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THE USE OF FILTERS IN TOPOLOGY
Title
THE USE OF FILTERS IN TOPOLOGY.
Creator
Dasser, Abdellatif, Richardson, Gary, University of Central Florida
Abstract / Description
Sequences are sufficient to describe topological properties in metric spaces or, more generally, topological spaces having a countable base for the topology. However, filters or nets are needed in more abstract spaces. Nets are more natural extension of sequences but are generally less friendly to work with since quite often two nets have distinct directed sets for domains. Operations involving filters are set theoretic and generally certain to filters on the same set. The concept of a filter...
Show moreSequences are sufficient to describe topological properties in metric spaces or, more generally, topological spaces having a countable base for the topology. However, filters or nets are needed in more abstract spaces. Nets are more natural extension of sequences but are generally less friendly to work with since quite often two nets have distinct directed sets for domains. Operations involving filters are set theoretic and generally certain to filters on the same set. The concept of a filter was introduced by H. Cartan in 1937 and an excellent treatment of the subject can be found in N. Bourbaki (1940).
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Date Issued
2004
Identifier
CFE0000202, ucf:46271
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0000202
CATEGORICAL PROPERTIES OF LATTICE-VALUED CONVERGENCE SPACES
Title
CATEGORICAL PROPERTIES OF LATTICE-VALUED CONVERGENCE SPACES.
Creator
Flores, Paul, Richardson, Gary, University of Central Florida
Abstract / Description
This work can be roughly divided into two parts. Initially, it may be considered a continuation of the very interesting research on the topic of Lattice-Valued Convergence Spaces given by Jäger [2001, 2005]. The alternate axioms presented here seem to lead to theorems having proofs more closely related to standard arguments used in Convergence Space theory when the Lattice is L=.Various Subcategories are investigated. One such subconstruct is shown to be isomorphic to the category of...
Show moreThis work can be roughly divided into two parts. Initially, it may be considered a continuation of the very interesting research on the topic of Lattice-Valued Convergence Spaces given by Jäger [2001, 2005]. The alternate axioms presented here seem to lead to theorems having proofs more closely related to standard arguments used in Convergence Space theory when the Lattice is L=.Various Subcategories are investigated. One such subconstruct is shown to be isomorphic to the category of Lattice Valued Fuzzy Convergence Spaces defined and studied by Jäger . Our principal category is shown to be a topological universe and contains a subconstruct isomorphic to the category of probabilistic convergence spaces discussed in Kent and Richardson when L=. Fundamental work in lattice-valued convergence from the more general perspective of monads can be found in Gähler . Secondly, diagonal axioms are defined in the category whose objects consist of all the lattice valued convergence spaces. When the latter lattice is linearly ordered, a diagonal condition is given which characterizes those objects in the category that are determined by probabilistic convergence spaces which are topological. Certain background information regarding filters, convergence spaces, and diagonal axioms with its dual are given in Chapter 1. Chapter 2 describes Probabilistic Convergence and associated Diagonal axioms. Chapter 3 defines Jäger convergence and proves that Jäger's construct is isomorphic to a bireflective subconstruct of SL-CS. Furthermore, connections between the diagonal axioms discussed and those given by Gähler are explored. In Chapter 4, further categorical properties of SL-CS are discussed and in particular, it is shown that SL-CS is topological, cartesian closed, and extensional. Chapter 5 explores connections between diagonal axioms for objects in the sub construct δ(PCS) and SL-CS. Finally, recommendations for further research are provided.
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Date Issued
2007
Identifier
CFE0001715, ucf:47292
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0001715
LATTICE-VALUED CONVERGENCE: QUOTIENT MAPS
Title
LATTICE-VALUED CONVERGENCE: QUOTIENT MAPS.
Creator
Boustique, Hatim, Richardson, Gary, University of Central Florida
Abstract / Description
The introduction of fuzzy sets by Zadeh has created new research directions in many fields of mathematics. Fuzzy set theory was originally restricted to the lattice , but the thrust of more recent research has pertained to general lattices. The present work is primarily focused on the theory of lattice-valued convergence spaces; the category of lattice-valued convergence spaces has been shown to possess the following desirable categorical properties: topological, cartesian-closed, and...
Show moreThe introduction of fuzzy sets by Zadeh has created new research directions in many fields of mathematics. Fuzzy set theory was originally restricted to the lattice , but the thrust of more recent research has pertained to general lattices. The present work is primarily focused on the theory of lattice-valued convergence spaces; the category of lattice-valued convergence spaces has been shown to possess the following desirable categorical properties: topological, cartesian-closed, and extensional. Properties of quotient maps between objects in this category are investigated in this work; in particular, one of our principal results shows that quotient maps are productive under arbitrary products. A category of lattice-valued interior operators is defined and studied as well. Axioms are given in order for this category to be isomorphic to the category whose objects consist of all the stratified, lattice-valued, pretopological convergence spaces. Adding a lattice-valued convergence structure to a group leads to the creation of a new category whose objects are called lattice-valued convergence groups, and whose morphisms are all the continuous homomorphisms between objects. The latter category is studied and results related to separation properties are obtained. For the special lattice , continuous actions of a convergence semigroup on convergence spaces are investigated; in particular, invariance properties of actions as well as properties of a generalized quotient space are presented.
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Date Issued
2008
Identifier
CFE0002369, ucf:47811
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0002369
Autoregressive Models
Title
Autoregressive Models.
Creator
Wade, William, Richardson, Gary, Pensky, Marianna, Li, Xin, University of Central Florida
Abstract / Description
Consider a sequence of random variables which obeys a first order autoregressive model with unknown parameter alpha. Under suitable assumptions on the error structure of the model, the limiting distribution of the normalized least squares estimator of alpha is discussed. The choice of the normalizing constant depends on whether alpha is less than one, equals one, or is greater than one in absolute value. In particular, the limiting distribution is normal provided that the absolute value of...
Show moreConsider a sequence of random variables which obeys a first order autoregressive model with unknown parameter alpha. Under suitable assumptions on the error structure of the model, the limiting distribution of the normalized least squares estimator of alpha is discussed. The choice of the normalizing constant depends on whether alpha is less than one, equals one, or is greater than one in absolute value. In particular, the limiting distribution is normal provided that the absolute value of alpha is less than one, but is a function of Brownian motion whenever the absolute value of alpha equals one. Some general remarks are made whenever the sequence of random variables is a first order moving average process.
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Date Issued
2012
Identifier
CFE0004276, ucf:49546
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004276
Functional Data Analysis and its application to cancer data
Title
Functional Data Analysis and its application to cancer data.
Creator
Martinenko, Evgeny, Pensky, Marianna, Tamasan, Alexandru, Swanson, Jason, Richardson, Gary, University of Central Florida
Abstract / Description
The objective of the current work is to develop novel procedures for the analysis of functional dataand apply them for investigation of gender disparity in survival of lung cancer patients. In particular,we use the time-dependent Cox proportional hazards model where the clinical information isincorporated via time-independent covariates, and the current age is modeled using its expansionover wavelet basis functions. We developed computer algorithms and applied them to the dataset which is...
Show moreThe objective of the current work is to develop novel procedures for the analysis of functional dataand apply them for investigation of gender disparity in survival of lung cancer patients. In particular,we use the time-dependent Cox proportional hazards model where the clinical information isincorporated via time-independent covariates, and the current age is modeled using its expansionover wavelet basis functions. We developed computer algorithms and applied them to the dataset which is derived from Florida Cancer Data depository data set (all personal information whichallows to identify patients was eliminated). We also studied the problem of estimation of a continuousmatrix-variate function of low rank. We have constructed an estimator of such functionusing its basis expansion and subsequent solution of an optimization problem with the Schattennormpenalty. We derive an oracle inequality for the constructed estimator, study its properties viasimulations and apply the procedure to analysis of Dynamic Contrast medical imaging data.
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Date Issued
2014
Identifier
CFE0005377, ucf:50447
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005377
Spatial Models with Specific Error Structures
Title
Spatial Models with Specific Error Structures.
Creator
Adu, Nathaniel, Richardson, Gary, Mohapatra, Ram, Song, Zixia, Lang, Sheau-Dong, University of Central Florida
Abstract / Description
The purpose of this dissertation is to study the first order autoregressive model in the spatial context with specific error structures. We begin by supposing that the error structure has a long memory in both the i and the j components. Whenever the model parameters alpha and beta equal one, the limiting distribution of the sequence of normalized Fourier coefficients of the spatial process is shown to be a function of a two parameter fractional Brownian sheet. This result is used to find the...
Show moreThe purpose of this dissertation is to study the first order autoregressive model in the spatial context with specific error structures. We begin by supposing that the error structure has a long memory in both the i and the j components. Whenever the model parameters alpha and beta equal one, the limiting distribution of the sequence of normalized Fourier coefficients of the spatial process is shown to be a function of a two parameter fractional Brownian sheet. This result is used to find the limiting distribution of the periodogram ordinate of the spatial process under the null hypothesis that alpha equals one and beta equals one. We then give the limiting distribution of the normalized Fourier coefficients of the spatial process for both a moving average and autoregressive error structure. Two cases of autoregressive errors are considered. The first error model is autoregressive in one component and the second is autoregressive in both components. We show that the normalizing factor needed to ensure convergence in distribution of the sequence of Fourier coefficients is different in the moving average case, and the two autoregressive cases. In other words, the normalizing factor differs in each of these three cases.Finally, a specific case of the functional central limit theorem in the spatial setting is stated and proved. The assumptions made here are placed on the autocovariance functions. We then discuss some specific examples and provide a test statistics based on the periodogram ordinate.
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Date Issued
2019
Identifier
CFE0007772, ucf:52385
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007772
Bayesian Model Selection for Classification with Possibly Large Number of Groups
Title
Bayesian Model Selection for Classification with Possibly Large Number of Groups.
Creator
Davis, Justin, Pensky, Marianna, Swanson, Jason, Richardson, Gary, Crampton, William, Ni, Liqiang, University of Central Florida
Abstract / Description
The purpose of the present dissertation is to study model selection techniques which are specifically designed for classification of high-dimensional data with a large number of classes. To the best of our knowledge, this problem has never been studied in depth previously. We assume that the number of components p is much larger than the number of samples n, and that only few of those p components are useful for subsequent classification. In what follows, we introduce two Bayesian models...
Show moreThe purpose of the present dissertation is to study model selection techniques which are specifically designed for classification of high-dimensional data with a large number of classes. To the best of our knowledge, this problem has never been studied in depth previously. We assume that the number of components p is much larger than the number of samples n, and that only few of those p components are useful for subsequent classification. In what follows, we introduce two Bayesian models which use two different approaches to the problem: one which discards components which have "almost constant" values (Model 1) and another which retains the components for which between-group variations are larger than within-group variation (Model 2). We show that particular cases of the above two models recover familiar variance or ANOVA-based component selection. When one has only two classes and features are a priori independent, Model 2 reduces to the Feature Annealed Independence Rule (FAIR) introduced by Fan and Fan (2008) and can be viewed as a natural generalization to the case of L (>) 2 classes. A nontrivial result of the dissertation is that the precision of feature selection using Model 2 improves when the number of classes grows. Subsequently, we examine the rate of misclassification with and without feature selection on the basis of Model 2.
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Date Issued
2011
Identifier
CFE0004097, ucf:49091
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004097
Estimation for the Cox Model with Various Types of Censored Data
Title
Estimation for the Cox Model with Various Types of Censored Data.
Creator
Riddlesworth, Tonya, Ren, Joan, Mohapatra, Ram, Richardson, Gary, Ni, Liqiang, Schott, James, University of Central Florida
Abstract / Description
In survival analysis, the Cox model is one of the most widely used tools. However, up to now there has not been any published work on the Cox model with complicated types of censored data, such as doubly censored data, partly-interval censored data, etc., while these types of censored data have been encountered in important medical studies, such as cancer, heart disease, diabetes, etc. In this dissertation, we first derive the bivariate nonparametric maximum likelihood estimator (BNPMLE) Fn(t...
Show moreIn survival analysis, the Cox model is one of the most widely used tools. However, up to now there has not been any published work on the Cox model with complicated types of censored data, such as doubly censored data, partly-interval censored data, etc., while these types of censored data have been encountered in important medical studies, such as cancer, heart disease, diabetes, etc. In this dissertation, we first derive the bivariate nonparametric maximum likelihood estimator (BNPMLE) Fn(t,z) for joint distribution function Fo(t,z) of survival time T and covariate Z, where T is subject to right censoring, noting that such BNPMLE Fn has not been studied in statistical literature. Then, based on this BNPMLE Fn we derive empirical likelihood-based (Owen, 1988) confidence interval for the conditional survival probabilities, which is an important and difficult problem in statistical analysis, and also has not been studied in literature. Finally, with this BNPMLE Fn as a starting point, we extend the weighted empirical likelihood method (Ren, 2001 and 2008a) to the multivariate case, and obtain a weighted empirical likelihood-based estimation method for the Cox model. Such estimation method is given in a unified form, and is applicable to various types of censored data aforementioned.
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Date Issued
2011
Identifier
CFE0004158, ucf:49051
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004158
Lattice-Valued T-Filters and Induced Structures
Title
Lattice-Valued T-Filters and Induced Structures.
Creator
Reid, Frederick, Richardson, Gary, Brennan, Joseph, Han, Deguang, Lang, Sheau-Dong, University of Central Florida
Abstract / Description
A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a T-convergence structure which is defined in terms of T-filters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of...
Show moreA complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a T-convergence structure which is defined in terms of T-filters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional.Subcategories of compact and of complete spaces are investigated. It is shown that each T-convergence space has a compactification with the extension property provided the frame is a Boolean algebra. T-Cauchy spaces are defined and sufficient conditions for the existence of a completion are given. T-uniform limit spaces are also defined and their completions are given in terms of the T-Cauchy spaces they induce. Categorical properties of these subcategories are also investigated. Further, for a fixed T-convergence space, under suitable conditions, it is shown that there exists an order preserving bijection between the set of all strict, regular, Hausdorff compactifications and the set of all totally bounded T-Cauchy spaces which induce the fixed space.
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Date Issued
2019
Identifier
CFE0007520, ucf:52586
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007520
HJB Equation and Statistical Arbitrage applied to High Frequency Trading
Title
HJB Equation and Statistical Arbitrage applied to High Frequency Trading.
Creator
Park, Yonggi, Yong, Jiongmin, Swanson, Jason, Richardson, Gary, Shuai, Zhisheng, University of Central Florida
Abstract / Description
In this thesis we investigate some properties of market making and statistical arbitrage applied to High Frequency Trading (HFT). Using the Hamilton-Jacobi-Bellman(HJB) model developed by Guilbaud, Fabien and Pham, Huyen in 2012, we studied how market making works to obtain optimal strategy during limit order and market order. Also we develop the best investment strategy through Moving Average, Exponential Moving Average, Relative Strength Index, Sharpe Ratio.
Date Issued
2013
Identifier
CFE0004907, ucf:49628
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004907
Extensions of S-spaces
Title
Extensions of S-spaces.
Creator
Losert, Bernd, Richardson, Gary, Mikusinski, Piotr, Dutkay, Dorin, Brennan, Joseph, Marinescu, Dan, University of Central Florida
Abstract / Description
Given a convergence space X, a continuous action of a convergence semigroup S on X and a compactification Y of X, under what conditions on X and the action on X is it possible to extend the action to a continuous action on Y. Similarly, given a Cauchy space X, a Cauchy continuous action of a Cauchy semigroup S on X and a completion Y of X, under what conditions on X and the action on X is it possible to extend the action to a Cauchy continuous action on Y. We answer the first question for...
Show moreGiven a convergence space X, a continuous action of a convergence semigroup S on X and a compactification Y of X, under what conditions on X and the action on X is it possible to extend the action to a continuous action on Y. Similarly, given a Cauchy space X, a Cauchy continuous action of a Cauchy semigroup S on X and a completion Y of X, under what conditions on X and the action on X is it possible to extend the action to a Cauchy continuous action on Y. We answer the first question for some particular compactifications like the one-point compactification and the star compactification as well as for the class of regular compactifications. We answer the second question for the class of regular strict completions. Using these results, we give sufficient conditions under which the pseudoquotient of a compactification/completion of a space is the compactification/completion of the pseudoquotient of the given space.
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Date Issued
2013
Identifier
CFE0004881, ucf:49661
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004881