956 resultados para resolvent convergence
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The system of coupled oscillators and its time-discretization (with constant stepsize h) are considered in this paper. Under some conditions, it is showed that the discrete systems have one-dimensional global attractors l(h) converging to l which is the global attractor of continuous system.
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Optimal Bayesian multi-target filtering is in general computationally impractical owing to the high dimensionality of the multi-target state. The Probability Hypothesis Density (PHD) filter propagates the first moment of the multi-target posterior distribution. While this reduces the dimensionality of the problem, the PHD filter still involves intractable integrals in many cases of interest. Several authors have proposed Sequential Monte Carlo (SMC) implementations of the PHD filter. However, these implementations are the equivalent of the Bootstrap Particle Filter, and the latter is well known to be inefficient. Drawing on ideas from the Auxiliary Particle Filter (APF), a SMC implementation of the PHD filter which employs auxiliary variables to enhance its efficiency was proposed by Whiteley et. al. Numerical examples were presented for two scenarios, including a challenging nonlinear observation model, to support the claim. This paper studies the theoretical properties of this auxiliary particle implementation. $\mathbb{L}_p$ error bounds are established from which almost sure convergence follows.
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In this thesis we consider smooth analogues of operators studied in connection with the pointwise convergence of the solution, u(x,t), (x,t) ∈ ℝ^n x ℝ, of the free Schrodinger equation to the given initial data. Such operators are interesting examples of oscillatory integral operators with degenerate phase functions, and we develop strategies to capture the oscillations and obtain sharp L^2 → L^2 bounds. We then consider, for fixed smooth t(x), the restriction of u to the surface (x,t(x)). We find that u(x,t(x)) ∈ L^2(D^n) when the initial data is in a suitable L^2-Sobolev space H^8 (ℝ^n), where s depends on conditions on t.
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Chapter I
Theories for organic donor-acceptor (DA) complexes in solution and in the solid state are reviewed, and compared with the available experimental data. As shown by McConnell et al. (Proc. Natl. Acad. Sci. U.S., 53, 46-50 (1965)), the DA crystals fall into two classes, the holoionic class with a fully or almost fully ionic ground state, and the nonionic class with little or no ionic character. If the total lattice binding energy 2ε1 (per DA pair) gained in ionizing a DA lattice exceeds the cost 2εo of ionizing each DA pair, ε1 + εo less than 0, then the lattice is holoionic. The charge-transfer (CT) band in crystals and in solution can be explained, following Mulliken, by a second-order mixing of states, or by any theory that makes the CT transition strongly allowed, and yet due to a small change in the ground state of the non-interacting components D and A (or D+ and A-). The magnetic properties of the DA crystals are discussed.
Chapter II
A computer program, EWALD, was written to calculate by the Ewald fast-convergence method the crystal Coulomb binding energy EC due to classical monopole-monopole interactions for crystals of any symmetry. The precision of EC values obtained is high: the uncertainties, estimated by the effect on EC of changing the Ewald convergence parameter η, ranged from ± 0.00002 eV to ± 0.01 eV in the worst case. The charge distribution for organic ions was idealized as fractional point charges localized at the crystallographic atomic positions: these charges were chosen from available theoretical and experimental estimates. The uncertainty in EC due to different charge distribution models is typically ± 0.1 eV (± 3%): thus, even the simple Hückel model can give decent results.
EC for Wurster's Blue Perchl orate is -4.1 eV/molecule: the crystal is stable under the binding provided by direct Coulomb interactions. EC for N-Methylphenazinium Tetracyanoquino- dimethanide is 0.1 eV: exchange Coulomb interactions, which cannot be estimated classically, must provide the necessary binding.
EWALD was also used to test the McConnell classification of DA crystals. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine: 7,7,8,8-Tetracyanoquinodimethan) EC = -4.0 eV while 2εo = 4.65 eV: clearly, exchange forces must provide the balance. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine:para-Chloranil) EC = -4.4 eV, while 2εo = 5.0 eV: again EC falls short of 2ε1. As a Gedankenexperiment, two nonionic crystals were assumed to be ionized: for (1:1)-(Hexamethyl- benzene:para-Chloranil) EC = -4.5 eV, 2εo = 6.6 eV; for (1:1)- (Napthalene:Tetracyanoethylene) EC = -4.3 eV, 2εo = 6.5 eV. Thus, exchange energies in these nonionic crystals must not exceed 1 eV.
Chapter III
A rapid-convergence quantum-mechanical formalism is derived to calculate the electronic energy of an arbitrary molecular (or molecular-ion) crystal: this provides estimates of crystal binding energies which include the exchange Coulomb inter- actions. Previously obtained LCAO-MO wavefunctions for the isolated molecule(s) ("unit cell spin-orbitals") provide the starting-point. Bloch's theorem is used to construct "crystal spin-orbitals". Overlap between the unit cell orbitals localized in different unit cells is neglected, or is eliminated by Löwdin orthogonalization. Then simple formulas for the total kinetic energy Q^(XT)_λ, nuclear attraction [λ/λ]XT, direct Coulomb [λλ/λ'λ']XT and exchange Coulomb [λλ'/λ'λ]XT integrals are obtained, and direct-space brute-force expansions in atomic wavefunctions are given. Fourier series are obtained for [λ/λ]XT, [λλ/λ'λ']XT, and [λλ/λ'λ]XT with the help of the convolution theorem; the Fourier coefficients require the evaluation of Silverstone's two-center Fourier transform integrals. If the short-range interactions are calculated by brute-force integrations in direct space, and the long-range effects are summed in Fourier space, then rapid convergence is possible for [λ/λ]XT, [λλ/λ'λ']XT and [λλ'/λ'λ]XT. This is achieved, as in the Ewald method, by modifying each atomic wavefunction by a "Gaussian convergence acceleration factor", and evaluating separately in direct and in Fourier space appropriate portions of [λ/λ]XT, etc., where some of the portions contain the Gaussian factor.
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This article investigates the convergence properties of iterative processes involving sequences of self-mappings of metric or Banach spaces. Such sequences are built from a set of primary self-mappings which are either expansive or non-expansive self-mappings and some of the non-expansive ones can be contractive including the case of strict contractions. The sequences are built subject to switching laws which select each active self-mapping on a certain activation interval in such a way that essential properties of boundedness and convergence of distances and iterated sequences are guaranteed. Applications to the important problem of stability of dynamic switched systems are also given.