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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6
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Publication Open Access A stochastic representation for mean curvature type geometric flows(Institute of Mathematical Statistics (IMS), 2003) Touzi, N.; Department of Mathematics; Soner, Halil Mete; Faculty Member; Department of Mathematics; College of Administrative Sciences and EconomicsA smooth solution {Gamma(t)}(tis an element of[0,T]) subset of R-d of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem. In this control problem the controller tries to steer the state process into a given deterministic set T with probability one. The reachability set, V(t), for the target problem is the set of all initial data x from which the state process X-X(v)(t) is an element of T for some control process v. This representation is proved by studying the squared distance function to Gamma(t). For the codimension k mean curvature flow, the state process is dX(t) = root2P dW(t), where W(t) is a d-dimensional Brownian motion, and the control P is any projection matrix onto a (d - k)-dimensional plane. Smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given.Publication Open Access Conditional law and occupation times of two-sided sticky Brownian motion(Elsevier, 2020) Department of Mathematics; Çağlar, Mine; Can, Buğra; Faculty Member; Department of Mathematics; College of Sciences; 105131; N/ASticky Brownian motion on the real line can be obtained as a weak solution of a system of stochastic differential equations. We find the conditional distribution of the process given the driving Brownian motion, both at an independent exponential time and at a fixed time t>0. As a classical problem, we find the distribution of the occupation times of a half-line, and at 0, which is the sticky point for the process.Publication Open Access Stickelberger elements and Kolyvagin systems(Duke University Press (DUP), 2011) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of SciencesIn this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is r >1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.Publication Open Access Perturbative analysis of spectral singularities and their optical realizations(American Physical Society (APS), 2012) Department of Mathematics; Mostafazadeh, Ali; Rostamzadeh, Saber; Faculty Member; Department of Mathematics; College of Sciences; 4231; N/AWe develop a perturbative method of computing spectral singularities of a Schrodinger operator defined by a general complex potential that vanishes outside a closed interval. These can be realized as zero-width resonances in optical gain media and correspond to a lasing effect that occurs at the threshold gain. Their time-reversed copies yield coherent perfect absorption of light that is also known as antilasing. We use our general results to establish the exactness of the nth-order perturbation theory for an arbitrary complex potential consisting of n delta functions, obtain an exact expression for the transfer matrix of these potentials, and examine spectral singularities of complex barrier potentials of arbitrary shape. In the context of optical spectral singularities, these correspond to inhomogeneous gain media.Publication Open Access Density of a random interval catch digraph family and its use for testing uniformity(National Statistical Institute (NSI), 2016) Department of Mathematics; Ceyhan, Elvan; Undergraduate Student; Faculty Member; Department of Mathematics; College of SciencesWe consider (arc) density of a parameterized interval catch digraph (ICD) family with random vertices residing on the real line. The ICDs are random digraphs where randomness lies in the vertices and are defined with two parameters, a centrality parameter and an expansion parameter, hence they will be referred as central similarity ICDs (CS-ICDs). We show that arc density of CS-ICDs is a U-statistic for vertices being from a wide family of distributions with support on the real line, and provide the asymptotic (normal) distribution for the (interiors of) entire ranges of centrality and expansion parameters for one dimensional uniform data. We also determine the optimal parameter values at which the rate of convergence (to normality) is fastest. We use arc density of CS-ICDs for testing uniformity of one dimensional data, and compare its performance with arc density of another ICD family and two other tests in literature (namely, Kolmogorov-Smirnov test and Neyman’s smooth test of uniformity) in terms of empirical size and power. We show that tests based on ICDs have better power performance for certain alternatives (that are symmetric around the middle of the support of the data).Publication Open Access Minimal number of singular fibers in a nonorientable Lefschetz fibration(Springer, 2022) Onaran, Sinem; Department of Mathematics; Özbağcı, Burak; Faculty Member; Department of Mathematics; College of Sciences; 29746We show that there exists an admissible nonorientable genus g Lefschetz fibration with only one singular fiber over a closed orientable surface of genus h if and only if g >= 4 and h >= 1.Publication Open Access On the distributions of sigma(N)/N and N/Phi(N)(Rocky Mountain Mathematics Consortium, 2013) Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803We prove that the distribution functions of sigma(n)/n and n/phi(n) both have super-exponential asymptotic decay when n ranges over certain subsets of integers, which, in particular, can be taken as the set of l-free integers not divisible by a thin subset of primes.Publication Open Access Examples of area-minimizing surfaces in 3-manifolds(Oxford University Press (OUP), 2012) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of SciencesIn this paper, we give some examples of area-minimizing surfaces to clarify some wellknown features of these surfaces in more general settings. The first example is about Meeks–Yau’s result on the embeddedness of the solution to the Plateau problem. We construct an example of a simple closed curve in R3 which lies in the boundary of a mean convex domain in R3, but the area-minimizing disk in R3 bounding this curve is not embedded. Our second example shows that White’s boundary decomposition theorem does not extend when the ambient space has nontrivial homology. Our last examples show that there are properly embedded absolutely area-minimizing surfaces in a mean convex 3-manifold M such that, while their boundaries are disjoint, they intersect each other nontrivially, unlike the area-minimizing disks case.Publication Open Access The 2nd symposium on multiscale, multiphase, multiphysics and turbulent flow simulations(American Institute of Physics (AIP) Publishing, 2018) Çelebi, Serdar; Department of Mathematics; Çağlar, Mine; Faculty Member; Department of Mathematics; College of Sciences; 105131Publication Open Access Application of stochastic flows to the sticky Brownian motion equation(Multidisciplinary Digital Publishing Institute (MDPI), 2017) Hajri, Hatem; Arnaudon, Marc; Department of Mathematics; Çağlar, Mine; Faculty Member; Department of Mathematics; College of Sciences; 105131We investigate the relationship between the levels of industry collaboration and entrepreneurial activities at universities and the employment choices of their science and engineering doctoral students. Using data from 176 U.S. universities over the period 1996-2005, we document that more interaction with industry at a university is typically associated with more of the university's doctoral students choosing industry employment. We also document a positive relationship between universities' licenses and startups and their graduates' post-doctoral study choices.