Unified Galerkin- and DAE-Based Approximation of Fractional Order Systems

We consider numerical solutions of nonlinear multiterm fractional integrodifferential equations, where the order of the highest derivative is fractional and positive but is otherwise arbitrary. Here, we extend and unify our previous work, where a Galerkin method was developed for efficiently approximating fractional order operators and where elements of the present differential algebraic equation (DAE) formulation were introduced. The DAE system developed here for arbitrary orders of the fractional derivative includes an added block of equations for each fractional order operator, as well as forcing terms arising from nonzero initial conditions. We motivate and explain the structure of the DAE in detail. We explain how nonzero initial conditions should be incorporated within the approximation. We point out that our approach approximates the system and not a specific solution. Consequently, some questions not easily accessible to solvers of initial value problems, such as stability analyses, can be tackled using our approach. Numerical examples show excellent accuracy. DOI: 10.1115/1.4002516]

Excitation and relaxation energies of trans-stilbene: Confined singlet, triplet, and charged bipolarons

The π-electronic excitations and excited-state geometries of trans-stilbene (tS) are found by combining exact solutions of the Pariser-Parr-Pople (PPP) model and semiempirical Parametric Method 3 (PM3) calculations. Comprehensive comparisons with tS spectra are obtained and related to the fluorescence and topological alternation of poly(paraphenylenevinylene) (PPV). The one-photon absorption and triplet of tS correspond, respectively, to singlet and triplet bipolarons confined to two phenyls, while the tS2- ground state is a confined charged bipolaron. Independent estimates of the relaxation energy between vertical and adiabatic excitation show the bipolaron binding energy to depend on both charge and spin, as expected for interacting π electrons in correlated or molecular states. Complete configuration interaction within the PPP model of tS accounts for the singlet-triplet gap, for the fine-structure constants and triplet-triplet spectra, for two-photon transitions and intensities, and for one-photon spectra and the radiative lifetime, although the relative position of nearly degenerate covalent and ionic singlets is not resolved. The planar PM3 geometry and low rotational barrier of tS agree with resolved rotational and vibrational spectra in molecular beams. PM3 excitation and relaxation energies for tS bipolarons are consistent with experiment and with PPP results. Instead of the exciton model, we interpret tS excitations in terms of states that are localized on each ring or extended over an alternating chain, as found exactly in Hückel theory, and find nearly degenerate transitions between extended and localized states in the singlet, triplet, and dianion manifolds. The large topological alternation of the extended system increases the ionicity and interchanges the order of the lowest one- and two-photon absorption of PPV relative to polyenes.

Top-spin analysis of new scalar and tensor interactions in e(+)e(-) collisions with beam polarization

We utilize top polarization in the process e(+)e(-) -> t (t) over bar at the International Linear Collider ( ILC) with transverse beam polarization to probe interactions of the scalar and tensor type beyond the standard model and to disentangle their individual contributions. Ninety percent confidence level limits on the interactions with realistic integrated luminosity are presented and are found to improve by an order of magnitude compared to the case when the spin of the top quark is not measured. Sensitivities of the order of a few times 10(-3) TeV-2 for real and imaginary parts of both scalar and tensor couplings at root s = 500 and 800 GeV with an integrated luminosity of 500 fb(-1) and completely polarized beams are shown to be possible. A powerful model-independent framework for inclusive measurements is employed to describe the spin-momentum correlations, and their C, P, and T properties are presented in a technical appendix.

Translational and rotational motion in molecular liquids: A computer simulation study of Lennard-Jones ellipsoids

In order to understand the translational and rotational motion in dense molecular liquids, detailed molecular dynamics simulations of Lennard-Jones ellipsoids have been carried out for three different values of the aspect ratio kappa. For ellipsoids with an aspect ratio equal to 2, the product of the translational diffusion coefficient (D-T) and the average orientational correlation time of the l-th rank harmonics (tau(lR)), converges to a nearly constant value at high density. Surprisingly, this density independent value of D-T tau(lR) is within 5% of the hydrodynamic prediction with the slip boundary condition. This is despite the fact that both D-T and tau(lR) themselves change nearly by an order of magnitude in the density range considered, and the rotational correlation function itself is strongly nonexponential. For small aspect ratios (kappa less than or equal to 1.5), the rotational correlation function remains largely Gaussian even at a very large density, while for a large aspect ratio (kappa greater than or equal to 3), the transition to the nematic liquid-crystalline phase precludes the hydrodynamic regime. Thus, the rotational dynamics of ellipsoids show great sensitivity to the aspect ratio. At low density, tau(lR) goes through a minimum value, indicating the role of interactions in enhancing the rate of orientational relaxation. (C) 1997 American Institute of Physics. [S0021-9606(97)50142-5].

Analysis of an Interior Penalty Method for Fourth Order Problems on Polygonal Domains

Error analysis for a stable C (0) interior penalty method is derived for general fourth order problems on polygonal domains under minimal regularity assumptions on the exact solution. We prove that this method exhibits quasi-optimal order of convergence in the discrete H (2), H (1) and L (2) norms. L (a) norm error estimates are also discussed. Theoretical results are demonstrated by numerical experiments.

Generalized distributive law for ML decoding of space-time block codes

The problem of designing good space-time block codes (STBCs) with low maximum-likelihood (ML) decoding complexity has gathered much attention in the literature. All the known low ML decoding complexity techniques utilize the same approach of exploiting either the multigroup decodable or the fast-decodable (conditionally multigroup decodable) structure of a code. We refer to this well-known technique of decoding STBCs as conditional ML (CML) decoding. In this paper, we introduce a new framework to construct ML decoders for STBCs based on the generalized distributive law (GDL) and the factor-graph-based sum-product algorithm. We say that an STBC is fast GDL decodable if the order of GDL decoding complexity of the code, with respect to the constellation size, is strictly less than M-lambda, where lambda is the number of independent symbols in the STBC. We give sufficient conditions for an STBC to admit fast GDL decoding, and show that both multigroup and conditionally multigroup decodable codes are fast GDL decodable. For any STBC, whether fast GDL decodable or not, we show that the GDL decoding complexity is strictly less than the CML decoding complexity. For instance, for any STBC obtained from cyclic division algebras which is not multigroup or conditionally multigroup decodable, the GDL decoder provides about 12 times reduction in complexity compared to the CML decoder. Similarly, for the Golden code, which is conditionally multigroup decodable, the GDL decoder is only half as complex as the CML decoder.

Overexpression of the Rv0805 phosphodiesterase elicits a cAMP-independent transcriptional response

The Rv0805 gene in Mycobacterium tuberculosis encodes a metallophosphoesterase which shows cAMP-hydrolytic activity. Overexpression of Rv0805 has been used as a tool to lower intracellular cAMP levels and thereby elucidate the roles of cAMP in mycobacteria. Here we show that levels of cAMP in M. tuberculosis were lowered by only similar to 30% following overexpression of Rv0805, and transcript levels of a number of genes, which include those associated with virulence and the methyl citrate cycle, were altered. The genes that showed altered expression were distinct from those differentially regulated in a strain deleted for the cAMP-receptor protein (CRPMt), consistent with the relatively low dependence on cAMP of CRPMt binding to DNA. Using mutants of Rv0805 we show that the transcriptional signature of Rv0805 overexpression is a combination of catalysis-dependent and independent effects, and that the structurally flexible C-terminus of Rv0805 is crucial for the catalysis-independent effects of the protein. Our study demonstrates the dissociation of Rv0805 and cAMP-regulated gene expression, and reveals alternate functions for this phosphodiesterase from M. tuberculosis.

Combining data sets of satellite-retrieved products for basin-scale water balance study: 2. Evaluation on the Mississippi Basin and closure correction model

In this study, we applied the integration methodology developed in the companion paper by Aires (2014) by using real satellite observations over the Mississippi Basin. The methodology provides basin-scale estimates of the four water budget components (precipitation P, evapotranspiration E, water storage change Delta S, and runoff R) in a two-step process: the Simple Weighting (SW) integration and a Postprocessing Filtering (PF) that imposes the water budget closure. A comparison with in situ observations of P and E demonstrated that PF improved the estimation of both components. A Closure Correction Model (CCM) has been derived from the integrated product (SW+PF) that allows to correct each observation data set independently, unlike the SW+PF method which requires simultaneous estimates of the four components. The CCM allows to standardize the various data sets for each component and highly decrease the budget residual (P - E - Delta S - R). As a direct application, the CCM was combined with the water budget equation to reconstruct missing values in any component. Results of a Monte Carlo experiment with synthetic gaps demonstrated the good performances of the method, except for the runoff data that has a variability of the same order of magnitude as the budget residual. Similarly, we proposed a reconstruction of Delta S between 1990 and 2002 where no Gravity Recovery and Climate Experiment data are available. Unlike most of the studies dealing with the water budget closure at the basin scale, only satellite observations and in situ runoff measurements are used. Consequently, the integrated data sets are model independent and can be used for model calibration or validation.

Measurement of Large Dipolar Couplings of a Liquid Crystal with Terminal Phenyl Rings and Estimation of the Order Parameters

NMR spectroscopy is a powerful means of studying liquid-crystalline systems at atomic resolutions. Of the many parameters that can provide information on the dynamics and order of the systems, H-1-C-13 dipolar couplings are an important means of obtaining such information. Depending on the details of the molecular structure and the magnitude of the order parameters, the dipolar couplings can vary over a wide range of values. Thus the method employed to estimate the dipolar couplings should be capable of estimating both large and small dipolar couplings at the same time. For this purpose, we consider here a two-dimensional NMR experiment that works similar to the insensitive nuclei enhanced by polarization transfer (INEPT) experiment in solution. With the incorporation of a modification proposed earlier for experiments with low radio frequency power, the scheme is observed to enable a wide range of dipolar couplings to be estimated at the same time. We utilized this approach to obtain dipolar couplings in a liquid crystal with phenyl rings attached to either end of the molecule, and estimated its local order parameters.

Tensionless Contact Of A Finite Beam Resting On Reissner Foundation

A more generalized model of a beam resting on a tensionless Reissner foundation is presented. Compared with the Winkler foundation model, the Reissner foundation model is a much improved one. In the Winkler foundation model, there is no shear stress inside the foundation layer and the foundation is assumed to consist of closely spaced, independent springs. The presence of shear stress inside Reissner foundation makes the springs no longer independent and the foundation to deform as a whole. Mathematically, the governing equation of a beam on Reissner foundation is sixth order differential equation compared with fourth order of Winkler one. Because of this order change of the governing equation, new boundary conditions are needed and related discussion is presented. The presence of the shear stress inside the tensionless Reissner foundation together with the unknown feature of contact area/length makes the problem much more difficult than that of Winkler foundation. In the model presented here, the effects of beam dimension, gap distance, loading asymmetry and foundation shear stress on the contact length are all incorporated and studied. As the beam length increases, the results of a finite beam with zero gap distance converge asymptotically to those obtained by the previous model for an infinitely long beam. (C) 2008 Elsevier Ltd. All rights reserved.

Changes in the size structure of the yellowfin tuna population of the tropical eastern Pacific Ocean from 1947 to 1955

ENGLISH: Morphometric studies by Godsil (1948), Godsil and Greenhood (1951), Royce (1953) and Schaefer (1952, 1955) have indicated that the yellowfin tuna of the Eastern Pacific are distinct from those of the Central Pacific. Tagging of yellowfin tuna by the California Department of Fish and Game, and by the Inter-American Tropical Tuna Commission in the Eastern Pacific, and by the Pacific Oceanic Fishery Investigations in the Central Pacific, have not yet revealed any migrations between these areas. Shimada and Schaefer (1956) have compared changes in population abundance and fishing intensity, considering the population in the Eastern Pacific as a separate entity. They conclude " ... the amount of fishing has had a real effect upon the stock of Eastern Pacific yellowfin tuna, taken in the aggregate, over the period studied. The evidence suggests also that for this species the intensity of fishing in some recent years has reached and might have even exceeded the level corresponding to the maximum equilibrium yield." Tagging experiments by the California Department of Fish and Game and by the Inter-American Tropical Tuna Commission have yielded returns in the order of one to five percent (Roedel 1954, and unpublished data of both agencies), a level much lower than that at which fishing intensity would be expected to noticeably affect the population size. These results are probably a reflection of the inadequacies of the present tagging methods, but they could lend doubt to the conclusions of Shimada and Schaefer. It is desirable, therefore, to examine other, independent, evidence as to the effects of fishing on the population. At the high levels of fishing intensity suggested by Shimada and Schaefer, in addition to changes in quantity, measurable changes would be expected to have occurred in the quality of the yellowfin tuna stocks, because the average age and size of the fish would have been reduced by the high mortality rates accompanying high fishing intensities. A continuing regular program of sampling catches and determining their length composition, to assess changes in the size composition of the stocks, was initiated by the Commission in 1954 but direct measurements are not available for the earlier, more dynamic period of growth of the fishery. Consequently, other, more general indications of possible changes in the size composition were sought. SPANISH: Los estudios morfométricos efectudos por Godsil (1948), Godsil y Greenhood (1951), Royce (1953) y Schaefer (1952, 1955), han demostrado que el atún aleta amarilla del Pacífico Oriental es distinto del que habita el PacÍfico Central. Los experimentos del Departamento de Pesca y Caza de California y de la Comisión Interamericana del Atún Tropical en el Pacífico Oriental, así como los de las Investigaciones Pesqueras del Océano Pacífico en el Pacífico Central,consistentes en la marcación de atunes aleta amarilla, aún no han puesto de manifiesto movimientos migratorios entre dichas áreas. Shimada y Schaefer (1956) han hecho estudios comparativos sobre la abundancia de la población y la intensidad de la pesca, considerando a la población del Pacífico Oriental como una entidad separada. Su conclusión es que " ... la intensidad de la pesca ha tenido un definido efecto sobre la población del atún aleta amarilla del Pacífico Oriental, tomada en conjunto, a lo largo del período estudiado. La evidencia de que se dispone sugiere así mismo que, por lo que hace a esta especie, la intensidad de la pesca en los últimos años ha alcanzado y quizás aún sobrepasado el nivel correspondiente a la máxima pesca de equilibrio". Los experimentos de mar•cación del Departamento de Pesca y Caza de California y de la Comisión Interamericana del Atún Tropical han producido recuperaciones ,entre el uno y el cinco por ciento (Roedel 1954 y datos inéditos de ambos organismos), lo que constituye un nivel mucho más bajo de aquél en que la intensidad de la pesca podría considerarse que afectaría notablemente el tamaño de la población. Estos resultados reflejan probablemente lo inadecuados que son aún los métodos de marcación, pero ellos podrían, quizá, poner en tela de juicio las conclusiones de Shimada y Schaefer. Por lo tanto,es deseable examinar otras fuentes de evidencia independientes, relacionadas con el efecto que la pesca tiene sobre la población. En efecto, si los altos índices de pesca sugeridos por Shimada y Schaefer son correctos, es de esperar que, además de los cambios en la magnitud de la población, se hayan producido otros, concomitantes y sensibles, en la calidad de los stocks de atún aleta amarilla, puesto que tanto el promedio de edad como el de tamaño de los individuos habrían disminuído debido a las elevadas tasas de mortalidad inherentes a las altas intensidades de pesca. En 1954 la Comisión inició un programa ininterrumpido para tomar muestras y determinar en ellas las frecuencias de tallas y evaluar de este modo los cambios correlativos que tuvieran lugar en los stocks pero, infortunadamente, este sistema de evaluación directa no fué practicado en el período anterior, que fué precisamente el de rápida expansión de la pesquería. En tal virtud, hubo de ser necesario buscar indicios más generales referentes a los cambios posibles en la composición de tamaños. (PDF contains 20 pages.)

Arithmetic of cyclotomic fields and Fermat-type equations

Let l be any odd prime, and ζ a primitive l-th root of unity. Let C_l be the l-Sylow subgroup of the ideal class group of Q(ζ). The Teichmüller character w : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity, and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l be l^h_3). The main result of this thesis is the following: For every n ≥ max( 1, h_3 ), the equation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ≠ xyz. The same result is also proven with n ≥ max(1,h_5), under the assumption that C_l^((5)) is a cyclic group of order l^h_5. Applications of the methods used to prove the above results to the second case of Fermat's last theorem and to a Fermat-like equation in four variables are given.

The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with a theorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa's main conjecture for cyclotomic fields by B. Mazur and A. Wiles [MW]. In [V1] Vandiver claimed that the first case of Fermat's Last Theorem held for l if l did not divide the class number h^+ of the maximal real subfield of Q(e^(2πi/i)). The crucial gap in Vandiver's attempted proof that has been known to experts is explained, and complete proofs of all the results used from his papers are given.

Methods of multiparameter inversion of seismic data using the acoustic and elastic born approximations

This thesis presents two different forms of the Born approximations for acoustic and elastic wavefields and discusses their application to the inversion of seismic data. The Born approximation is valid for small amplitude heterogeneities superimposed over a slowly varying background. The first method is related to frequency-wavenumber migration methods. It is shown to properly recover two independent acoustic parameters within the bandpass of the source time function of the experiment for contrasts of about 5 percent from data generated using an exact theory for flat interfaces. The independent determination of two parameters is shown to depend on the angle coverage of the medium. For surface data, the impedance profile is well recovered.

The second method explored is mathematically similar to iterative tomographic methods recently introduced in the geophysical literature. Its basis is an integral relation between the scattered wavefield and the medium parameters obtained after applying a far-field approximation to the first-order Born approximation. The Davidon-Fletcher-Powell algorithm is used since it converges faster than the steepest descent method. It consists essentially of successive backprojections of the recorded wavefield, with angular and propagation weighing coefficients for density and bulk modulus. After each backprojection, the forward problem is computed and the residual evaluated. Each backprojection is similar to a before-stack Kirchhoff migration and is therefore readily applicable to seismic data. Several examples of reconstruction for simple point scatterer models are performed. Recovery of the amplitudes of the anomalies are improved with successive iterations. Iterations also improve the sharpness of the images.

The elastic Born approximation, with the addition of a far-field approximation is shown to correspond physically to a sum of WKBJ-asymptotic scattered rays. Four types of scattered rays enter in the sum, corresponding to P-P, P-S, S-P and S-S pairs of incident-scattered rays. Incident rays propagate in the background medium, interacting only once with the scatterers. Scattered rays propagate as if in the background medium, with no interaction with the scatterers. An example of P-wave impedance inversion is performed on a VSP data set consisting of three offsets recorded in two wells.

I. Spectroscopic evidence for slow vibrational relaxation in excited electronic states of diatomics in rare gas solids. II. Static crystal field effects in the electronic spectra of isotopically mixed benzene crystals

Part I

Studies of vibrational relaxation in excited electronic states of simple diatomic molecules trapped in solid rare-gas matrices at low temperatures are reported. The relaxation is investigated by monitoring the emission intensity from vibrational levels of the excited electronic state to vibrational levels of the ground electronic state. The emission was in all cases excited by bombardment of the doped rare-gas solid with X-rays.

The diatomics studied and the band systems seen are: N₂, Vegard-Kaplan and Second Positive systems; O₂, Herzberg system; OH and OD, A ²Σ⁺ - X²II_i system. The latter has been investigated only in solid Ne, where both emission and absorption spectra were recorded; observed fine structure has been partly interpreted in terms of slightly perturbed rotational motion in the solid. For N₂, OH, and OD emission occurred from v' > 0, establishing a vibrational relaxation time in the excited electronic state of the order, of longer than, the electronic radiative lifetime. The relative emission intensity and decay times for different v' progressions in the Vegard-Kaplan system are found to depend on the rare-gas host and the N₂ concentration, but are independent of temperature in the range 1.7°K to 30°K.

Part II

Static crystal field effects on the absorption, fluorescence, and phosphorescence spectra of isotopically mixed benzene crystals were investigated. Evidence is presented which demonstrate that in the crystal the ground, lowest excited singlet, and lowest triplet states of the guest deviate from hexagonal symmetry. The deviation appears largest in the lowest triplet state and may be due to an intrinsic instability of the ³B_1u state. High resolution absorption and phospho- rescence spectra are reported and analyzed in terms of site-splitting of degenerate vibrations and orientational effects. The guest phosphorescence lifetime for various benzene isotopes in C₆D₆ and sym-C₆H₃D₃ hosts is presented and discussed.

Towards understanding the mixing characteristics of turbulent buoyant flows

This work proposes a new simulation methodology in which variable density turbulent flows can be studied in the context of a mixing layer with or without the presence of gravity. Specifically, this methodology is developed to probe the nature of non-buoyantly-driven (i.e. isotropically-driven) or buoyantly-driven mixing deep inside a mixing layer. Numerical forcing methods are incorporated into both the velocity and scalar fields, which extends the length of time over which mixing physics can be studied. The simulation framework is designed to allow for independent variation of four non-dimensional parameters, including the Reynolds, Richardson, Atwood, and Schmidt numbers. Additionally, the governing equations are integrated in such a way to allow for the relative magnitude of buoyant energy production and non-buoyant energy production to be varied.

The computational requirements needed to implement the proposed configuration are presented. They are justified in terms of grid resolution, order of accuracy, and transport scheme. Canonical features of turbulent buoyant flows are reproduced as validation of the proposed methodology. These features include the recovery of isotropic Kolmogorov scales under buoyant and non-buoyant conditions, the recovery of anisotropic one-dimensional energy spectra under buoyant conditions, and the preservation of known statistical distributions in the scalar field, as found in other DNS studies.

This simulation methodology is used to perform a parametric study of turbulent buoyant flows to discern the effects of varying the Reynolds, Richardson, and Atwood numbers on the resulting state of mixing. The effects of the Reynolds and Atwood numbers are isolated by looking at two energy dissipation rate conditions under non-buoyant (variable density) and constant density conditions. The effects of Richardson number are isolated by varying the ratio of buoyant energy production to total energy production from zero (non-buoyant) to one (entirely buoyant) under constant Atwood number, Schmidt number, and energy dissipation rate conditions. It is found that the major differences between non-buoyant and buoyant turbulent flows are contained in the transfer spectrum and longitudinal structure functions, while all other metrics are largely similar (e.g. energy spectra, alignment characteristics of the strain-rate tensor). Also, despite the differences noted between fully buoyant and non-buoyant turbulent fields, the scalar field, in all cases, is unchanged by these. The mixing dynamics in the scalar field are found to be insensitive to the source of turbulent kinetic energy production (non-buoyant vs. buoyant).