Mathematics Commons^™

A Study Of Operators On The Discrete Analogue Of Hardy Spaces On Homogeneous Trees And On Other Structures., P. Muthukumar Dr.

Doctoral Theses

In analytic function theory, the study of multiplication and composition operators has a rich structure for various analytic function spaces of the unit disk D = {z ∈ C : |z| < 1} such as the Hardy spaces Hp, the Bergman spaces Ap and the Bloch space B. This theory connects the operator theoretic properties such as boundedness, compactness, spectrum, invertibility, isometry with that of the function theoretic properties of the inducing map (symbol) such as bijectivity, boundary behaviour and vise versa In Chapter 2, we define discrete analogue of generalized Hardy spaces (Tp) and their separable subspaces (Tp,0) on a homogenous rooted tree and study some of their properties such as completeness, inclusion relations with other spaces, separability and growth estimate for functions in these spaces and their consequences. In Chapter 3, we obtain equivalent conditions for multiplication operators Mψ on Tp and Tp,0 to be bounded and compact. Furthermore, we discuss point spectrum, approximate point spectrum and spectrum of multiplication operators and discuss when a multiplication operator is an isometry. In Chapter 4, we give an equivalent conditions for the composition operator Cφ to be bounded on Tp and on Tp,0 spaces and compute their operator norms. We have considered the composition operators induced by special symbols such as univalent and multivalent maps and automorphism of a homogenous tree. We also characterize invertible composition operators and isometric composition operators on Tp and on Tp,0 spaces. Also, we discuss the compactness of Cφ on Tp spaces and finally we prove that there are no compact composition operators on Tp,0 spaces. In Chapter 5, we consider the composition operators on the Hardy-Dirichlet space H2, the space of Dirichlet series with square summable coefficients. By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on H2 , for the affine-like inducing symbol ϕ(s) = c1 + cqq −s , where q ≥ 2 is a fixed integer. We also give an estimate for approximation numbers of a composition operators in our H2 setting. In Chapter 6, we study the weighted composition operators preserving the class Pα. Some of its consequences and examples of certain special cases are presented. Furthermore, we discuss about the fixed points of weighted composition operators.

A Problem Of Solute Transport In A Cylindrical Porous Media With A Fractal Structure Taking Into Account Adsorption Phenomena, B. X. Xujayorov, J. M. Maxmudov, A. I. Usmanov, B. O. Saidov

Scientific Journal of Samarkand University

The process of anomalous solute transport in a coaxial cylindrical porous media is modelled by differential equations with a fractional derivative. The problem of solute transport in a two-zone cylindrical media consisting of macro- and micropores taking into account adsorption effectshas been numericallyposed and solved. The concentration profiles of suspended particles and the adsorbed solute in the macropore and micropore, the surface of the local concentration in the micropore are determined. The influence of adsorption phenomena and the order of the derivative with respect to the coordinate, i.e. fractal dimension of the media, on the characteristics of the solute transport …

Math 220p Foundations Of Mathematics, Nicholas Vlamis

Open Educational Resources

No abstract provided.

On A Two-Speed Mathematical Model Of Two-Fluid Medium With One Pressure, X. X. Imomnazarov, S. B. Kuyliyev

Scientific Journal of Samarkand University

An overdetermined stationary system of second-order differential equations is obtained. For the two-dimensional system, a variational statement of the problem is established. It is shown that the variational problem for the system of equations of two-speed hydrodynamics is well-posed in the corresponding Sobolev space

The Cauchy Problem For The System Of The Elasticity, I. E. Niyozov, O. Karshiboyev

Scientific Journal of Samarkand University

In this paper we consider the problem of analytical continuation of solutions to the system of the elasticity in a bounded domain from their values and values of their strains on a part of the boundary of this domain, i.e., we study the Cauchy problem

A Discontinuous Galerkin Method For The Stolz–Adams Approximate Deconvolution Model For Turbulent Flows, Monika Neda, Béatrice Rivière

Mathematical Sciences Faculty Research

We consider the zeroth order model of the family of approximate deconvolution models of Stolz and Adams. We propose and analyze fully discrete schemes using discontinuous finite elements. Optimal error estimates are derived. The dependence of these estimates with respect to the Reynolds number Re is O(ReeRe), which is an improvement with respect to the classical continuous finite element method where the dependence is O(ReeRe3), Layton [1].

On Estimates For The Damped Oscillatory Integrals, I. A. Ikromov, Sh. A. Muranov

Scientific Journal of Samarkand University

In this paper we consider estimates of the Fourier transform measures, concentrated on analytic hypersurfaces containing the of damping factor. The paper presents the solution of the problem S.D.Soggi and I.M. Stein about the optimal decay of the transformation Fourier measures with a damping factor for any analytic surfaces in three-dimensional Euclidean space.

Of The Cauchy Problem For Thelaplace Equations On A Plane, D. S. Shodiyev

Scientific Journal of Samarkand University

In this paper, a uniqueness theorem is proved, and an estimate of conditional stability is obtained, and an approximate solution is constructed by the method of quasi-derivation and Tikhonov regularization.

The Problem Of Integral Geometry In A Strip With Weight Function, A. X. Begmatov, A. S. Ismoilov

Scientific Journal of Samarkand University

In this work we consider the problem of reconstructing a function from a family of parabolas in the upper half-plane with a weight function having a singularity. The uniqueness of theorem for the solution of equation is proved and the inversion formula is derived. It is shown that the solution of the problem posed is weakly ill-posed, that is, stability estimates are obtained in spaces of finite smoothness.

Uniqueness And Stability Problem Of Integral Geometry With Indignation, Akram Begmatov, Zarifjon Ochilov, Abduroziq Xusanov

Scientific Journal of Samarkand University

Рассматривается новый класс задач интегральной геометрии типа Вольтерра с функцией специального веса. Доказаны теоремы единственности и существования решения, получены оценки устойчивости и формула обращения в пространствах Соболева, что свидетельствует о слабой некорректности решения задачи интегральной геометрии. Постановка задачи интегральной геометрии с возмущением на семейство парабол в полосе считается. Теорема единственности ее решения доказана в классе дважды непрерывно дифференцируемых функций с компактным носителем, а оценки устойчивости получены в пространствах конечной гладкости.

A Short Note On Extreme Points Of Certain Polytopes, Lei Cao, Ariana Hall, Selcuk Koyuncu

Mathematics Faculty Articles

We give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly substochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

Arbitrarily High-Order Unconditionally Energy Stable Schemes For Thermodynamically Consistent Gradient Flow Models, Yuezheng Gong, Jia Zhao, Qi Wang

Mathematics and Statistics Faculty Publications

We present a systematic approach to developing arbitrarily high-order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization method, we formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic …

The Continuous, The Discrete And The Infinitesimal In Philosophy And Mathematics (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reviewed Title: The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics by John L. Bell. Springer, 2019. 313 pp. ISBN: 9783030187064.

From Dyck Paths To Standard Young Tableaux, Juan B. Gil, Peter R. W. Mcnamara, Jordan O. Tirrell, Michael D. Weiner

Faculty Journal Articles

We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength n that is shown to be in bijection with the set of all SYT with n boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some …

Towards A Novel Generalized Chinese Remainder Algorithm For Extended Rabin Cryptosystem, Justin Zhan, Peter J. Shiue, Shen C. Huang, Benjamin J. Lowe

Mathematical Sciences Faculty Research

This paper proposes a number of theorems and algorithms for the Chinese Remainder Theorem, which is used to solve a system of linear congruences, and the extended Rabin cryptosystem, which accepts a key composed of an arbitrary finite number of distinct primes. This paper further proposes methods to relax the condition on the primes with trade-offs in the time complexity. The proposed algorithms can be used to provide ciphertext indistinguishability. Finally, this paper conducts extensive experimental analysis on six large data sets. The experimental results show that the proposed algorithms are asymptotically tight to the existing decryption algorithm in the …

A General Setting For Functions Of Fueter Variables: Differentiability, Rational Functions, Fock Module And Related Topics, Daniel Alpay, Ismael L. Paiva, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

We develop some aspects of the theory of hyperholomorphic functions whose values are taken in a Banach algebra over a field—assumed to be the real or the complex numbers—and which contains the field. Notably, we consider Fueter expansions, Gleason’s problem, the theory of hyperholomorphic rational functions, modules of Fueter series, and related problems. Such a framework includes many familiar algebras as particular cases. The quaternions, the split quaternions, the Clifford algebras, the ternary algebra, and the Grassmann algebra are a few examples of them.

Art, Artfulness, Or Artifice?: A Review Of The Art Of Statistics: How To Learn From Data, By David Spiegelhalter, Jason Makansi

Numeracy

David Spiegelhalter. 2019. The Art of Statistics: How to Learn From Data. (London: The Penguin Group). 444 pp. ISBN 978-1541618510

The author successfully eases the reader away from the rigor of statistical methods and calculations and into the realm of statistical thinking. Despite an engaging style and attention-grabbing examples, the reader of The Art of Statistics will need more than a casual grounding in statistics to get what Spiegelhalter, I believe, intends from his book. It should be viewed as a companion to a more rigorous textbook on statistical methods but not necessarily a book that makes statistics any …

Locally Recoverable Codes From Planar Graphs, Kathryn Haymaker, Justin O'Pella

Kanbar College Faculty Papers

In this paper we apply Kadhe and Calderbank’s definition of LRCs from convex polyhedra and planar graphs [4] to analyze the codes resulting from 3-connected regular and almost regular planar graphs. The resulting edge codes are locally recoverable with availability two. We prove that the minimum distance of planar graph LRCs is equal to the girth of the graph, and we also establish a new bound on the rate of planar graph edge codes. Constructions of regular and almost regular planar graphs are given, and their associated code parameters are determined. In certain cases, the code families meet the rate …

On Possibility Of Binary Companion Of The Sun: A Serendipity Finding And Comparison With Uvs Model Of Solar System, Victor Christianto, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

While we completely understood that a binary dwarf companion of the Sun has not been accepted by majority of astronomers, allow us to present some new arguments, along with our own serendipitous encounter with such a binary companion of the Sun. We hope that the present note will be found useful for further investigations, in relation to Planet Nine and such a dwarf star companion of the Sun (sometimes dubbed as Nemesis). Nonetheless, this article is not an exhaustive review of such dwarf companion star theories.

Finite Element Analysis Of An Arbitrary Lagrangian–Eulerian Method For Stokes/Parabolic Moving Interface Problem With Jump Coefficients, Rihui Lan, Michael J. Ramirez, Pengtao Sun

Mathematical Sciences Faculty Research

In this paper, a type of arbitrary Lagrangian–Eulerian (ALE) finite element method in the monolithic frame is developed for a linearized fluid–structure interaction (FSI) problem — an unsteady Stokes/parabolic interface problem with jump coefficients and moving interface, where, the corresponding mixed finite element approximation in both semi- and fully discrete scheme are developed and analyzed based upon one type of ALE formulation and a novel H1- projection technique associated with a moving interface problem, and the stability and optimal convergence properties in the energy norm are obtained for both discretizations to approximate the solution of a transient Stokes/parabolic interface problem …