A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

Aplicação das fórmulas de Vincenty nos cálculos das correções dos efeitos do relevo na gravidade e na altura geoidal.

O presente trabalho apresenta a aplicação das fórmulas de Vincenty nos cálculos das correções do terreno e do efeito indireto, que desempenham papel relevante na construção de cartas geoidais. Implementa-se um programa de processamento que realiza a integração numérica sobre o modelo digital do terreno, discretizado em células triangulares de Delaunay. O sistema foi desenvolvido com a linguagem de programação FORTRAN, para a execução de intensos algoritmos numéricos usando compiladores livres e robustos. Para o cálculo do efeito indireto, considera-se a redução gravimétrica efetuada com base no segundo método de condensação de Helmert, face ao pequeno valor de efeito indireto no cálculo do geóide, em função da mudança que este produz no potencial da gravidade devido ao deslocamento da massa topográfica. Utiliza-se, o sistema geodésico SIRGAS 2000 como sistema de referência para o cômputo das correções. Simplificando o exame dos resultados alcançados, distingue-se o processamento e desenvolvimento do trabalho em etapas como a escolha de ferramentas geodésicas para máxima precisão dos resultados, elaboração de subrotinas e comparação de resultados com cálculos anteriores. Os resultados encontrados foram de geração sadia e satisfatória e podem ser perfeitamente empregados no cálculo do geóide em qualquer área do globo.

Some properties of the coefficients of cyclotomic polynomials

An explicit formula is obtained for the coefficients of the cyclotomic polynomial F_n(x), where n is the product of two distinct odd primes. A recursion formula and a lower bound and an improvement of Bang’s upper bound for the coefficients of F_n(x) are also obtained, where n is the product of three distinct primes. The cyclotomic coefficients are also studied when n is the product of four distinct odd primes. A recursion formula and upper bounds for its coefficients are obtained. The last chapter includes a different approach to the cyclotomic coefficients. A connection is obtained between a certain partition function and the cyclotomic coefficients when n is the product of an arbitrary number of distinct odd primes. Finally, an upper bound for the coefficients is derived when n is the product of an arbitrary number of distinct and odd primes.

Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation

An analytical formula for the cross-spectral density matrix of the electric field of anisotropic electromagnetic Gaussian-Schell model beams propagating in free space is derived by using a tensor method. The effects of coherence on those beams are studied. It is shown that two anisotropic stochastic electromagnetic beams that propagate from the source plane z = 0 into the half-space z > 0 may have different beam shapes (i.e., spectral density) and states of polarization in the half-space, even though they have the same beam shape and states of polarization in the source plane. This fact is due to a difference in the coherence properties of the field in the source plane. (C) 2007 Optical Society of America.

Incremental Control Synthesis for Robotics in the Presence of Temporal Logic Specifications

This thesis presents methods for incrementally constructing controllers in the presence of uncertainty and nonlinear dynamics. The basic setting is motion planning subject to temporal logic specifications. Broadly, two categories of problems are treated. The first is reactive formal synthesis when so-called discrete abstractions are available. The fragment of linear-time temporal logic (LTL) known as GR(1) is used to express assumptions about an adversarial environment and requirements of the controller. Two problems of changes to a specification are posed that concern the two major aspects of GR(1): safety and liveness. Algorithms providing incremental updates to strategies are presented as solutions. In support of these, an annotation of strategies is developed that facilitates repeated modifications. A variety of properties are proven about it, including necessity of existence and sufficiency for a strategy to be winning. The second category of problems considered is non-reactive (open-loop) synthesis in the absence of a discrete abstraction. Instead, the presented stochastic optimization methods directly construct a control input sequence that achieves low cost and satisfies a LTL formula. Several relaxations are considered as heuristics to address the rarity of sampling trajectories that satisfy an LTL formula and demonstrated to improve convergence rates for Dubins car and single-integrators subject to a recurrence task.

一种步进扫描投影光刻机承片台不平度检测新技术

提出一种步进扫描投影光刻机承片台不平度检测新技术。在晶圆与承片台存在不同偏移量时,利用线性差分传感器在线测量晶圆上不同点的局部高度;通过建立临时边界条件,以递推法消除晶圆面形影响,并逐行计算出承片台的相对不平度;通过逐行计算的结果递推相邻行之间的高度差,并将该高度差叠加到每一行,以消除临时边界条件的限制,得到处于同一高度上的承片台不平度;将计算的结果作为初始值,根据最小二乘原理,以邻近的四个测量点作为参考,逐步逼近得到承片台的真实不平度。计算机仿真结果验证了该检测方法的正确性,计算结果逐步收敛并逼近真实值

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.

I. Efforts toward the synthesis of aliphatic iodonium salts. II. Flourine-19 nuclear magnetic resonance spectroscopy of cyclic and bicyclic systems

The synthesis of iodonium salts of the general formula [C₆H₅IR]⁺X^-, where R is an alkyl group and x- is a stabilizing anion, was attempted. For the choice of R three groups were selected, whose derivatives are known to be sluggish in S_N1 and S_N2 substitutions: cyclopropyl, 7, 7 -dimethyl-1-norbornyl, and 9 -triptycyl. The synthetic routes followed along classical lines which have been exploited in recent years by Beringer and students. Ultimately, the object of the present study was to study the reactions of the above salts with nucleophiles. In none of the three cases, however, was it possible to isolate a stable salt. A thermodynamic argument suggests that this must be due to kinetic instability rather than thermodynamic instability. Only iodocyclopropane and 1-iodoapocamphane formed isolable iododichlorides.

Several methylated 2, 2-difluoronorbornanes were prepared with the intent of correlating fluorine -19 chemical shifts with geometric features in a rigid system. The effect of a methyl group on the shielding of a β -fluorine is dependent upon the dihedral angle; the maximum effect (an upfield shift of the resonance) occurs at 0° and 180°, whereas almost no effect is felt at a dihedral angle of 120°. The effect of a methyl group on a γ -fluorine is to strongly shift the resonance downfield when fluorine and methyl group are in a 1, 3 - diaxial-like relationship. Molecular orbital calculations of fluorine shielding in a variety of molecules were carried out using the formalism developed by Pople; the results are, at best, in modest agreement with experiment.

Synthesis and structural characterisation of [Cu2[(PDC)2(bpa)(H2O)2]•nH2O•mDMF (n=3 and m=1 for 1, n=7 and m=0 for 2, PDC=2,5-pyrindinedicarboxylate and bpa= 1,2-di(4-pyridil)ethane))

[EN] Combination of polycarboxylate anions and dipyridyl ligands is an effective strategy to produce solid coordination frameworks (SCF) which are crystalline materials based on connections between metal ions through organic ligands. In this sense, combination of polycarboxylate anions and dipyridyl ligands is an effective strategy to produce extended structures. In this context, this work is focused on two novel CuII-based SCFs exhibiting PDC (2,5-pyridinedicarboxylate) and bpa (1,2-di(4-pyridyl)ethane), being the first structures reported in literature containing both ligands. Chemical formula are [Cu2[(PDC)2(bpa)(H2O)2]•3H2O•DMF (1), and [Cu2(PDC)2(bpa)(H2O)2]•7H2O (2), where DMF is dimethylformamide. Compounds 1 and 2 have been characterized by means of XRD, IR, TG/DTG, and DTA analysis.

Influence of local geology on earthquake ground motion

As a simplified approach for estimating theoretically the influence of local subsoils upon the ground motion during an earthquake, the problem of an idealized layered system subjected to vertically incident plane body waves was studied. Both the technique of steady-state analysis and the technique of transient analysis have been used to analyze the problem.

In the steady-state analysis, a recursion formula has been derived for obtaining the response of a layered system to sinusoidally steady-state input. Several conclusions are drawn concerning the nature of the amplification spectrum of a nonviscous layered system having its layer stiffnesses increasing with depth. Numerical examples are given to demonstrate the effect of layer parameters on the amplification spectrum of a layered system.

In the transient analysis, two modified shear beam models have been established for obtaining approximately the response of a layered system to earthquake-like excitation. The method of continuous modal analysis was adopted for approximate analysis of the models, with energy dissipation in the layers, if any, taken into account. Numerical examples are given to demonstrate the accuracy of the models and the effect of a layered system in modifying the input motion.

Conditions are established, under which the theory is applicable to predict the influence of local subsoils on the ground motion during an earthquake. To demonstrate the applicability of the models to actual cases, three examples of actually recorded earthquake events are examined. It is concluded that significant modification of the incoming seismic waves, as predicted by the theory, is likely to occur in well defined soft subsoils during an earthquake, provided that certain conditions concerning the nature of the incoming seismic waves are satisfied.

Quantificação da área ocupada pela guta-percha e pelo Resilon em canais achatados após termoplastificação

O presente trabalho teve por objetivo quantificar, comparativamente, a área de preenchimento de dois materiais obturadores sólidos, cones de guta-percha (GP) e cones de Resilon (R), no terço apical de incisivos inferiores humanos, ex vivo, obturados pela técnica da onda contínua de condensação. Os espécimes foram submetidos a um protocolo, desde a cirurgia de acesso até o final do preparo químico-mecânico e divididos aleatoriamente em dois grupos, de 21 dentes cada, de acordo com o material utilizado. Não foi utilizado cimento endodôntico em nenhuma das amostras. Após a obturação, as amostras foram seccionadas transversalmente em dois níveis, a 3 e a 5mm do ápice, e subdivididas em grupos de acordo com a altura de corte e do material obturador, sendo estabelecido: GP3 (guta-percha com corte a 3mm), GP5 (guta-percha com corte a 5mm), R3 (Resilon com corte a 3mm) e R5 (Resilon com corte a 5mm). Posteriormente, as amostras foram submetidas a um processo de lixamento e polimento e examinadas em microscópio óptico por reflexão com aumento de 50x a 100x. Para a análise e processamento digital das imagens, foi utilizado o sistema de imagens Axio Vision 4.6 para Windows, sendo obtidas as medidas para cada área observada em micrômetros (μm); uma da área da cavidade, e outra da área de material obturador. Foi aferido o grau de circularidade de cada amostra, por uma fórmula matemática utilizada automaticamente pelo programa, onde 1 (um) é considerado o círculo perfeito e, quanto mais achatado o canal, mais tendente a 0 (zero) nesta escala. Obteve-se a área do canal, a circularidade de 0 a 1, a área preenchida pelo material obturador e, a porcentagem da área de preenchimento do material obturador em relação à área do canal. Foi realizado o cruzamento dos grupos dois a dois pelo teste t de Student, sendo verificada diferença estatisticamente significante entre os grupos GP3 e R3, tendo o grupo R3 apresentado maior porcentagem de área do canal radicular preenchida pelo material obturador em suas amostras (p<0,05). Na relação da circularidade com a quantidade de preenchimento, com o teste de Correlação de Pearson, não foi observada forte correlação entre a forma final do canal (relação de circularidade) e a quantidade de preenchimento do canal radicular pelos materiais obturadores testados. Conclui-se que houve grande variação de preenchimento mínimo e máximo em todos os grupos testados e o Resilon apresentou maior porcentagem de preenchimento de área do canal radicular em suas amostras.

Museos y turismo: expectativas y realidades

Cap. 1. La interrelación entre los sistemas turístico y patrimonial: más allá de los discursos apologéticos y las prácticas reduccionistas. Iñaki Arrieta Urtizberea Cap. 2. Turistas y museos. Apocalípticos e integrados. José Antonio Donaire. Cap. 3. Turismo cultural. Ficciones sobre realidades, realidades sobre invenciones. Agustín Santana Talavera, Pablo Díaz Rodríguez y Alberto Jonay Rodríguez Darias. Cap. 4. ¿Museos a la deriva o continentes a la deriva?: consecuencias de la crisis financiera para los museos de América del Norte, Yves Bergeron. Cap. 5. Patrimonio histórico, turismo, economía: ¿un desafío o una alianza? El caso de Populonia (Toscana, Italia). Daniele Manacorda. Cap. 6. Diagnóstico posrevolucionario en Túnez: delirio turístico, fiebre museística y la locura del jazmín. Habib Saidi. Cap. 7. Patrimonio etnológico: ¿recurso socioeconómico o instrumento sociopolítico? El caso de los Astilleros Nereo de Málaga. Esther Fernández de Paz. Cap. 8. De Rampas y Pasarelas: los museos Guggenheim como espacios artísticos genéricos. Sophia Carmen Vackimes. Cap. 9. El patrimonio como fuente de desarrollo sostenible en las regiones del interior norte de Portugal: el caso del municipio de Vieira do Minho. Eduardo Jorge Duque. Cap. 10. Museos, turismo y desarrollo local: el caso de Belmonte, Portugal. Luís Silva. Cap. 11. ¿Existen razones de eficiencia económica en las decisiones de cierre parcial de algunos museos locales? Análisis del caso del Museo Darder (Banyoles) en el contexto de los museos de Cataluña. Gabriel Alcalde, Josep Burch, Modest Fluvià, Ricard Rigall i Torrent y Albert Saló.

Geometrical-transformation approach to optical two-dimensional beam shaping: comment

I show that the research reported by Arieli et al. [Appl. Opt. 86, 9129 (1997)] has two serious mistakes: One is that an important factor is lost in the formula used in that study to determine the x-direction coordinate transformation; the other is the conclusion that the geometrical-transformation approach given by Arieli et al. can provide a smooth phase distribution. A potential research direction for obtaining a smooth phase distribution for a generic two-dimensional beam-shaping problem is stated. (C) 1998 Optical Society of America.

光通过纳米颗粒随机散射体透射光强的计算及分析

光通过随机散射体后透射光强的估算,对于提取散射体内部无法直接测量或者无法直接观察的信息至关重要。在分析平行光束通过纳米圆形颗粒随机散射体出射面上光强组成的基础上推导了透射光强的计算公式,并指出多重散射理论、一阶多重散射理论和朗伯比尔定律在一定近似程度上可以相对精确地估算透射光强。这三种方法的估算结果之间会出现四种相对关系。根据这些相对关系,分析了光在散射体内部传输时散射过程的特征以及各种散射过程对出射面光强的贡献大小。

柯林斯公式的逆运算及其在波面重构中的应用

给出描述光波通过傍轴光学系统衍射的柯林斯公式及其逆运算表达式，然后将数字全息检测研究中的物平面及CCD探测平面视为ABCD系统的输入平面及输出平面，提出通过柯林斯公式的逆运算进行波面重构的一种计算方法。为便于该方法的实际应用，研究了参考光是球面波时全息图的频谱结构，并从理论上证明，全息图的频率空问中存在可以通过滤波器分离的物光及共轭物光的频谱。最后．通过理论模拟及实验研究证明了使用柯林斯公式的逆运算重构波面的可能性。