On the distribution of the signs of the conjugates of the cyclotomic units in the maximal real subfield of the qth cyclotomic field, q A prime

Let F = Ǫ(ζ + ζ ^–1) be the maximal real subfield of the cyclotomic field Ǫ(ζ) where ζ is a primitive qth root of unity and q is an odd rational prime. The numbers u₁=-1, u_k=(ζ^k-ζ^-k)/(ζ-ζ^-1), k=2,…,p, p=(q-1)/2, are units in F and are called the cyclotomic units. In this thesis the sign distribution of the conjugates in F of the cyclotomic units is studied.

Let G(F/Ǫ) denote the Galoi's group of F over Ǫ, and let V denote the units in F. For each σϵ G(F/Ǫ) and μϵV define a mapping sgn_σ: V→GF(2) by sgn_σ(μ) = 1 iff σ(μ) ˂ 0 and sgn_σ(μ) = 0 iff σ(μ) ˃ 0. Let {σ₁, ... , σ_p} be a fixed ordering of G(F/Ǫ). The matrix M_q=(sgn_σj(v_i) ) , i, j = 1, ... , p is called the matrix of cyclotomic signatures. The rank of this matrix determines the sign distribution of the conjugates of the cyclotomic units. The matrix of cyclotomic signatures is associated with an ideal in the ring GF(2) [x] / (x^p+ 1) in such a way that the rank of the matrix equals the GF(2)-dimension of the ideal. It is shown that if p = (q-1)/ 2 is a prime and if 2 is a primitive root mod p, then M_q is non-singular. Also let p be arbitrary, let ℓ be a primitive root mod q and let L = {i | 0 ≤ i ≤ p-1, the least positive residue of defined by ℓⁱ mod q is greater than p}. Let H_q(x) ϵ GF(2)[x] be defined by H_q(x) = g. c. d. ((Σ xⁱ/I ϵ L) (x+1) + 1, x^p + 1). It is shown that the rank of M_q equals the difference p - degree H_q(x).

Further results are obtained by using the reciprocity theorem of class field theory. The reciprocity maps for a certain abelian extension of F and for the infinite primes in F are associated with the signs of conjugates. The product formula for the reciprocity maps is used to associate the signs of conjugates with the reciprocity maps at the primes which lie above (2). The case when (2) is a prime in F is studied in detail. Let T denote the group of totally positive units in F. Let U be the group generated by the cyclotomic units. Assume that (2) is a prime in F and that p is odd. Let F₍₂₎ denote the completion of F at (2) and let V₍₂₎ denote the units in F₍₂₎. The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is non-singular. 2) U∩T = U². 3) U∩F²₍₂₎ = U². 4) V₍₂₎/ V₍₂₎² = ˂v₁ V₍₂₎²˃ ʘ…ʘ˂v_p V₍₂₎²˃ ʘ ˂3V₍₂₎²˃.

The rank of M_q was computed for 5≤q≤929 and the results appear in tables. On the basis of these results and additional calculations the following conjecture is made: If q and p = (q -1)/ 2 are both primes, then M_q is non-singular.

Modelagem, simulação e otimização operacional de unidades de síntese de metanol

A indústria de processos químicos tem sofrido consideráveis transformações devido ao acirramento da competitividade. Importantes progressos tecnológicos têm sido atingidos através de técnicas de modelagem, simulação e otimização visando o aumento da lucratividade e melhoria contínua nos processos industriais. Neste contexto, as plantas de metanol, um dos mais importantes produtos petroquímicos, podem ser destacadas. Atualmente, a principal matéria-prima para obtenção de metanol é o gás natural. A produção do metanol é caracterizada por três etapas: geração de gás de síntese, conversão do gás de síntese em metanol (unidade de síntese ou loop de síntese) e purificação do produto na especificação requerida. Os custos fixos e variáveis da unidade de síntese são fortemente dependentes das variáveis operacionais, como temperatura, pressão, razão de reciclo e composição da carga. Desta forma, foi desenvolvido um conjunto de modelos e algoritmos computacionais para representar matematicamente unidades de síntese de metanol. O modelo apresenta operações unitárias associadas aos seguintes equipamentos: divisores de correntes, misturadores de correntes, compressores, trocadores de calor, vasos de flash e reatores. Inicialmente, foi proposto um simulador estacionário, que serviu como base para um pseudo-estacionário, o qual contempla a desativação do catalisador por sinterização térmica. Os simuladores foram criados segundo uma arquitetura seqüencial modular e empregou-se o método de substituição sucessiva para a convergência dos reciclos. O estudo envolveu dois fluxogramas típicos, um constituído por reatores adiabáticos em série, enquanto o outro constituído por um reator tipo quench. Uma análise do efeito das principais variáveis operacionais foi realizada para o reator e para o loop de síntese. Estudou-se também o efeito da desativação do catalisador ao longo do tempo. Uma ferramenta de otimização operacional foi empregada para alcançar a máxima produção manipulando as injeções de carga fria na entrada dos leitos catalíticos. Formulou-se também um problema de maximização do lucro em função da otimização de paradas da unidade para troca do catalisador. Os resultados obtidos apontam que a ferramenta desenvolvida é promissora para a compreensão e otimização da unidade objeto deste estudo

Generation of double pulses in-line by using reflective Dammann gratings

Doubled femtosecond laser pulses in-line are needed in the collinear pump-probe technique, collinear second harmonic generation frequency-resolved optical gating (SHG FROG) and the spectral phase interferometry for direct electric-field reconstruction (SPIDER), etc. Normally, it is generated by using a Michelson's structure. In this paper, we proposed a novel structure with two-layered reflective Dammann gratings and the reflective mirrors to generate doubled femtosecond laser pulses in line without transmission optical elements. Angular dispersion and spectral spatial walk-off are both compensated. In addition, this structure can also compress the positive chirped pulse, which cannot be realized with a Michelson's structure. By adopting triangular grating and blazed gratings, the efficiency of the system would in principle be increased as the Michelson's scheme. Experiments demonstrated that this method should be an alternative approach for generation of the double compressed pulses of femtosecond laser for practical applications. (c) 2006 Elsevier GmbH. All rights reserved.

Tsunamis - a model of their generation and propagation

A general solution is presented for water waves generated by an arbitrary movement of the bed (in space and time) in a two-dimensional fluid domain with a uniform depth. The integral solution which is developed is based on a linearized approximation to the complete (nonlinear) set of governing equations. The general solution is evaluated for the specific case of a uniform upthrust or downthrow of a block section of the bed; two time-displacement histories of the bed movement are considered.

An integral solution (based on a linear theory) is also developed for a three-dimensional fluid domain of uniform depth for a class of bed movements which are axially symmetric. The integral solution is evaluated for the specific case of a block upthrust or downthrow of a section of the bed, circular in planform, with a time-displacement history identical to one of the motions used in the two-dimensional model.

Since the linear solutions are developed from a linearized approximation of the complete nonlinear description of wave behavior, the applicability of these solutions is investigated. Two types of non-linear effects are found which limit the applicability of the linear theory: (1) large nonlinear effects which occur in the region of generation during the bed movement, and (2) the gradual growth of nonlinear effects during wave propagation.

A model of wave behavior, which includes, in an approximate manner, both linear and nonlinear effects is presented for computing wave profiles after the linear theory has become invalid due to the growth of nonlinearities during wave propagation.

An experimental program has been conducted to confirm both the linear model for the two-dimensional fluid domain and the strategy suggested for determining wave profiles during propagation after the linear theory becomes invalid. The effect of a more general time-displacement history of the moving bed than those employed in the theoretical models is also investigated experimentally.

The linear theory is found to accurately approximate the wave behavior in the region of generation whenever the total displacement of the bed is much less than the water depth. Curves are developed and confirmed by the experiments which predict gross features of the lead wave propagating from the region of generation once the values of certain nondimensional parameters (which characterize the generation process) are known. For example, the maximum amplitude of the lead wave propagating from the region of generation has been found to never exceed approximately one-half of the total bed displacement. The gross features of the tsunami resulting from the Alaskan earthquake of 27 March 1964 can be estimated from the results of this study.

Compliance verification methodology for renewable generation integration. Application to island power grids

261 p.

Phase-shifting apodizer for next-generation digital versatile disk

The next generation digital versatile disk (DVD) using blue lasers will have a capacity of 13 to 15 Gbytes. Compared with current DVD, the wavelength will be shorter and the numerical aperture (NA) will be higher. But with the increase of NA and decrease of wave length, the depth of focus (DOF) decrease rapidly, which makes it hard for the servo-system to track. We propose an optimized three-portion phase-shifting apodizer to increase the depth of focus and at the same time minimize the spot size, which makes the DOF of next generation DVD comparable to current DVD. The simulation result shows that an optical system with this apodizer also has a good defocus characteristic. (C) 2001 Society of Photo-Optical Instrumentation Engineers.

Second-harmonic generation in Ge20As25S55 glass irradiated by an electron beam

Second-harmonic generation was observed in Ge(20)AS(25)S(55) chalcogenide glass irradiated by an electron beam. The second-harmonic intensity increased with increasing electron-beam current and accelerating voltage. The second-harmonic generation in Ge20As25S55 glass was caused by the space-charge electrostatic field that was generated by irradiation of an electron beam. Second-order nonlinearity chi ((2)) as great as 0.8 pm/V was obtained. The results of measurements of thermally stimulated depolarization current indicated that the glass was poled in the thin layers of its surface (several micrometers) and that the nonlinearity was stable. (C) 2001 Optical Society of America.

Instances of combinatorial optimization problems: complexity and generation

138 p.