Department: College of Arts and Sciences / Department of Mathematical Science ![[clear]](close-x.png "Remove this limiter")
33 matches in the database.
These are records: 1 - 30.
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1.
Abramov, Vilen.
Stopping Times Related to Trading Strategies.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2008, Kent State University
► We use CUSUM procedure to analyze trading the line strategy. Closed form…
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▼ We use CUSUM procedure to analyze trading the line strategy. Closed form expressions concerning probabilistic characteristics of the CUSUM stopping time and stopped process were obtained in discrete time setting for a wide class of processes. This class of discrete processes was recently defined by K. M. Khan and R. A. Khan. In continuous time, the CUSUM procedure applied to the processes driven by a particular stochastic differential equation was studied. As a result the joint Laplace transform of the maximum process and CUSUM stopping time was derived. Finally, the trading the line strategy was studied for the process driven by the fractional Brownian motion. As in regular Brownian motion case, the Laplace transform was linked to the partial differential equation. Although the lack of optional sampling theorem in this case prevents us from getting a closed form expression, the structure of the Laplace transform is derived. By using these results we point some of the subtle features of the trading the line strategy.
Advisors/Committee Members: Khan, Kazim.
Subjects: Mathematics
Keywords: Stochastic Process; CUSUM Stopping Time; Brownian Motion; fractional Brownian Motion; Laplace Transform; Stochastic Differential Equation; Trading the Line Strategy
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2.
Aziziheris, Kamal.
Determining Group Structure From the Sets of Character Degrees.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2010, Kent State University
► In 1998, Mark Lewis posed a question which would strengthen the connection…
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▼ In 1998, Mark Lewis posed a question which would strengthen the connection between the structure of a finite solvable group G and the set of its character degrees. Specifically, Lewis asked the following question: Lewis’ Question: Let G be a solvable group with cd(G)={1, a, b, c, ab, ac}, where a, b, and c are pairwise relatively prime positive integers. Must G = A × B, where cd(A)={1, a} and cd(B)={1, b, c}? To lend credibility to his question, Lewis verified it if a, b, and c are distinct primes. In this dissertation, we work on the structure of finite solvable groups whose character degree sets are in the form {1, a, b, c, ab, ac}, where a, b, and c are pairwise coprime integers. If p is a prime number and m is a positive integer greater than 1, then we say that the ordered pair (p,m) is a strongly coprime pair if m is not divisible by p and also p does not divide u-1, where 1 < u < m is any proper prime power divisor of m. We prove that: THEOREM. Let G be a solvable group with cd(G)={1, a, b, c, ab, ac}, where a, b, and c are pairwise relatively prime positive integers. Then dl(G)≤ 4, and if a is prime such that the pairs (a,b) and (a,c) are strongly coprime pairs, then one of the following holds: 1. G = A × B, where cd(A)={1, a} and cd(B)={1, b, c}. 2. There is a prime t such that G has a normal Sylow t-subgroup T with cd(T)={1, tl} for some integer l ≥ 2, ttl in {b, c}, and the Fitting height of G is at most 3.
Advisors/Committee Members: Lewis, Mark.
Subjects: Mathematics
Keywords: Character degree; Solvable group; Strongly coprime pair; Direct Product; Fitting subgroup
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3.
Beil, Joel S.
Geometric Properties of Orbits of Integral Operators.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2010, Kent State University
► This dissertation addresses some of the geometric properties of orbits of integral…
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▼ This dissertation addresses some of the geometric properties of orbits of integral operators on the Banach spaces C[0, 1] and Lp[0, 1]. It will be shown that, under very general conditions on the starting element, an orbit of the Volterra operator cannot be a Schauder basis for its closed linear span. However, lacunary subsequences of the orbit will be seen to be Schauder bases for their closed linear span. Bounds on the norm of the iterates and a monotonicity result for a certain class of functions will be established. Moreover, exact asymptotic constants arising from the analysis will be exhibited.
Advisors/Committee Members: Enflo, Per.
Subjects: Mathematics
Keywords: integral operators; Schauder bases; operator orbits; lacunary subsequences; asymptotic analysis
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4.
Bhaduri, Sudipta.
Finding A Maximum Clique of A Chordal Graph by Removing Vertices of Minimum Degree.
Degree: MS, College of Arts and Sciences / Department of Mathematical Science, 2008, Kent State University
► A graph is chordal if each of its cycles of four or…
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▼ A graph is chordal if each of its cycles of four or more nodes has a chord, which is an edge joining two nodes that are not adjacent in the cycle. Chordal graphs are also called triangulated graphs. A graph is chordal if and only if it has a perfect elimination ordering. A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex v, v and neighbors of v that occur later than v in the order form a clique. In their paper Rose, Tarjan and Lueker have shown that a perfect elimination ordering of a chordal graph can be found efficiently using an algorithm lexicographic breadth first search. This algorithm maintains a partition of the vertices of the graph into a sequence of sets; initially this sequence consists of a single set with all vertices. The algorithm repeatedly chooses a vertex v from the earliest set in the sequence that contains previously not chosen vertices, and splits each set S of the sequence into two smaller subsets, the first consisting of the neighbors of v in S and the second consisting of the non-neighbors. When this splitting process is done for all the vertices, the sequence of the sets has one vertex per set, and is the reverse of a perfect elimination ordering. In his paper Fanica Gavril finds a maximum clique and a minimum coloring of a chordal graph using this perfect elimination ordering.In this paper, we will give an algorithm for finding a maximum clique of a chordal graph by removing vertices of minimum degree. We will also show how to color a chordal graph with minimum number of colors. We give a linear time algorithm to find a maximum clique of a chordal graph and also will show that the size of a maximum clique of a chordal graph equals the chromatic number of the graph, i.e., we can color the whole graph with the number of colors equal to the size of a maximum clique.
Advisors/Committee Members: Dragan, Feodor F.
Subjects: Computer science
Keywords: Minimum Degree Ordering, M.D.O, Maximum Clique
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5.
De Kock, Mienie.
Absolute continuity and on the range of a vector measure.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2008, Kent State University
► Let Ω be a compact Hausdorff space1 with Borel σ-field Σ and…
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▼ Let Ω be a compact Hausdorff space1 with Borel σ-field Σ and let μ and ν be regular Borel probabilities on Ω. Then the following are equivalent: (a) [C(Ω) ↪ L1(μ)] ≪ [C(Ω) ↪ L1(ν)] (b) μ ≪ ν (c) [B(Σ) ↪ L1(μ)] ≪ [B(Σ) ↪ L1(ν)] where B(Σ) is the Banach space of all bounded Borel measurable functions equipped with the supremum norm. We extend this result to vector-valued cases. 2. To which Banach spaces X is it so that if C is a countable subset of X that lies in the range of a countably additive X∗∗-valued measure with the same σ-field domain, then there is an X-valued countably additive measure with a σ-field domain, whose range also contains C? We give a partial solution to the problem.
Advisors/Committee Members: Diestel, Joseph.
Subjects: Mathematics
Keywords: Absolute continiuty; Range of a vector measure
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6.
Dugan, Carrie T.
Solvable Groups Whose Character Degree Graphs Have Diameter Three.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2007, Kent State University
► In 2001 Lewis constructed a solvable group whose character degree graph has…
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▼ In 2001 Lewis constructed a solvable group whose character degree graph has diameter three. In the construction of this group he used specific primes 2, 3, and 5. At the time it was known that the diameter of the degree graph of a solvable group must be at most three. Until then, however, no examples of solvable groups with diameter three degree graphs had materialized. It was even conjectured that no such group existed. My results expand on this original result by Lewis which left open the question as to whether or not more solvable groups exist with degree graphs of diameter three. Here we choose primes with certain divisibility conditions. The construction is the same as that of Lewis, which was a construction used by I.M. Isaacs. In general any primes can be used with this construction. However, the need arises for our restrictions on the primes so that the actions are coprime, and in some cases the actions are Frobenius. The result is that the character degree graphs for this family of groups is, in fact, isomorphic to the graph provided by Lewis.
Advisors/Committee Members: Lewis, Mark L.
Subjects: Mathematics
Keywords: Irr; αi; Lemma
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7.
Fenta, Aderaw Workneh.
Lacunary Power Sequences and Extremal Vectors.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2008, Kent State University
► This dissertation has two parts. The first four chapters deal with…
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▼ This dissertation has two parts. The first four chapters deal with lacunary power sequences. In 1966, V.I. Gurariy and V.I. Matsaev showed that a sequence {tλk} is a basic sequence in the spaces C[0, 1] and Lp[0, 1], (1 ≤ p < ∞) if and only if {λk} is a lacunary sequence. Here, we use various methods to generalize this result to sequences {hλkf} in the spaces C[a, b] and Lp[a, b], where 1 ≤ p < ∞ and 0 ≤ a < b. The fifth chapter is on extremal vectors. In 1996 P. Enflo introduced backward minimal vectors to study invariant subspaces. If a bounded linear operator T on a Hilbert space H has dense range, then for each non-zero element x0 of H, each positive number epsilon; with ε ≤ ‖x0‖ and each natural number n, there exists a unique vector yε = y(x0 , ε , n), called backward minimal vector, such that ‖Tnyε - x0‖ ≤ ε and y = inf{‖y‖ : ‖Tny - x0‖ ≤ ε}. Here, we investigate rectifiability properties of the curve γ : ε → Tyε for the multiplication operator T on L2[0, 1].
Advisors/Committee Members: Enflo, Per.
Subjects: Mathematics
Keywords: Schauder basis; Basic sequence; Lacunary sequence; Extremal vectr; Backward minimal vector; Rectifiable curve
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8.
Fontes, Ramiro C.
Applications of Allouba's Differentiation Theory and Semi-SPDEs.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2010, Kent State University
► Recently in [1], Allouba introduced a new stochastic differentiation theory for Itô's…
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▼ Recently in [1], Allouba introduced a new stochastic differentiation theory for Itô's calculus. His approach accomplished two major fundamental objectives: it gave-for the first time in over sixty years-a true differentiation theory counterpart to Itô's stochastic integral calculus; and it provided a fundamental building block for a new generalized theory of stochastic calculus, developed in [2] and followup papers, with wide ranging applications and ramifications beyond the classical processes setting. In the first part of this dissertation, we show that the Allouba differentiation theory for the Itô setting allows us to formulate and prove-in a simple manner-new variants of famous representation formulas from stochastic analysis like the Clark-Ocone formula, the Clark-Ocone formula under change of measure, and Stroock's formula. The setting and the proofs for these variants are naturally linked to Itô's calculus, without the need for Malliavin calculus, Hida's white noise analysis, distributional or Sobolev-type derivatives. In addition, the invariance of the Allouba derivative under change of measure allows us to obtain simpler formulas under change of measure than the ones obtained via the other approaches. By applying these formulas to the well known digital option of mathematical finance (e.g., see [11]), we show that these variants apply even when their original classical versions do not, and we discuss the difference between our conditions versus the original classical-often more restrictive-ones. We note that there are other approaches that relax the original conditions and extend the classical formulas to near their applicability limit (covering digital options and more). These approaches, however, involve either “classical" stochastic analysis in terms of the non-anticipating Radon-Nikodym-type stochastic derivative or the adoption of the technical Hida white-noise analysis combined with Malliavin's classical theory (see e.g., [11]). In the second part, we use the space-time change of measure theorem formulated and proved in [5, 6] to investigate semi-SPDEs bond models that are driven by the time-derivative of the two-parameter Brownian sheet (where the second parameter represents maturity time). In [7], it was shown that such models admit no-arbitrage; and the space-time change of measure was used to obtain the risk-neutral measure. We first show that, in this semi-SPDE setting, any arbitrary portfolio can be hedged; i.e., this two-parameter-noise-model is complete. This means that the risk-neutral measure found in [7] is unique. In addition, guided by the original one parameter Black-Scholes-Merton (BSM) PDE and its derivation (e.g., [29]) and by Allouba's connection of Brownian-sheet and hyperbolic SPDEs to PDEs systems [10], we derive a BSM-type system of PDEs associated with the semi-SPDE bond model. The BSM system involves a Burgers nonlinearity in the volatility. This system may also be regarded as a single BSM nonlinear PDE for this bond option model. Finally, the Allouba derivative is used to introduce the concept of stochastic Greeks.
Advisors/Committee Members: Allouba, Hassan.
Subjects: Mathematics
Keywords: ALLOUBA; dWt; ds; FT
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9.
Garcia, Francisco Javier.
THREE NON-LINEAR PROBLEMS ON NORMED SPACES.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2007, Kent State University
► In this dissertation, we will study the following three non-linear problems: 1.…
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▼ In this dissertation, we will study the following three non-linear problems: 1. The lineability problem for functionals. 2. The minimum-norm problem for translations. 3. The Banach-Mazur conjecture for rotations. As far as we know, all of them are currently open, and we believe that any approach to their solutions will constitute a work of great interest to the mathematical community. In this dissertation, we obtain progresses that lead to partial solutions of these problems.
Advisors/Committee Members: Aron, Richard Martin.
Subjects: Mathematics
Keywords: lineability, separable quotient, minimum norm, norm-attaining functional, transitivity, Banach-Mazur
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10.
Haiduc, Florian.
ALTERNATIVE APPROACHES TO CLASSICAL ELEMENTARY AND ELLIPTIC FUNCTIONS.
Degree: MS, College of Arts and Sciences / Department of Mathematical Science, 2007, Kent State University
► A careful examination of most calculus textbooks reveals a gap in the…
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▼ A careful examination of most calculus textbooks reveals a gap in the foundational treatment of the trigonometric functions; for pedagogical reasons, they rely on geometric axioms and intuition. By contrast, in a later course in real variables, only the axioms for the real numbers are allowed in the development of the trigonometric functions, although this means that higher structures (such as power series or the Fundamental Theorem of Calculus) must first be developed. In the first part of this thesis we examine five of the most common alternative definitions of the classic elementary functions (radical/power, exponential/logarithmic, and trigonometric/inverse-trigonometric functions). In particular we show how they may be developed using recurrence relations and only the basic axioms for the real numbers, and we also discuss the more commonly seen alternative developments using power series, inverse of integrals, functional equations, and differential equations. The primary idea is to show that, by starting from any of these definitions, each of the main properties of a function may be deduced. To illustrate the extension of these ideas more exotic functions (so-called special functions), in the second part of the thesis we examine various historial approaches to elliptic functions due to Abel, Gauss, Jacobi, and Weierstrass.
Advisors/Committee Members: Davidson, Morley.
Keywords: Elementary Functions; Elliptical Functions
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11.
Hanchin, Terence G.
On Sylvester's Theorem.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2010, Kent State University
► Variation-diminishing convultion transforms have proven useful in a variety of areas of…
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▼ Variation-diminishing convultion transforms have proven useful in a variety of areas of mathematics. A theorem of J.J. Sylvester on sums of shifted monomials, when extended to the context of convolution on the real line, shows that monomial kernels exhibit properties that are nearly variation-diminishing. We call these related properties even variation-diminishing and odd variation-diminishing. We demonstrate methods for generating further examples of even and odd variation-diminishing kernels, and ultimately provide a characterization of such kernels in terms of their translation determinants. We also show how the even and odd variation-diminishing properties of a particular class of kernels lead to the fact that convolution on the circle with certain generalized de la Vallée Poussin polynomials is a cyclic variation-diminishing linear transformation.
Advisors/Committee Members: Cavaretta, Dr. Alfred.
Subjects: Mathematics
Keywords: variation diminishing; convolution; de la vallee poussin
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12.
Hearn, Tristan A.
Numerical Methods For Ill-Posed Problems With Applications.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2012, Kent State University
► Several new methods for object detection, denoising, and deblurring of digital images…
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▼ Several new methods for object detection, denoising, and deblurring of digital images are presented. Among these are a new method for fast computation of convolution operations, a new framework for denoising via adaptive thresholding of wavelet coefficients based upon high order statistics of the residual image, and an extension of a non-iterative blind deconvolution algorithm to non-periodic boundary conditions. Each presented method is applicable to real-world problems, substantiated through extensive numerical experimentation.
Advisors/Committee Members: Reichel, Lothar.
Subjects: Applied Mathematics
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13.
Hongcheng, Li.
Multivariate Extensions of CUSUM Procedure.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2007, Kent State University
► In quality control, to use the recent history data of the process,…
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▼ In quality control, to use the recent history data of the process, Page(1954) introduced the CUSUMprocedure in the univariate case. It has been proved(Moustakides,1986)that (when the process is out of control) the CUSUMprocedure has the smallest expected run length among all procedures with the same in-control ARL. In this dissertation, we investigate the multivariate extension of Page's CUSUMprocedure. The expectation and the variance of the run length for various multivariate distributions are studied. Both analytical and simulation results of the ARLand variance are given. The exact expression for the ARLof a trinomial model for any decision intervals are given. Computer programs computing the ARLand variance for any given decision intervals are also given.
Advisors/Committee Members: Khan, Mohammad Kazim.
Subjects: Mathematics
Keywords: ARL, CUSUM, quality control, trinomial model, variance
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14.
Ignatyev, Oleksiy.
The Compact Support Property for Hyperbolic SPDEs: Two Contrasting Equations.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2008, Kent State University
► In this Dissertation we investigate the compact support property of the solutions…
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▼ In this Dissertation we investigate the compact support property of the solutions of two hyperbolic stochastic partial differential equations (SPDEs) whose initial condition function is deterministic and compactly supported. First, we consider the hyperbolic semi-SPDE treated by Allouba and Goodman and others in Financial mathematics modelling. This is an SPDE in both time and space in which all the derivatives (including in the noise) are only with respect to the time parameter, and hence the name semi-SPDE. It turns out that, under appropriate conditions on the diffusion coefficient, the semi-SPDE preserves the compact support property. I.e., starting from a compactly supported initial solution u0(x), the solution u(t,x) is compactly supported in x for all times t>0. Second, we consider a rotated wave SPDE in time-space considered by Allouba. Our approach here is to use the Allouba stochastic differential-difference equations (SDDE) approach. In this approach, we start by discretizing space, leaving time continuous, thereby obtaining a simpler version of the SPDE under question. We then resolve the question for the SDDE (or SPDE on the spatial lattice) and then use a limiting argument – as the mesh size of the spatial lattice goes to zero – to transfer regularity results to the associated SPDE. We also prove a noncompact support result for the SPDE. It turns out that in the rotated wave SPDE case, the compact support property is not preserved with positive probability. The contrast between the two SPDEs’ behaviors is due to the extra differentiation in space in the second SPDE which plays a crucial role in the behavior of solutions.
Advisors/Committee Members: Allouba, Prof. Hassan.
Subjects: Mathematics
Keywords: Stochastic Partial Differential Equations; Compact Support Property
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15.
Kiteu, Marco M.
Orbits of operators on Hilbert space and some classes of Banach spaces.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2012, Kent State University
► We have studied diagonal operators on Hilbert space and some classes of…
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▼ We have studied diagonal operators on Hilbert space and some classes of Banach spaces.On the Hilbert space l2 we give the necessary and sufficient conditions for the diagonal operators to have hyperful orbits, that is,orbits where every subsequence spans the whole space. A consequence of this characterization is that for diagonal operators, either all cyclic vectors have hyperful orbits or none of the cyclic vectors has a hyperful orbit. Then, we extend some of these results to the spaces c0, C(0; 1), and L2[0; 1].On L2[0; 1] we consider a Multiplication Operator in place of a Diagonal Operator. We also study the case with complex scalars, we show that on every separable Banach space there are operators with hyperful orbits.
Advisors/Committee Members: Enflo, Dr.Per.
Subjects: Mathematics
Keywords: Hyperful orbits
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16.
Kolomiyets, Yuriy V.
Asymptotic Behavior of Randomly Perturbed Dynamical Systems.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2006, Kent State University
► We consider systems of random differential equations. The coefficients of the equations…
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▼ We consider systems of random differential equations. The coefficients of the equations depend on a small parameter. The first equation, “slow” component, Ordinary Differential Equation (ODE), has unbounded highly oscillating in space variable coefficients and random disturbances, which are described by the second equation, “fast” component, with periodic coefficients. Sufficient conditions for weak convergence, as the small parameter goes to zero of the solutions of the “slow” components to the certain random process, are proved. The Classical Diffusion Approximation Theorem (DAT) says that the drift coefficient, of the approximated Stochastic Differential Equation (SDE), includes a derivative with respect to a space variable of the unbounded coefficients (see, e.g. the monograph of A. V. Skorokhod [13], and the bibliography). So, we cannot apply the classical DAT because of the highly oscillating character of dependency on the small parameter of the unbounded coefficient of the “slow” component. On other hand, we cannot apply the Limit Theorem for SDE’s (in the sense of G. L. Kulinich [8], N. I. Portenko [11], M. Freidlin, A. D. Wentzell [4], S. Ya. Makhno [9]), because the “slow” component is an ODE, and consequently has no nonzero diffusion coefficient (the presence of strongly positive diffusion coefficient is a necessary conditions for such kind of the theorems).
Advisors/Committee Members: Allouba, Hassan.
Subjects: Mathematics
Keywords: Diffusion approximation; random equations; asymptotic behavior; hidh oscillation
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17.
Korte, Robert A.
Inference in Power Series Distributions.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2012, Kent State University
► Historically research on inference in power series distributions has focused on maximum…
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▼ Historically research on inference in power series distributions has focused on maximum likelihood estimation technique due to its asymptotically optimal properties. This work discusses asymptotic results for the uniformly minimum variance unbiased (UMVU) estimation technique. We present its asymptotic distribution. The limiting mean deviation for the UMVU estimator and its rate of convergence are presented. Criteria for the existence of moments of the estimator and the limiting rth moment of the absolute deviating for 1 ≦ r < 2 are given. We also present applications to inference in censored power series distributions and a few links to some problems in approximation theory. test
Advisors/Committee Members: Khan, M. Kazim.
Subjects: Mathematics; Statistics
Keywords: Power Series Distributions; UMVU; UMVUE; Asymptotic Properties; Unbiased Estimation
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18.
Kracht, Darci L.
Applications of the Artin-Hasse Exponential Series and Its Generalizations to Finite Algebra Groups.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2011, Kent State University
► If F is a finite field of characteristic p and order q…
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▼ If F is a finite field of characteristic p and order q and J is a finite-dimensional nilpotent associative F-algebra, then we call the finite p-group G = 1+J an F-algebra group. A subgroup H of G is called an algebra subgroup if H = 1+A for some subalgebra A of J. A subgroup K of G is said to be strong if the order of the intersection of K with H is a power of q for all algebra subgroups H of G. If Jp = 0, then the ordinary exponential series can be used to show that normalizers of algebra subgroups are strong. If Jp is not equal to 0, then the exponential series does not make sense, but a generalization, the Artin-Hasse exponential series, is defined. We use the Artin-Hasse series to determine when normalizers of algebra subgroups are strong and when counter-examples exist. In addition, we give a description of strong subgroups in terms of stringent power series, that is, power series whose linear coefficient and constant term are both 1.
Advisors/Committee Members: Gagola, Stephen M.
Subjects: Mathematics
Keywords: algebra group; Artin-Hasse exponential series; strong subgroup
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19.
Kratky, Joseph J.
SERIES EXPANSION FOR SEMI-SPDES WITH REMARKS ON HYPERBOLIC SPDES ON THE LATTICE.
Degree: MS, College of Arts and Sciences / Department of Mathematical Science, 2011, Kent State University
► The goal in this Thesis is to give a Taylor-Ito expansion result…
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▼ The goal in this Thesis is to give a Taylor-Ito expansion result of the semi-SPDE (1.1.1) that has generated interest recently in finance. This equation is called a semi-SPDE because all of the partial derivatives are in terms of one variable, namely time. The Taylor-Ito expansion represented in this Thesis is in integral form. It generalizes both the integral form of Ito and the integral version of Taylor's formula. We will also note that this form can be reduced to represent the simpler SDE case (1.1.8). Furthermore, we will express a bound on the remainder of the expansion. Finally, we will discuss the more difficult SPDE case (1.1.12) on the lattice. More specifically, we will use Allouba's SDDE approach to SPDEs in which the SPDE of interest is spatially discretized in a specific way, and remark on some of the difficulties in naively translating our expansion from the semi-SPDE to this hyperbolic SPDE case. We also give brief remarks on some of Allouba's recent related work.
Advisors/Committee Members: Allouba, Hassan.
Subjects: Mathematics
Keywords: Kratky; Allouba; SPDE; SDDE; semi-SPDE; Taylor; Ito; SDE
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20.
Mackey, Benjamin James.
Convex Bodies with SO(2) Congruent Projections.
Degree: MS, College of Arts and Sciences / Department of Mathematical Science, 2012, Kent State University
► Suppose two convex bodies K and L in three dimensional Euclidean space…
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▼ Suppose two convex bodies K and L in three dimensional Euclidean space have the property that every orthogonal projection of K is SO(2) congruent to the corresponding orthogonal projection of L. The goal of this research is to prove that such bodies must themselves be congruent. After introducing several tools of convex geometry and tomography, we present a theorem which states that if the orthogonal projections of L can be translated into the corresponding projection of K, then K can be obtained by a translation of L. The rest of the thesis is spent attacking the issue of rotationally congruent projections. We present the proof found in Vladamir Golubyatnikov's book "Uniqueness Questions in Reconstruction of Multidimensional Objects from Tomography-Type Data" that, assuming no projection has a nontrivial SO(2) symmetry, the bodies K and L are either parallel or L can be obtained by reflecting K about some point. A deep lemma of Golubyatnikov's for which no symmetry assumption is necessary is also proven, as well as an analogous result about bodies which have SO(2) congruent sections rather than projections. Using the notion of polar duality, a new special case of the problem with no symmetry assumptions is considered, and it is proven the bodies K and L must coincide or be symmetric about the origin in this setting.
Advisors/Committee Members: Ryabogin, Dmitry.
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21.
Martin, David Royce.
Quadrature Approximation of Matrix Functions, with Applications.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2012, Kent State University
► This dissertation concerns the development of iterative methods for the approximation of…
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▼ This dissertation concerns the development of iterative methods for the approximation of functions of a large matrix, motivated by the development of efficient algorithms for the analysis of large-scale linear discrete ill-posed problems and complex networks. The connection between the Lanczos process and Gauss-type quadrature rules has been exploited for several decades in the development of iterative methods for bounding matrix functions; this work further elaborates on the application of Gauss-type quadrature to matrix computations. In addition, methods based on the singular value decomposition (SVD) and eigenvalue decomposition are presented. First we consider the minimization of a functional over a set of approximate solutions of a linear discrete ill-posed problem, in order to compute confidence intervals for components of the solution of such a problem. Our iterative method, using the link between Lanczos bidiagonalization and quadrature rules to bound certain matrix functions, greatly reduces large-scale work relative to available methods. Next we consider methods based on quadrature rules computed via block Lanczos algorithms. We derive anti-Gauss quadrature rules for the symmetric and nonsymmetric block Lanczos algorithms. These methods are applied to the computation of functions of an adjacency matrix. Moreover, novel quantities are introduced which characterize the nodes of a complex network, and which can be computed cheaply via our block methods. The remaining numerical methods presented rely on bounds based on the use of a partial SVD or partial eigenvalue decomposition of a large matrix. Such methods are competitive when solving many problems with a fixed matrix A but different data vectors. An alternative is provided to the aforementioned quadrature-based algorithm for computing confidence intervals for ill-posed problems, which is favorable when confidence intervals for many components are desired. Regarding complex networks, we use a partial eigenvalue decomposition to bound elements of the exponential of a symmetric adjacency matrix, and develop an algorithm which identifies important nodes in a complex network.
Advisors/Committee Members: Reichel, Lothar.
Subjects: Mathematics
Keywords: matrix; numerical analysis; quadrature; ill-posed; Lanczos; network
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22.
Meinke, Ashley Marie.
Fibonacci Numbers and Associated Matrices.
Degree: MS, College of Arts and Sciences / Department of Mathematical Science, 2011, Kent State University
► MEINKE, ASHLEY MARIE, M.S. AUGUST 2011 MATHEMATICS FIBONACCI NUMBERS AND ASSOCIATED MATRICES…
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▼ MEINKE, ASHLEY MARIE, M.S. AUGUST 2011 MATHEMATICS FIBONACCI NUMBERS AND ASSOCIATED MATRICES (43 pp.) Director of Thesis: Aloysius Bathi Kasturiarachi In this thesis, we approach the study of Fibonacci numbers using the theory of matrices. Fibonacci numbers are widely studied and the formulas to derive them are well-known. Such formulas include Binet's formula and Cassini's formula. We use the theory of diagonalizing a matrix and examine eigenvalues of certain 2 x 2 generating matrices to derive Binet-type formulas for the Lucas, generalized Fibonacci and generalized weighted Fibonacci numbers. We derive Cassini-type formulas for the Lucas, generalized Fibonacci and generalized weighted Fibonacci numbers by computing determinants of certain matrices. We extend these results to Tribonacci and generalized Tribonacci numbers using a similar 3 x 3 matrix approach. In all cases, we do a thorough analysis of the recursive sequences versus the derived Binet-type formulas. Throughout the thesis, we make notes of important historical information and examine Fibonacci's life and mathematical work. We also discuss the presence of these numbers even before Fibonacci's time by looking at the role that Indian mathematics played in the “so-called” Fibonacci numbers. In addition, we provide a sampling of some of the interesting properties that Fibonacci numbers possess.
Advisors/Committee Members: Kasturiarachi, Aloysius Bathi.
Subjects: Mathematics
Keywords: Fibonacci numbers; Lucas numbers; Generalized Fibonnaci numbers; Generalized weighted Fibonacci numbers; Tribonacci numbers; Generalized Tribonacci numbers; Matrix theory; Hemachandra
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23.
Milker, Joseph Alan.
Move-Count Means with Cancellation and Word Selection Problems in Rubik's Cube Solution Approaches.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2012, Kent State University
► In this dissertation we solve two fundamental, and until now only partially…
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▼ In this dissertation we solve two fundamental, and until now only partially treated, problems associated with multi-step solution approaches for the Rubik's Cube: the average effect of partial commutativity and cancellation of moves at the step interfaces, and the minimization with respect to certain group-theoretic constraints the number of required move sequences comprising the step look-up tables. The results may be adapted to other permutation groups to some extent, although portions of this work depend explicitly on the interplay between the geometry of the hexahedron and the structure of the Rubik's Cube group.
Advisors/Committee Members: Davidson, Morley.
Subjects: Mathematics
Keywords: Rubik's Cube; combinatorial group theory
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24.
Miller, Barbara L.
Grammar Efficiency of Parts-of-Speech Systems.
Degree: MS, College of Arts and Sciences / Department of Mathematical Science, 2011, Kent State University
► In Linguistic Typology, Parts-of-Speech system classification is a classification of languages which…
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▼ In Linguistic Typology, Parts-of-Speech system classification is a classification of languages which is based on syntactic properties, specifically the word classes present in a language and how these word classes function in syntactic slots, also known as propositional slots. The two-number Parts-of-Speech system classification introduced by Vulanović in 2008, which is a modification of a classification by Hengeveld et al. in 2004, is a mathematical approach to this linguistic classification. In this study the validity of the two-number classification was tested with regard to how the complexity and efficiency of the Parts-of-Speech systems relate to the two-number labeling system, l.n, where l is the number of word classes and n is the number of propositional slots present in the Parts-of-Speech system type. The minimum efficiency and maximum efficiency of the theoretically possible Parts-of-Speech system types were determined using an efficiency formula based on machine efficiency. Intervals of efficiency were determined for each system type and a linear and a non-linear equation were fit to this data. It was shown that a correlation exists between the two-number labeling and the efficiency of 17 theoretically possible Parts-of-Speech system types, 10 of which are attested by natural languages.
Advisors/Committee Members: Vulanović, Relja.
Subjects: Linguistics; Mathematics
Keywords: Parts-of-Speech; POS; Grammar Efficiency; mathematical linguistics
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25.
Mlaiki, Nabil M.
A central series associated with V(G).
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2011, Kent State University
► In this dissertation, we generalize Lewis’s result about a central series associated…
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▼ In this dissertation, we generalize Lewis’s result about a central series associated with the vanishing off subgroup. We write V1 = V(G) for the vanishing off subgroup of G, and Vi = [Vi-1,G] for the terms in this central series. Let the Gi be members of the lower central series of G: Lewis proved that there exists a positive integer n such that if V3 < G3, then |G : V1| = |G′ : V2|2 = p2n. Let D3/V3 = CG/V3(G′/V3). He also showed in [8] that if V3 < G3, then either |G : D3| = pn or D3 = V1. We show that if Vi < Gi for i ≥ 4; where Gi is the i-th term in the lower central series of G, then |Gi-1 : Vi-1| = |G : D3|: Also, we prove that this index depends on the size of the centralizer of G′. We say that Gk is H1, if for every subgroup N such that Vk ≤ N < Gk, we have Vk-1(G/N) = (Gk-1/N) ∩ Yk(G/N) where Yk/Vk = Z(G/Vk): We show that for every Gi > 1, we have Gi is H1. Also, we have a few results about Camina triples. Camina triples are a generalization of Camina pairs. In the last chapter of this dissertation, we have some future research questions, for example: What can we prove about the index of Vi-1 in Gi-1; without the hypothesis G′/Vi is abelian?
Advisors/Committee Members: Lewis, Mark.
Subjects: Mathematics
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26.
Neuman, Arthur James III.
Regularization Methods for Ill-posed Problems.
Degree: MS, College of Arts and Sciences / Department of Mathematical Science, 2010, Kent State University
► This thesis examines solution methods for large linear systems of equations with…
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▼ This thesis examines solution methods for large linear systems of equations with a matrix of ill-determined rank and an error-contaminated right-hand side. The numerical solution is delicate, because the matrix is very ill-conditioned and may be singular. To solve such systems, one replaces the system with one that is less sensitive to error a process known as regularization. This thesis focuses on the regularization method known as truncated iteration. A new algorithm is presented and compared to other existing methods.
Advisors/Committee Members: Reichel, Lothar.
Subjects: Mathematics
Keywords: Regularization methods; iterative methods; range restricted GMRES; RRGMRES; krylov subspace method
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27.
Richards, Gregory P.
Macroscopic Modeling of the Smectic-CG Phase Formed By Bent-Core Liquid Crystals.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2010, Kent State University
► Equilibrium states of liquid crystalline materials are commonly found via minimizationof free…
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▼ Equilibrium states of liquid crystalline materials are commonly found via minimizationof free energy functionals, a key part of which is the free energy density due to distortional elasticity. The most famous of these elastic energy densities, used when studying nematic liquid crystal behavior, is the Frank-Oseen model. The proper method to develop an elastic energy density for each liquid crystal phase has not yet been standardized. Examples of the various techniques used can be seen when comparing the development of the smectic-C elastic energy density of the Orsay Group to that found in the book by Stewart, for example. We present here a unifying theory for the development of an elastic energy density for phenomenological macroscopic liquid crystal models. The developed technique is then used to build an elastic energy density for the smectic-CG phase formed by bent-core molecules. A unique characteristic of the smectic-CG elastic energy density, the existence of a nonconstant term that is zeroth order in the gradient tensors of the orientational vector fields, is discussed. A method to quantify this unique term, via lattice packing calculations, is proposed. Finally we end by applying our findings to model natural forming fibers of bentcore molecules.
Advisors/Committee Members: Gartland, Eugene C.
Subjects: Materials science; Mathematics
Keywords: liquid crystals; elastic energy density; fiber modeling; smectic-CG
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28.
Ryabtseva, Elena.
Analysis of a Magnetic-Field-Induced Periodic Instability in a Liquid Crystal Film.
Degree: MS, College of Arts and Sciences / Department of Mathematical Science, 2009, Kent State University
► The orientational properties of liquid crystal films are governed by three competing…
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▼ The orientational properties of liquid crystal films are governed by three competing influences: elasticity, boundary conditions, and external fields (electric or magnetic). A Freedericksz transition is a basic instability in which an abrupt change in orientational order is triggered as the strength of the external field increases over a critical threshold. The nature of the Freedericksz transition depends on the type of the boundary conditions and the orientation of the external field. For one such transition, the so called “bend Freedericksz geometry,” it was observed in experiments many years ago that a secondary transition to a spatially periodic structure (a so-called “stripe phase”) can take place at a higher threshold. Subsequent two-dimensional numerical modeling of the particular branch revealed that it reconnected to the uniformly distorted branch at a still higher threshold. In this thesis, we use a combination of analytical two-dimensional and numerical one-dimensional techniques to determine the onset of the periodic instability as well as the point of reconnection to the primary branch. Tools required are calculus of variations, nonlinear differential equations and first integrals, stability theory and generalized algebraic eigenvalue problems.
Advisors/Committee Members: Gartland, Dr. Eugene C.
Subjects: Mathematics
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29.
Sbeih, Reema.
NON-LINEAR MAPS BETWEEN SUBSETS OF BANACH SPACES.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2009, Kent State University
► There is an extensive literature on linear maps and operators on Banach…
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▼ There is an extensive literature on linear maps and operators on Banach spaces. However, a corresponding non-linear theory is still in its beginning. We study non-linear maps between subsets of the classical Banach spaces and give estimates for the Banach-Mazur Lipschitz norm for some of the natural non-linear maps between unit spheres of these spaces. We show that the Banach-Mazur norm of these maps is much larger than the corresponding linear norms. We also study projections onto unit spheres of Banach spaces, we show that the scaling down projection is the best projection in L1(0,1), and we give estimates for the Lp-spaces, p>1.
Advisors/Committee Members: Enflo, Per.
Subjects: Mathematics
Keywords: d2; SCALING DOWN MAPS; ln; d1; b2; scaling down projection
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30.
Shyshkov, Andriy.
NUMERICAL SOLUTION OF ILL-POSED PROBLEMS.
Degree: PhD, College of Arts and Sciences / Department of Mathematical Science, 2009, Kent State University
► In ill-posed problems, small changes in the data can cause arbitrarily large…
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▼ In ill-posed problems, small changes in the data can cause arbitrarily large changes in the solutions. Many efficient methods have been proposed in order to remove this type of difficulties. In this work existent methods are reviewed and also several new developments are presented.
Advisors/Committee Members: Reichel, Lothar.
Subjects: Mathematics
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