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Item Open Access A class of infinite convex geometries(2015) Adaricheva, Kira; Nation, J.B.Various characterizations of finite convex geometries are well known. This note provides similar characterizations for possibly infinite convex geometries whose lattice of closed sets is strongly coatomic and lower continuous. Some classes of examples of such convex geometries are givenItem Open Access A Lumped-Parameter Model for Nonlinear Waves in Graphene(2015) Wei, Dongming; Hazim, Hamad; Elgindi, Mohamed; Soukiassian, YeranA lumped-parameter nonlinear spring-mass model which takes into account the third-order elastic sti ness constant is considered for mod- eling the free and forced axial vibrations of a graphene sheet with one xed end and one free end with a mass attached. It 's demonstrated through this simple model that, in free vibration, within certain initial energy level and depending upon its length and the nonlinear elas- tic constants, there exist bounded periodic solutions which are non- sinusoidal, and that for each xed energy level, there is a bifurcation point depending upon material constants, beyond which the periodic solutions disappear. The amplitude, frequency, and the corresponding wave solutions for both free and forced harmonic vibrations are cal- culated analytically and numerically. Energy sweep is also performed for resonance applications.Item Open Access A Lumped-Parameter Model for Nonlinear Waves in Graphene(World Journal of Engineering and Technology, 2015) Hazim, Hamad; Wei, Dongming; Elgindi, Mohamed B. M.; Soukiassian, YeranA lumped-parameter nonlinear spring-mass model which takes into account the third-order elastic stiffness constant is considered for modeling the free and forced axial vibrations of a graphene sheet with one fixed end and one free end with a mass attached. It is demonstrated through this simple model that, in free vibration, within certain initial energy level and depending upon its length and the nonlinear elastic constants, that there exist bounded periodic solutions which are non-sinusoidal, and that for each fixed energy level, there is a bifurcation point depending upon material constants, beyond which the periodic solutions disappear. The amplitude, frequency, and the corresponding wave solutions for both free and forced harmonic vibrations are calculated analytically and numerically. Energy sweep is also performed for resonance applicationsItem Open Access A New Weibull–Pareto Distribution: Properties and Applications(Communications in Statistics: Simulation and Computation, 2016-11-25) Tahir, M. H.; Cordeiro, Gauss M.; Alzaatreh, Ayman; Mansoor, M.; Zubair, M.Many distributions have been used as lifetime models. In this article, we propose a new three-parameter Weibull–Pareto distribution, which can produce the most important hazard rate shapes, namely, constant, increasing, decreasing, bathtub, and upsidedown bathtub. Various structural properties of the new distribution are derived including explicit expressions for the moments and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, and generating and quantile functions. The Rényi and q entropies are also derived. We obtain the density function of the order statistics and their moments. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of two real datasets on Wheaton river flood and bladder cancer. In the two applications, the new model provides better fits than the Kumaraswamy–Pareto, beta-exponentiated Pareto, beta-Pareto, exponentiated Pareto, and Pareto models.Item Open Access A Penalty Method for Approximations of the Stationary Power-Law Stokes Problem(Electronic journal of differential equations, 2001) Lefton, Lew; Wei, DongmingWe study approximations of the steady state Stokes problem governed by the power-law model for viscous incompressible non-Newtonian flow using the penalty formulation. We establish convergence and find error estimates.Item Open Access Adaptive cross approximation for ill-posed problems(Journal of Computational and Applied Mathematics, 2016-09-01) Mach, Thomas; Reichel, Lothar; Van Barel, Marc; Vandebril, R.Integral equations of the first kind with a smooth kernel and perturbed right-hand side, which represents available contaminated data, arise in many applications. Discretization gives rise to linear systems of equations with a matrix whose singular values cluster at the origin. The solution of these systems of equations requires regularization, which has the effect that components in the computed solution connected to singular vectors associated with small singular values are damped or ignored. In order to compute a useful approximate solution typically approximations of only a fairly small number of the largest singular values and associated singular vectors of the matrix are required. The present paper explores the possibility of determining these approximate singular values and vectors by adaptive cross approximation. This approach is particularly useful when a fine discretization of the integral equation is required and the resulting linear system of equations is of large dimensions, because adaptive cross approximation makes it possible to compute only fairly few of the matrix entries.Item Open Access Algebraic convex geometries revisited(2014) Adaricheva, KiraRepresentation of convex geometry as an appropriate join of compatible orderings of the base set can be achieved, when closure operator of convex geometry is algebraic, or finitary. This bears to the finite case proved by P. Edelman and R. Jamison to the greater extent than was thought earlierItem Open Access Algebraic numbers, hyperbolicity, and density modulo one(2011) Kadyrov, Shirali; Gorodnik, A.Item Open Access Amount of failure of upper-semicontinuity of entropy in noncompact rank one situations, and hausdorff dimension(2012) Kadyrov, Shirali; Pohl, A.Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal ows on homogeneous spaces nG, where G is any connected semisimple Lie group of real rank 1 with nite center, and is any nonuniform lattice in G. We show that this bound is sharp, and apply the methods used to establish bounds for the Hausdorff dimension of the set of points which diverge on average.Item Open Access An extended Hessenberg form for Hamiltonian matrices(Calcolo, 2016-06-01) Ferranti, Micol; Iannazzo, Bruno; Mach, Thomas; Vandebril, RafA unitary symplectic similarity transformation for a special class of Hamiltonian matrices to extended Hamiltonian Hessenberg form is presented. Whereas the classical Hessenberg form links to Krylov subspaces, the extended Hessenberg form links to extended Krylov subspaces. The presented algorithm generalizes thus the classic reduction to Hamiltonian Hessenberg form and offers more freedom in the choice of Hamiltonian condensed forms, to be used within an extended Hamiltonian QR algorithm. Theoretical results identifying the structure of the extended Hamiltonian Hessenberg form and proofs of uniqueness of the reduction process are included. Numerical experiments confirm the validity of the approach.Item Open Access Analyzing Chaos in Higher Order Disordered Quartic-Sextic Klein-Gordon Lattices Using q-Statistics(arXiv, 2018-03-19) Antonopoulos, Chris G.; Skokos, Charalampos; Bountis, Tassos; Flach, SergejIn the study of subdiffusive wave-packet spreading in disordered Klein- Gordon (KG) nonlinear lattices, a central open question is whether the motion continues to be chaotic despite decreasing densities, or tends to become quasi-periodic as nonlinear terms become negligible. In a recent study of such KG particle chains with quartic (4th order) anharmonicity in the on-site potential it was shown that q−Gaussian probability distribu- tion functions of sums of position observables with q > 1 always approach pure Gaussians (q = 1) in the long time limit and hence the motion of the full system is ultimately “strongly chaotic”. In the present paper, we show that these results continue to hold even when a sextic (6th order) term is gradually added to the potential and ultimately prevails over the 4th order anharmonicity, despite expectations that the dynamics is more “regular”, at least in the regime of small oscillations. Analyzing this sys- tem in the subdiffusive energy domain using q-statistics, we demonstrate that groups of oscillators centered around the initially excited one (as well as the full chain) possess strongly chaotic dynamics and are thus far from any quasi-periodic torus, for times as long as t = 10Item Metadata only Analyzing chaos in higher order disordered quartic-sextic Klein-Gordon lattices using q-statistics(Chaos, Solitons & Fractals, 2017-11-01) Antonopoulos, Chris G.; Skokos, Charalampos; Bountis, Tassos; Flach, Sergej; Chris G., AntonopoulosAbstract In the study of subdiffusive wave-packet spreading in disordered Klein–Gordon (KG) nonlinear lattices, a central open question is whether the motion continues to be chaotic despite decreasing densities, or tends to become quasi-periodic as nonlinear terms become negligible. In a recent study of such KG particle chains with quartic (4th order) anharmonicity in the on-site potential it was shown that q−Gaussian probability distribution functions of sums of position observables with q > 1 always approach pure Gaussians (q=1) in the long time limit and hence the motion of the full system is ultimately “strongly chaotic”. In the present paper, we show that these results continue to hold even when a sextic (6th order) term is gradually added to the potential and ultimately prevails over the 4th order anharmonicity, despite expectations that the dynamics is more “regular”, at least in the regime of small oscillations. Analyzing this system in the subdiffusive energy domain using q-statistics, we demonstrate that groups of oscillators centered around the initially excited one (as well as the full chain) possess strongly chaotic dynamics and are thus far from any quasi-periodic torus, for times as long as t=109.Item Open Access The Asymmetric Active Coupler: Stable Nonlinear Supermodes and Directed Transport(Scientific Reports, 2016-09-19) Kominis, Yannis; Bountis, Tassos; Flach, SergejWe consider the asymmetric active coupler (AAC) consisting of two coupled dissimilar waveguides with gain and loss. We show that under generic conditions, not restricted by parity-time symmetry, there exist finite-power, constant-intensity nonlinear supermodes (NS), resulting from the balance between gain, loss, nonlinearity, coupling and dissimilarity. The system is shown to possess non-reciprocal dynamics enabling directed power transport functionality.Item Open Access Bernstein-walsh inequalities in higherdimensions over exponential curves(2011) Kadyrov, Shirali; Lawrence, MarkLet x = (x1; : : : ; xd) 2 [1; 1]d be linearly independent over Z, set K = f(ez; ex1z; ex2z : : : ; exdz) : jzj 1g:We prove sharp estimates for the growth of a polynomial of degree n, in terms of En(x) := supfkPk d+1 : P 2 Pn(d + 1); kPkK 1g; where d+1 is the unit polydisk. For all x 2 [1; 1]d with linearly independent entries, we have the lower estimate logEn(x) nd+1 (d 1)!(d + 1) log n O(nd+1); for Diophantine x, we have logEn(x) nd+1 (d 1)!(d + 1) log n + O(nd+1): In particular, this estimate holds for almost all x with respect to Lebesgue measure. The results here generalize those of [6] for d = 1, without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.Item Open Access Bornological projective limits of inductive limits of normed spaces(Functiones et Approximatio, Commentarii Mathematici, 2011) Bonet, José; Wegner, Sven AkeWe establish a criterion to decide when a countable projective limit of countable inductive limits of normed spaces is bornological. We compare the conditions occurring within our criterion with well-known abstract conditions from the context of homological algebra and with conditions arising within the investigation of weighted PLB-spaces of continuous functions.Item Open Access Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices(Electronic Transactions on Numerical Analysis, 2014) Mach, Thomas; Pranić, Miroslav S.; Vandebril, RafIt has been shown that approximate extended Krylov subspaces can be computed, under certain assumptions, without any explicit inversion or system solves. Instead, the vectors spanning the extended Krylov space are retrieved in an implicit way, via unitary similarity transformations, from an enlarged Krylov subspace. In this paper this approach is generalized to rational Krylov subspaces, which aside from poles at infinity and zero, also contain finite non-zero poles. Furthermore, the algorithms are generalized to deal with block rational Krylov subspaces and techniques to exploit the symmetry when working with Hermitian matrices are also presented. For each variant of the algorithm numerical experiments illustrate the power of the new approach. The experiments involve matrix functions, Ritz-value computations, and the solutions of matrix equations.Item Open Access Computing the eigenvalues of symmetric H2-matrices by slicing the spectrum(Computing and Visualization in Science, 2015-03-04) Benner, Peter; Börm, Steffen; Mach, Thomas; Reimer, KnutThe computation of eigenvalues of large-scale matrices arising from finite element discretizations has gained significant interest in the last decade (Knyazev et al. in Numerical solution of PDE eigenvalue problems, vol 56. Mathematisches Forschungsinstitut, Oberwolfach, 2013). Here we present an new algorithm based on slicing the spectrum that takes advantage of the rank structure of resolvent matrices in order to compute (Formula presented.) eigenvalues of the generalized symmetric eigenvalue problem in (Formula presented.) operations, where (Formula presented.) is a small constant.Item Metadata only Conical square functions associated with Bessel, Laguerre and Schrödinger operators in UMD Banach spaces(Journal of Mathematical Analysis and Applications, 2017-03-01) Betancor, Jorge J.; Castro, Alejandro J.; Fariña, Juan C.; Rodríguez-Mesa, L.; Jorge J., BetancorAbstract In this paper we consider conical square functions in the Bessel, Laguerre and Schrödinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hytönen, van Neerven and Portal [36], in order to define our conical square functions, we use γ-radonifying operators. We obtain new equivalent norms in the Lebesgue–Bochner spaces Lp((0,∞),B) and Lp(Rn,B), 1Item Open Access Controllable asymmetric phase-locked states of the fundamental active photonic dimer(Physical Review, 2017-10-16) Kominis, Yannis; Kovanis, Vassilios; Bountis, TassosCoupled semiconductor lasers are systems possessing complex dynamics that are interesting for numerous applications in photonics. In this work, we investigate the existence and the stability of asymmetric phase-locked states of the fundamental active photonic dimer consisting of two coupled lasers.We showthat stable phase-locked states of arbitrary asymmetry exist for extended regions of the parameter space of the system and that their field amplitude ratio and phase difference can be dynamically controlled by appropriate current injection. The model includes the important role of carrier density dynamics and shows that the phase-locked state asymmetry is related to operation conditions providing, respectively, gain and loss in the two lasers.Item Open Access Controllable Asymmetric Phase-Locked States of the Fundamental Active Photonic Dimer(arXiv, 2018-03-07) Kominis, Yannis; Kovanis, Vassilios; Bountis, TassosCoupled semiconductor lasers are systems possessing complex dynamics that are interesting for numerous applications in photonics. In this work, we investigate the existence and the stability of asymmetric phase-locked states of the fundamental active photonic dimer consisting of two coupled lasers. We show that stable phase-locked states of arbitrary asymmetry exist for extended regions of the parameter space of the system and that their field amplitude ratio and phase difference can be dynamically controlled by appropriate current injection. The model includes the important role of carrier density dynamics and shows that the phase-locked state asymmetry is related to operation conditions providing, respectively, gain and loss in the two lasers