Publications with Fulltext
Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6
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Publication 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 Perfect broadband invisibility in isotropic media with gain and loss(The Optical Society (OSA) Publishing, 2017) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We offer a simple route to perfect omnidirectional invisibility in a spectral band of desired width. Our approach is based on the observation that in two dimensions a complex potential v(x; y) is invisible for incident plane waves with a wavenumber not exceeding a pre-assigned value a, provided that its Fourier transform with respect to y, which we denote by v (x; R-y), vanishes for R-y <= 2a. We can fulfill this condition for potentials modeling the permittivity profile of an optical slab. Such a slab is perfectly invisible for any transverse electric wave whose wavenumber is in the range [0; a]. Our results also apply to transverse magnetic waves propagating in a medium with a relative permittivity epsilon (x; y) that is a smooth bounded function with a positive real part.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 Nonlinear scattering and its transfer matrix formulation in one dimension(Springer, 2019) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231We present a systematic formulation of the scattering theory for nonlinear interactions in one dimension and develop a nonlinear generalization of the transfer matrix that has a composition property similar to that of its linear analog. We offer alternative characterizations of spectral singularities, unidirectional reflectionlessness and invisibility, and nonreciprocal transmission for nonlinear scattering systems, and examine the application of our general results in addressing the scattering problem for nonlinear single- and double-δ-function potentials.Publication Open Access Variational formulas and disorder regimes of random walks in random potentials(International Statistical Institute (ISI), 2017) Rassoul-Agha, Firas; Seppalainen, Timo; Department of Mathematics; Yılmaz, Atilla; Faculty Member; Department of Mathematics; College of Sciences; 26605We give two variational formulas (qVar1) and (qVar2) for the quenched free energy of a random walk in random potential (RWRP) when (i) the underlying walk is directed or undirected, (ii) the environment is stationary and ergodic, and (iii) the potential is allowed to depend on the next step of the walk which covers random walk in random environment (RWRE). In the directed i.i.d. case, we also give two variational formulas (aVar1) and (aVar2) for the annealed free energy of RWRP. These four formulas are the same except that they involve infima over different sets, and the first two are modified versions of a previously known variational formula (qVar0) for which we provide a short alternative proof. Then, we show that (qVar0) always has a minimizer, (aVar2) never has any minimizers unless the RWRP is an RWRE, and (aVar1) has a minimizer if and only if the RWRP is in the weak disorder regime. In the latter case, the minimizer of (aVar1) is unique and it is also the unique minimizer of (qVar1), but (qVar2) has no minimizers except for RWRE. In the case of strong disorder, we give a sufficient condition for the nonexistence of minimizers of (qVar1) and (qVar2) which is satisfied for the log-gamma directed polymer with a sufficiently small parameter. We end with a conjecture which implies that (qVar1) and (qVar2) have no minimizers under very strong disorder.Publication Open Access Approximation of stability radii for large-scale dissipative Hamiltonian systems(Springer, 2020) Aliyev, Nicat; Mehrmann, Volker; Department of Mathematics; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; 113760A linear time-invariant dissipative Hamiltonian (DH) system (x) over dot = (J-R)Qx, with a skew-Hermitian J, a Hermitian positive semidefinite R, and a Hermitian positive definite Q, is always Lyapunov stable and under further weak conditions even asymptotically stable. By exploiting the characterizations from Mehl et al. (SIAM J. Matrix Anal. Appl. 37(4), 1625-1654, 2016), we focus on the estimation of two stability radii for large-scale DH systems, one with respect to non-Hermitian perturbations of R in the form R + B Delta C-H for given matrices B, C, and another with respect to Hermitian perturbations in the form R + B Delta B-H, Delta = Delta(H). We propose subspace frameworks for both stability radii that converge at a superlinear rate in theory. The one for the non-Hermitian stability radius benefits from the DH structure-preserving model order reduction techniques, whereas for the Hermitian stability radius we derive subspaces yielding a Hermite interpolation property between the full and projected problems. With the proposed frameworks, we are able to estimate the two stability radii accurately and efficiently for large-scale systems which include a finite-element model of an industrial disk brake.Publication Open Access On the regularity of the solution map of the Euler-Poisson system(TÜBİTAK, 2019) Department of Mathematics; İnci, Hasan; Department of Mathematics; College of Sciences; 274184In this paper we consider the Euler-Poisson system (describing a plasma consisting of positive ions with a negligible temperature and massless electrons in thermodynamical equilibrium) on the Sobolev spaces H-s(R-3) , s > 5/2. Using a geometric approach we show that for any time T > 0 the corresponding solution map, (rho(0), u(0)) bar right arrow (rho(T), u(T)) , is nowhere locally uniformly continuous. On the other hand it turns out that the trajectories of the ions are analytic curves in R-3.Publication Open Access Censoring distances based on labeled cortical distance maps in cortical morphometry(Frontiers, 2013) Nishino, Tomoyuki; Alexopolous, Dimitrios; Todd, Richard D.; Botteron, Kelly N.; Miller, Michael I.; Ratnanather, J. Tilak; Department of Mathematics; Ceyhan, Elvan; Undergraduate Student; Faculty Member; Department of Mathematics; College of SciencesIt has been demonstrated that shape differences in cortical structures may be manifested in neuropsychiatric disorders. Such morphometric differences can be measured by labeled cortical distance mapping (LCDM) which characterizes the morphometry of the laminar cortical mantle of cortical structures. LCDM data consist of signed/labeled distances of gray matter (GM) voxels with respect to GM/white matter (VW) surface. Volumes and other summary measures for each subject and the pooled distances can help determine the morphometric differences between diagnostic groups, however they do not reveal all the morphometric information contained in LCDM distances. To extract more information from LCDM data, censoring of the pooled distances is introduced for each diagnostic group where the range of LCDM distances is partitioned at a fixed increment size; and at each censoring step, the distances not exceeding the censoring distance are kept. Censored LCDM distances inherit the advantages of the pooled distances but also provide information about the location of morphometric differences which cannot be obtained from the pooled distances. However, at each step, the censored distances aggregate, which might confound the results. The influence of data aggregation is investigated with an extensive Monte Carlo simulation analysis and it is demonstrated that this influence is negligible. As an illustrative example, GM of ventral medial prefrontal cortices (VMPFCs) of subjects with major depressive disorder (MDD), subjects at high risk (HR) of MDD, and healthy control (Ctrl) subjects are used. A significant reduction in laminar thickness of the VMPFC in MDD and HR subjects is observed compared to Ctrl subjects. Moreover, the GM LCDM distances (i.e., locations with respect to the GM/WM surface) for which these differences start to occur are determined. The methodology is also applicable to LCDM-based morphometric measures of other cortical structures affected by disease.Publication Open Access Number of Least area planes in gromov hyperbolic 3-spaces(American Mathematical Society (AMS), 2010) Department of Mathematics; Coşkunüzer, Barış; Faculty Member; Department of Mathematics; College of SciencesWe show that for a generic simple closed curve Γ in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exists a unique least area plane Σ in X such that ∂∞Σ = Γ. This result has interesting topological applications for constructions of canonical 2-dimensional objects in Gromov hyperbolic 3-manifolds.Publication Open Access A subspace method for large-scale eigenvalue optimization(Society for Industrial and Applied Mathematics (SIAM), 2018) Meerbergen, Karl; Michiels, Wim; Department of Mathematics; Kangal, Fatih; Mengi, Emre; Faculty Member; Department of Mathematics; College of Sciences; Graduate School of Sciences and Engineering; N/A; 113760We consider the minimization or maximization of the Jth largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi, Yildirim, and Kilic [SIAM T. Matrix Anal. Appl., 35, pp. 699-724, 2014]. This work addresses the setting when the matrix-valued function involved is very large. We describe subspace procedures that convert the original problem into a small-scale one by means of orthogonal projections and restrictions to certain subspaces, and that gradually expand these subspaces based on the optimal solutions of small-scale problems. Global convergence and superlinear rate-of-convergence results with respect to the dimensions of the subspaces are presented in the infinite dimensional setting, where the matrix-valued function is replaced by a compact operator depending on parameters. In practice, it suffices to solve eigenvalue optimization problems involving matrices with sizes on the scale of tens, instead of the original problem involving matrices with sizes on the scale of thousands.