Modal liquid crystal microaxicon array

A novel tunable liquid crystal microaxicon array is proposed and experimentally demonstrated. The proposed structure is capable of generating tunable axicons (thousands of elements) of micrometric size, with simple control (four control voltages) and low voltage, and is totally reconfigurable. Depending on the applied voltages, control over the diameter, as well as the effective wedge angle, can be achieved. Controls over the diameter ranging from 107 to 77 μm have been demonstrated. In addition, a control over the phase profile tunability, from 12π to 24π radians, has been demonstrated. This result modifies the effective cone angle. The diameter tunability, as well the effective cone angle, results in a control over the nondiffractive Bessel beam distance. The RMS wavefront deviation from the ideal axicon is only λ∕3. The proposed device has several advantages over the existing microaxicon arrays, including being simple having a low cost. The device could contribute to developing new applications and to reducing the fabrication costs of current devices.

A framework for implementing outsourcing schemes

En esta tesis se aborda el problema de la externalización segura de servicios de datos y computación. El escenario de interés es aquel en el que el usuario posee datos y quiere subcontratar un servidor en la nube (“Cloud”). Además, el usuario puede querer también delegar el cálculo de un subconjunto de sus datos al servidor. Se presentan dos aspectos de seguridad relacionados con este escenario, en concreto, la integridad y la privacidad y se analizan las posibles soluciones a dichas cuestiones, aprovechando herramientas criptográficas avanzadas, como el Autentificador de Mensajes Homomórfico (“Homomorphic Message Authenticators”) y el Cifrado Totalmente Homomórfico (“Fully Homomorphic Encryption”). La contribución de este trabajo es tanto teórica como práctica. Desde el punto de vista de la contribución teórica, se define un nuevo esquema de externalización (en lo siguiente, denominado con su término inglés Outsourcing), usando como punto de partida los artículos de [3] y [12], con el objetivo de realizar un modelo muy genérico y flexible que podría emplearse para representar varios esquemas de ”outsourcing” seguro. Dicho modelo puede utilizarse para representar esquemas de “outsourcing” seguro proporcionando únicamente integridad, únicamente privacidad o, curiosamente, integridad con privacidad. Utilizando este nuevo modelo también se redefine un esquema altamente eficiente, construido en [12] y que se ha denominado Outsourcinglin. Este esquema permite calcular polinomios multivariados de grado 1 sobre el anillo Z2k . Desde el punto de vista de la contribución práctica, se ha construido una infraestructura marco (“Framework”) para aplicar el esquema de “outsourcing”. Seguidamente, se ha testado dicho “Framework” con varias implementaciones, en concreto la implementación del criptosistema Joye-Libert ([18]) y la implementación del esquema propio Outsourcinglin. En el contexto de este trabajo práctico, la tesis también ha dado lugar a algunas contribuciones innovadoras: el diseño y la implementación de un nuevo algoritmo de descifrado para el esquema de cifrado Joye-Libert, en colaboración con Darío Fiore. Presenta un mejor comportamiento frente a los algoritmos propuestos por los autores de [18];la implementación de la función eficiente pseudo-aleatoria de forma amortizada cerrada (“amortized-closed-form efficient pseudorandom function”) de [12]. Esta función no se había implementado con anterioridad y no supone un problema trivial, por lo que este trabajo puede llegar a ser útil en otros contextos. Finalmente se han usado las implementaciones durante varias pruebas para medir tiempos de ejecución de los principales algoritmos.---ABSTRACT---In this thesis we tackle the problem of secure outsourcing of data and computation. The scenario we are interested in is that in which a user owns some data and wants to “outsource” it to a Cloud server. Furthermore, the user may want also to delegate the computation over a subset of its data to the server. We present the security issues related to this scenario, namely integrity and privacy and we analyse some possible solutions to these two issues, exploiting advanced cryptographic tools, such as Homomorphic Message Authenticators and Fully Homomorphic Encryption. Our contribution is both theoretical and practical. Considering our theoretical contribution, using as starting points the articles of [3] and [12], we introduce a new cryptographic primitive, called Outsourcing with the aim of realizing a very generic and flexible model that might be employed to represent several secure outsourcing schemes. Such model can be used to represent secure outsourcing schemes that provide only integrity, only privacy or, interestingly, integrity with privacy. Using our new model we also re-define an highly efficient scheme constructed in [12], that we called Outsourcinglin and that is a scheme for computing multi-variate polynomials of degree 1 over the ring Z2k. Considering our practical contribution, we build a Framework to implement the Outsourcing scheme. Then, we test such Framework to realize several implementations, specifically the implementation of the Joye-Libert cryptosystem ([18]) and the implementation of our Outsourcinglin scheme. In the context of this practical work, the thesis also led to some novel contributions: the design and the implementation, in collaboration with Dario Fiore, of a new decryption algorithm for the Joye-Libert encryption scheme, that performs better than the algorithms proposed by the authors in [18]; the implementation of the amortized-closed-form efficient pseudorandom function of [12]. There was no prior implementation of this function and it represented a non trivial work, which can become useful in other contexts. Finally we test the implementations to execute several experiments for measuring the timing performances of the main algorithms.

Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined, for a system $\cP$ of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from $\cP$ through derivations and multiplications by Laurent monomials. To start, through derivations, a system $\ps(\cP)$ of $L$ polynomials in $L-1$ algebraic variables is obtained, which is non sparse in the order of derivation. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in $\ps(\cP)$, to obtain polynomials in the differential elimination ideal generated by $\cP$. The formulas obtained are multiples of the sparse differential resultant defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case.

Calibración geométrica de cámaras no métricas. Estudio de metodologías y modelos matemáticos de distorsión

La Fotogrametría, como ciencia y técnica de obtención de información tridimensional del espacio objeto a partir de imágenes bidimensionales, requiere de medidas de precisión y en ese contexto, la calibración geométrica de cámaras ocupa un lugar importante. El conocimiento de la geometría interna de la cámara es fundamental para lograr mayor precisión en las medidas realizadas. En Fotogrametría Aérea se utilizan cámaras métricas (fabricadas exclusivamente para aplicaciones cartográficas), que incluyen objetivos fotográficos con sistemas de lentes complejos y de alta calidad. Pero en Fotogrametría de Objeto Cercano se está trabajando cada vez con más asiduidad con cámaras no métricas, con ópticas de peor calidad que exigen una calibración geométrica antes o después de cada trabajo. El proceso de calibración encierra tres conceptos fundamentales: modelo de cámara, modelo de distorsión y método de calibración. El modelo de cámara es un modelo matemático que aproxima la transformación proyectiva original a la realidad física de las lentes. Ese modelo matemático incluye una serie de parámetros entre los que se encuentran los correspondientes al modelo de distorsión, que se encarga de corregir los errores sistemáticos de la imagen. Finalmente, el método de calibración propone el método de estimación de los parámetros del modelo matemático y la técnica de optimización a emplear. En esta Tesis se propone la utilización de un patrón de calibración bidimensional que se desplaza en la dirección del eje óptico de la cámara, ofreciendo así tridimensionalidad a la escena fotografiada. El patrón incluye un número elevado de marcas, lo que permite realizar ensayos con distintas configuraciones geométricas. Tomando el modelo de proyección perspectiva (o pinhole) como modelo de cámara, se realizan ensayos con tres modelos de distorsión diferentes, el clásico de distorsión radial y tangencial propuesto por D.C. Brown, una aproximación por polinomios de Legendre y una interpolación bicúbica. De la combinación de diferentes configuraciones geométricas y del modelo de distorsión más adecuado, se llega al establecimiento de una metodología de calibración óptima. Para ayudar a la elección se realiza un estudio de las precisiones obtenidas en los distintos ensayos y un control estereoscópico de un panel test construido al efecto. ABSTRACT Photogrammetry, as science and technique for obtaining three-dimensional information of the space object from two-dimensional images, requires measurements of precision and in that context, the geometric camera calibration occupies an important place. The knowledge of the internal geometry of the camera is fundamental to achieve greater precision in measurements made. Metric cameras (manufactured exclusively for cartographic applications), including photographic lenses with complex lenses and high quality systems are used in Aerial Photogrammetry. But in Close Range Photogrammetry is working increasingly more frequently with non-metric cameras, worst quality optical components which require a geometric calibration before or after each job. The calibration process contains three fundamental concepts: camera model, distortion model and method of calibration. The camera model is a mathematical model that approximates the original projective transformation to the physical reality of the lenses. The mathematical model includes a series of parameters which include the correspondents to the model of distortion, which is in charge of correcting the systematic errors of the image. Finally, the calibration method proposes the method of estimation of the parameters of the mathematical modeling and optimization technique to employ. This Thesis is proposing the use of a pattern of two dimensional calibration that moves in the direction of the optical axis of the camera, thus offering three-dimensionality to the photographed scene. The pattern includes a large number of marks, which allows testing with different geometric configurations. Taking the projection model perspective (or pinhole) as a model of camera, tests are performed with three different models of distortion, the classical of distortion radial and tangential proposed by D.C. Brown, an approximation by Legendre polynomials and bicubic interpolation. From the combination of different geometric configurations and the most suitable distortion model, brings the establishment of a methodology for optimal calibration. To help the election, a study of the information obtained in the various tests and a purpose built test panel stereoscopic control is performed.

Decision boundary for discrete Bayesian network classifiers

Bayesian network classifiers are a powerful machine learning tool. In order to evaluate the expressive power of these models, we compute families of polynomials that sign-represent decision functions induced by Bayesian network classifiers. We prove that those families are linear combinations of products of Lagrange basis polynomials. In absence of V -structures in the predictor sub-graph, we are also able to prove that this family of polynomials does indeed characterize the specific classifier considered. We then use this representation to bound the number of decision functions representable by Bayesian network classifiers with a given structure.

On the classification of binary shifts of finite commutant index

We provide a complete classification up to conjugacy of the binary shifts of finite commutant index on the hyperfinite II1, factor. There is a natural correspondence between the conjugacy classes of these shifts and polynomials over GF(2) satisfying a certain duality condition.

Do cyanobacteria swim using traveling surface waves?

Bacteria that swim without the benefit of flagella might do so by generating longitudinal or transverse surface waves. For example, swimming speeds of order 25 microns/s are expected for a spherical cell propagating longitudinal waves of 0.2 micron length, 0.02 micron amplitude, and 160 microns/s speed. This problem was solved earlier by mathematicians who were interested in the locomotion of ciliates and who considered the undulations of the envelope swept out by ciliary tips. A new solution is given for spheres propagating sinusoidal waveforms rather than Legendre polynomials. The earlier work is reviewed and possible experimental tests are suggested.

Método da propagação de feixe de ângulo largo para análise de guias de ondas ópticos não-lineares

Este trabalho propõe uma extensão do método de propagação de feixe (BPM - Beam Propagation Method) para a análise de guias de ondas ópticos e acopladores baseados em materiais não-lineares do tipo Kerr. Este método se destina à investigação de estruturas onde a utilização da equação escalar de Helmholtz (EEH) em seu limite paraxial não mais se aplica. Os métodos desenvolvidos para este fim são denominados na literatura como métodos de propagação de feixe de ângulo largo. O formalismo aqui desenvolvido é baseado na técnica das diferenças finitas e nos esquemas de Crank-Nicholson (CN) e Douglas generalizado (GD). Estes esquemas apresentam como característica o fato de apresentarem um erro de truncamento em relação ao passo de discretização transversal, Δx, proporcional a O(Δx2) para o primeiro e O(Δx4). A convergência do método em ambos esquemas é otimizada pela utilização de um algoritmo interativo para a correção do campo no meio não-linear. O formalismo de ângulo largo é obtido pela expansão da EEH para os esquemas CN e GD em termos de polinômios aproximantes de Padé de ordem (1,0) e (1,1) para CN e GD, e (2,2) e (3,3) para CN. Os aproximantes de ordem superior a (1,1) apresentam sérios problemas de estabilidade. Este problema é eliminado pela rotação dos aproximantes no plano complexo. Duas condições de contorno nos extremos da janela computacional são também investigadas: 1) (TBC - Transparent Boundary Condition) e 2) condição de contorno absorvente (TAB - Transparent Absorbing Boundary). Estas condições de contorno possuem a facilidade de evitar que reflexões indesejáveis sejam transmitidas para dentro da janela computacional. Um estudo comparativo da influência destas condições de contorno na solução de guias de ondas ópticos não-lineares é também abordada neste trabalho.

Modelos de mapas simpléticos para o movimento de deriva elétrica com efeitos de raio de Larmor finito

Mapas simpléticos têm sido amplamente utilizados para modelar o transporte caótico em plasmas e fluidos. Neste trabalho, propomos três tipos de mapas simpléticos que descrevem o movimento de deriva elétrica em plasmas magnetizados. Efeitos de raio de Larmor finito são incluídos em cada um dos mapas. No limite do raio de Larmor tendendo a zero, o mapa com frequência monotônica se reduz ao mapa de Chirikov-Taylor, e, nos casos com frequência não-monotônica, os mapas se reduzem ao mapa padrão não-twist. Mostramos como o raio de Larmor finito pode levar à supressão de caos, modificar a topologia do espaço de fases e a robustez de barreiras de transporte. Um método baseado na contagem dos tempos de recorrência é proposto para analisar a influência do raio de Larmor sobre os parâmetros críticos que definem a quebra de barreiras de transporte. Também estudamos um modelo para um sistema de partículas onde a deriva elétrica é descrita pelo mapa de frequência monotônica, e o raio de Larmor é uma variável aleatória que assume valores específicos para cada partícula do sistema. A função densidade de probabilidade para o raio de Larmor é obtida a partir da distribuição de Maxwell-Boltzmann, que caracteriza plasmas na condição de equilíbrio térmico. Um importante parâmetro neste modelo é a variável aleatória gama, definida pelo valor da função de Bessel de ordem zero avaliada no raio de Larmor da partícula. Resultados analíticos e numéricos descrevendo as principais propriedades estatísticas do parâmetro gama são apresentados. Tais resultados são então aplicados no estudo de duas medidas de transporte: a taxa de escape e a taxa de aprisionamento por ilhas de período um.

Estimating and forecasting generalized fractional Long memory stochastic volatility models

In recent years fractionally differenced processes have received a great deal of attention due to its flexibility in financial applications with long memory. This paper considers a class of models generated by Gegenbauer polynomials, incorporating the long memory in stochastic volatility (SV) components in order to develop the General Long Memory SV (GLMSV) model. We examine the statistical properties of the new model, suggest using the spectral likelihood estimation for long memory processes, and investigate the finite sample properties via Monte Carlo experiments. We apply the model to three exchange rate return series. Overall, the results of the out-of-sample forecasts show the adequacy of the new GLMSV model.

On the existence of fractal strings whose set of dimensions of fractality is not perfect

In this paper we give an example of a nonlattice self-similar fractal string such that the set of real parts of their complex dimensions has an isolated point. This proves that, in general, the set of dimensions of fractality of a fractal string is not a perfect set.

A note on the real projection of the zeros of partial sums of Riemann zeta function

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function ζn(s):=∑nk=11ks,n>2 , is an accumulation point of the set {Res : ζ n (s) = 0}.

Subwavelength beams with polarization singularities in plasmonic metamaterials

We investigated the diffraction behavior of plasmonic Bessel beams propagating in metal-dielectric stratified materials and wire media. Our results reveal various regimes in which polarization singularities are selectively maintained. This polarization-pass effect can be controlled by appropriately setting the filling factor of the metallic inclusions and its internal periodic distribution. These results may have implications in the development of devices at the nanoscale level for manipulation of polarization and angular momentum of cylindrical vector beams.

On the Analytic Solutions of the Functional Equations w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0

In this paper, it is showed that, given an integer number n ≥ 2, each zero of an exponential polynomial of the form w1az1+w2az2+⋯+wnazn, with non-null complex numbers w 1,w 2,…,w n and a 1,a 2,…,a n , produces analytic solutions of the functional equation w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0 on certain domains of C, which represents an extension of some existing results in the literature on this functional equation for the case of positive coefficients a j and w j.

Computing the zeros of the partial sums of the Riemann zeta function

In this paper, we introduce a formula for the exact number of zeros of every partial sum of the Riemann zeta function inside infinitely many rectangles of the critical strips where they are situated.