Analysis Commons^™

Contemporary Mathematical Approaches To Computability Theory, Luis Guilherme Mazzali De Almeida

Undergraduate Student Research Internships Conference

In this paper, I present an introduction to computability theory and adopt contemporary mathematical definitions of computable numbers and computable functions to prove important theorems in computability theory. I start by exploring the history of computability theory, as well as Turing Machines, undecidability, partial recursive functions, computable numbers, and computable real functions. I then prove important theorems in computability theory, such that the computable numbers form a field and that the computable real functions are continuous.

Hessian Formulas And Estimates For Parabolic Schrödinger Operators, Xue-Mei Li

Journal of Stochastic Analysis

No abstract provided.

Remembering Kunita-San, Ken-Iti Sato

Journal of Stochastic Analysis

No abstract provided.

The Life And Scientific Work Of Hiroshi Kunita, Yasushi Ishikawa

Journal of Stochastic Analysis

No abstract provided.

Memories Of Professor Hiroshi Kunita, Ichiro Shigkeawa

Journal of Stochastic Analysis

No abstract provided.

On The Works Of Hiroshi Kunita In The Sixties, Masatoshi Fukushima

Journal of Stochastic Analysis

No abstract provided.

Personal Memories Of Hiroshi Kunita, David Elworthy

Journal of Stochastic Analysis

No abstract provided.

Preface, Shigeki Aida, David Applebaum, Yasushi Ishikawa, Arturo Kohatsu-Higa, Nicolas Privault

Journal of Stochastic Analysis

No abstract provided.

Trilinear Smoothing Inequalities And A Variant Of The Triangular Hilbert Transform, Michael Christ, Polona Durcik, Joris Roos

Mathematics, Physics, and Computer Science Faculty Articles and Research

Lebesgue space inequalities are proved for a variant of the triangular Hilbert transform involving curvature. The analysis relies on a crucial trilinear smoothing inequality developed herein, and on bounds for an anisotropic variant of the twisted paraproduct.

The trilinear smoothing inequality also leads to Lebesgue space bounds for a corresponding maximal function and a quantitative nonlinear Roth-type theorem concerning patterns in the Euclidean plane.

Holomorphic Functions, Relativistic Sum, Blaschke Products And Superoscillations, Daniel Alpay, Fabrizio Colombo, Stefano Pinton, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The notion of superoscillation is a particular case of that one of supershift. In the recent years, superoscillating functions, that appear for example in weak values in quantum mechanics, have become an interesting and independent field of research in complex analysis and in the theory of infinite order differential operators. The aim of this paper is to study some infinite order differential operators acting on entire functions which naturally arise in the study of superoscillating functions. Such operators are of particular interest because they are associated …

An Introduction To Fractal Analysis, Lucas Yong

Rose-Hulman Undergraduate Mathematics Journal

Classical analysis is not able to treat functions whose domain is fractal. We present an introduction to analysis on a particular class of fractals known as post-critically finite (PCF) self-similar sets that is suitable for the undergraduate reader. We develop discrete approximations of PCF self-similar sets, and construct discrete Dirichlet forms and corresponding discrete Laplacians that both preserve self-similarity and are compatible with a notion of harmonic functions that is analogous to a classical setting. By taking the limit of these discrete Laplacians, we construct continuous Laplacians on PCF self-similar sets. With respect to this continuous Laplacian, we also construct …

Irrational Philosophy? Kronecker's Constructive Philosophy And Finding The Real Roots Of A Polynomial, Richard B. Schneider

Rose-Hulman Undergraduate Mathematics Journal

The prominent mathematician Leopold Kronecker (1823 – 1891) is often relegated to footnotes and mainly remembered for his strict philosophical position on the foundation of mathematics. He held that only the natural numbers are intuitive, thus the only basis for all mathematical objects. In fact, Kronecker developed a complete school of thought on mathematical foundations and wrote many significant algebraic works, but his enigmatic writing style led to his historical marginalization. In 1887, Kronecker published an extended version of his paper, “On the Concept of Number,” translated into English in 2010 for the first time by Edward T. Dean, who …

Dynamic Parameter Estimation From Partial Observations Of The Lorenz System, Eunice Ng

Theses and Dissertations

Recent numerical work of Carlson-Hudson-Larios leverages a nudging-based algorithm for data assimilation to asymptotically recover viscosity in the 2D Navier-Stokes equations as partial observations on the velocity are received continuously-in-time. This "on-the-fly" algorithm is studied both analytically and numerically for the Lorenz equations in this thesis.

Smooth Global Approximation For Continuous Data Assimilation, Kenneth R. Brown

Theses and Dissertations

This thesis develops the finite element method, constructs local approximation operators, and bounds their error. Global approximation operators are then constructed with a partition of unity. Finally, an application of these operators to data assimilation of the two-dimensional Navier-Stokes equations is presented, showing convergence of an algorithm in all Sobolev topologies.

Interpolation And Sampling In Analytic Tent Spaces, Caleb Parks

Graduate Theses and Dissertations

Introduced by Coifman, Meyer, and Stein, the tent spaces have seen wide applications in harmonic analysis. Their analytic cousins have seen some applications involving the derivatives of Hardy space functions. Moreover, the tent spaces have been a recent focus of research. We introduce the concept of interpolating and sampling sequences for analytic tent spaces analogously to the same concepts for Bergman spaces. We then characterize such sequences in terms of Seip's upper and lower uniform density. We accomplish this by exploiting a kind of Mobius invariance for the tent spaces.

Berry-Esseen Bounds For Approximate Maximum Likelihood Estimators In The Α-Brownian Bridge, Khalifa Es-Sebaiy, Jabrane Moustaaid, Idir Ouassou

Journal of Stochastic Analysis

No abstract provided.

On A Polyanalytic Approach To Noncommutative De Branges–Rovnyak Spaces And Schur Analysis, Daniel Alpay, Fabrizio Colombo, Kamal Diki, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we begin the study of Schur analysis and of de Branges–Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows us to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, …

Numeric And Dynamic B-Stability, Exact-Monotone And Asymptotic Two-Point Behavior Of Theta Methods For Stochastic Differential Equations, Henri Schurz

Journal of Stochastic Analysis

No abstract provided.

On Distributions Of Self-Adjoint Extensions Of Symmetric Operators, Franco Fagnola, Zheng Li

Journal of Stochastic Analysis

No abstract provided.

On Digital Metric Space Satisfying Certain Rational Inequalities, Krati Shukla

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we have established some new results by extending some existing theorems in the setting of Digital Metric Space. We also proved some results in Digital Metric Space which were established earlier in the context of Complete Metric Space by different authors.