Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator

A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2^n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.

In Chapter 2, we apply the graded-commutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the graded-commutative algebra has to be generalized: the "signs" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.1-2.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of "sign". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free graded-commutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.

A reconstruction of the Beilinson-Bloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.

For the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and Connes-Karoubi character.

Free algebras in Von Neumann-Bernays-Gӧdel set theory and positive elementary inductions in reasonable structures

This thesis consists of two independent chapters. The first chapter deals with universal algebra. It is shown, in von Neumann-Bernays-Gӧdel set theory, that free images of partial algebras exist in arbitrary varieties. It follows from this, as set-complete Boolean algebras form a variety, that there exist free set-complete Boolean algebras on any class of generators. This appears to contradict a well-known result of A. Hales and H. Gaifman, stating that there is no complete Boolean algebra on any infinite set of generators. However, it does not, as the algebras constructed in this chapter are allowed to be proper classes. The second chapter deals with positive elementary inductions. It is shown that, in any reasonable structure ᶆ, the inductive closure ordinal of ᶆ is admissible, by showing it is equal to an ordinal measuring the saturation of ᶆ. This is also used to show that non-recursively saturated models of the theories ACF, RCF, and DCF have inductive closure ordinals greater than _ω.

Two-step digit-set-restricted modified signed-digit addition-subtraction algorithm and its optoelectronic implementation

A novel, to our knowledge, two-step digit-set-restricted modified signed-digit (MSD) addition-subtraction algorithm is proposed. With the introduction of the reference digits, the operand words are mapped into an intermediate carry word with all digits restricted to the set {(1) over bar, 0} and an intermediate sum word with all digits restricted to the set {0, 1}, which can be summed to form the final result without carry generation. The operation can be performed in parallel by use of binary logic. An optical system that utilizes an electron-trapping device is suggested for accomplishing the required binary logic operations. By programming of the illumination of data arrays, any complex logic operations of multiple variables can be realized without additional temporal latency of the intermediate results. This technique has a high space-bandwidth product and signal-to-noise ratio. The main structure can be stacked to construct a compact optoelectronic MSD adder-subtracter. (C) 1999 Optical Society of America.

One-step digit-set-restricted modified signed-digit adder using an incoherent correlator based on a shared content-addressable memory

An efficient one-step digit-set-restricted modified signed-digit (MSD) adder based on symbolic substitution is presented. In this technique, carry propagation is avoided by introducing reference digits to restrict the intermediate carry and sum digits to {1,0} and {0,1}, respectively. The proposed technique requires significantly fewer minterms and simplifies system complexity compared to the reported one-step MSD addition techniques. An incoherent correlator based on an optoelectronic shared content-addressable memory processor is suggested to perform the addition operation. In this technique, only one set of minterms needs to be stored, independent of the operand length. (C) 2002 society or Photo-Optical Instrumentation Engineers.

Efficient two-step digit-set-restricted modified signed-digit algorithm based on optoelectronic shared content-addressable memory

A two-step digit-set-restricted modified signed-digit (MSD) adder based on symbolic substitution is presented. In the proposed addition algorithm, carry propagation is avoided by using reference digits to restrict the intermediate MSD carry and sum digits into {(1) over bar ,0} and {0, 1}, respectively. The algorithm requires only 12 minterms to generate the final results, and no complementarity operations for nonzero outputs are involved, which simplifies the system complexity significantly. An optoelectronic shared content-addressable memory based on an incoherent correlator is used for experimental demonstration. (c) 2005 Society of Photo-Optical Instrumentation Engineers.

Efficient two-step digit-set-restricted modified signed-digit algorithm based on optoelectronic shared content-addressable memory

A two-step digit-set-restricted modified signed-digit (MSD) adder based on symbolic substitution is presented. In the proposed addition algorithm, carry propagation is avoided by using reference digits to restrict the intermediate MSD carry and sum digits into {(1) over bar ,0} and {0, 1}, respectively. The algorithm requires only 12 minterms to generate the final results, and no complementarity operations for nonzero outputs are involved, which simplifies the system complexity significantly. An optoelectronic shared content-addressable memory based on an incoherent correlator is used for experimental demonstration. (c) 2005 Society of Photo-Optical Instrumentation Engineers.

Algebraic Techniques in Coding Theory: Entropy Vectors, Frames, and Constrained Coding

The study of codes, classically motivated by the need to communicate information reliably in the presence of error, has found new life in fields as diverse as network communication, distributed storage of data, and even has connections to the design of linear measurements used in compressive sensing. But in all contexts, a code typically involves exploiting the algebraic or geometric structure underlying an application. In this thesis, we examine several problems in coding theory, and try to gain some insight into the algebraic structure behind them.

The first is the study of the entropy region - the space of all possible vectors of joint entropies which can arise from a set of discrete random variables. Understanding this region is essentially the key to optimizing network codes for a given network. To this end, we employ a group-theoretic method of constructing random variables producing so-called "group-characterizable" entropy vectors, which are capable of approximating any point in the entropy region. We show how small groups can be used to produce entropy vectors which violate the Ingleton inequality, a fundamental bound on entropy vectors arising from the random variables involved in linear network codes. We discuss the suitability of these groups to design codes for networks which could potentially outperform linear coding.

The second topic we discuss is the design of frames with low coherence, closely related to finding spherical codes in which the codewords are unit vectors spaced out around the unit sphere so as to minimize the magnitudes of their mutual inner products. We show how to build frames by selecting a cleverly chosen set of representations of a finite group to produce a "group code" as described by Slepian decades ago. We go on to reinterpret our method as selecting a subset of rows of a group Fourier matrix, allowing us to study and bound our frames' coherences using character theory. We discuss the usefulness of our frames in sparse signal recovery using linear measurements.

The final problem we investigate is that of coding with constraints, most recently motivated by the demand for ways to encode large amounts of data using error-correcting codes so that any small loss can be recovered from a small set of surviving data. Most often, this involves using a systematic linear error-correcting code in which each parity symbol is constrained to be a function of some subset of the message symbols. We derive bounds on the minimum distance of such a code based on its constraints, and characterize when these bounds can be achieved using subcodes of Reed-Solomon codes.

On the set of eigenvalues of a class of equimodular matrices

The structure of the set ϐ(A) of all eigenvalues of all complex matrices (elementwise) equimodular with a given n x n non-negative matrix A is studied. The problem was suggested by O. Taussky and some aspects have been studied by R. S. Varga and B.W. Levinger.

If every matrix equimodular with A is non-singular, then A is called regular. A new proof of the P. Camion-A.J. Hoffman characterization of regular matrices is given.

The set ϐ(A) consists of m ≤ n closed annuli centered at the origin. Each gap, ɤ, in this set can be associated with a class of regular matrices with a (unique) permutation, π(ɤ). The association depends on both the combinatorial structure of A and the size of the a_ii. Let A be associated with the set of r permutations, π₁, π₂,…, π_r, where each gap in ϐ(A) is associated with one of the π_k. Then r ≤ n, even when the complement of ϐ(A) has n+1 components. Further, if π(ɤ) is the identity, the real boundary points of ɤ are eigenvalues of real matrices equimodular with A. In particular, if A is essentially diagonally dominant, every real boundary point of ϐ(A) is an eigenvalues of a real matrix equimodular with A.

Several conjectures based on these results are made which if verified would constitute an extension of the Perron-Frobenius Theorem, and an algebraic method is introduced which unites the study of regular matrices with that of ϐ(A).

On the character of the phytoplankton of the Yakimvarskii Gulf (Lake Ladoga) [Translation from: Botanicheskii Zhurnal, T. 44, No. 9, 1311-1314, 1959]

The Laboratory of Lake Studies of the Academy of Sciences of the USSR in 1956 undertook this complicated study of Lake Ladoga. The plan of the laboratory made allowance for the detailed investigation of the phytoplankton of the lake during a series of years. In the present article are given the preliminary results of observations on the phytoplankton of the Yakimvarskii Bay from 1956 to 1957. Details are also given on the concentration of dissolved gases and biogenic substances of the bay.

Locally convex Riesz spaces and Archimedean quotient spaces

A Riesz space with a Hausdorff, locally convex topology determined by Riesz seminorms is called a locally convex Riesz space. A sequence {x_n} in a locally convex Riesz space L is said to converge locally to x ϵ L if for some topologically bounded set B and every real r ˃ 0 there exists N (r) and n ≥ N (r) implies x – x_n ϵ r^b. Local Cauchy sequences are defined analogously, and L is said to be locally complete if every local Cauchy sequence converges locally. Then L is locally complete if and only if every monotone local Cauchy sequence has a least upper bound. This is a somewhat more general form of the completeness criterion for Riesz – normed Riesz spaces given by Luxemburg and Zaanen. Locally complete, bound, locally convex Riesz spaces are barrelled. If the space is metrizable, local completeness and topological completeness are equivalent.

Two measures of the non-archimedean character of a non-archimedean Riesz space L are the smallest ideal A_o (L) such that quotient space is Archimedean and the ideal I (L) = { x ϵ L: for some 0 ≤ v ϵ L, n |x| ≤ v for n = 1, 2, …}. In general A_o (L) ᴝ I (L). If L is itself a quotient space, a necessary and sufficient condition that A_o (L) = I (L) is given. There is an example where A_o (L) ≠ I (L).

A necessary and sufficient condition that a Riesz space L have every quotient space Archimedean is that for every 0 ≤ u, v ϵ L there exist u₁ = sup (inf (n v, u): n = 1, 2, …), and real numbers m₁ and m₂ such that m₁ u₁ ≥ v₁ and m₂ v₁ ≥ u₁. If, in addition, L is Dedekind σ – complete, then L may be represented as the space of all functions which vanish off finite subsets of some non-empty set.

Feedback communication using orthogonal signals

This research is concerned with block coding for a feedback communication system in which the forward and feedback channels are independently disturbed by additive white Gaussian noise and average power constrained. Two coding schemes are proposed in which the messages to be coded for transmission over the forward channel are realized as a set of orthogonal waveforms. A finite number of forward and feedback transmissions (iterations) per message is made. Information received over the feedback channel is used to modify the waveform transmitted on successive forward iterations in such a way that the expected value of forward signal energy is zero on all iterations after the first. Similarly, information is sent over the feedback channel in such a way that the expected value of feedback signal energy is also zero on all iterations after the first. These schemes are shown to achieve a lower probability of error than the best one-way coding scheme at all rates up to the forward channel capacity, provided only that the feedback channel capacity be greater than the forward channel capacity. These schemes make more efficient use of the available feedback power than existing feedback coding schemes, and therefore require less feedback power to achieve a given error performance.

A novel three-dimensional imaging array recording and computational method by means of coded cameras reconstruction

In this paper, we propose a novel three-dimensional imaging method by which the object is captured by a coded cameras array (CCA) and computationally reconstructed as a series of longitudinal layered surface images of the object. The distribution of cameras in array, named code pattern, is crucial for reconstructed images fidelity when the correlation decoding is used. We use DIRECT global optimization algorithm to design the code patterns that possess proper imaging property. We have conducted primary experiments to verify and test the performance of the proposed method with a simple discontinuous object and a small-scale CCA including nine cameras. After certain procedures such as capturing, photograph integrating, computational reconstructing and filtering, etc., we obtain reconstructed longitudinal layered surface images of the object with higher signal-to-noise ratio. The results of experiments show that the proposed method is feasible. It is a promising method to be used in fields such as remote sensing, machine vision, etc. (c) 2006 Elsevier GmbH. All rights reserved.

A multiple scattering problem

The present work deals with the problem of the interaction of the electromagnetic radiation with a statistical distribution of nonmagnetic dielectric particles immersed in an infinite homogeneous isotropic, non-magnetic medium. The wavelength of the incident radiation can be less, equal or greater than the linear dimension of a particle. The distance between any two particles is several wavelengths. A single particle in the absence of the others is assumed to scatter like a Rayleigh-Gans particle, i.e. interaction between the volume elements (self-interaction) is neglected. The interaction of the particles is taken into account (multiple scattering) and conditions are set up for the case of a lossless medium which guarantee that the multiple scattering contribution is more important than the self-interaction one. These conditions relate the wavelength λ and the linear dimensions of a particle a and of the region occupied by the particles D. It is found that for constant λ/a, D is proportional to λ and that |Δχ|, where Δχ is the difference in the dielectric susceptibilities between particle and medium, has to lie within a certain range.

The total scattering field is obtained as a series the several terms of which represent the corresponding multiple scattering orders. The first term is a single scattering term. The ensemble average of the total scattering intensity is then obtained as a series which does not involve terms due to products between terms of different orders. Thus the waves corresponding to different orders are independent and their Stokes parameters add.

The second and third order intensity terms are explicitly computed. The method used suggests a general approach for computing any order. It is found that in general the first order scattering intensity pattern (or phase function) peaks in the forward direction Θ = 0. The second order tends to smooth out the pattern giving a maximum in the Θ = π/2 direction and minima in the Θ = 0 , Θ = π directions. This ceases to be true if ka (where k = 2π/λ) becomes large (> 20). For large ka the forward direction is further enhanced. Similar features are expected from the higher orders even though the critical value of ka may increase with the order.

The first order polarization of the scattered wave is determined. The ensemble average of the Stokes parameters of the scattered wave is explicitly computed for the second order. A similar method can be applied for any order. It is found that the polarization of the scattered wave depends on the polarization of the incident wave. If the latter is elliptically polarized then the first order scattered wave is elliptically polarized, but in the Θ = π/2 direction is linearly polarized. If the incident wave is circularly polarized the first order scattered wave is elliptically polarized except for the directions Θ = π/2 (linearly polarized) and Θ = 0, π (circularly polarized). The handedness of the Θ = 0 wave is the same as that of the incident whereas the handedness of the Θ = π wave is opposite. If the incident wave is linearly polarized the first order scattered wave is also linearly polarized. The second order makes the total scattered wave to be elliptically polarized for any Θ no matter what the incident wave is. However, the handedness of the total scattered wave is not altered by the second order. Higher orders have similar effects as the second order.

If the medium is lossy the general approach employed for the lossless case is still valid. Only the algebra increases in complexity. It is found that the results of the lossless case are insensitive in the first order of k_imD where k_im = imaginary part of the wave vector k and D a linear characteristic dimension of the region occupied by the particles. Thus moderately extended regions and small losses make (k_imD)² ≪ 1 and the lossy character of the medium does not alter the results of the lossless case. In general the presence of the losses tends to reduce the forward scattering.

Trajetórias de jovens de origem popular rumo à carreira acadêmica: mobilidade social, identidade e conflitos

Esta dissertação tem como objetivo analisar a trajetória escolar de jovens de origem popular e oriundos de um bairro de periferia da cidade do Rio de Janeiro, que ingressaram em cursos de pós-graduação stricto sensu de importantes universidades públicas. Tais trajetórias serão aqui analisadas a partir das narrativas dos próprios atores sociais em questão, buscando evidenciar as marcas de sua socialização e formação escolar em um contexto social que, do ponto de vista de algumas análises no campo da sociologia da educação, não favoreceriam a aquisição do capital cultural e social necessários ao ingresso na carreira acadêmica. Desta perspectiva, além de não serem muito comuns nos meios populares, essas trajetórias também não seriam reconhecidas pelo grupo de origem que, na maioria das vezes, identifica a formação escolar como porta de entrada no mundo do trabalho, tendo no curso superior o ponto máximo de uma formação escolar bem sucedida. Busca-se aqui, a partir da análise do conjunto dessas trajetórias, apreender os elementos e experiências sociais que possibilitaram esse prolongamento na formação escolar, e, sobretudo, o impacto e o significado da inserção na carreira acadêmica, tanto do ponto de vista da mobilidade social quanto dos conflitos decorrentes dessa experiência subjetiva, quer em relação às expectativas familiares ou do grupo social de origem. O trabalho de campo de caráter etnográfico, baseado na observação participante e realização de entrevistas aprofundadas constituíram as ferramentas metodológicas básicas a partir das quais essa pesquisa foi desenvolvida. Com relação à discussão teórica aqui proposta tomou-se como referência os trabalhos de Pierre Bourdieu, Jean-Claude Passeron, Bernard Lahire, Jailson de Souza e Silva e Maria da Graça Jacintho Setton, entre outros