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Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order
The thermodynamical formalism has been developed by the authors for a very general class of transcendental meromorphic functions. A
function
In the present manuscript we first improve upon our earlier paper
in providing a systematic account of the thermodynamical formalism for such a meromorphic function
Then we provide various, mainly geometric, applications
of this theory. Indeed, we examine the finer fractal structure of the radial (in fact non-escaping) Julia set by developing the
multifractal analysis of Gibbs states. In particular, the Bowen’s formula for the Hausdorff dimension of the radial Julia set from our
earlier paper is reproved. Moreover, the multifractal spectrum function is proved to be convex, real-analytic and to be the Legendre
transform conjugate to the temperature function. In the last chapter we went even further by showing that, for a analytic family
satisfying a symmetric version of the growth condition (1.1) in a uniform way, the multifractal spectrum function is real-analytic also
with respect to the parameter. Such a fact, up to our knowledge, has not been so far proved even for hyperbolic rational functions nor
even for the quadratic family
eBook
Fourier Series in Several Variables with Applications to Partial Differential Equations
Discussing many results and studies from the literature, this work illustrates the value of Fourier series methods in solving difficult nonlinear PDEs. Using these methods, the author presents results for stationary Navier-Stokes equations, nonlinear reaction-diffusion systems, and quasilinear elliptic PDEs and resonance theory. He also establishes the connection between multiple Fourier series and number theory, presents the periodic Ca-theory of Calderon and Zygmund, and explores the extension of Fatou's famous work on antiderivatives and nontangential limits to higher dimensions. The importance of surface spherical harmonic functions is emphasized throughout.
eBook
Advanced Calculus of Several Variables
ADVANCED CALCULUS OF SEVERAL VARIABLES covers important topics of Transformations and topology on Euclidean in n-space Rn Functions of several variables, Differentiation in Rn, Multiple integrals and Integration in Rn. The topics have been presented in a simple clear and coherent style with a number of examples and exercises. Proofs have been made direct and simple. Unsolved problems just after relevant articles in the form of exercises and typical problems followed by suggestions have been given. This book will help the reader work on the problems of Numerical Analysis, Operations Research, Differential Equations and Engineering applications.
eBook
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\\mathbb F_p(t)$, when $p$ is prime and $r\\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\\mathbb F_q(t^1/d)$.
eBook
Generalized Mercer Kernels and Reproducing Kernel Banach Spaces
This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing
kernel Hilbert spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be
well-posed and of easy implementation. First we verify many advanced properties of the general RKBSs such as density, continuity,
separability, implicit representation, imbedding, compactness, representer theorem for learning methods, oracle inequality, and
universal approximation. Then, we develop a new concept of generalized Mercer kernels to construct
eBook
On Productiveness and Complexity in Computable Analysis Through Rice-Style Theorems for Real Functions
This paper investigates the complexity of real functions through proof techniques inspired by formal language theory. Productiveness, which is a stronger form of non-recursive enumerability, is employed to analyze the complexity of various problems related to real functions. Our work provides a deep reexamination of Hilbert’s tenth problem and the equivalence to the identically 0 function problem, extending the undecidability results of these problems into the realm of productiveness. Additionally, we study the complexity of the equivalence to the identically 0 function problem over different domains. We then construct highly efficient many-one reductions to establish Rice-style theorems for the study of real functions. Specifically, we show that many predicates, including those related to continuity, differentiability, uniform continuity, right and left differentiability, semi-differentiability, and continuous differentiability, are as hard as the equivalence to the identically 0 function problem. Due to their high efficiency, these reductions preserve nearly any level of complexity, allowing us to address both complexity and productiveness results simultaneously. By demonstrating these results, which highlight a more nuanced and potentially more intriguing aspect of real function theory, we provide new insights into how various properties of real functions can be analyzed.
Journal Article