DIFFERENTIABILITY OF CONVEX FUNCTIONS AND ASPLUND SPACES

Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years effort, and a characteristic theorem is given for Banach spaces which are (weak) Asplund spaces.

Differentiability of Convex Functions in Locally Convex Spaces（Ⅰ）──Continuous Convex Functions and Gateaux Differentiability

ＤｉｆｆｅｒｅｎｔｉａｂｉｌｉｔｙｏｆＣｏｎｖｅｘＦｕｎｃｔｉｏｎｓｉｎＬｏｃａｌｌｙＣｏｎｖｅｘＳｐａｃｅｓ（Ⅰ）──ＣｏｎｔｉｎｕｏｕｓＣｏｎｖｅｘＦｕｎｃｔｉｏｎｓａｎｄＧａｔｅａｕｘＤｉｆｆｅｒｅｎｔｉａｂｉｌｉｔｙＷＵＣｏｎｇｘｉｎＣＨＥＮ...

ON TERM BY TERM DYADIC DIFFERENTIABILITY OF WALSH-KACZMARZ SERIES

The aim of this paper is to prove the following theorem concerning the term by term differentiation the-orem of Walsh-Kaczmarz series. Let (ck) be a decreasing real sequence withare integrable functions and f(x) is a. e. dyadic (or Butzer and Wagner) differentiate withThe function Kk means the kth Walsh-Kaczmarz function.

ON THE PRODUCT OF GTEAUX DIFFERENTIABILITY LOCALLY CONVEX SPACES

A locally convex space is said to be a Gateaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in .D.This paper shows that the product of a GDS and a family of separable Prechet spaces is a GDS,and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.

HOLDER CONTINUITY AND DIFFERENTIABILITY ON CONVERGING SUBSEQUENCES

It is shown that an arbitrary function from D Rn to Rm will become C0,a-continuous in almost every x∈ D after restriction to a certain subset with limit pointx. For n 〉 m differentiability can be obtained. Examples show the Ho1der exponent a=min{1,n/m}is optimal.

ON THE CONTINUITY AND DIFFERENTIABILITY OF AKIND OF FRACTAL INTERPOLATION FUNCTION

The sufficient conditions of Holder continuity of two kinds of fractal interpolation functions defined by IFS (Iterated Function System) were obtained. The sufficient and necessary condition for its differentiability was proved. Its derivative was a fractal interpolation function generated by the associated IFS, if it is differentiable.

The differentiability of solutions for elliptic equations which degenerate on part of the boundary of a convex domain

In this paper,we study the differentiability of solutions on the boundary for equartions of type L;u=;u/x;+|x|;;u/y;=f（x,y）,where λ is an arbitrary positive number. By introducing a proper metric that is related to the elliptic operator L;, we prove the differentiability on the boundary when some well-posed boundary conditions are satisfied. The main difficulty is the construction of new barrier functions in this article.

Smulyan Lemma and Differentiability of the Support Function

The purpose of this paper is to verify the Smulyan lemma for the support function, and also the Gateaux differentiability of the support function is studied on its domain. Moreover, we provide a characterization of Frechet differentiability of the support function on the extremal points.

Differentiability of Convex Functions in Locally Convex Spaces(Ⅱ)──Frechet

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DIFFERENTIABILITY FOR THE HIGH DIMENSIONAL POLYNOMIAL-LIKE ITERATIVE EQUATION

This paper studies the smoothness of solutions of the higher dimensional polynomial-like iterative equation. The methods given by Zhang Weinian([7]) and by Kulczvcki M, Tabor j.([3]) are improved by constructing a new operator for the structure of the equation in order to apply fixed point theorems. Existence, uniqueness and stability of continuously differentiable solutions are given.

Higher Order Fractional Differentiability for the Stationary Stokes System

This paper focuses on the higher order fractional differentiability of weak solution pairs to the following nonlinear stationary Stokes system{div A(x-Du)-■π=divF,inΩdivu=0,inΩ.In terms of the difference quotient method,our first result reveals that if F∈B_(p,q.loc)^(β)(Ω,R^(n))for p=2 and 1≤q≤2n/n-2β,then such extra Besov regularity can transfer to the symmetric gradient Du and its pressureπwith no losses under a suitable fractional differentiability assumption on x■A(x,ξ).Furthermore,when the vector field A(x,Du)is simplified to the full gradient■u,we improve the aforementioned Besov regularity for all integrability exponents p and q by establishing a new Campanato-type decay estimates for(■u,π).

Astrocytic endothelin-1 overexpression impairs learning and memory ability in ischemic stroke via altered hippocampal neurogenesis and lipid metabolism

Vascular etiology is the second most prevalent cause of cognitive impairment globally.Endothelin-1,which is produced and secreted by endothelial cells and astrocytes,is implicated in the pathogenesis of stroke.However,the way in which changes in astrocytic endothelin-1 lead to poststroke cognitive deficits following transient middle cerebral artery occlusion is not well understood.Here,using mice in which astrocytic endothelin-1 was overexpressed,we found that the selective overexpression of endothelin-1 by astrocytic cells led to ischemic stroke-related dementia(1 hour of ischemia;7 days,28 days,or 3 months of reperfusion).We also revealed that astrocytic endothelin-1 overexpression contributed to the role of neural stem cell proliferation but impaired neurogenesis in the dentate gyrus of the hippocampus after middle cerebral artery occlusion.Comprehensive proteome profiles and western blot analysis confirmed that levels of glial fibrillary acidic protein and peroxiredoxin 6,which were differentially expressed in the brain,were significantly increased in mice with astrocytic endothelin-1 overexpression in comparison with wild-type mice 28 days after ischemic stroke.Moreover,the levels of the enriched differentially expressed proteins were closely related to lipid metabolism,as indicated by Kyoto Encyclopedia of Genes and Genomes pathway analysis.Liquid chromatography-mass spectrometry nontargeted metabolite profiling of brain tissues showed that astrocytic endothelin-1 overexpression altered lipid metabolism products such as glycerol phosphatidylcholine,sphingomyelin,and phosphatidic acid.Overall,this study demonstrates that astrocytic endothelin-1 overexpression can impair hippocampal neurogenesis and that it is correlated with lipid metabolism in poststroke cognitive dysfunction.

Spatial-temporal differentiation and influencing factors of rural settlements in mountainous areas: an example of Liangshan Yi Autonomous Prefecture, Southwestern China

Rural settlement is the basic spatial unit for compact communities in rural area. Scientific exploration of spatial-temporal differentiation and its influencing factors is the premise of spatial layout rationalization. Based on land use data of Liangshan Yi Autonomous Prefecture(hereinafter referred to as Liangshan Prefecture) in Sichuan Province, China from 1980 to 2020, compactness index, fractal dimension, imbalance index, location entropy and the optimal parameters-based geographical detector(OPGD) model are used to analyze the spatial-temporal evolution of the morphological characteristics of rural settlements, and to explore the influence of natural geographical factors, socioeconomic factors, and policy factors on the spatial differentiation of rural settlements. The results show that:(1) From 1980 to 2020, the rural settlements area in Liangshan Prefecture increased by 15.96 km^(2). In space, the rural settlements are generally distributed in a local aggregation, dense in the middle and sparse around the periphery. In 2015, the spatial density and expansion index of rural settlements reached the peak.(2) From 1980 to 2020, the compactness index decreased from 0.7636 to 0.7496, the fractal dimension increased from 1.0283 to 1.0314, and the fragmentation index decreased from 0.1183 to 0.1047. The spatial morphological structure of rural settlements tended to be loose, the shape contour tended to be complex, the degree of fragmentation decreased, and the spatial distribution was significantly imbalanced.(3) The results of OPGD detection in 2015 show that the influence of each factor is slope(0.2371) > traffic accessibility(0.2098) > population(0.1403) > regional GDP(0.1325) > elevation(0.0987) > poverty alleviation(0). The results of OPGD detection in 2020 show that the influence of each factor is slope(0.2339) > traffic accessibility(0.2198) > population(0.1432) > regional GDP(0.1219) > poverty alleviation(0.0992) > elevation(0.093). Natural geographical factors(slope and elevation) are the basic factors affecting the spatial distribution of rural settlements, and rural settlements are widely distributed in the river valley plain and the second half mountain area. Socioeconomic factors(traffic accessibility, population, and regional GDP) have a greater impact on the spatial distribution of rural settlements, which is an important factor affecting the spatial distribution of rural settlements. Policy factors such as poverty alleviation relocation have an indispensable impact on the spatial distribution of rural settlements. The research results can provide decisionmaking basis for the spatial arrangement of rural settlements in Liangshan Prefecture, and optimize the implementation of rural revitalization policies.

Transcriptional regulation in the development and dysfunction of neocortical projection neurons

Glutamatergic projection neurons generate sophisticated excitatory circuits to integrate and transmit information among different cortical areas,and between the neocortex and other regions of the brain and spinal cord.Appropriate development of cortical projection neurons is regulated by certain essential events such as neural fate determination,proliferation,specification,differentiation,migration,survival,axonogenesis,and synaptogenesis.These processes are precisely regulated in a tempo-spatial manner by intrinsic factors,extrinsic signals,and neural activities.The generation of correct subtypes and precise connections of projection neurons is imperative not only to support the basic cortical functions(such as sensory information integration,motor coordination,and cognition)but also to prevent the onset and progression of neurodevelopmental disorders(such as intellectual disability,autism spectrum disorders,anxiety,and depression).This review mainly focuses on the recent progress of transcriptional regulations on the development and diversity of neocortical projection neurons and the clinical relevance of the failure of transcriptional modulations.

An inverse analysis of fluid flow through granular media using differentiable lattice Boltzmann method

This study presents a method for the inverse analysis of fluid flow problems.The focus is put on accurately determining boundary conditions and characterizing the physical properties of granular media,such as permeability,and fluid components,like viscosity.The primary aim is to deduce either constant pressure head or pressure profiles,given the known velocity field at a steady-state flow through a conduit containing obstacles,including walls,spheres,and grains.The lattice Boltzmann method(LBM)combined with automatic differentiation(AD)(AD-LBM)is employed,with the help of the GPU-capable Taichi programming language.A lightweight tape is used to generate gradients for the entire LBM simulation,enabling end-to-end backpropagation.Our AD-LBM approach accurately estimates the boundary conditions for complex flow paths in porous media,leading to observed steady-state velocity fields and deriving macro-scale permeability and fluid viscosity.The method demonstrates significant advantages in terms of prediction accuracy and computational efficiency,making it a powerful tool for solving inverse fluid flow problems in various applications.

Cell replacement with stem cell-derived retinal ganglion cells from different protocols

Glaucoma,characterized by a degenerative loss of retinal ganglion cells,is the second leading cause of blindness worldwide.There is currently no cure for vision loss in glaucoma because retinal ganglion cells do not regenerate and are not replaced after injury.Human stem cell-derived retinal ganglion cell transplant is a potential therapeutic strategy for retinal ganglion cell degenerative diseases.In this review,we first discuss a 2D protocol for retinal ganglion cell differentiation from human stem cell culture,including a rapid protocol that can generate retinal ganglion cells in less than two weeks and focus on their transplantation outcomes.Next,we discuss using 3D retinal organoids for retinal ganglion cell transplantation,comparing cell suspensions and clusters.This review provides insight into current knowledge on human stem cell-derived retinal ganglion cell differentiation and transplantation,with an impact on the field of regenerative medicine and especially retinal ganglion cell degenerative diseases such as glaucoma and other optic neuropathies.

Metabolic and proteostatic differences in quiescent and active neural stem cells

Adult neural stem cells are neurogenesis progenitor cells that play an important role in neurogenesis.Therefore,neural regeneration may be a promising target for treatment of many neurological illnesses.The regenerative capacity of adult neural stem cells can be chara cterized by two states:quiescent and active.Quiescent adult neural stem cells are more stable and guarantee the quantity and quality of the adult neural stem cell pool.Active adult neural stem cells are chara cterized by rapid proliferation and differentiation into neurons which allow for integration into neural circuits.This review focuses on diffe rences between quiescent and active adult neural stem cells in nutrition metabolism and protein homeostasis.Furthermore,we discuss the physiological significance and underlying advantages of these diffe rences.Due to the limited number of adult neural stem cells studies,we refe rred to studies of embryonic adult neural stem cells or non-mammalian adult neural stem cells to evaluate specific mechanisms.

Novel Investigation of Stochastic Fractional Differential Equations Measles Model via the White Noise and Global Derivative Operator Depending on Mittag-Leffler Kernel

Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.

DNA methylation in poultry:a review

As an important epigenetic modification,DNA methylation is involved in many biological processes such as animal cell differentiation,embryonic development,genomic imprinting and sex chromosome inactivation.As DNA methylation sequencing becomes more sophisticated,it becomes possible to use it to solve more zoological problems.This paper reviews the characteristics of DNA methylation,with emphasis on the research and application of DNA methylation in poultry.

Present-day Upper-crustal Strain Rate Field in Southeastern Tibet and its Geodynamic Implications:Constraints from GPS Measurements with ABIC Method

The Earth’s surface kinematics and deformation are fundamental to understanding crustal evolution.An effective research approach is to estimate regional motion field and deformation fields based on modern geodetic networks.If the discrete observed velocity field is obtained,the velocity related fields,such as dilatation rate and maximum shear strain rate,can be estimated by applying varied mathematical approaches.This study applied Akaike's Bayesian Information Criterion(ABIC)method to calculate strain rate fields constrained by GPS observations in the southeast Tibetan Plateau.Comparison with results derived from other three methods revealed that our ABIC-derived strain rate fields were more precise.The maximum shear strain rate highlighted the Xianshuihe–Xiaojiang fault system as the main boundary for the outward migration of material in southeastern Tibet,indicating rotation of eastern Tibet material around the eastern Himalaya rather than whole extrusion along a fixed channel.Additionally,distinct dilatation rate patterns in the northeast and southwest regions of the fault system were observed.The northeast region,represented by the Longmenshan area,exhibited negative dilatational anomalies;while the southwest region,represented by the Jinsha River area north of 29°N,displayed positive dilatational anomalies.This indicates compression in the former and extension in the latter.Combined with deep geophysical observations,we believe that the upper and lower crusts of the Jinsha River area north of 29°N are in an entire expanding state,probably caused by the escape-drag effect of material.The presence of a large,low-viscosity region south of 29°N may not enable the entire escape of the crust,but instead result in a differential escape of the lower crust faster than the upper crust.