found. The dependence of the spatial period of localization of plastic deformation (the length of the localization autowave) on the Ni content in the studied alloys is established. Keywords: localization of plastic deformation, autowaves, copper-nickel alloys, structure, mechanical properties, deformation and impurity hardening.

Duality Approach to the Regularity Problems for the Navier-Stokes Equations

In this note, we describe a way to study local regularity of a weak solution to the Navier-Stokes equations, satisfying the simplest scale-invariant restriction, with the help of zooming and duality approach to the corresponding mild bounded ancient solution.

Global centralised and structured discriminative non‐negative matrix factorisation for hyperspectral unmixing

AbstractAdvances have been achieved in hyperspectral unmixing using the existing manifold Non‐negative Matrix Factorisation methods, although most of these methods only exploit the preliminary structural information, that is, the nearest neighbour graph. Consequently, the performance of these methods would be degraded when considering only the geometrical structure due to the diverse distribution of the hyperspectral data, that is, the close pixels could belong to different categories or the distant points could be sampled from the same classes. In this context, the present study worked from the perspective of both global and local data relationships to develop and propose a novel approach—the Global centralised and Structured discriminative Non‐negative Matrix Factorisation (GSNMF)—to achieve a further effective representation of hyperspectral unmixing. GSNMF involved maintaining the global centralised clustering and the local structured discriminative regularisation, based on which it could perfectly mine the structure information and drive a discriminative representation of the data. Experiments comparing the application of GSNMF and the state‐of‐the‐art methods to synthetic data demonstrated the superiority of GSNMF. In addition, the consistency of the fractional abundances obtained using GSNMF with the real distributions of spectral data was evaluated on two real‐world datasets.

Local Regularity Criteria in Terms of One Velocity Component for the Navier–Stokes Equations

This paper is devoted to presenting new interior regularity criteria in terms of one velocity component for weak solutions to the Navier–Stokes equations in three dimensions. It is shown that the velocity is regular near a point z if its scaled $$L^p_tL^q_x$$ -norm of some quantities related to the velocity field is finite and the scaled $$L^p_tL^q_x$$ -norm of one velocity component is sufficiently small near z.

Lattice local integrable regularization of the Sine-Gordon model

We study the local lattice integrable regularization of the Sine-Gordon model written down in terms of the lattice Bose-operators. We show that the local spin Hamiltonian obtained from the six-vertex model with alternating inhomogeneities in fact leads to the Sine-Gordon in the low-energy limit. We show that the Bethe Ansatz results for this model lead to the correct general relations for different critical exponents of the coupling constant.

On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients

We consider fractional operators of the form Hs=(∂t-divx(A(x,t)∇x))s,(x,t)∈Rn×R,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {H}}^s=(\\partial _t -\ ext {div}_{x} ( A(x,t)\ abla _{x}))^s,\\ (x,t)\\in {\\mathbb {R}}^n\ imes {\\mathbb {R}}, \\end{aligned}$$\\end{document}where sin (0,1) and A=A(x,t)={A_{i,j}(x,t)}_{i,j=1}^{n} is an accretive, bounded, complex, measurable, ntimes n-dimensional matrix valued function. We study the fractional operators {{mathcal {H}}}^s and their relation to the initial value problem (λ1-2su′)′(λ)=λ1-2sHu(λ),λ∈(0,∞),u(0)=u,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\begin{aligned} (\\lambda ^{1-2s}\ extrm{u}')'(\\lambda )&=\\lambda ^{1-2s}{\\mathcal {H}}\ extrm{u}(\\lambda ), \\quad \\lambda \\in (0, \\infty ), \\\\ \ extrm{u}(0)&= u, \\end{aligned} \\end{aligned}$$\\end{document}in {mathbb {R}}_+times {mathbb {R}}^ntimes {mathbb {R}}. Exploring the relation, and making the additional assumption that A=A(x,t)={A_{i,j}(x,t)}_{i,j=1}^{n} is real, we derive some local properties of solutions to the non-local Dirichlet problem Hsu=(∂t-divx(A(x,t)∇x))su=0for(x,t)∈Ω×J,u=ffor(x,t)∈Rn+1\\(Ω×J).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {H}}^su=(\\partial _t -\ ext {div}_{x} ( A(x,t)\ abla _{x}))^su&=0\\hbox { for}\\ (x,t)\\in \\Omega \ imes J,\ onumber \\\\ u&=f \ ext{ for } (x,t)\\in {\\mathbb {R}}^{n+1}\\setminus (\\Omega \ imes J). \\end{aligned}$$\\end{document}Our contribution is that we allow for non-symmetric and time-dependent coefficients.

Nonautonomous semilinear Wentzell problems in fractal domains

We study a nonautonomous semilinear parabolic problem with a dynamical boundary condition in an irregular domain with fractal boundary. Local existence, uniqueness and regularity results for the mild solution are established via a family of evolution operators. A sufficient condition on the initial datum for global existence is given.

Neural Basis of the Implicit Learning of Complex Artificial Grammar with Nonadjacent Dependencies.

The capacity for the implicit learning/processing of complex grammar with nonadjacent dependencies is an important feature of human language learning. In this fMRI study, using an implicit AGL paradigm, we explored the neural basis of the implicit learning of the nonadjacent dependency rule, disentangling from sequence-based chunk knowledge (i.e., local sequential regularities or substring) by focusing on the low chunk strength items (which were naturally less similar to training strings), based on tracking neural responses during training and test phases. After listening to and memorizing a series of strings of 10 syllables generated from nonadjacent artificial grammar in the training phase, participants implicitly acquired the knowledge of grammar and chunks. Regarding grammaticality, Broca's area was specifically related to low chunk strength grammatical strings relative to nongrammatical strings in the test phase. This region showed decreased activity with time in the training phase, and a lesser decrease in activity was associated with higher performance in grammar learning. Furthermore, Broca's area showed significantly higher strength of functional connectivity with the left superior temporal gyrus in the low chunk strength grammatical string compared with nongrammatical strings, and this functional connectivity increased with the training time. For the chunks, the performance of accurate discrimination of high chunk strength from low chunk strength nongrammatical strings was predicted by hippocampal activity in the training phase. Converging evidence from the training and test phases showed that Broca's area and its functional connectivity with the left superior temporal gyrus were engaged in the implicit learning/processing of the nonadjacent dependency rule, separating the effects of chunks.

A local regularity result for the relaxed micromorphic model based on inner variations

In this paper we study local higher regularity properties of a linear elliptic system that is coupled with a system of Maxwell-type. The regularity result is proved by means of a modified finite difference argument. These modified finite differences are based on inner variations combined with a Piola-type transformation in order to preserve the curl-structure in the Maxwell system. The result is applied to the relaxed micromorphic model.

Multifractal analysis for improved osteoporosis classification

Osteoporosis is characterized by decrease of bone mineral density and alterations of the bone microarchitecture, leading to bone fragility and an increased risk of fractures. Discrimination between osteoporotic and healthy subjects using X-ray images presents a major challenge for medical image analysis and classification. This work introduces a new approach based on multifractal texture analysis for the characterization of the trabecular bone microarchitecture to distinguish between Control Subjects (CS) and Osteoporotic Patients (OP). The proposed method enables a description of both local and global regularity and roughness of the trabecular bone on radiographic images. To this end, the multifractal spectrum is exploited to extract new texture attributes and reveal bone microarchitecture changes due to osteoporosis. To assess the proposed approach effectiveness, tests were carried out for the classification of two populations (CS and OP) using a logistic regression model. A classification rate of 98.01% was reached showing that the proposed approach presents a promising potential as a complementary tool for osteoporosis diagnosis.

Local Temporal Regularities in Child-Directed Speech in Spanish.

The purpose of this study is to characterize the local (utterance-level) temporal regularities of child-directed speech (CDS) that might facilitate phonological development in Spanish, classically termed a syllable-timed language. Eighteen female adults addressed their 4-year-old children versus other adults spontaneously and also read aloud (CDS vs. adult-directed speech [ADS]). We compared CDS and ADS speech productions using a spectrotemporal model (Leong & Goswami, 2015), obtaining three temporal metrics: (a) distribution of modulation energy, (b) temporal regularity of stressed syllables, and (c) syllable rate. CDS was characterized by (a) significantly greater modulation energy in the lower frequencies (0.5-4 Hz), (b) more regular rhythmic occurrence of stressed syllables, and (c) a slower syllable rate than ADS, across both spontaneous and read conditions. CDS is characterized by a robust local temporal organization (i.e., within utterances) with amplitude modulation bands aligning with delta and theta electrophysiological frequency bands, respectively, showing greater phase synchronization than in ADS, facilitating parsing of stress units and syllables. These temporal regularities, together with the slower rate of production of CDS, might support the automatic extraction of phonological units in speech and hence support the phonological development of children. https://doi.org/10.23641/asha.21210893.

Multidimensional fully adaptive lattice Boltzmann methods with error control based on multiresolution analysis

Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fine mesh is unnecessary, still ensuring the overall quality of the numerical solution through error control. This work provides a possible answer to this interesting question, by connecting, for the first time, the field of lattice-Boltzmann Methods (LBM) to the adaptive multiresolution (MR) approach based on wavelets. To this end, we employ a MR multi-scale transform to adapt the mesh as the solution evolves in time according to its local regularity. The collision phase is not affected due to its inherent local nature and because we do not modify the speed of the sound, contrarily to most of the LBM/Adaptive Mesh Refinement (AMR) strategies proposed in the literature, thus preserving the original structure of any LBM scheme. Besides, an original use of the MR allows the scheme to resolve the proper physics by efficiently controlling the accuracy of the transport phase. We carefully test our method to conclude on its adaptability to a wide family of existing lattice Boltzmann schemes, treating both hyperbolic and parabolic systems of equations, thus being less problem-dependent than the AMR approaches, which have a hard time guaranteeing an effective control on the error. The ability of the method to yield a very efficient compression rate and thus a computational cost reduction for solutions involving localized structures with loss of regularity is also shown, while guaranteeing a precise control on the approximation error introduced by the spatial adaptation of the grid. The numerical strategy is implemented on a specific open-source platform called SAMURAI with a dedicated data-structure relying on set algebra.

Foundations of indigenous knowledge on disasters due to natural hazards: lessons from the outlook on floods among the Bayira of the Rwenzori region.

The role of indigenous knowledge in increasing context specificity and exposing blind spots in scientific understanding is widely evidenced in disaster studies. This paper aims to structure the processes that shape indigenous knowledge production and its optimisation using the case of floods. An inductive analytical approach is applied among riparian indigenous communities (focus on the Bayira) of the Rwenzori region of Uganda where plenty of indigenous flood practices have been recorded. Indigenous knowledge of floods is found to be based on intimate comprehension of local hydrometeorological regularities. Insofar as these regularities follow natural dynamics, indigenous socio-epistemic processes are noted to be consistent with the laws of nature. Coupled with regular open sociocultural deliberations, the conceptualisation of hydrometeorological regularities induces an indigenous ontology and empiricist epistemology. This, together with the techniques used, is the driver of crucial epistemic virtues which enable indigenous knowledge to provide disaster solutions that are adapted, pragmatic, and holistic.

ScEMAIL: Universal and Source-free Annotation Method for scRNA-seq Data with Novel Cell-type Perception

Current cell-type annotation tools for single-cell RNA sequencing (scRNA-seq) data mainly utilize well-annotated source data to help identify cell types in target data. However, on account of privacy preservation, their requirements for raw source data may not always be satisfied. In this case, achieving feature alignment between source and target data explicitly is impossible. Additionally, these methods are barely able to discover the presence of novel cell types. A subjective threshold is often selected by users to detect novel cells. We propose a universal annotation framework for scRNA-seq data called scEMAIL, which automatically detects novel cell types without accessing source data during adaptation. For new cell-type identification, a novel cell-type perception module is designed with three steps. First, an expert ensemble system measures uncertainty of each cell from three complementary aspects. Second, based on this measurement, bimodality tests are applied to detect the presence of new cell types. Third, once assured of their presence, an adaptive threshold via manifold mixup partitions target cells into “known” and “unknown” groups. Model adaptation is then conducted to alleviate the batch effect. We gather multi-order neighborhood messages globally and impose local affinity regularizations on “known” cells. These constraints mitigate wrong classifications of the source model via reliable self-supervised information of neighbors. scEMAIL is accurate and robust under various scenarios in both simulation and real data. It is also flexible to be applied to challenging single-cell ATAC-seq data without loss of superiority. The source code of scEMAIL can be accessed at https://github.com/aster-ww/scEMAIL and https://ngdc.cncb.ac.cn/biocode/tools/BT007335/releases/v1.0.

Local regularity estimates for general discrete dynamic programming equations

We obtain an analytic proof for asymptotic Hölder estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.