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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 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 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 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 Motivic cohomology of fat points in milnor range(Deutsche Mathematiker-Vereinigung (DMV), 2018) Park, Jinhyun; Department of Mathematics; Ünver, Sinan; Faculty Member; Department of Mathematics; College of Sciences; 177871We introduce a new algebraic-cycle model for the motivic cohomology theory of truncated polynomials k[t]/(t(m)) in one variable. This approach uses ideas from the deformation theory and non-archimedean analysis, and is distinct from the approaches via cycles with modulus. We prove that the groups in the Milnor range give the Milnor K-groups of k[t]I/(t(m)), when the base field is of characteristic 0. Its relative part is the sum of the absolute Kahler differential forms.Publication Open Access The intersection problem for PBD(5*,3)s(Elsevier, 2008) Department of Mathematics; Küçükçifçi, Selda; Faculty Member; Department of Mathematics; College of Sciences; 105252For every v equivalent to 5 (mod 6) there exists a pairwise balanced design (PBD) of order v with exactly one block of size 5 and the rest of size 3. We will refer to such a PBD as a PBD(5*, 3). A flower in a PBD(5*, 3) is the set of all blocks containing a given point. If (S, B) is a PBD(5*, 3) and F is a flower, we will write F* to indicate that F contains the block of size 5. The intersection problem for PBD(5*, 3)s is the determination of all pairs (v, k) such that there exists a pair of PBD(5*, 3)s (S, B-1) and (S, B-2) of order v containing the same block b of size 5 such that vertical bar(B-1\b) boolean AND (B-2\b)vertical bar = k. The flower intersection problem for PBD(5*, 3)s is the determination of all pairs (v, k) such that there exists a pair of PBD(5*, 3)s (S, B-1) and (S, B-2) of order v having a common flower F* such that vertical bar(B-1\F*) boolean AND (B-2\F*)vertical bar = k. In this paper we give a complete solution of both problems.Publication Open Access Optimal obstacle placement with disambiguations(Institute of Mathematical Statistics (IMS), 2012) Aksakalli, Vural; Department of Mathematics; Ceyhan, Elvan; Undergraduate Student; Faculty Member; Department of Mathematics; College of SciencesWe introduce the optimal obstacle placement with disambiguations problem wherein the goal is to place true obstacles in an environment cluttered with false obstacles so as to maximize the total traversal length of a navigating agent (NAVA). Prior to the traversal, the NAVA is given location information and probabilistic estimates of each disk-shaped hindrance (hereinafter referred to as disk) being a true obstacle. The NAVA can disambiguate a disk's status only when situated on its boundary. There exists an obstacle placing agent (OPA) that locates obstacles prior to the NAVA's traversal. The goal of the OPA is to place true obstacles in between the clutter in such a way that the NAVA's traversal length is maximized in a game-theoretic sense. We assume the OPA knows the clutter spatial distribution type, but not the exact locations of clutter disks. We analyze the traversal length using repeated measures analysis of variance for various obstacle number, obstacle placing scheme and clutter spatial distribution type combinations in order to identify the optimal combination. Our results indicate that as the clutter becomes more regular (clustered), the NAVA's traversal length gets longer (shorter). On the other hand, the traversal length tends to follow a concave-down trend as the number of obstacles increases. We also provide a case study on a real-world maritime minefield data set.Publication Open Access Exact solvability of stochastic differential equations driven by finite activity levy processes(Multidisciplinary Digital Publishing Institute (MDPI), 2012) Ünal G.; Department of Mathematics; İyigünler, İsmail; Çağlar, Mine; Faculty Member; Department of Mathematics; Graduate School of Sciences and Engineering; College of Sciences; N/A; 105131We consider linearizing transformations of the one-dimensional nonlinear stochastic differential equations driven by Wiener and compound Poisson processes, namely finite activity Levy processes. We present linearizability criteria and derive the required transformations. We use a stochastic integrating factor method to solve the linearized equations and provide closed-form solutions. We apply our method to a number ofstochastic differential equations including Cox-Ingersoll-Ross short-term interest rate model, log-mean reverting asset pricing model and geometric Ornstein- Uhlenbeck equation all with additional jump terms. We use their analytical solutions to illustrate the accuracy of the numerical approximations obtained from Euler and Maghsoodi discretization schemes. The means of the solutions are estimated through Monte Carlo method.Publication Open Access A review on computational modeling tools for MOF-based mixed matrix membranes(Multidisciplinary Digital Publishing Institute (MDPI), 2019) Department of Chemical and Biological Engineering; Keskin, Seda; Faculty Member; Department of Chemical and Biological Engineering; College of Engineering; 40548Computational modeling of membrane materials is a rapidly growing field to investigate the properties of membrane materials beyond the limits of experimental techniques and to complement the experimental membrane studies by providing insights at the atomic-level. In this study, we first reviewed the fundamental approaches employed to describe the gas permeability/selectivity trade-off of polymer membranes and then addressed the great promise of mixed matrix membranes (MMMs) to overcome this trade-off. We then reviewed the current approaches for predicting the gas permeation through MMMs and specifically focused on MMMs composed of metal organic frameworks (MOFs). Computational tools such as atomically-detailed molecular simulations that can predict the gas separation performances of MOF-based MMMs prior to experimental investigation have been reviewed and the new computational methods that can provide information about the compatibility between the MOF and the polymer of the MMM have been discussed. We finally addressed the opportunities and challenges of using computational studies to analyze the barriers that must be overcome to advance the application of MOF-based membranes.