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

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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.

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.

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.

Imprinting at the KBTBD6 locus involves species-specific m ternal methylation and monoallelic expression in livestock animals

Background The primary differentially methylated regions(DMRs) which are maternally hypermethylated serve as imprinting control regions(ICRs) that drive monoallelic gene expression, and these ICRs have been investigated due to their implications in mammalian development. Although a subset of genes has been identified as imprinted, in-depth comparative approach needs to be developed for identification of species-specific imprinted genes. Here, we examined DNA methylation status and allelic expression at the KBTBD6 locus across species and tissues and explored potential mechanisms of imprinting.Results Using whole-genome bisulfite sequencing and RNA-sequencing on parthenogenetic and normal porcine embryos, we identified a maternally hypermethylated DMR between the embryos at the KBTBD6 promoter Cp G island and paternal monoallelic expression of KBTBD6. Also, in analyzed domesticated mammals but not in humans, non-human primates and mice, the KBTBD6 promoter Cp G islands were methylated in oocytes and/or allelically methyl-ated in tissues, and monoallelic KBTBD6 expression was observed, indicating livestock-specific imprinting. Further analysis revealed that these Cp G islands were embedded within transcripts in porcine and bovine oocytes which coexisted with an active transcription mark and DNA methylation, implying the presence of transcription-dependent imprinting.Conclusions In this study, our comparative approach revealed an imprinted expression of the KBTBD6 gene in domesticated mammals, but not in humans, non-human primates, and mice which implicates species-specific evolution of genomic imprinting.

Spatiotemporal dynamics of the social structure of Indo-Pacific humpback dolphins(Sousa chinensis)in Xiamen waters from 2007 to 2019

As highly social animals,Indo-Pacific humpback dolphins(Sousa chinensis)exhibit community differentiation.Nevertheless,our understanding of the external and internal factors influencing these dynamics,as well as their spatiotemporal variations,is still limited.In the present study,variations in the social structure of an endangered Indo-Pacific humpback dolphin population in Xiamen Bay,China,were monitored over two distinct periods(2007–2010 and 2017–2019)to analyze the effects of habitat utilization and the composition of individuals within the population.In both periods,the population demonstrated a strikingly similar pattern of social differentiation,characterized by the division of individuals into two main clusters and one small cluster.Spatially,the two primary clusters occupied the eastern and western waters,respectively,although the core distribution area of the eastern cluster shifted further eastward between the two periods.Despite this distribution shift,the temporal stability of the social structure and inter-associations within the eastern cluster remained unaffected.A subset of 16individuals observed in both periods,comprising 51.6%and 43.2%of the population in each respective period,emerged as a foundational element of the social structure and may be responsible for sustaining social structure stability,especially during the 2007–2010 period.These observations suggest that the composition of dominant individuals,an internal factor,had a more substantial influence on the formation of the social network than changes in habitat use,an external factor.Consequently,the study proposes distinct conservation measures tailored to each of the two main clusters.

Salinity Acclimation Induces Reduced Energy Metabolism,Osmotic Pressure Regulation and Transcriptional Reprogramming in Hypotrichida Ciliate Gastrostyla setifera

Coastal and estuarine protists are frequently exposed to salinity undulation.While the tolerance and stress responses of microalgae to salinity have been extensively studied,there have been scarce studies on the physiological response of heterotrophic protists to salinity stressing.In this study,we investigated the physiological response of the heterotrophic ciliate Gastrostyla setifera to a salinity of 3,via a transcriptomic approach.The first transcriptome of genus Gastrostyla was obtained utilizing a group of manually isolated ciliate individuals(cells)and RNA-seq technique.The completeness of the transcriptome was verified.Differentially expressed gene(DEG)analysis was performed among the transcriptomes of G.setifera acclimated in saline water(salinity 3)and those cultured in fresh water.The results demonstrated a significant alternation in gene transcription,in which the ciliate exhibits a transcripttomic acclimation in responding salinity stressing.The up-regulated DEGs were enriched in the pathways of cytoskeleton proteins,membrane trafficking,protein kinases and protein phosphatases.These may represent enhanced functions of ion transport,stress response and cell protections.Pathways involved in energy metabolism and biosynthesis were markedly down-regulated,reflecting decreased cell activity.Particularly,we detected significantly down-regulated genes involved in several pathways of amino acid catabolism,which may lead to accumulation of amino acids in the ciliate cell.Amino acid could act as compatible solutes in the cytoplasm to maintain the osmotic balance in saline water.Overall,this work is an initial exploration to the molecular basis of the heterotrophic protist responding to salinity stressing.The result sheds light on the mechanisms of enhancement of cell protection,reduction of cell activity,and osmotic pressure regulation in ciliates acclimated to salinity.