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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 Multidimensional wave packet dynamics within the fluid dynamical formulation of the Schrodinger equation(American Institute of Physics (AIP) Publishing, 1998) Rabitz, H.; Department of Mathematics; Aşkar, Attila; Dey, Bijoy K.; Faculty Member; Faculty Member; Department of Mathematics; College of Sciences; 178822; N/AThis paper explores the quantum fluid dynamical (QFD) representation of the time-dependent Schrodinger equation for the motion of a wave packet in a high dimensional space. A novel alternating direction technique is utilized to single our each of the many dimensions in the QFD equations. This technique is used to solve the continuity equation for the density and the equation for the convection of the flux for the quantum particle. The ability of the present scheme to efficiently and accurately describe the dynamics of a quantum particle is demonstrated in four dimensions where analytical results are known. We also apply the technique to the photodissociation of NOCl and NO2 where the systems are reduced to two coordinates by freezing the angular variable at its equilibrium value.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 Generalized adiabatic product expansion: a nonperturbative method of solving the time-dependent Schrodinger equation(American Institute of Physics (AIP) Publishing, 1999) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field ℰ→(t) through the Stark Hamiltonian H = λ(J→ · ℰ→)2.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 Spectral singularities of complex scattering potentials and ınfinite reflection and transmission coefficients at real energies(American Physical Society (APS), 2009) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Researcher; Department of Mathematics; Department of Physics; College of SciencesSpectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a waveguide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.Publication Open Access Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions(American Physical Society (APS), 2021) Loran, Farhang; Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a 2 x 2 matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. We show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices, each related to the time-evolution operator for an effective nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for delta-function potentials in 2D and 3D and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.Publication Open Access Quantum fluid dynamics in the Lagrangian representation and applications to photodissociation problems(American Institute of Physics (AIP) Publishing, 1999) Rabitz, H. A.; Department of Mathematics; Aşkar, Attila; Faculty Member; Department of Mathematics; College of Sciences; N/A; 178822This paper considers the practical utility of quantum fluid dynamics (QFD) whereby the time-dependent Schrodinger's equation is transformed to observing the dynamics of an equivalent "gas continuum." The density and velocity of this equivalent gas continuum are respectively the probability density and the gradient of the phase of the wave function. The numerical implementation of the QFD equations is carried out within the Lagrangian approach, which transforms the solution of Schrodinger's equation into following the trajectories of a set of mass points, i.e., subparticles, obtained by discretization of the continuum equations. The quantum dynamics of the subparticles which arise in the present formalism through numerical discretization are coupled by the density and the quantum potential. Numerical illustrations are performed for photodissociation of nocl and NO2 treated as two-dimensional models. The dissociation cross sections sigma(omega) are evaluated in the dramatically short CPU times of 33 s for nocl and 40 s for NO2 on a Pentium-200 mhz PC machine. The computational efficiency comes from a combination of (a) the QFD representation dealing with the near monotonic amplitude and phase as dependent variables, (b) the Lagrangian description concentrating the computation effort at all times into regions of highest probability as an optimal adaptive grid, and (c) the use of an explicit time integrator whereby the computational effort grows only linearly with the number of discrete points.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).