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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; Department of Mathematics; Soner, Halil Mete; Faculty Member; 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; Department of Mathematics; Aşkar, Attila; Dey, Bijoy K.; Faculty Member; Faculty Member; 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 Generalized adiabatic product expansion: a nonperturbative method of solving the time-dependent Schrodinger equation(American Institute of Physics (AIP) Publishing, 1999) Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; 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; Department of Mathematics; Mostafazadeh, Ali; Rostamzadeh, Saber; Faculty Member; 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 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; Department of Mathematics; Aşkar, Attila; Faculty Member; 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 Minimal number of singular fibers in a nonorientable Lefschetz fibration(Springer, 2022) Onaran, Sinem; Department of Mathematics; Department of Mathematics; Özbağcı, Burak; Faculty Member; 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; Department of Mathematics; Alkan, Emre; Faculty Member; 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; Department of Mathematics; Çağlar, Mine; Faculty Member; College of Sciences; 105131Publication Open Access Differential realization of pseudo-Hermiticity: a quantum mechanical analog of Einstein's field equation(American Institute of Physics (AIP) Publishing, 2006) Department of Mathematics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; College of Sciences; 4231For a given pseudo-Hermitian Hamiltonian of the standard form: H=p(2)/2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator eta satisfying H(dagger)=eta H eta(-1) to the solution of a differential equation. If the configuration space is R, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of eta. We apply our general results to calculate eta for the PT-symmetric square well, an imaginary scattering potential, and a class of imaginary delta-function potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general eta up to second-order terms in the coupling constants.Publication Open Access Singularity links with exotic Stein fillings(Worldwide Center of Mathematics, 2014) Akhmedov, Anar; Department of Mathematics; Department of Mathematics; Özbağcı, Burak; Faculty Member; College of Sciences; 29746In [4], it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifert fibered 3-manifolds with many singular fibers and observe that these 3-manifolds are singularity links. In addition, we prove that the contact structures induced by the Stein fillings are the canonical contact structures on these singularity links. As a consequence, we verify a prediction of András Némethi by providing examples of isolated complex surface singularities whose links with their canonical contact structures admitting infinitely many exotic simply-connected Stein fillings. Moreover, for infinitely many of these contact singularity links and for each positive integer n, we also construct an infinite family of exotic Stein fillings with fixed fundamental group ? ? ?n.