Bernstein's problem on weighted polynomial approximation

We formulate a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure [lletra "mu" minúscula de l'alfabet grec] on the real line we give a criterion for density of polynomials in Lp[lletra "mu" minúscula de l'alfabet grec entre parèntesis].

Bandwidth allocation based on real time calculations using the convolution approach

This paper focuses on one of the methods for bandwidth allocation in an ATM network: the convolution approach. The convolution approach permits an accurate study of the system load in statistical terms by accumulated calculations, since probabilistic results of the bandwidth allocation can be obtained. Nevertheless, the convolution approach has a high cost in terms of calculation and storage requirements. This aspect makes real-time calculations difficult, so many authors do not consider this approach. With the aim of reducing the cost we propose to use the multinomial distribution function: the enhanced convolution approach (ECA). This permits direct computation of the associated probabilities of the instantaneous bandwidth requirements and makes a simple deconvolution process possible. The ECA is used in connection acceptance control, and some results are presented

On concentrators and related approximation constants

Pippenger [Pi77] showed the existence of (6m,4m,3m,6)-concentrator for each positive integer m using a probabilistic method. We generalize his approach and prove existence of (6m,4m,3m,5.05)-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation of almost additive set functions by additive set functions from 44.5 (established by Kalton and Roberts in [KaRo83] to 39. We show a more direct connection of the latter problem to the Whitney type estimate for approximation of continuous functions on a cube in &b&R&/b&&sup&d&/sup& by linear functions, and improve the estimate of this Whitney constant from 802 (proved by Brudnyi and Kalton in [BrKa00] to 73.

Automatic generation of active coordinates for quantum dynamics calculations: application to the dynamics of benzene photochemistry

A new practical method to generate a subspace of active coordinates for quantum dynamics calculations is presented. These reduced coordinates are obtained as the normal modes of an analytical quadratic representation of the energy difference between excited and ground states within the complete active space self-consistent field method. At the Franck-Condon point, the largest negative eigenvalues of this Hessian correspond to the photoactive modes: those that reduce the energy difference and lead to the conical intersection; eigenvalues close to 0 correspond to bath modes, while modes with large positive eigenvalues are photoinactive vibrations, which increase the energy difference. The efficacy of quantum dynamics run in the subspace of the photoactive modes is illustrated with the photochemistry of benzene, where theoretical simulations are designed to assist optimal control experiments

Determination of vibrational polarizabilities and hyperpolarizabilities using field-induced coordinates

An analytical set of field-induced coordinates is defined and is used to show that the vibrational degrees of freedom required to completely describe nuclear relaxation polarizabilities and hyperpolarizabilities is reduced from 3N-6 to a relatively small number. As this number does not depend upon the size of the molecule, the process provides computational advantages. A method is provided to separate anharmonic contributions from harmonic contributions as well as effective mechanical from electrical anharmonicity. The procedures are illustrated by Hartree-Fock calculations, indicating that anharmonicity can be very important

Nuclear relaxation contribution to static and dynamic (infinite frequency approximation) nonlinear optical properties by means of electrical property expansions: Application to HF, CH4, CF4, and SF6

Electrical property derivative expressions are presented for the nuclear relaxation contribution to static and dynamic (infinite frequency approximation) nonlinear optical properties. For CF4 and SF6, as opposed to HF and CH4, a term that is quadratic in the vibrational anharmonicity (and not previously evaluated for any molecule) makes an important contribution to the static second vibrational hyperpolarizability of CF4 and SF6. A comparison between calculated and experimental values for the difference between the (anisotropic) Kerr effect and electric field induced second-harmonic generation shows that, at the Hartree-Fock level, the nuclear relaxation/infinite frequency approximation gives the correct trend (in the series CH4, CF4, SF6) but is of the order of 50% too small

Variational calculation of static and dynamic vibrational nonlinear optical properties

The vibrational configuration interaction method used to obtain static vibrational (hyper)polarizabilities is extended to dynamic nonlinear optical properties in the infinite optical frequency approximation. Illustrative calculations are carried out on H2 O and N H3. The former molecule is weakly anharmonic while the latter contains a strongly anharmonic umbrella mode. The effect on vibrational (hyper)polarizabilities due to various truncations of the potential energy and property surfaces involved in the calculation are examined

Nuclear magnetic resonance chemical shifts with the statistical average of orbital-dependent model potentials in Kohn-Sham density functional theory

The performance of the SAOP potential for the calculation of NMR chemical shifts was evaluated. SAOP results show considerable improvement with respect to previous potentials, like VWN or BP86, at least for the carbon, nitrogen, oxygen, and fluorine chemical shifts. Furthermore, a few NMR calculations carried out on third period atoms (S, P, and Cl) improved when using the SAOP potential

Modelització de les propietats òptiques de partícules metàl•liques en matriu dielèctrica

En el marc del projecte "Modelització de les propietats òptiques de partícules metàl•liques en matriu dielèctrica" s'han desenvolupat un conjunt d'eines numèriques que permeten avançar en l'ús de l'espectroscòpia òptica per a l'obtenció d'informació morfològica de materials compostos consistents en partícules metàl•liques en matriu dielèctrica. S'han implementat esquemes numèrics per a calcular les propietats òptiques de materials compostos on les partícules poden presentar una distribució de mides i formes i diferent graus d'ordenament espacial. Les simulacions s'han realitzat a dos nivells: i) amb l’aproximació quasi-estàtica, que permet descriure el comportament d'aquests materials en termes de constants òptiques efectives i ii) amb càlculs electrodinàmics exactes, que han servit per avaluar la validesa de l’aproximació anterior i que han permès d'estudiar la interacció de partícules amb feixos de llum focalitzats o amb polarització no homogènia. A través de l’anàlisi d'aquestes simulacions, s'han desenvolupat models senzills que permeten parametritzar la influència de diferents quantitats físiques en el comportament òptic del material. Aquests models s'han implementat en un programari de càlcul que permeten trobar el valor òptim dels paràmetres físics d'interès mitjançant l'ajust d'espectres òptics. Els models s'han avaluat amb l'anàlisi de dades experimentals subministrades per altres laboratoris.

Equivalence of piecewise-linear approximation and Lagrangian relaxation for network revenue management

The network revenue management (RM) problem arises in airline, hotel, media,and other industries where the sale products use multiple resources. It can be formulatedas a stochastic dynamic program but the dynamic program is computationallyintractable because of an exponentially large state space, and a number of heuristicshave been proposed to approximate it. Notable amongst these -both for their revenueperformance, as well as their theoretically sound basis- are approximate dynamic programmingmethods that approximate the value function by basis functions (both affinefunctions as well as piecewise-linear functions have been proposed for network RM)and decomposition methods that relax the constraints of the dynamic program to solvesimpler dynamic programs (such as the Lagrangian relaxation methods). In this paperwe show that these two seemingly distinct approaches coincide for the network RMdynamic program, i.e., the piecewise-linear approximation method and the Lagrangianrelaxation method are one and the same.

Calibration of stochastic volatility models via second order approximation: the Heston model case

Using a suitable Hull and White type formula we develop a methodology to obtain asecond order approximation to the implied volatility for very short maturities. Using thisapproximation we accurately calibrate the full set of parameters of the Heston model. Oneof the reasons that makes our calibration for short maturities so accurate is that we alsotake into account the term-structure for large maturities. We may say that calibration isnot "memoryless", in the sense that the option's behavior far away from maturity doesinfluence calibration when the option gets close to expiration. Our results provide a wayto perform a quick calibration of a closed-form approximation to vanilla options that canthen be used to price exotic derivatives. The methodology is simple, accurate, fast, andit requires a minimal computational cost.

On the closed-form approximation of short-time random strike options

In this paper we propose a general technique to develop first and second order closed-form approximation formulas for short-time options withrandom strikes. Our method is based on Malliavin calculus techniques andallows us to obtain simple closed-form approximation formulas dependingon the derivative operator. The numerical analysis shows that these formulas are extremely accurate and improve some previous approaches ontwo-assets and three-assets spread options as Kirk's formula or the decomposition mehod presented in Alòs, Eydeland and Laurence (2011).