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In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

The authors consider the nonlinear equation -\\frac 1m=z+Sm with a parameter z in the complex upper half plane \\mathbb H , where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in \\mathbb H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on \\mathbb R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z\\in \\mathbb H, including the vicinity of the singularities.

The construction of the $p$-adic local Langlands correspondence for $\\mathrm{GL}_2(\\mathbf{Q}_p)$ uses in an essential way Fontaine's theory of cyclotomic $(\\varphi ,\\Gamma )$-modules. Here cyclotomic means that $\\Gamma = \\mathrm {Gal}(\\mathbf{Q}_p(\\mu_{p^\\infty})/\\mathbf{Q}_p)$ is the Galois group of the cyclotomic extension of $\\mathbf Q_p$. In order to generalize the $p$-adic local Langlands correspondence to $\\mathrm{GL}_{2}(L)$, where $L$ is a finite extension of $\\mathbf{Q}_p$, it seems necessary to have at our disposal a theory of Lubin-Tate $(\\varphi ,\\Gamma )$-modules. Such a generalization has been carried out, to some extent, by working over the $p$-adic open unit disk, endowed with the action of the endomorphisms of a Lubin-Tate group. The main idea of this article is to carry out a Lubin-Tate generalization of the theory of cyclotomic $(\\varphi ,\\Gamma )$-modules in a different fashion. Instead of the $p$-adic open unit disk, the authors work over a character variety that parameterizes the locally $L$-analytic characters on $o_L$. They study $(\\varphi ,\\Gamma )$-modules in this setting and relate some of them to what was known previously.

We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin derivatives explicitly. Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of the initial data, in particular upper bounds for their Among other tools, we use methods from harmonic analysis. As a by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the optimality of the bound we established.

This volume is dedicated to the memory of Björn Jawerth. It contains original research contributions and surveys in several of the areas of mathematics to which Björn made important contributions. Those areas include harmonic analysis, image processing, and functional analysis, which are of course interrelated in many significant and productive ways.Among the contributors are some of the world's leading experts in these areas. With its combination of research papers and surveys, this book may become an important reference and research tool.This book should be of interest to advanced graduate students and professional researchers in the areas of functional analysis, harmonic analysis, image processing, and approximation theory. It combines articles presenting new research with insightful surveys written by foremost experts.

This book contains the proceedings of the 14th International Conference on p-adic Functional Analysis, held from June 30-July 4, 2016, at the Universit d'Auvergne, Aurillac, France. Articles included in this book feature recent developments in various areas of non-Archimedean analysis: summation of p-adic series, rational maps on the projective line over \\mathbb{Q}p, non-Archimedean Hahn-Banach theorems, ultrametric Calkin algebras, G-modules with a convex base, non-compact Trace class operators and Schatten-class operators in p-adic Hilbert spaces, algebras of strictly differentiable functions, inverse function theorem and mean value theorem in Levi-Civita fields, ultrametric spectra of commutative non-unital Banach rings, classes of non-Archimedean K the spaces, p-adic Nevanlinna theory and applications, and sub-coordinate representation of p-adic functions. Moreover, a paper on the history of p-adic analysis with a comparative summary of non-Archimedean fields is presented. Through a combination of new research articles and a survey paper, this book provides the reader with an overview of current developments and techniques in non-Archimedean analysis as well as a broad knowledge of some of the sub-areas of this exciting and fast-developing research area.

The estimation of large functional and longitudinal data, which refers to the estimation of mean function, estimation of covariance function, and prediction of individual trajectory, is one of the most challenging problems in the field of high-dimensional statistics. Functional Principal Components Analysis (FPCA) and Functional Linear Mixed Model (FLMM) are two major statistical tools used to address the estimation of large functional and longitudinal data; however, the former suffers from a dramatically increasing computational burden while the latter does not have clear asymptotic properties. In this paper, we propose a computationally effective estimator of large functional and longitudinal data within the framework of FLMM, in which all the parameters can be automatically estimated. Under certain regularity assumptions, we prove that the mean function estimation and individual trajectory prediction reach the minimax lower bounds of all nonparametric estimations. Through numerous simulations and real data analysis, we show that our new estimator outperforms the traditional FPCA in terms of mean function estimation, individual trajectory prediction, variance estimation, covariance function estimation, and computational effectiveness.

This book presents a wide panorama of methods to investigate the spectral properties of block operator matrices. Particular emphasis is placed on classes of block operator matrices to which standard operator theoretical methods do not readily apply: non-self-adjoint block operator matrices, block operator matrices with unbounded entries, non-semibounded block operator matrices, and classes of block operator matrices arising in mathematical physics.

Gastrodia elata , also named Tianma, is a valuable traditional Chinese herbal medicine. It has numerous important pharmacological roles such as in sedation and lowering blood pressure and as anticonvulsant and anti-aging, and it also has effects on the immune and cardiovascular systems. The whole genome sequencing of G. elata has been completed in recent years, which provides a strong support for the construction of the G. elata gene functional analysis platform. Therefore, in our research, we collected and processed 39 transcriptome data of G. elata and constructed the G. elata gene co-expression networks, then we identified functional modules by the weighted correlation network analysis (WGCNA) package. Furthermore, gene families of G. elata were identified by tools including HMMER, iTAK, PfamScan, and InParanoid. Finally, we constructed a gene functional analysis platform for G. elata 1 . In our platform, we introduced functional analysis tools such as BLAST, gene set enrichment analysis (GSEA), and cis- elements (motif) enrichment analysis tool. In addition, we analyzed the co-expression relationship of genes which might participate in the biosynthesis of gastrodin and predicted 19 mannose-binding lectin antifungal proteins of G. elata . We also introduced the usage of the G. elata gene function analysis platform (GelFAP) by analyzing CYP51G1 and GFAP4 genes. Our platform GelFAP may help researchers to explore the gene function of G. elata and make novel discoveries about key genes involved in the biological processes of gastrodin.

Sphaeropleales is a diverse group with over one thousand species described, which are found in a wide range of habitats showed strong environmental adaptability. This study presented comprehensive genomic analyses of seven newly sequenced Sphaeropleales strains with BUSCO completeness exceeding 90%, alongside comparative assessments with previously sequenced genomes. The genome sizes of Sphaeropleales species ranged from 39.8 Mb to 151.9 Mb, with most having a GC content around 56%. Orthologous analysis revealed unique gene families in each strain, comprising 2 to 3.5% of all genes. Comparative functional analysis indicated that transporters, genes encoding pyrroline-5-carboxylate reductase and antioxidant enzymes played a crucial role in adaptation to environmental stressors like salinity, cold, heavy metals and varying nutrient conditions. Additionally, Sphaeropleales species were found to be B 12 auxotrophy, acquiring this vitamin or its precursors through a symbiotic relationship with bacteria. Phylogenetic studies based on 18S rDNA and the low copy othologues confirmed species identification and the relationships inside core Chlorophyta and between prasinophytes. Evolutionary analyses demonstrated all the species exhibited a large count of gene family expansions and contraction, with rapidly evolving and positive selected genes identified in terrestrial Bracteacoccus species, which contributed to their adaptation to terrestrial habitat. These findings enriched the genomic data for Sphaeropleales, particularly the genus Bracteacoccus , which can help in understanding the ecological adaptations, evolutionary relationships, and biotechnological applications of this group of algae.