UMD Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/3
New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a given thesis/dissertation in DRUM.
More information is available at Theses and Dissertations at University of Maryland Libraries.
Browse
10 results
Search Results
Item Towards a Classification of Almost Complex and Spin^h Manifolds(2024) Mills, Keith; Rosenberg, Jonathan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We show that all homotopy CP^ns, smooth closed manifolds with the oriented homotopy type of CP^n, admit almost complex structures for 3 ≤ n ≤ 6, and classify these structures by their Chern classes for n=4, 6. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy CP^4s. We also show that all homotopy RP^(2k+1)s admit stably almost complex structures. Spin^h manifolds are the quaternionic analogue to spin^c manifolds. At the prime 2 we compute the spin^h bordism groups by proving a structure theorem for the cohomology of the spin^h bordism spectrum MSpin^h as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of MSpin^h as a wedge sum of familiar spectra. We also compute the decomposition of H^*(MSpin^h; Z/2Z) explicitly in degrees up through 30 via a counting process.Item A Pedagogical Approach to Ramsey Multiplicity(2023) Brady, Robert; Gasarch, William; Computer Science; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)It is well known that for all 2-colorings of the edges of $K_6$ there is amonochromatic triangle. Less well known is that there are two monochromatic triangles. More generally, for all 2-colorings of the edges of $K_n$ there are roughly $\ge n^3/24$ monochromatic triangles. Another way to state this is that the density of monochromatic triangles is at least $1/4$. The Ramsey Multiplicity of $k$ is (asymptotically) the greatest $\alpha$ such that for every coloring of $K_n$ the density of monochromatic $K_k$'s is $\alpha$. This concept has been studied for many years. We survey the area and provide proofs that are more complete, more motivated, and using modern notation.Item On the Difficulty of Breaking Substitution Ciphers(2021) Wertheimer, Phil; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We analyze different methods of attacking substitution ciphers using $m$-gram frequency analysis. For $m=1$ this amounts to studying symbol counts in random strings, and for $m\geq 2$ we use the Markov Chain Monte Carlo method introduced by Diaconis \cite{mcmcr}. Our study includes both numerical simulations of the English language and theoretical analysis of random alphabets, which are probabilistic constructions for studying the distribution of $m$-grams in random strings. We present several results in the direction of explaining why the $2$-gram method performs the best in breaking the substitution ciphers.Item Cluster Algebras and Polylogarithm Relations(2021) Greenberg, Zachary; Zickert, Christian; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We seek to illuminate the connection between multiple polylogarithm relations and cluster algebras in two ways. First, we give a uniform description of the cluster modular group of affine and doubly extended cluster algebras. This will be critical for the future work of extracting polylogarithm relations from infinite type cluster algebras. Second, we introduce a differential one form, ωn, associated to each multiple polylogarithm, which can be used to compute multiple polylogarithm relations. This form satisfies a clean recurrence relation, mirroring the inductive definition of multiple polylogarithms. We are able to use this recurrence to find several families of “small” polylogarithm relations that hold in any weight. Finally for small values of n, we extract polylogarithm relations from type An and Dn cluster algebras.Item Class groups of characteristic-p function field analogues of Q(n^(1/p))(2021) Reich, Steven; Washington, Lawrence; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In the theory of cyclotomic function fields, the Carlitz module $\Lambda_M$ associated to a polynomial $M$ in a global function field of characteristic $p$ provides a strong analogy to the roots of unity $\mu_p$ in a number field. In this work, we consider a natural extension of this theory to give a compatible analogue of the $p$-th root of an integer $n$. The most fundamental case, and the one which most closely mimics the number field situation, is when the Carlitz module is defined by a linear polynomial (which can be assumed to be $T$) in $k={\mathbb F}_q(T)$. The Carlitz module $\Lambda_T$ generates a degree-$(q-1)$ extension $k(\Lambda_T)$ which shares many properties with the field ${\mathbb Q}(\mu_p)$, where $\mu_p$ is the module of $p$-th roots of unity. To form the analogue of ${\mathbb Q}(\sqrt[p]{n})$, we define a degree-$q$ extension $F/k$ associated to a polynomial $P(T) \in k$, for which the normal closure is formed by adjoining $\Lambda_T$. In the introduction, we describe in detail the parallels between this construction and that in the number field setting. We then compute the class number $h_F$ for a large number of such fields. The remainder of the work is concerned with proving results about the class groups and class numbers of this family of fields. These are:\begin{itemize} \item a formula relating the class number of $F$ to that of its normal closure, along with a theorem about the structure of the class group of the normal closure \item a formula relating the class number of a compositum of such $F$ to the class numbers of the constituent fields \item conditions on $P(T)$ for when the characteristic, $p$, of $F$ divides its class number, along with bounds on the rank of the $p$-part of the class group. \end{itemize}Item Generalizations of Schottky groups(2017) Burelle, Jean-Philippe; Goldman, William M; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Schottky groups are classical examples of free groups acting properly discontinuously on the complex projective line. We discuss two different applications of similar constructions. The first gives examples of 3-dimensional Lorentzian Kleinian groups which act properly discontinuously on an open dense subset of the Einstein universe. The second gives a large class of examples of free subgroups of automorphisms groups of partially cyclically ordered spaces. We show that for a certain cyclic order on the Shilov boundary of a Hermitian symmetric space, this construction corresponds exactly to representations of fundamental groups of surfaces with boundary which have maximal Toledo invariant.Item Algorithms and Generalizations for the Lovasz Local Lemma(2015) Harris, David; Srinivasan, Aravind; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The Lovasz Local Lemma (LLL) is a cornerstone principle of the probabilistic method for combinatorics. This shows that one can avoid a large of set of “bad-events” (forbidden configurations of variables), provided the local conditions are satisfied. The original probabilistic formulation of this principle did not give efficient algorithms. A breakthrough result of Moser & Tardos led to an framework based on resampling variables which turns nearly all applications of the LLL into efficient algorithms. We extend and generalize the algorithm of Moser & Tardos in a variety of ways. We show tighter bounds on the complexity of the Moser-Tardos algorithm, particularly its parallel form. We also give a new, faster parallel algorithm for the LLL. We show that in some cases, the Moser-Tardos algorithm can converge even thoughthe LLL itself does not apply; we give a new criterion (comparable to the LLL) for determining when this occurs. This leads to improved bounds for k-SAT and hypergraph coloring among other applications. We describe an extension of the Moser-Tardos algorithm based on partial resampling, and use this to obtain better bounds for problems involving sums of independent random variables, such as column-sparse packing and packet-routing. We describe a variant of the partial resampling algorithm specialized to approximating column-sparse covering integer programs, a generalization of set-cover. We also give hardness reductions and integrality gaps, showing that our partial resampling based algorithm obtains nearly optimal approximation factors. We give a variant of the Moser-Tardos algorithm for random permutations, one of the few cases of the LLL not covered by the original algorithm of Moser & Tardos. We use this to develop the first constructive algorithms for Latin transversals and hypergraph packing, including parallel algorithms. We analyze the distribution of variables induced by the Moser-Tardos algorithm. We show it has a random-like structure, which can be used to accelerate the Moser-Tardos algorithm itself as well as to cover problems such as MAX k-SAT in which we only partially avoid bad-events.Item Investigations of Highly Irregular Primes and Associated Ray Class Fields(2014) Stern, Morgan Benjamin; Washington, Lawrence; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We investigate properties of the class number of certain ray class fields of prime conductor lying above imaginary quadratic fields. While most previous work in this area restricted to the case of imaginary quadratic fields of class number 1, we deal almost exclusively with class number 2. Our main results include finding 5 counterexamples to a generalization of the famous conjecture of Vandiver that the class number of the pth real cyclotomic field is never divisible by p. We give these counterexamples the name highly irregular primes due to the fact that any counterexample of classical Vandiver is an irregular prime. In addition we explore whether several consequences of Vandiver's conjecture still hold for these highly irregular primes, including the cyclicity of certain class groups.Item Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps(2014) Contreras, Fabian Elias; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation consists of two parts. In the first part, we consider a piecewise expanding unimodal map (PEUM) $f:[0,1] \to [0,1]$ with $\mu=\rho dx$ the (unique) SRB measure associated to it and we show that $\rho$ has a Taylor expansion in the Whitney sense. Moreover, we prove that the set of points where $\rho$ is not differentiable is uncountable and has Hausdorff dimension equal to zero. In the second part, we consider a family $f_t:[0,1] \to [0,1]$ of PEUMs with $\mu_t$ the correspoding SRB measure and we present a new proof of \cite{BS1} when considering the observables in $C^1[0,1]$ . That is, $\Gamma(t)=\int \phi d\mu_t$ is differentiable at $t=0$, with $\phi \in C^1[0,1]$, when assuming $J(c)=\sum_{k=0}^{\infty} \frac{v(f^k(c))}{Df^k(f(c))}$ is zero. Furthermore, we show that in fact $\Gamma(t)$ is never differentiable when $J(c)$ is not zero and we give the exact modulus of continuity.Item Performance enhancement of heat exchanger coolers with evaporative cooling(2014) Popli, Sahil; Radermacher, Reinhard; Mechanical Engineering; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Air or water cooled heat exchangers (HX) are typically utilized as condensers or coolers for air-conditioning, refrigeration or process cooling applications in both commercial and industrial sector. However, air cooled heat exchanger performance degrades considerably with rise in ambient air temperature and water cooled coolers require considerable pumping power, a cooling tower and may consume a significant amount of water which may come from fresh water sources. Evaporative cooling offers a unique solution to this problem, where a small amount of wetting water evaporates on HX surface to boost performance in high ambient air temperature conditions. In this study, several evaporative cooling technologies were applied to three wavy-fin HXs to quantify capacity enhancement ratio (CER) and air-side pressure drop penalty ratio (PRΔPa) compared to respective dry case baseline values. Effect of varying wetting water flow rate, air velocity, fin spacing, hydrophilic coatings, spray orientation and inlet air temperature and relative humidity was investigated on hybrid heat exchanger performance. Several new performance comparison parameters were defined to compare different evaporative cooling approaches. Deluge cooling achieved overall highest CER but at a PRΔPa that was similar in magnitude to the CER. This limitation was found to be inherent to the nature of wetting water distribution method itself. Although front spray cooling tests indicated PRΔPa~1, front spray evaporative cooling technology was found to have up to 23-75 % lower CER at 60-100% lower PRΔPa compared to deluge cooling. In order to understand the wetting behavior a novel visualization method was proposed and implemented, which consisted of borescope assisted flow mapping of water distribution within the HX core as a function of air velocities and wetting water flow rates. It was found that up to 85% of HX volume remained dry during front spray cooling which accounted for lower capacity enhancement and deluge cooling forms non-uniform and thick water film which causes bridging and increased PRΔPa, A larger component level testing with HX size similar to commercial units allowed to identify constraints of different evaporative cooling methods, which would not be possible if tests were performed at a smaller segment or fin level. A novel spray cooling technology utilizing internal jet spray cooling within HX volume was both proposed and implemented and a provisional patent # 61/782,825 was obtained. Compared to front spray cooling at a given spray rate, internal spray cooling could either achieve up to 35% higher HX cooler capacity, or obtain same HX cooler capacity at approximately three times lower air-side pressure drop. Alternatively, at same air-side pressure drop wetting water savings of up to 68-97% are achieved. Internal spraying combines advantages of conventional technologies and overcomes the drawbacks, by getting CER of approx. 3.8, without film carryover and at PRΔPa=1, while getting maximum wetting uniformity. Intermittent cooling combined with internal spraying could reduce water consumption as evaporative cooling sustains though the brief period when spray is turned off. Thus, potential for significant energy and water savings, targeted cooling, and retrofit design offers significant commercialization opportunity for future hybrid evaporative coolers. Discussions are underway for the inclusion of this technology into product line up of a leading HX manufacturing company.