Interpretation of the pressuremeter test using numerical models based on deformation tensor equations

The pressuremeter test in boreholes has proven itself as a useful tool in geotechnical explorations, especially comparing its results with those obtained from a mathematical model ruled by a soil representative constitutive equation. The numerical model shown in this paper is aimed to be the reference framework for the interpretation of this test. The model analyses variables such as: the type of response, the initial state, the drainage regime and the constitutive equations. It is a model of finite elements able to work with a mesh without deformation or one adapted to it.

Secure optical networks based on quantum key distribution and weakly trusted repeaters

Abstract—In this paper we explore how recent technologies can improve the security of optical networks. In particular, we study how to use quantum key distribution(QKD) in common optical network infrastructures and propose a method to overcome its distance limitations. QKD is the first technology offering information theoretic secretkey distribution that relies only on the fundamental principles of quantum physics. Point-to-point QKDdevices have reached a mature industrial state; however, these devices are severely limited in distance, since signals at the quantum level (e.g., single photons) are highly affected by the losses in the communication channel and intermediate devices. To overcome this limitation, intermediate nodes (i.e., repeaters) are used. Both quantum-regime and trusted, classical repeaters have been proposed in the QKD literature, but only the latter can be implemented in practice. As a novelty, we propose here a new QKD network model based on the use of not fully trusted intermediate nodes, referred to as weakly trusted repeaters. This approach forces the attacker to simultaneously break several paths to get access to the exchanged key, thus improving significantly the security of the network. We formalize the model using network codes and provide real scenarios that allow users to exchange secure keys over metropolitan optical networks using only passive components. Moreover, the theoretical framework allows one to extend these scenarios not only to accommodate more complex trust constraints, but also to consider robustness and resiliency constraints on the network.

On the generative equations of fractal self-similarity in granular media and the related PSD models

The physical appearance of granular media suggests the existence of geometrical scale invariance. The paper discuss how this physico-empirical property can be mathematically encoded leading to different generative models: a smooth one encoded by a differential equation and another encoded by an equation coming from a measure theoretical property.

Advances in global instability computations: from incompressible to hypersonic flow

Esta tesis constituye un gran avance en el conocimiento del estudio y análisis de inestabilidades hidrodinámicas desde un punto de vista físico y teórico, como consecuencia de haber desarrollado innovadoras técnicas para la resolución computacional eficiente y precisa de la parte principal del espectro correspondiente a los problemas de autovalores (EVP) multidimensionales que gobiernan la inestabilidad de flujos con dos o tres direcciones espaciales inhomogéneas, denominados problemas de estabilidad global lineal. En el contexto del trabajo de desarrollo de herramientas computacionales presentado en la tesis, la discretización mediante métodos de diferencias finitas estables de alto orden de los EVP bidimensionales y tridimensionales que se derivan de las ecuaciones de Navier-Stokes linealizadas sobre flujos con dos o tres direcciones espaciales inhomogéneas, ha permitido una aceleración de cuatro órdenes de magnitud en su resolución. Esta mejora de eficiencia numérica se ha conseguido gracias al hecho de que usando estos esquemas de diferencias finitas, técnicas eficientes de resolución de problemas lineales son utilizables, explotando el alto nivel de dispersión o alto número de elementos nulos en las matrices involucradas en los problemas tratados. Como más notable consecuencia cabe destacar que la resolución de EVPs multidimensionales de inestabilidad global, que hasta la fecha necesitaban de superordenadores, se ha podido realizar en ordenadores de sobremesa. Además de la solución de problemas de estabilidad global lineal, el mencionado desarrollo numérico facilitó la extensión de las ecuaciones de estabilidad parabolizadas (PSE) lineales y no lineales para analizar la inestabilidad de flujos que dependen fuertemente en dos direcciones espaciales y suavemente en la tercera con las ecuaciones de estabilidad parabolizadas tridimensionales (PSE-3D). Precisamente la capacidad de extensión del novedoso algoritmo PSE-3D para el estudio de interacciones no lineales de los modos de estabilidad, desarrollado íntegramente en esta tesis, permite la predicción de transición en flujos complejos de gran interés industrial y por lo tanto extiende el concepto clásico de PSE, el cuál ha sido empleado exitosamente durante las pasadas tres décadas en el mismo contexto para problemas de capa límite bidimensional. Típicos ejemplos de flujos incompresibles se han analizado en este trabajo sin la necesidad de recurrir a restrictivas presuposiciones usadas en el pasado. Se han estudiado problemas vorticales como es el caso de un vórtice aislado o sistemas de vórtices simulando la estela de alas, en los que la homogeneidad axial no se impone y así se puede considerar la difusión viscosa del flujo. Además, se ha estudiado el chorro giratorio turbulento, cuya inestabilidad se utiliza para mejorar las características de funcionamiento de combustores. En la tesis se abarcan adicionalmente problemas de flujos compresibles. Se presenta el estudio de inestabilidad de flujos de borde de ataque a diferentes velocidades de vuelo. También se analiza la estela formada por un elemento rugoso aislado en capa límite supersónica e hipersónica, mostrando excelentes comparaciones con resultados obtenidos mediante simulación numérica directa. Finalmente, nuevas inestabilidades se han identificado en el flujo hipersónico a Mach 7 alrededor de un cono elíptico que modela el vehículo de pruebas en vuelo HIFiRE-5. Los resultados comparan favorablemente con experimentos en vuelo, lo que subraya aún más el potencial de las metodologías de análisis de estabilidad desarrolladas en esta tesis. ABSTRACT The present thesis constitutes a step forward in advancing the frontiers of knowledge of fluid flow instability from a physical point of view, as a consequence of having been successful in developing groundbreaking methodologies for the efficient and accurate computation of the leading part of the spectrum pertinent to multi-dimensional eigenvalue problems (EVP) governing instability of flows with two or three inhomogeneous spatial directions. In the context of the numerical work presented in this thesis, the discretization of the spatial operator resulting from linearization of the Navier-Stokes equations around flows with two or three inhomogeneous spatial directions by variable-high-order stable finite-difference methods has permitted a speedup of four orders of magnitude in the solution of the corresponding two- and three-dimensional EVPs. This improvement of numerical performance has been achieved thanks to the high-sparsity level offered by the high-order finite-difference schemes employed for the discretization of the operators. This permitted use of efficient sparse linear algebra techniques without sacrificing accuracy and, consequently, solutions being obtained on typical workstations, as opposed to the previously employed supercomputers. Besides solution of the two- and three-dimensional EVPs of global linear instability, this development paved the way for the extension of the (linear and nonlinear) Parabolized Stability Equations (PSE) to analyze instability of flows which depend in a strongly-coupled inhomogeneous manner on two spatial directions and weakly on the third. Precisely the extensibility of the novel PSE-3D algorithm developed in the framework of the present thesis to study nonlinear flow instability permits transition prediction in flows of industrial interest, thus extending the classic PSE concept which has been successfully employed in the same context to boundary-layer type of flows over the last three decades. Typical examples of incompressible flows, the instability of which was analyzed in the present thesis without the need to resort to the restrictive assumptions used in the past, range from isolated vortices, and systems thereof, in which axial homogeneity is relaxed to consider viscous diffusion, as well as turbulent swirling jets, the instability of which is exploited in order to improve flame-holding properties of combustors. The instability of compressible subsonic and supersonic leading edge flows has been solved, and the wake of an isolated roughness element in a supersonic and hypersonic boundary-layer has also been analyzed with respect to its instability: excellent agreement with direct numerical simulation results has been obtained in all cases. Finally, instability analysis of Mach number 7 ow around an elliptic cone modeling the HIFiRE-5 flight test vehicle has unraveled flow instabilities near the minor-axis centerline, results comparing favorably with flight test predictions.

Accurate Parabolic Navier-Stokes solutions of the supersonic flow around and elliptic cone

Flows of relevance to new generation aerospace vehicles exist, which are weakly dependent on the streamwise direction and strongly dependent on the other two spatial directions, such as the flow around the (flattened) nose of the vehicle and the associated elliptic cone model. Exploiting these characteristics, a parabolic integration of the Navier-Stokes equations is more appropriate than solution of the full equations, resulting in the so-called Parabolic Navier-Stokes (PNS). This approach not only is the best candidate, in terms of computational efficiency and accuracy, for the computation of steady base flows with the appointed properties, but also permits performing instability analysis and laminar-turbulent transition studies a-posteriori to the base flow computation. This is to be contrasted with the alternative approach of using order-of-magnitude more expensive spatial Direct Numerical Simulations (DNS) for the description of the transition process. The PNS equations used here have been formulated for an arbitrary coordinate transformation and the spatial discretization is performed using a novel stable high-order finite-difference-based numerical scheme, ensuring the recovery of highly accurate solutions using modest computing resources. For verification purposes, the boundary layer solution around a circular cone at zero angle of attack is compared in the incompressible limit with theoretical profiles. Also, the recovered shock wave angle at supersonic conditions is compared with theoretical predictions in the same circular-base cone geometry. Finally, the entire flow field, including shock position and compressible boundary layer around a 2:1 elliptic cone is recovered at Mach numbers 3 and 4

A dynamic programming approach to the formulation and solution of finite element equations

A method for formulating and algorithmically solving the equations of finite element problems is presented. The method starts with a parametric partition of the domain in juxtaposed strips that permits sweeping the whole region by a sequential addition (or removal) of adjacent strips. The solution of the difference equations constructed over that grid proceeds along with the addition removal of strips in a manner resembling the transfer matrix approach, except that different rules of composition that lead to numerically stable algorithms are used for the stiffness matrices of the strips. Dynamic programming and invariant imbedding ideas underlie the construction of such rules of composition. Among other features of interest, the present methodology provides to some extent the analyst's control over the type and quantity of data to be computed. In particular, the one-sweep method presented in Section 9, with no apparent counterpart in standard methods, appears to be very efficient insofar as time and storage is concerned. The paper ends with the presentation of a numerical example

Development of AMREM: a Maxwell's equations solver with Adaptive Mesh Refinement

El principal objetivo de la presente tesis es el de desarrollar y probar un código capaz de resolver las ecuaciones de Maxwell en el dominio del tiempo con Malla Refinada Adaptativa (AMR por sus siglas en inglés). AMR es una técnica de cálculo basada en dividir el dominio físico del problema en distintas mallas rectangulares paralelas a las direcciones cartesianas. Cada una de las mallas tendrá distinta resolución y aquellas con mayor resolución se sitúan allí dónde las ondas electromagnéticas se propagan o interaccionan con los materiales, es decir, dónde mayor precisión es requerida. Como las ondas van desplazándose por todo el dominio, las mayas deberán seguirlas. El principal problema al utilizar esta metodología se puede encontrar en las fronteras internas, dónde las distintas mallas se unen. Ya que el método más corrientemente utilizado para resolver las ecuaciones de Maxwell es el de las diferencias finitas en el dominio del tiempo (FDTD por sus siglas en inglés) , el trabajo comenzó tratando de adaptar AMR a FDTD. Tras descubrirse que esta interacción resultaba en problemas de inestabilidades en las fronteras internas antes citadas, se decidió cambiar a un método basado en volúmenes finitos en el dominio del tiempo (FVTD por sus siglas en inglés). Este se basa en considerar la forma en ecuaciones de conservación de las ecuaciones de Maxwell y aplicar a su resolución un esquema de Godunov. Se ha probado que es clave para el correcto funcionamiento del código la elección de un limitador de flujo que proteja los extremos de la onda de la disipación típica de los métodos de este tipo. Otro problema clásico a la hora de resolver las ecuaciones de Maxwell es el de tratar con las condiciones de frontera física cuando se simulan dominios no acotados, es decir, dónde las ondas deben salir del sistema sin producir ninguna reflexión. Normalmente la solución es la de disponer una banda absorbente en las fronteras físicas. En AMREM se ha desarrollado un nuevo método basado en los campos característicos que con menor requisito de CPU funcina suficientemente bien incluso en los casos más desfaborables. El código ha sido contrastado con soluciones analíticas de diferentes problemas y también su velocidad ha sido comparada con la de Meep, uno de los programas más conocidos del ámbito. También algunas aplicaciones han sido simuladas con el fin de demostrar el amplio espectro de campos en los que AMREM puede funcionar como una útil herramienta.

Networks based on QKD and weakly trusted repeaters

We study how to use quantum key distribution (QKD) in common optical network infrastructures and propose a method to overcome its distance limitations. QKD is the first technology offering information theoretic secret-key distribution that relies only on the fundamental principles of quantum physics. Point-to-point QKD devices have reached a mature industrial state; however, these devices are severely limited in distance, since signals at the quantum level (e.g. single photons) are highly affected by the losses in the communication channel and intermediate devices. To overcome this limitation, intermediate nodes (i.e. repeaters) are used. Both, quantum-regime and trusted, classical, repeaters have been proposed in the QKD literature, but only the latter can be implemented in practice. As a novelty, we propose here a new QKD network model based on the use of not fully trusted intermediate nodes, referred as weakly trusted repeaters. This approach forces the attacker to simultaneously break several paths to get access to the exchanged key, thus improving significantly the security of the network. We formalize the model using network codes and provide real scenarios that allow users to exchange secure keys over metropolitan optical networks using only passive components.

Overtopping of harbour breakwaters: a comparison of semi-empirical equations, neural networks, and physical model tests

This paper reports extensive tests of empirical equations developed by different authors for harbour breakwater overtopping. First, the existing equations are compiled and evaluated as tools for estimating the overtopping rates on sloping and vertical breakwaters. These equations are then tested using the data obtained in a number of laboratory studies performed in the Centre for Harbours and Coastal Studies of the CEDEX, Spain. It was found that the recommended application ranges of the empirical equations typically deviate from those revealed in the experimental tests. In addition, a neural network model developed within the European CLASH Project is tested. The wind effects on overtopping are also assessed using a reduced scale physical model

Ricci flows on surfaces related to the Einstein weyl and Abelian vortex equations

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve these equations is preserved by the Ricci flow. The equations are solved explicitly, and among the metrics obtained are all steady gradient Ricci solitons (e.g. the cigar soliton) and the sausage metric; there are found other examples of eternal, ancient, and immortal Ricci flows, as well as some Ricci flows with conical singularities.

Analysis of human cytochrome P450 3A4 cooperativity: Construction and characterization of a site-directed mutant that displays hyperbolic steroid hydroxylation kinetics

Cytochrome P450 3A4 is generally considered to be the most important human drug-metabolizing enzyme and is known to catalyze the oxidation of a number of substrates in a cooperative manner. An allosteric mechanism is usually invoked to explain the cooperativity. Based on a structure–activity study from another laboratory using various effector–substrate combinations and on our own studies using site-directed mutagenesis and computer modeling of P450 3A4, the most likely location of effector binding is in the active site along with the substrate. Our study was designed to test this hypothesis by replacing residues Leu-211 and Asp-214 with the larger Phe and Glu, respectively. These residues were predicted to constitute a portion of the effector binding site, and the substitutions were designed to mimic the action of the effector by reducing the size of the active site. The L211F/D214E double mutant displayed an increased rate of testosterone and progesterone 6β-hydroxylation at low substrate concentrations and a decreased level of heterotropic stimulation elicited by α-naphthoflavone. Kinetic analyses of the double mutant revealed the absence of homotropic cooperativity with either steroid substrate. At low substrate concentrations the steroid 6β-hydroxylase activity of the wild-type enzyme was stimulated by a second steroid, whereas L211F/D214E displayed simple substrate inhibition. To analyze L211F/D214E at a more mechanistic level, spectral binding studies were carried out. Testosterone binding by the wild-type enzyme displayed homotropic cooperativity, whereas substrate binding by L211F/D214E displayed hyperbolic behavior.

Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Hyers–Ulam stability of functional equations with a square-symmetric operation

The stability of the functional equation f(x ○ y) = H(f(x), f(y)) (x, y ∈ S) is investigated, where H is a homogeneous function and ○ is a square-symmetric operation on the set S. The results presented include and generalize the classical theorem of Hyers obtained in 1941 on the stability of the Cauchy functional equation.

Shedding light on the dark and weakly fluorescent states of green fluorescent proteins

Recent experiments on various similar green fluorescent protein (GFP) mutants at the single-molecule level and in solution provide evidence of previously unknown short- and long-lived “dark” states and of related excited-state decay channels. Here, we present quantum chemical calculations on cis-trans photoisomerization paths of neutral, anionic, and zwitterionic GFP chromophores in their ground and first singlet excited states that explain the observed behaviors from a common perspective. The results suggest that favorable radiationless decay channels can exist for the different protonation states along these isomerizations, which apparently proceed via conical intersections. These channels are suggested to rationalize the observed dramatic reduction of fluorescence in solution. The observed single-molecule fast blinking is attributed to conversions between the fluorescent anionic and the dark zwitterionic forms whereas slow switching is attributed to conversions between the anionic and the neutral forms. The predicted nonadiabatic crossings are seen to rationalize the origins of a variety of experimental observations on a common basis and may have broad implications for photobiophysical mechanisms in GFP.