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1 
SIMPLE TWOSIDED RATIONAL VECTOR SPACES OF RANK TWOHart, John Walker 25 March 2010 (has links)
<p>The purpose of this thesis is to find sufficient conditions under which a noncommutative version of the polynomial ring in two variables exists. The noncommutative rings we construct are noncommutative symmetric algebras over a twosided vector space. After reviewing the definition of a twosided vector space and giving some examples, we briefly recall the theory of simple twosided vector spaces. We then assume k is a field of characteristic zero and t is transcendental over k and we find sufficient conditions under which a simple kcentral twosided vector space V over k(t) has left and right dimension two. Given such a V, and letting <sup>*</sup>V and V<sup>*</sup> denote the left and right duals we find conditions under which (V<sup>i*</sup>,V<sup>(i+1)*</sup>,V<sup>(i+2)*</sup> ) has a simultaneous for all i, i an integer. This condition implies the noncommutative symmetric algebra over V can be constructed. We conclude by exhibiting a fivedimensional family of simple kcentral twosided vector spaces over k(t) of left and right dimension two who noncommutative symmetric algebras exist.</p>

2 
REGULARIZATION PARAMETER SELECTION METHODS FOR ILL POSED POISSON IMAGING PROBLEMSGoldes, John 02 August 2010 (has links)
A common problem in imaging science is to estimate some underlying true image given noisy measurements of image intensity. When image intensity is measured by the counting of incident photons emitted by the object of interest, the datanoise is accurately modeled by a Poisson distribution, which motivates the use of Poisson maximum likelihood estimation. When the underlying model equation is illposed, regularization must be employed. I will present a computational framework for solving such problems, including statistically motivated methods for choosing the regularization parameter. Numerical examples will be included.

3 
Comparison of Trend Detection MethodsGray, Katharine Lynn 26 September 2007 (has links)
Trend estimation is important in many fields, though arguably the most important applications appear in ecology. Trend is difficult to quantify; in fact, the term itself is not welldefined. Often, trend is quantified by estimating the slope coefficient in a regression model where the response variable is an index of population size, and time is the explanatory variable. Linear trend is often unrealistic for biological populations; in fact, many critical environmental changes occur abruptly as a result of very rapid changes in human activities. My PhD research has involved formulating methods with greater flexibility than those currently in use. Penalized spline regression provides a flexible technique for fitting a smooth curve. This method has proven useful in many areas including environmental monitoring; however, inference is more difficult than with ordinary linear regression because so many parameters are estimated. My research has focused on developing methods of trend detection and comparing these methods to other methods currently in use. Attention is given to comparing estimated Type I error rates and power across several trend detection methods. This was accomplished through an extensive simulation study. Monte Carlo simulations and randomization tests were employed to construct an empirical sampling distribution for the test statistic under the null hypothesis of no trend. These methods are superior over smoothing methods over other smoothing methods of trend detection with respect to achieving the designated Type I error rate. The likelihood ratio test using a mixed effects model had the most power for detecting linear trend while a test involving the first derivative was the most powerful for detecting nonlinear trend for small sample sizes.

4 
Randomness In Tree Ensemble MethodsElias, Joran 15 October 2009 (has links)
Tree ensembles have proven to be a popular and powerful tool for predictive modeling tasks. The theory behind several of these methods (e.g. boosting) has received considerable attention. However, other tree ensemble techniques (e.g. bagging, random forests) have attracted limited theoretical treatment. Specifically, it has remained somewhat unclear as to why the simple act of randomizing the tree growing algorithm should lead to such dramatic improvements in performance. It has been suggested that a specific type of tree ensemble acts by forming a locally adaptive distance metric [Lin and Jeon, 2006]. We generalize this claim to include all tree ensembles methods and argue that this insight can help to explain the exceptional performance of tree ensemble methods. Finally, we illustrate the use of tree ensemble methods for an ecological niche modeling example involving the presence of malaria vectors in Africa.

5 
REGULARIZATION METHODS FOR ILLPOSED POISSON IMAGINGLaobeul, N'Djekornom Dara 15 January 2009 (has links)
<P>The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative logarithm of the Poisson likelihood in place of the ubiquitous least squares ttodata. However, if the underlying mathematical model is assumed to have the form z = Au, where A is a linear, compact operator, the problem of minimizing the negative logPoisson likelihood function is illposed, and hence some form of regularization is required. In this work, it involves solving a variational problem of the form u def = arg min u0 `(Au; z) + J(u); where ` is the negativelog of a Poisson likelihood functional, and J is a regularization functional. The main result of this thesis is a theoretical analysis of this variational problem for four dierent regularization functionals. In addition, this work presents an ecient computational method for its solution, and the demonstration of the eectiveness of this approach in practice by applying the algorithm to simulated astronomical imaging data corrupted by the CCD camera noise model mentioned above.</P>

6 
Computational Methods for Support Vector Machine Classification and LargeScale Kalman FilteringHoward, Marylesa 17 July 2013 (has links)
The first half of this dissertation focuses on computational methods for solving the constrained quadratic program (QP) within the support vector machine (SVM) classifier. One of the SVM formulations requires the solution of bound and equality constrained QPs. We begin by describing an augmented Lagrangian approach which incorporates the equality constraint into the objective function, resulting in a bound constrained QP. Furthermore, all constraints may be incorporated into the objective function to yield an unconstrained quadratic program, allowing us to apply the conjugate gradient (CG) method. Lastly, we adapt the scaled gradient projection method of [10] to the SVM QP and compare the performance of these methods with the stateoftheart sequential minimal optimization algorithm and MATLAB's built in constrained QP solver, quadprog. The augmented Lagrangian method outperforms other stateoftheart methods on three image test cases.
The second half of this dissertation focuses on computational methods for largescale Kalman filtering applications. The Kalman filter (KF) is a method for solving a dynamic, coupled system of equations. While these methods require only linear algebra, standard KF is often infeasible in largescale implementations due to the storage requirements and inverse calculations of large, dense covariance matrices. We introduce the use of the CG and Lanczos methods into various forms of the Kalman filter for lowrank approximations of the covariance matrices, with lowstorage requirements. We also use CG for efficient Gaussian sampling within the ensemble Kalman filter method. The CGbased KF methods perform similarly in rootmeansquare error when compared to the standard KF methods, when the standard implementations are feasible, and outperform the limitedmemory BroydenFletcherGoldfarbShanno approximation method.

7 
PeripherallyMultiplicative Spectral Preservers Between Function AlgebrasJohnson, Jeffrey Verlyn 17 July 2013 (has links)
General sufficient conditions are established for maps between function algebras to be composition or weighted composition operators, which extend previous results regarding spectral conditions for maps between uniform algebras. Let X and Y be a locally compact Hausdorff spaces, where A \subset C(X) and B \subset C(Y) are function algebras, not necessarily with unit. Also let \partial A be the Shilov boundary of A, \delta A the Choquet boundary of A, and p(A) the set of ppoints of A. A map T \colon A \to B is called weakly peripherallymultiplicative if the peripheral spectra of fg and TfTg have nonempty intersection for all f,g in A. (i.e. \sigma_{pi}( fg ) \cap \sigma_{pi}(TfTg ) \neq \emptyset for all f,g in A) The map is said to be almost peripherallymultiplicative if the peripheral spectrum of fg is contained in the peripheral spectrum of TfTg (or if the peripheral spectrum of TfTg is contained in the peripheral spectrum of fg) for all f,g in A.
Let X be a locally compact Hausdorff space and A \subset C(X) be a dense subalgebra of a function algebra, not necessarily with unit, such that \delta A = p(A). We show that if T\colon A \to B is a surjective map onto a function algebra B\subset C(Y) that is almost peripherallymultiplicative, then there is a homeomorphism \psi\colon \delta B\to\delta A and a function \alpha on \delta B so that (Tf)(y)=\a(y)\,f(\psi(y)) for all f \in A and y \in\delta B, i.e. T is a weighted composition operator where the weight function is a signum function.
We also show that if T is weakly peripherallymultiplicative, and either \sigma_{pi}(f)\subset \sigma_{pi}(Tf) for all f in A, or, alternatively,
\sigma_{pi}(Tf) \subset \sigma_{pi}(f) for all f in A, then (Tf)(y)=f(\psi(y)) for all f \in A and y \in \delta B. In particular, if A and B are uniform algebras and T \colon A \to B is a weak peripherallymultiplicative operator, that has a limit, say b, at some a in A with a^2=1, then (Tf)(y)=b(y)\,a(\psi(y))\, f(\psi(y)) for every f in A and y in \delta B.
Also, we show that if a weak peripherallymultiplicative map preserving peaking functions in the sense \mathcal{P}(B) \subset T[ \mathbb{T} \cdot \mathcal{P}(A)] or T[\mathcal{P}(A)] \subset \mathbb{T} \cdot \mathcal{P}(B) then T is a weighted composition operator with a signum weight function. Finally, for function algebras containing sufficiently many peak functions, including function algebras on metric spaces, it is shown that weak peripherallymultiplicative maps are necessarily composition operators.

8 
The Szegï¿½ Kernel for NonPseudoconvex Domains in â<sup>2</sup>Gilliam, Michael 05 August 2011 (has links)
<p>There are many operators associated with a domain Î© â â<sup>n</sup> with smooth boundary âÎ©. There are two closely related projections that are of particular interest. The <i>Bergman projection</i> <b><i>B</b></i> is the orthogonal projection of L<sup>2</sup>(Î©) onto the closed subspace L<sup>2</sup>(Î©)â©O(Î©), where O(Î©)is the space of all holomorphic functions on â¦. The <i>Szegï¿½ projection</i> <b><i>S</b></i> is the orthogonal projection of L<sup>2</sup>(ââ¦) onto the space H<sup>2</sup>(Î©) of boundary values of elements of O(Î©). These projection operators have integral representations</p>
<p><b><i>B</b></i>[f](z) = <big>â«</big><sub>â¦,</sub>f(w)<b><i>B</b></i>(z,w)dw, <b><i>S</b></i>[f](z) = <big>â«</big><sub>ââ¦,</sub>f(w)<b><i>S</b></i>(z,w)do(w).</p>
<p>The distributions <b><i>B</b></i> and <b><i>S</b></i> are known respectively as the Bergman and Szegï¿½ kernels. In an attempt to prove that <b><i>B</b></i> and <b><i>S</b></i> are bounded operators on L<sup>p</sup>, 1 < p < â, many authors have obtained size estimates for the kernels B and S for <i>pseudoconvex</i> domains in â<sup>n</sup>.</p>
<p>In this thesis, we restrict our attention to the Szegï¿½ kernel for a large class of domains of the form 1 Such a domain fails to be pseudoconvex precisely when b is not convex on all of R. In an influential paper, Nagel, Rosay, Stein, and Wainger obtain size estimates for both kernels and sharp mapping properties for their respective operators in the convex setting. Consequently, if b is a convex polynomial, the Szegï¿½ kernel S is absolutely convergent off the diagonal only. Carracino proves that the Szegï¿½ kernel has singularities on <i>and off</i> the diagonal for a specific nonsmooth, <i>{nonconvex</i> piecewise defined quadratic b. Her results are novel since very little is known for the Szegï¿½ kernel for nonpseudoconvex domains 2. I take b to be an arbitrary evendegree polynomial with positive leading coefficient and identify the set in 3 on which the Szegï¿½ kernel is absolutely convergent. For a polynomial b, we will see that the Szegï¿½ kernel is smooth off the diagonal if and only if b is convex. These results provide an incremental step toward proving the projection S is bounded on 4, for a large class of nonpseudoconvex domains â¦.</p>

9 
Existence and stability of solutions to the equations of fibre suspension flowsMunganga, Justin Manango Wazute January 1999 (has links)
Includes bibliographical references. / A popular approach to formulating the initialboundary value problem for fibre suspension flows is that in which fibre orientation is accounted for in an averaged sense, through the introduction of a secondorder orientation tensor A. This variable, together with the velocity and pressure, then constitutes the set of unknown variables for the problem. The governing equations are balance of linear momentum, the incompressibility condition, an evolution equation for A, and a constitutive equation for the stress. The evolution equation contains a fourthorder orientation tensor A, and it is necessary to approximate A as a function of A, through a closure relation. The purpose of this these is to examine the wellposedness of the equations governing fibre fibre suspension ﬂows, for various closure relations. It has previously been shown by GP Galdi and BD Reddy that, for the linear closure, the problem is wellposed provided that the particle number, a material constant, is less than a critical value. The work by Galdi and Reddy made of a model in which rotary diffusivity is a function of the ﬂow. This thesis reexamines these issues in two different ways. First, the second law of thermodynamics is used to establish the constraints that the constitutive equations have to satisfy in order to be compatible with this law. This investigation is carried out for a variety of closure rules. The second contribution of the thesis concerns the existence and uniqueness of solutions to the governing equations, for the linear and quadratic closures; for a model in which the rotary diffusivity is treated as a constant, local and global existence of solutions are established, for sufficiently small data, and in the case of the linear closure, for admissible values of the particle number. The existence theory uses a Schauder fixed point approach.

10 
Perturbation field theory methods for calculating expectation valuesAhmed, Samah 22 March 2017 (has links)
No description available.

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