Studies in non-Gaussian analysis

This thesis presents general methods in non-Gaussian analysis in infinite dimensional spaces. As main applications we study Poisson and compound Poisson spaces. Given a probability measure μ on a co-nuclear space, we develop an abstract theory based on the generalized Appell systems which are bi-orthogonal. We study its properties as well as the generated Gelfand triples. As an example we consider the important case of Poisson measures. The product and Wick calculus are developed on this context. We provide formulas for the change of the generalized Appell system under a transformation of the measure. The L² structure for the Poisson measure, compound Poisson and Gamma measures are elaborated. We exhibit the chaos decomposition using the Fock isomorphism. We obtain the representation of the creation, annihilation operators. We construct two types of differential geometry on the configuration space over a differentiable manifold. These two geometries are related through the Dirichlet forms for Poisson measures as well as for its perturbations. Finally, we construct the internal geometry on the compound configurations space. In particular, the intrinsic gradient, the divergence and the Laplace-Beltrami operator. As a result, we may define the Dirichlet forms which are associated to a diffusion process. Consequently, we obtain the representation of the Lie algebra of vector fields with compact support. All these results extends directly for the marked Poisson spaces.

Efeitos de acoplamentos biquadráticos e campos aleatórios em vidros de spins

In this work we have studied the effects of random biquadratic and random fields in spin-glass models using the replica method. The effect of a random biquadratic coupling was studied in two spin-1 spin-glass models: in one case the interactions occur between pairs of spins, whereas in the second one the interactions occur between p spins and the limit p > oo is considered. Both couplings (spin glass and biquadratic) have zero-mean Gaussian probability distributions. In the first model, the replica-symmetric assumption reveals that the system presents two pha¬ses, namely, paramagnetic and spin-glass, separated by a continuous transition line. The stability analysis of the replica-symmetric solution yields, besides the usual instability associated with the spin-glass ordering, a new phase due to the random biquadratic cou¬plings between the spins. For the case p oo, the replica-symmetric assumption yields again only two phases, namely, paramagnetic and quadrupolar. In both these phases the spin-glass parameter is zero. Besides, it is shown that they are stable under the Almeida-Thouless stability analysis. One of them presents negative entropy at low temperatures. We developed one step of replica simmetry breaking and noticed that a new phase, the biquadratic glass phase, emerge. In this way we have obtained the correct phase diagram, with.three first-order transition lines. These lines merges in a common triple point. The effects of random fields were studied in the Sherrington-Kirkpatrick model consi¬dered in the presence of an external random magnetic field following a trimodal distribu¬tion {P{hi) = p+S(hi - h0) +Po${hi) +pS(hi + h0))- It is shown that the border of the ferromagnetic phase may present, for conveniently chosen values of p0 and hQ, first-order phase transitions, as well as tricritical points at finite temperatures. It is verified that the first-order phase transitions are directly related to the dilution in the fields: the extensions of these transitions are reduced for increasing values of po- In fact, the threshold value pg, above which all phase transitions are continuous, is calculated analytically. The stability analysis of the replica-symmetric solution is performed and the regions of validity of such a solution are identified

Prolapse-free relativistic Gaussian basis sets for the superheavy elements up to Uuo (Z=118) and Lr (Z=103)

Prolapse-free basis sets suitable for four-component relativistic quantum chemical calculations are presented for the superheavy elements UP to (118)Uuo ((104)Rf, (105)Db, (106)Sg, (107)Bh, (108)Hs, (109)Mt, (110)Ds, (111)Rg, (112)Uub, (113)Uut, (114)Uuq, (115)Uup, (116)Uuh, (117)Uus, (118)Uuo) and Lr-103. These basis sets were optimized by minimizing the absolute values of the energy difference between the Dirac-Fock-Roothaan total energy and the corresponding numerical value at a milli-Hartree order of magnitude, resulting in a good balance between cost and accuracy. Parameters for generating exponents and new numerical data for some superheavy elements are also presented. (c) 2007 Elsevier B.V. All rights reserved.

Gaussian quadrature rules with simple node-weight relations

In this paper, we consider the symmetric Gaussian and L-Gaussian quadrature rules associated with twin periodic recurrence relations with possible variations in the initial coefficient. We show that the weights of the associated Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 4. We also show that the weights of the associated L-Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 5. Special cases of these quadrature rules are given. Finally, an easy to implement procedure for the evaluation of the nodes is described.

Rational approximations of the Arrhenius integral using Jacobi fractions and gaussian quadrature

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Fluid dynamics of air in a packed bed: velocity profiles and the continuum model assumption

Air flow through packed beds was analyzed experimentally under conditions ranging from those that reinforce the effect of the wall on the void fraction to those that minimize it. The packing was spherical particles, with a tube-to-particle diameter ratio (D/dp) between 3 and 60. Air flow rates were maintained between 1.3 and 4.44 m3/min, and gas velocity was measured with a Pitot tube positioned above the bed exit. Measurements were made at various radial and angular coordinate values, allowing the distribution of air flow across the bed to be described in detail. Comparison of the experimentally observed radial profiles with those derived from published equations revealed that at high D/dp ratios the measured and calculated velocity profiles behaved similarly. At low ratios, oscillations in the velocity profiles agreed with those in the voidage profiles, signifying that treating the porous medium as a continuum medium is questionable in these cases.

Gaussian basis sets to the theoretical study of the electronic structure of perovskite (LaMnO3)

The Generator Coordinate Hartree-Fock (GCHF) method is applied to generate extended (20s14p), (30s19p13d), and (31s23p18d) Gaussian basis sets for the 0, Mn, and La atoms, respectively. The role of the weight functions (WFs) in the assessment of the numerical integration range of the GCHF equations is shown. These basis sets are then contracted to [5s3p] and [11s6p6d] for 0 and Mn atoms, respectively, and [17s11p7d] for La atom by a standard procedure. For quality evaluation of contracted basis sets in molecular calculations, we have accomplished calculations of total and orbital energies in the Hartree-Fock-Roothaan (HFR) method for (MnO1+)-Mn-5 and (LaO1+)-La-1 fragments. The results obtained with the contracted basis sets are compared with values obtained with the extended basis sets. The addition of one d polarization function in the contracted basis set for 0 atom and its utilization with the contracted basis sets for Mn and La atoms leads to the calculations of dipole moment and total atomic charges of perovskite (LaMnO3). The calculations were performed at the HFR level with the crystal [LaMnO3](2) fragment in space group C-2v the values of dipole moment, total energy, and total atomic charges showed that it is reasonable to believe that LaMnO3 presents behaviour of piezoelectric material. (C) 2003 Elsevier B.V. All rights reserved.

The generator coordinate Hartree-Fock method as strategy for building Gaussian basis sets to ab initio study of the process of adsorption of sulfur on platinum (200) surface

The scheme named generator coordinate Hartree-Fock method (GCHF) is used to build (22s14p) and (33s22p16d9f) gaussian basis sets to S ((3)P) and Pt ((3)D) atoms, respectively. Theses basis sets are contracted to [13s10p] and [19s13p9d5f] through of Dunning's segmented contraction scheme and are enriched with d and g polarization functions, [13s10p1d] and [19s13p9d5flg]. Finally, the [19s13p9d5f1g] basis Set to Pt ((3)D) was supplemented with s and d diffuse functions, [20s13p10d5flg], and used in combination with [13s10p1d] to study the effects of adsorption of S ((3)D) atom on a pt ((3)D) atom belonged to infinite Pt (200) surface. Atom-atom overlap population, bond order, and infrared spectrum of [pt(_)S](2 -) were calculated properties and were carried out at Hartree-Fock-Roothaan level. The results indicate that the process of adsorption of S ((3)P) on pt ((3)D) in the infinite Pt (200) surface is mainly caused by a strong contribution of sigma between the 3p(z) orbital of S ((3)P) and the 6s orbital of pt ((3)D). (c) 2004 Elsevier B.V. All rights reserved.

Design of Gaussian basis sets to the theoretical interpretation of IR-spectrum of hexaaquachromium (III) ion, tetraoxochromium (IV) ion, and tetraoxochromium (VI) ion

The Generator Coordinate Hartree-Fock (GCHF) method is employed to design 16s, 16s10p, 24s17p13d, 25s17p13d, and 26s17p Gaussian basis sets for the H ((2)S), O ((3)P), O(2-) ((1)S), Cr(3+) ((4)F), Cr(4+) ((3)F), and Cr(6+) ((1)S) atomic species. These basis sets are then contracted to (4s) for H ((2)S), (6s4p) for O ((3)P), and O(2-) ((1)S), (986p3d) for Cr(3+) ((4)F), (10s8p3d) for Cr(4+) ((3)F), and (13s7p) for Cr(6+) (1S) by a standard procedure. For evaluation of the quality of those basis sets in molecular calculations, we have accomplished studies of total and orbital (HOMO and HOMO-1) energies at the HF-Roothaan level for the molecular species of our interest. The results obtained with the contracted basis sets are compared to the values obtained with our extended basis sets and to the standard 6-311G basis set from literature. Finally, the contracted basis sets are enriched with polarization function and then utilized in the theoretical interpretation of IR-spectrum of hexaaquachromium (III) ion, [Cr(H(2)O)(6)](3+), tetraoxochromium (IV) ion, [CrO(4)](4-), and tetraoxochromium (VI) ion, [CrO(4)](2-). The respective theoretical harmonic frequencies and IR-intensities were computed at the density functional theory (DFT) level. In the DFT calculations we employed the Becke's 1988 functional using the LYP correlation functional. The comparison between the results obtained and the corresponding experimental values indicates a very good description of the IR-spectra of the molecular ions studied, and that the GCHF method is still a legitimate alternative for selection of Gaussian basis sets. (C) 2003 Elsevier B.V. All rights reserved.

Designing Gaussian basis sets to the theoretical study of the piezoelectric effect of perovskite (BaTiO3)

Gaussian basis sets (24s14p, 30s19p14d, and 33s21p14d for O (P-3), Ti (S-5), and Ba (S-1) atoms, respectively), are designed with the strategy of the Generator Coordinate Hartree-Fock method. The basis sets are then contracted to [6s4p], [10s5p4d], and [16s9p5d] to O, Ti, and Ba atoms, respectively, and used in calculations of total and orbital energies of (TiO+2)-Ti-1 and (BaO)-Ba-1 fragments for quality evaluation in molecular studies. For O atom, the [6s4p] basis set is enriched with d polarization function and used along with the [10s5p4d] and [16s9p5d] basis sets for the theoretical study of the piezoelectric effect of perovskite (BaTiO3). The results of this work evidence that the piezoelectric properties in BaTiO3 can be caused by electrostatic interactions. (C) 2003 Elsevier B.V. All rights reserved.

Gaussian basis sets by generator coordinate Hartree-Fock method to ab initio calculations of electron affinities of enolates

The Generator Coordinate Hartree-Fock (GCHF) method is employed to generate uncontracted 15s and 18s11p gaussian basis sets for the H, C and O atoms, respectively. These basis sets are then contracted to 3s and 4s H atom and 6s5p, for C and O atoms by a standard procedure. For quality evaluation of contracted basis sets in molecular calculations, we have accomplished calculations of total and orbital energies in the Hartree-Fock-Roothaaii (HFR) approach for CH, C(2) and CO molecules. The results obtained with the uncontracted basis sets are compared with values obtained with the standard D95, 6-311G basis sets and with values reported in the literature. The 4s and 6s5p basis sets are enriched with polarization and diffuse functions for atoms of the parent neutral systems and of the enolates anions (cycloheptanone enolate, 2,5-dimethyleyelopentanone enolate, 4-heptanone enolate, and di-isopropyl ketone enolate) from the literature, in order to assess their performance in ab initio molecular calculations, and applied for calculations of electron affinities of the enolates. The calculations were performed at the DFT (BLYP and B3LYP) and HF levels and compared with the corresponding experimental values and with those obtained by using other 6-3 1 + +G((*)) and 6-311 + +G((*)) basis sets from literature. For the enolates studied, the differences between the electron affinities obtained with GCHF basis sets, at the B3LYP level, and the experimental values are -0.001, -0,014, -0.001, and -0.001 eV. (C) 2002 Elsevier B.V. B.V. All rights reserved.

A pulsating Gaussian wave packet

A pulsating Gaussian wave packet for the harmonic oscillator is explicitly constructed. The mechanism for that behaviour is pointed out. © 1999 IOP Publishing Ltd.