Department of Mathematics
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Item Curved surfaces of right cones(University of Alabama Libraries, 1919) Milner, Robert Toombs; University of Alabama TuscaloosaIt is the purpose of this article to derive a formula for finding the curved surface of a right cone with any plain curve as a base.Item A compiler for the Bama-Bell floating point interpretive system(University of Alabama Libraries, 1962) Gray, William J.; University of Alabama TuscaloosaItem Topological transformation groups with a fixed end point(University of Alabama Libraries, 1965) Gray, William Jesse; University of Alabama TuscaloosaItem On the Theory of Structures in Sets(University of Alabama Libraries, 1970) Hall, Japheth Jr.; University of Alabama TuscaloosaIn many branches of mathematics the property of being a subspace receives considerable attention. The subspaces of a given space might be regarded as values of a structure in a set, that is, a function P on the subsets of a set V such that P(X) ⊆ V for all X ⊆ V. Thus, structures in sets form a basis for an abstract treatment of the property of being a subspace.Item Mathematics: Selected Reference Sources(1980-08-04) Sandy, John H.Item Rational Modules and Higher Order Cauchy Transforms(Hindawi Publishing Corporation, 1981) Wang, James Li-MingWe apply the higher order Cauchy transforms to describe the closures of rational modules with respect to the L^P norms, the uniform norm and different Lipschitz norms on a compact set in the plane.Item ON Q-ALGEBRAS(Wiley, 2001) Neggers, Joseph; Ahn, Sun Shin; Kim, Hee SikWe introduce a new notion, called a Q-algebra, which is a generalization of the idea of BCH /BCI /BCK -algebras and we generalize some theorems discussed in BCI -algebras. Moreover, we introduce the notion of “quadratic” Q-algebra, and show that every quadratic Q-algebra (X; ∗, e), e ∈ X, has a product of the form x ∗ y = x − y + e, where x, y ∈ X when X is a field with |X| ≥ 3. 2000 Mathematics Subject Classification. 06F35, 03G25.Item Managing risk with short term futures contracts(University of Alabama Libraries, 2009) Yu, Chunhui; Wu, Zhijian; University of Alabama Tuscaloosa[NOTE: Text or symbols not renderable in plain text are indicated by [...]. See PDF document for full abstract.] In this dissertation, we search for an optimal strategy to reduce the running risk in hedging a long-term supply commitment with short-term futures contracts under a certain constraint on the terminal risk, which leads to a class of intrinsic optimization problems. Motivated by a simple model initially discussed in [1] Culp and Miller, [8] Mello and Parsons, [4] Glasserman and a best strategy model by [5] Larcher and Leobacher, we studied an optimization problem as following: Under the condition that [...] Which measurable function [...] minimizes the value of [...]? We will show that a unique solution to this general problem always exists. By solving it numerically, we obtain a general dynamic solution. Furthermore, we will discuss properties of the solution and give analytic solutions in some special cases.Item A graph theoretical model for the analysis of the game of football and a discussion of applications thereof(University of Alabama Libraries, 2009) Taylor, Patrick; Neggers, Joseph; University of Alabama TuscaloosaIn this dissertation an epidemiological approach is used to develop a graph theoretical model for the game of football. This model is a preliminary model due to the limitation of available resources. Even in its preliminary form, it is evident that significant information is obtained and easily displayed using graphs and adjacency matrices. A similar approach may be used in other game-like situations where coaches (manipulators) make decisions about strategy and tactics in order to prevail over opponents. In our case, the goal is to create a package of tools for the working professional in the field, i.e., the football coach and his assistants. As a study, this paper discusses its construction and methods, including procedures used to collect the data, analysis of the data, conclusions drawn, and commentary on future designs.Item Three assets model for portfolio selection under a constrained consumption rate process(University of Alabama Libraries, 2009) Thagunna, Karan Singh; Wu, Zhijian; University of Alabama Tuscaloosa[NOTE: Text or symbols not renderable in plain text are indicated by [...]. See PDF document for full abstract.] In this dissertation, we consider a particular case of an optimal consumption and portfolio selection problem for an infinitely lived investor whose consumption rate process is subject to downside constraint. We also suppose that the wealth dynamics is composed of three assets (i) risklessassets (ii) risky assets (iii) hedge assets. We consider the investor's wealth process, interpreted in the sense of the Itô integral as [...]. Our work aims to find the optimal policies which maximize the expected discount utility function given by [...]. Furthermore, we obtain the optimal policies in an explicit form for the log utility function which is a special case (γ → 1) of the general utility(CRRA) function, using the martingale method and applying the Legendre transform formula and the Feynman-kac formula. We derive some numerical results for the optimal policies and illustrated graphically.Item The hidden subgroup problem for generalized quaternions(University of Alabama Libraries, 2009) Upton, Julia Tumasova; Corson, Jon M.; University of Alabama TuscaloosaThe hidden subgroup problem is a pivotal problem in quantum computation since it reflects the structure of tasks for which quantum algorithms significantly outperform classical algorithms. In this dissertation, a quantum algorithm that solves the hidden subgroup problem over the generalized quaternion group is developed. The algorithm employs the abelian quantum Fourier transform and Kuperberg sieve to reveal the hidden subgroup.Item A corona theorem for certain subalgebras of H∞(D)(University of Alabama Libraries, 2009) Ryle, Julie; Trent, Tavan T.; University of Alabama Tuscaloosa[NOTE: Text or symbols not renderable in plain text are indicated by [...]. See PDF document for full abstract.] The corona theorem for the space of bounded analytic functions on the unit disk, [...], which was proven by Carleson in 1962, states that D is dense in the maximal ideal space of [...]. This theorem can be reduced to the following result: [...]. Furthermore, if we have the additional condition that [...]. In this dissertation, we prove that the corona theorem holds for certain subalgebras of [...], and we provide estimates for the sizes of the given solutions. Among the algebras we consider are those which contain bounded analytic functions whose kth derivatives vanish at 0 for all k in K, a subset of the natural numbers, which we call [...]. We give several properties the set K must have in order for [...] to be an algebra. We then prove the corona theorem in both the vector and matrix cases for these algebras. In fact, in the vector case, we prove the corona theorem using two different techniques. Each gives a unique estimate, and one extends our findings to more general algebras. We also settle a conjecture of Mortini, Sasane, and Wick involving the algebra C+BH∞(D), where B is a Blaschke product. We prove the corona theorem in C+BH∞(D) holds for an infinite number of functions. We end with a few suggestions for future research.Item Volatility analysis for high frequency financial data(University of Alabama Libraries, 2009) Zheng, Xiaohua; Wu, Zhijian; University of Alabama TuscaloosaMeasuring and modeling financial volatility are key steps for derivative pricing and risk management. In financial markets, there are two kinds of data: low-frequency financial data and high-frequency financial data. Most research has been done based on low-frequency data. In this dissertation we focus on high-frequency data. In theory, the sum of squares of log returns sampled at high frequency estimates their variance. For log price data following a diffusion process without noise, the realized volatility converges to its quadratic variation. When log price data contain market microstructure noise, the realized volatility explodes as the sampling interval converges to 0. In this dissertation, we generalize the fundamental Ito isometry and analyze the speed with which stochastic processes approach to their quadratic variations. We determine the difference between realized volatility and quadratic variation under mean square constraints for Brownian motion and general case. We improve the estimation for quadratic variation. The estimators found by us converge to quadratic variation at a higher rate.Item The Corona Theorem for the multiplier algebras on weighted Dirichlet spaces(University of Alabama Libraries, 2009) Kidane, Berhanu Tekle; Trent, Tavan T.; University of Alabama TuscaloosaIn this dissertation we give a proof of "The Corona Theorem for Infinitely Many Functions for the Multiplier Algebras on Weighted Dirichlet Spaces", and we obtain explicit estimates on the size of the solution. We denote the open unit disc of the complex plane by D, and for α in (0, 1) we denote by Dα the Weighted Dirichlet Spaces of all holomorphic functions on D, and byItem P-algebras and Q-algebras(University of Alabama Libraries, 2010) Mahawanniarachchi, Padmal Sathyajith; Neggers, Joseph; University of Alabama TuscaloosaWe define two classes of algebras P- and Q-, which are derived from the definitions of BCK- and BCI- algebras. The birth of P-algebras is based on the symmetric difference in set theory. We prove that the class of P-algebras is a variety, and the definition of P-algebras is an alternative definition for groups of exponent 2, which we call P-groups. The class of Q-algebras consists of a combination of three axioms of BCK- and P- algebras. We study the relationship among P-, Q- and BCI- algebras. The theory of P- and Q- algebras is developed parallel to the theory of BCK- and BCI- algebras.Item On Integral Operators with Operator-Valued Kernels(Hindawi, 2010) Shahmurov, Rishad; Ministry of National Education - Turkey; Okan University; University of Alabama TuscaloosaHere, we study the continuity of integral operators with operator-valued kernels. Particularly we get L(q) (S;X) -> L(p) (T;Y) estimates under some natural conditions on the kernel k : T x S -> B (X, Y), where X and Y are Banach spaces, and (T, Sigma(T), mu) and (S, Sigma(S), nu) are positive measure spaces: Then, we apply these results to extend the well- known Fourier Multiplier theorems on Besov spaces.Item Matched interface and boundary enhanced multiresolution time-domain algorithm for electromagnetic simulations(University of Alabama Libraries, 2011) Yao, Pengfei; Zhao, Shan; University of Alabama TuscaloosaThe present work introduces a new boundary closure treatment for the wavelet based multiresolution time-domain (MRTD) solution of Maxwell's equations [1]. Accommodating nontrivial boundary conditions, such as the Robin condition or time dependent condition, has been a challenging issue in the MRTD analysis of wave scattering, radiation, and propagation. A matched interface and boundary multiresolution time-domain (MIBMRTD) method is introduced to overcome this difficulty. Several numerical benchmark tests are carried out to valid the MIB-MRTD method. Dispersion and stability analysis for the MIB-MRTD method are conducted and compared with the high-order finite difference time-domain (FDTD) method . The proposed boundary treatment can also be applied to other high order approaches, such as the dispersion-relation-preserving (DRP) method. The MIB boundary scheme greatly enhances the feasibility for applying the MRTD methods to more complicated electromagnetic structures.Item A non-geometric Brownian motion model estimated by Markov chain approximation(University of Alabama Libraries, 2011) Chang, Hung Yu; Wang, Pu; University of Alabama TuscaloosaThe pricing of most contingent claims is continuously monitored the movement of the underlying assets that follow geometric Brownian motion. However, for exotic options, the pricing of the underlying assets is difficult to be obtained analytically. In reality, numerical methods are employed to monitor discretized path-dependent options since complexity of exotic options increases the difficulty of obtaining the closed-form solutions. In this dissertation, we propose a Markov chain method to discretely monitor the underlying asset pricing of an European knock-out call option with time-varying barriers. Markov chain method provides some advantages in computation since the discretized time step can be partitioned to match with the number of the underlying non-dividend paying asset prices. Compared to Monte Carlo simulation, Markov chain method can not only efficiently handle the case where the initial asset price is close to a barrier level but also effectively improve the accuracy of obtaining the price of a barrier option. We study an European knock-out call option with either constant or time-varying barriers. Under risk-neural measure, the movement of the underlying stock price is said to follow a non-geometric Brownian motion. Furthermore, we are interested to estimate the parameter p value that generates optimal payoff of a knock-out option with time-varying barriers. However, implied volatility is an essential factor that affects the movement of the underlying asset price and determines whether the barrier option is knocked out or not during the lifetime of the option.Item Capping the variance of cash flow of hedging strategy(University of Alabama Libraries, 2011) Ginting, Maydison; Wu, Zhijian; University of Alabama TuscaloosaThis dissertation consolidates previous research on an optimal strategy to reduce the running risk in hedging a long-term supply commitment with short-dated futures contracts. By introducing a cap function, this dissertation defines scenarios of running risk over the hedging horizon. We introduce a linear cap function and wish to find a hedging strategy G with the smallest constant F such that the variance of the cumulative cash flow is less than or equal the multiplication of a cap function and the constant F. The objective is to seek the best function G(s) to cap the variance of cash flow under a given non-negative cap function. We also implement the result in MATLAB by creating a Graphical User Interface application that enables the user to see the various results of the variance of cash flow of the best hedging scenario.Item Performance evaluation of inexact GMRES(University of Alabama Libraries, 2011) Winkles, Nathan; Sidje, Roger B.; University of Alabama TuscaloosaIterative methods are aimed at sparse linear systems that arise in many applications (e.g., PDEs, biology, computer science, technology, engineering, etc). These applications give rise to matrices that differ in terms of structure and characteristics, and these ultimately impact the solvers. The Generalized Minimal Residual Method (GMRES) is a widely used solver due to its robustness. The inexact GMRES algorithm is a variant of the GMRES algorithm where matrix-vector products are performed inexactly, either out of necessity or deliberately, in view of trading accuracy for speed. Recent studies have shown that relaxing matrix-vector products this way can be justified theoretically and experimentally. Research so far has focused on decreasing the workload per iteration without significantly affecting the accuracy. But relaxing the accuracy per iteration is susceptible to increase the number of iterations, thereby increasing the overall runtime, which could potentially end up greater than that of the exact GMRES if there were not enough savings in the matrix-vector products. In this dissertation, we assess the benefit of the inexact approach in terms of actual CPU time derived from realistic problems, and we provide cases that shed instructive insights into results affected by the buildup of the inexactness. Such information is of vital importance to practitioners who need to decide if switching their vested work-flow to the inexact approach is worth the effort and the risk that might come with it. Our assessment is drawn from extensive numerical experiments on the Alabama Supercomputing Facility that gauge the effectiveness of the inexact scheme and its suitability to certain problems depending on how much inexactness is allowed in the matrix-vector products. We present many different applications throughout this dissertation and we show different structures and characteristics of matrices which are useful in the sense that linear system solvers sometimes do not converge to the correct solution if the matrices do not have specific properties. We apply some incomplete preconditioning techniques to our inexact scheme and we show that we could accelerate the convergence or even recover convergence that was lost from the restarted GMRES.