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Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6
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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 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 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.Publication Open Access Nonlinear spectral singularities for confined nonlinearities(American Physical Society (APS), 2013) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We introduce a notion of spectral singularity that applies for a general class of nonlinear Schrodinger operators involving a confined nonlinearity. The presence of the nonlinearity does not break the parity-reflection symmetry of spectral singularities but makes them amplitude dependent. Nonlinear spectral singularities are, therefore, associated with a resonance effect that produces amplified waves with a specific amplitude-wavelength profile. We explore the consequences of this phenomenon for a complex delta-function potential that is subject to a general confined nonlinearity.Publication Open Access Embedding partial Latin squares in Latin squares with many mutually orthogonal mates(Elsevier, 2020) Donovan, Diane; Grannell, Mike; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; Department of Mathematics; College of Sciences; 27432In this paper it is shown that any partial Latin square of order n can be embedded in a Latin square of order at most 16n2 which has at least 2n mutually orthogonal mates. Further, for any t⩾2, it is shown that a pair of orthogonal partial Latin squares of order n can be embedded in a set of t mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to n. A consequence of the constructions is that, if N(n) denotes the size of the largest set of MOLS of order n, then N(n2)⩾N(n)+2. In particular, it follows that N(576)⩾9, improving the previously known lower bound N(576)⩾8.Publication Open Access Maximum drawdown and drawdown duration of spectrally negative Lévy processes decomposed at extremes(Springer Nature, 2021) Vardar-Acar, Ceren; Avram, Florin; Department of Mathematics; Çağlar, Mine; Faculty Member; Department of Mathematics; College of Sciences; 105131Path decomposition is performed to characterize the law of the pre-/post-supremum, post-infimum and the intermediate processes of a spectrally negative Lévy process taken up to an independent exponential time T. As a result, mainly the distributions of the supremum of the post-infimum process and the maximum drawdown of the pre-/post-supremum, post-infimum processes and the intermediate processes are obtained together with the law of drawdown durations.Publication Open Access Pair correlation of sums of rationals with bounded height(De Gruyter, 2010) Xiong, Maosheng; Zaharescu, Alexandru; Department of Mathematics; Alkan, Emre; Faculty Member; Department of Mathematics; College of Sciences; 32803For each positive integer Q, let F(Q) denote the Farey sequence of order Q. We prove the existence of the pair correlation measure associated to the sum F(Q) + F(Q) modulo 1, as Q tends to infinity, and compute the corresponding limiting pair correlation function.Publication Open Access Symplectic and lagrangian surfaces in 4-Manifolds(Rocky Mountain Mathematics Consortium, 2008) Department of Mathematics; Etgü, Tolga; Faculty Member; Department of Mathematics; College of Sciences; 16206This is a brief summary of recent examples of isotopically different symplectic and Lagrangian surfaces representing a fixed homology class in a simply-connected symplectic 4-manifold.Publication Open Access Estimates of the coverage of parameter space by Latin hypercube and orthogonal array-based sampling(Elsevier, 2018) Donovan, D.; Burrage, K.; Burrage, P.; McCourt, T. A.; Thompson, B.; Department of Mathematics; Yazıcı, Emine Şule; Faculty Member; Department of Mathematics; College of Sciences; 27432In this paper we use counting arguments to prove that the expected percentage coverage of a d dimensional parameter space of size n when performing k trials with either Latin Hypercube sampling or Orthogonal Array-based Latin Hypercube sampling is the same. We then extend these results to an experimental design setting by projecting onto a t < d dimensional subspace. These results are confirmed by simulations. The theory presented has both theoretical and practical significance in modelling and simulation science when sampling over high dimensional spaces.