SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.

Investigation on Damage Regularity of Mikania micrantha

[Objectives]The paper was to understand the occurrence and damage regularity of the invasive plant Mikania micrantha in Huadu District of Guangzhou.[Methods]The damage status of M.micranthFa in different forest lands and its annual growth dynamics were investigated by field investigation.[Results]With the change of canopy density from low to high,the occurrence degree of M.micrantha changed from high to low.The occurrence degree of M.micrantha in different forest land types was:abandoned orchard>wasteland>roadside greenbelt>waterside>forest edge>normally managed orchard.[Conclusions]M.micrantha enters the rapid growth period from March to May in spring,with the growth rate gradually slowing down after June.The result provides a theoretical basis and practical guidance for the prevention and control of M.micrantha.

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.

GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY

This paper studies the global regularity of 2D incompressible anisotropic magnetomicropolar fluid equations with partial viscosity. Ma [22](Ma L. Nonlinear Anal: Real World Appl, 2018, 40: 95–129) examined the global regularity of the 2D incompressible magnetomicropolar fluid system for 21 anisotropic partial viscosity cases. He proved the global existence of a classical solution for some cases and established the conditional global regularity for some other cases. In this paper, we also investigate the global regularity of 12 cases in [22]and some other new partial viscosity cases. The global regularity is established by providing new regular conditions. Our work improves some results in [22] in this sense of weaker regular criteria.

THE REGULARITY CRITERIA OF WEAK SOLUTIONS TO 3D AXISYMMETRIC INCOMPRESSIBLE BOUSSINESQ EQUATIONS

In this paper,we obtain new regularity criteria for the weak solutions to the three dimensional axisymmetric incompressible Boussinesq equations.To be more precise,under some conditions on the swirling component of vorticity,we can conclude that the weak solutions are regular.

THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM

This paper examines the existence of weak solutions to a class of the high-order Korteweg-de Vries(KdV)system in Rn.We first prove,by the Leray-Schauder principle and the vanishing viscosity method,that any initial data N-dimensional vector value function u0(x)in Sobolev space H^(s)(R^(n))(s≥1)leads to a global weak solution.Second,we investigate some special regularity properties of solutions to the initial value problem associated with the KdV type system in R^(2)and R^(3).

THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA

A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L^(1)∩BV class was constructed in[1].In the current paper,we will continue to study the uniqueness and regularity of the constructed solution.The key ingredients are the Holder continuity estimates of the heat kernel in both spatial and time variables.With these finer estimates,we obtain higher order regularity of the constructed solution to Navier-Stokes equation,so that all of the derivatives in the equation of conservative form are in the strong sense.Moreover,this regularity also allows us to identify a function space such that the stability of the solutions can be established there,which eventually implies the uniqueness.

THEORETICAL RESULTS ON THE EXISTENCE,REGULARITY AND ASYMPTOTIC STABILITY OF ENHANCED PULLBACK ATTRACTORS:APPLICATIONS TO 3D PRIMITIVE EQUATIONS

Several new concepts of enhanced pullback attractors for nonautonomous dynamical systems are introduced here by uniformly enhancing the compactness and attraction of the usual pullback attractors over an infinite forward time-interval under strong and weak topologies.Then we provide some theoretical results for the existence,regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which can be applied to a large class of PDEs.The existence of these enhanced attractors is harder to obtain than the backward case[33],since it is difficult to uniformly control the long-time pullback behavior of the systems over the forward time-interval.As applications of our theoretical results,we consider the famous 3D primitive equations modelling the large-scale ocean and atmosphere dynamics,and prove the existence,regularity and asymptotic stability of the enhanced pullback attractors in V×V and H^(2)×H^(2) for the time-dependent forces which satisfy some weak conditions.

Lipschitz Regularity of Viscosity Solutions to the Infinity Laplace Equation

In this paper, we study the viscosity solutions of the Neumann problem in a bounded C2 domain Ω, where ΔN∞ is called the normalized infinity Laplacian. The normalized infinity Laplacian was first studied by Peres, Shramm, Sheffield and Wilson from the point of randomized theory named tug-of-war, which has wide applications in optimal mass transportation, financial option price problems, digital image processing, physical engineering, etc. We give the Lipschitz regularity of the viscosity solutions of the Neumann problem. The method we adopt is to choose suitable auxiliary functions as barrier functions and combine the perturbation method and viscosity solutions theory. .

Ground fissure development regularity and formation mechanism of shallow buried coal seam mining with Karst landform in Jiaozi coal mine: a case study

A comprehensive study was undertaken at Jiaozi coal mine to investigate the development regularity of ground fissures in shallow buried coal seam mining with Karst landform,shedding light on the development type,geographical distribution,dynamic development process,and failure mechanism of these ground fissures by employing field monitoring,numerical simulation,and theoretical analysis.The findings demonstrate that ground fissure development has an obvious feature of subregion,and its geographical distribution is significantly affected by topography.Tensile type,open type,and stepped type are three different categories of ground fissure.Ground fissures emerge dynamically as the panel advances,and they typically develop with a distance of less than periodic weighting step distance in advance of panel advancing position.Ground fissures present the dynamic development feature,temporary fissure has the ability of self-healing.The dynamic development process of ground fissure with closed-distance coal seam repeated mining is expounded,and the development scale is a dynamic development stage of“closure→expansion→stabilized”on the basis of the original development scale.From the perspective of topsoil deformation,the computation model considering two points movement vectors towards two directions of the gob and the ground surface is established,the development criterion considering the critical deformation value of topsoil is obtained.The mechanical model of hinged structure of inclined body is proposed to clarify the ground fissure development,and the interaction between slope activity and ground fissure development is expounded.These research results fulfill the gap of ground fissures about development regularity and formation mechanism,and can contribute to ground fissure prevention and treatment with Karst landform.

Regularity classification and corresponding analysis method requirements of horizontally curved bridges with unequal pier heights

The seismic behavior of horizontally curved bridges,particularly with unequal height piers,is more complicated than that of straight bridges due to their geometric properties.In this study,the seismic responses of several horizontally curved single-column-bent viaducts with various degrees of curvature and different pier heights have been investigated,employing three different analysis approaches:namely,modal pushover analysis,uniform load method,and nonlinear time history analysis.Considering the investigated bridge configurations and utilizing the most common regularity indices,the results indicate that viaducts with 45-degree and 90-degree deck subtended angles can be categorized as regular and moderately irregular,respectively,while the bridges with 180-degree deck subtended angle are found to be highly irregular.Furthermore,the viaducts whose pier heights are asymmetric may be considered as irregular for almost all ranges of the deck subtended angles.The effects of higher transverse and longitudinal modes are discussed and the minimum analysis requirements are identified to assess the seismic response of such bridge configurations for design purposes.Although the Regularity Indices used here are useful tools to distinguish between regular and irregular bridges,further studies are needed to improve their reliability.

Distributed Optimal Formation Control for Unmanned Surface Vessels by a Regularized Game-Based Approach

Dear Editor,This letter explores optimal formation control for a network of unmanned surface vessels(USVs).By designing an individual objective function for each USV,the optimal formation problem is transformed into a noncooperative game.Under this game theoretic framework,the optimal formation is achieved by seeking the Nash equilibrium of the regularized game.A modular structure consisting of a distributed Nash equilibrium seeker and a regulator is proposed.

THE GLOBAL EXISTENCE AND ANALYTICITY OF A MILD SOLUTION TO THE 3D REGULARIZED MHD EQUATIONS

In this paper,we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms.We establish the global existence of mild solutions to this system with small initial data.In addition,we also obtain the Gevrey class regularity and the temporal decay rate of the solution.

Composition Analysis and Identification of Ancient Glass Products Based on L1 Regularization Logistic Regression

In view of the composition analysis and identification of ancient glass products, L1 regularization, K-Means cluster analysis, elbow rule and other methods were comprehensively used to build logical regression, cluster analysis, hyper-parameter test and other models, and SPSS, Python and other tools were used to obtain the classification rules of glass products under different fluxes, sub classification under different chemical compositions, hyper-parameter K value test and rationality analysis. Research can provide theoretical support for the protection and restoration of ancient glass relics.

Enhanced Differentiable Architecture Search Based on Asymptotic Regularization

In differentiable search architecture search methods,a more efficient search space design can significantly improve the performance of the searched architecture,thus requiring people to carefully define the search space with different complexity according to various operations.Meanwhile rationalizing the search strategies to explore the well-defined search space will further improve the speed and efficiency of architecture search.With this in mind,we propose a faster and more efficient differentiable architecture search method,AllegroNAS.Firstly,we introduce a more efficient search space enriched by the introduction of two redefined convolution modules.Secondly,we utilize a more efficient architectural parameter regularization method,mitigating the overfitting problem during the search process and reducing the error brought about by gradient approximation.Meanwhile,we introduce a natural exponential cosine annealing method to make the learning rate of the neural network training process more suitable for the search procedure.Moreover,group convolution and data augmentation are employed to reduce the computational cost.Finally,through extensive experiments on several public datasets,we demonstrate that our method can more swiftly search for better-performing neural network architectures in a more efficient search space,thus validating the effectiveness of our approach.

Low-Rank Multi-View Subspace Clustering Based on Sparse Regularization

Multi-view Subspace Clustering (MVSC) emerges as an advanced clustering method, designed to integrate diverse views to uncover a common subspace, enhancing the accuracy and robustness of clustering results. The significance of low-rank prior in MVSC is emphasized, highlighting its role in capturing the global data structure across views for improved performance. However, it faces challenges with outlier sensitivity due to its reliance on the Frobenius norm for error measurement. Addressing this, our paper proposes a Low-Rank Multi-view Subspace Clustering Based on Sparse Regularization (LMVSC- Sparse) approach. Sparse regularization helps in selecting the most relevant features or views for clustering while ignoring irrelevant or noisy ones. This leads to a more efficient and effective representation of the data, improving the clustering accuracy and robustness, especially in the presence of outliers or noisy data. By incorporating sparse regularization, LMVSC-Sparse can effectively handle outlier sensitivity, which is a common challenge in traditional MVSC methods relying solely on low-rank priors. Then Alternating Direction Method of Multipliers (ADMM) algorithm is employed to solve the proposed optimization problems. Our comprehensive experiments demonstrate the efficiency and effectiveness of LMVSC-Sparse, offering a robust alternative to traditional MVSC methods.

A New Result on Regular Designs under Baseline Parameterization

The study on designs for the baseline parameterization has aroused attention in recent years. This paper focuses on two-level regular designs for the baseline parameterization. A general result on the relationship between K-aberration and word length pattern is developed.

The Convergence Rate of Fréchet Distribution under the Second-Order Regular Variation Condition

In this article we consider the asymptotic behavior of extreme distribution with the extreme value index γ>0 . The rates of uniform convergence for Fréchet distribution are constructed under the second-order regular variation condition.

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner product defined on both the unit ball and the unit sphere, construct the kernel-regularized learning algorithm from the view of semi-supervised learning and bound the upper bounds for the learning rates. The theory analysis shows that the learning algorithm has better uniform convergence according to the number of samples. The research can be regarded as an application of kernel-regularized semi-supervised learning.

Genetically engineered systems revealed the roles of basal ganglia in sleep-wake regulation

The basal ganglia(BG) act as a cohesive functional unit that regulates motor function,habit formation,and reward/addictive behaviors. However,it is still not well understood how the BG maintains wakefulness and suppresses sleep to achieve al these fundamental functions until genetical y engineered systems developed these years. Significant research efforts have recently been directed at developing genetic-molecular tools to achieve reversible and cell-type specific in vivo silencing or activation of neurons in behaving animals. Optogenetic tools can be used both to specifically activate or inhibit neurons of interest and identify functional synaptic connectivity between specific neuronal populations,both in vivo and in brain slices. Another recently developed system by Roth and colleagues permits the selective and ″remote″ manipulation(activation and silencing) of neuronal activity via all 3 major GPCR signaling pathways(G_i,G_s and G_q). These so-called ″ designer receptors exclusively activated by designer drugs″(DREADD) involve mutant GPCRs that do not respond to their endogenous ligands but are responsive to otherwise inert biological compounds. Recently,we demonstrated the essential roles and the neural pathways of the neurons expressing adenosine A_(2A) receptors or dopamine D_1 receptors in the BG for sleep-wake regulation using the genetically engineered systems including optogenetics and DREADD. We proposed a plausible model in which the caudate-putamen and the nucleus accumbens integrates behavioral processes with sleep/wakefulness through adenosine and dopamine receptors.