Mathematical Analysis Ii, 2016 Payame Noor University

Generalizations Of The Cauchy And Fujiwara Bounds For Products Of Zeros Of A Polynomial, 2016 University of Guelph

#### Generalizations Of The Cauchy And Fujiwara Bounds For Products Of Zeros Of A Polynomial, Rajesh Pereira, Mohammad Ali Vali

*Electronic Journal of Linear Algebra*

The Cauchy bound is one of the best known upper bounds for the modulus of the zeros of a polynomial. The Fujiwara bound is another useful upper bound for the modulus of the zeros of a polynomial. In this paper, compound matrices are used to derive a generalization of both the Cauchy bound and the Fujiwara bound. This generalization yields upper bounds for the modulus of the product of $m$ zeros of the polynomial.

On Logarithmic Sobolev Inequality And A Scalar Curvature Formula For Noncommutative Tori, 2016 The University of Western Ontario

#### On Logarithmic Sobolev Inequality And A Scalar Curvature Formula For Noncommutative Tori, Sajad Sadeghi

*Electronic Thesis and Dissertation Repository*

In the first part of this thesis, a noncommutative analogue of Gross' logarithmic Sobolev inequality for the noncommutative 2-torus is investigated. More precisely, Weissler's result on the logarithmic Sobolev inequality for the unit circle is used to propose that the logarithmic Sobolev inequality for a positive element $a= \sum a_{m,n} U^{m} V^{n} $ of the noncommutative 2-torus should be of the form $$\tau(a^{2} \log a)\leqslant \underset{(m,n)\in \mathbb{Z}^{2}}{\sum} (\vert m\vert + \vert n\vert) \vert a_{m,n} \vert ^{2} + \tau (a^{2})\log ( \tau (a^2))^{1 ...

Moduli Space And Deformations Of Special Lagrangian Submanifolds With Edge Singularities, 2016 The University of Western Ontario

#### Moduli Space And Deformations Of Special Lagrangian Submanifolds With Edge Singularities, Josue Rosario-Ortega

*Electronic Thesis and Dissertation Repository*

Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this thesis we use elliptic theory for edge- degenerate differential operators on singular manifolds to study general deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.

An Algorithm For The Machine Calculation Of Minimal Paths, 2016 East Tennessee State University

#### An Algorithm For The Machine Calculation Of Minimal Paths, Robert Whitinger

*Electronic Theses and Dissertations*

Problems involving the minimization of functionals date back to antiquity. The mathematics of the calculus of variations has provided a framework for the analytical solution of a limited class of such problems. This paper describes a numerical approximation technique for obtaining machine solutions to minimal path problems. It is shown that this technique is applicable not only to the common case of finding geodesics on parameterized surfaces in R^{3}, but also to the general case of finding minimal functionals on hypersurfaces in R^{n} associated with an arbitrary metric.

Extension Theorems On Matrix Weighted Sobolev Spaces, 2016 University of Tennessee, Knoxville

#### Extension Theorems On Matrix Weighted Sobolev Spaces, Christopher Ryan Loga

*Doctoral Dissertations*

Let D a subset of R^{n} [R n] be a domain with Lipschitz boundary and 1 ≤ p < ∞ [1 less than or equal to p less than infinity]. Suppose for each x in R^{n} that W(x) is an m x m [m by m] positive definite matrix which satisfies the matrix A_{p} [A p] condition. For k = 0, 1, 2, 3;... define the matrix weighted, vector valued, Sobolev space [L p k of D,W] with

[the weighted L p k norm of vector valued f over D to the p power equals the sum over all alpha with order less than k of the integral over D of the the pth power ...

Regression Model Fitting With Quadratic Term Leads To Different Conclusion In Economic Analysis Of Washington State Smoking Ban, 2016 Pennsylvania Department of Health, Harrisburg, PA

#### Regression Model Fitting With Quadratic Term Leads To Different Conclusion In Economic Analysis Of Washington State Smoking Ban, Marshal Ma, Scott Mcclintock

*Scott McClintock*

No abstract provided.

On The Group Invertibility Of Operators, 2016 South China Normal University,

#### On The Group Invertibility Of Operators, Chunyuan Deng

*Electronic Journal of Linear Algebra*

The main topic of this paper is the group invertibility of operators in Hilbert spaces. Conditions for the existence of the group inverses of products of two operators and the group invertibility of anti-triangular block operator matrices are studied. The equivalent conditions related to the reverse order law for the group inverses of operators are obtained.

Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, 2016 Rose-Hulman Institute of Technology

#### Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces, Sean A. Broughton

*Mathematical Sciences Technical Reports (MSTR)*

Two Riemann surfaces *S*_{1} and *S*_{2} with conformal *G*-actions have topologically equivalent actions if there is a homeomorphism *h :* *S _{1} -> S_{2} *which intertwines the actions. A weaker equivalence may be defined by comparing the representations of

*G*on the spaces of holomorphic

*q-*differentials

*H*and

^{q}(S_{1})*H*In this note we study the differences between topological equivalence and

^{q}(S_{2}).*H*equivalence of prime cyclic actions, where

^{q}*S*and

_{1}/G*S*have genus zero.

_{2}/GUniform Approximation On Riemann Surfaces, 2016 The University of Western Ontario

#### Uniform Approximation On Riemann Surfaces, Fatemeh Sharifi

*Electronic Thesis and Dissertation Repository*

This thesis consists of three contributions to the theory of complex approximation on

Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is usually not possible to approximate f uniformly by functions holomorphic on all of R. Firstly, we show, however, that for every open Riemann surface R and every closed subset E of R; there is closed subset F of E, which approximates E extremely well, such that every function holomorphic on F can be approximated much ...

On The Matrix Square Root Via Geometric Optimization, 2016 Massachusetts Institute of Technology

#### On The Matrix Square Root Via Geometric Optimization, Suvrit Sra

*Electronic Journal of Linear Algebra*

This paper is triggered by the preprint [P. Jain, C. Jin, S.M. Kakade, and P. Netrapalli. Computing matrix squareroot via non convex local search. Preprint, arXiv:1507.05854, 2015.], which analyzes gradient-descent for computing the square root of a positive definite matrix. Contrary to claims of Jain et al., the author’s experiments reveal that Newton-like methods compute matrix square roots rapidly and reliably, even for highly ill-conditioned matrices and without requiring com-mutativity. The author observes that gradient-descent converges very slowly primarily due to tiny step-sizes and ill-conditioning. The paper derives an alternative first-order method based on geodesic convexity ...

Henri Lebesgue And The Development Of The Integral Concept, 2016 Colorado State University-Pueblo

#### Henri Lebesgue And The Development Of The Integral Concept, Janet Heine Barnett

*Analysis*

No abstract provided.

Why Be So Critical? Nineteenth Century Mathematical And The Origins Of Analysis, 2016 Colorado State University-Pueblo

#### Why Be So Critical? Nineteenth Century Mathematical And The Origins Of Analysis, Janet Heine Barnett

*Analysis*

No abstract provided.

Norm Retrievable Frames In $\Mathbb{R}^N$, 2016 Department of Mathematics Vali-e-Asr University of Rafsanjan

#### Norm Retrievable Frames In $\Mathbb{R}^N$, Mohammad Ali Hasankhani Fard

*Electronic Journal of Linear Algebra*

This paper is concerned with the norm retrievable frames in $\mathbb{R}^n$. We present some equivalent conditions to the norm retrievable frames in $\mathbb{R}^n$. We will also show that the property of norm retrievability is stable under enough small perturbation of the frame set only for phase retrievable frames.

The Convergence Of V-Cycle Multigrid Algorithms For Axisymmetric Laplace And Maxwell Equations, 2016 Portland State University

#### The Convergence Of V-Cycle Multigrid Algorithms For Axisymmetric Laplace And Maxwell Equations, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity ...

Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, 2016 Portland State University

#### Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

The finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither Hermitian nor definite when the coefficient function is complex valued. For such problems, under a uniqueness assumption, we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero. Then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number, providing an efficient solver for computing discrete approximations. Numerical experiments, while confirming the theory ...

Multigrid For The Mortar Finite Element Method, 2016 Portland State University

#### Multigrid For The Mortar Finite Element Method, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

A multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed. These problems are posed on domains partitioned into subdomains, each of which is independently triangulated in a multilevel fashion. The multilevel mortar finite element spaces based on such triangulations (which need not align across subdomain interfaces) are in general not nested. Suitable grid transfer operators and smoothers are developed which lead to a variable Vcycle preconditioner resulting in a uniformly preconditioned algebraic system. Computational results illustrating the theory are also presented.

Quasioptimality Of Some Spectral Mixed Methods, 2016 Portland State University

#### Quasioptimality Of Some Spectral Mixed Methods, Jay Gopalakrishnan, Leszek Demkowicz

*Jay Gopalakrishnan*

In this paper, we construct a sequence of projectors into certain polynomial spaces satisfying a commuting diagram property with norm bounds independent of the polynomial degree. Using the projectors, we obtain quasioptimality of some spectralmixed methods, including the Raviart–Thomas method and mixed formulations of Maxwell equations. We also prove some discrete Friedrichs type inequalities involving curl.

Error Analysis Of Variable Degree Mixed Methods For Elliptic Problems Via Hybridization, 2016 University of Minnesota - Twin Cities

#### Error Analysis Of Variable Degree Mixed Methods For Elliptic Problems Via Hybridization, Bernardo Cockburn, Jay Gopalakrishnan

*Jay Gopalakrishnan*

A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new ...

Mortar Estimates Independent Of Number Of Subdomains, 2016 Portland State University

#### Mortar Estimates Independent Of Number Of Subdomains, Jay Gopalakrishnan

*Jay Gopalakrishnan*

The stability and error estimates for the mortar finite element method are well established. This work examines the dependence of constants in these estimates on shape and number of subdomains. By means of a Poincar´e inequality and some scaling arguments, these estimates are found not to deteriorate with increase in number of subdomains.