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We simulate freely jointed chains to investigate how knotting affects the overall shapes of freely fluctuating circular polymeric chains. To characterize the shapes of knotted polygons, we construct enveloping ellipsoids that minimize volume while containing the entire polygon. The lengths of the three principal axes of the enveloping ellipsoids are used to define universal size and shape descriptors analogous to the squared radius of gyration and the inertial asphericity and prolateness. We observe that polymeric chains forming more complex knots are more spherical and also more prolate than chains forming less complex knots with the same number of edges. We compare the shape measures, determined by the enveloping ellipsoids, with those based on constructing inertial ellipsoids and explain the differences between these two measures of polymer shape. [ABSTRACT FROM AUTHOR]

The depletion interactions of ellipsoidal colloidal particles in a solution of long polymer chains are analyzed. Of primary concern are the limiting cases in which the ellipsoid reduces to a cylinder of infinite length and finite radius and a “needle” of finite length and vanishing radius. Relations are obtained between the polymer effects induced by a needle that is much shorter than the polymer size and by a cylinder with radius much smaller than the polymer size. These imply exact quantitative results for the orientation-dependent depletion interaction between a short needle and a wall. Qualitative differences between the needle and thin disk are discussed. [ABSTRACT FROM AUTHOR]

We have carried out NPT molecular dynamics simulations of isolated ellipsoids in a glass forming binary mixture to gain insight into the nature of orientational relaxation (OR) in a viscous liquid. At high pressures when the liquid is highly viscous, the OR is found to occur mainly via correlated hopping, sometimes involving participation of several neighboring atoms, placed in a ring like tunnel. In the glassy state, hopping is found to be accompanied by larger fluctuations in the total energy and the volume of the system. Both orientational and translational hopping are found to be gated, restricted primarily by the entropic bottlenecks, with the orientational motion becoming increasingly slower than the translation as the pressure is increased. Orientational relaxation is found to occur with a wide distribution of decay times. [ABSTRACT FROM AUTHOR]

Abstract Forward attractors, especially invariant ones, of non-autonomous and random dynamical systems have not been as well studied as pullback attractors. This is mainly due to the different role that time plays in each case. Pullback attractors involve convergence at each instant of current time as the initial data is pulled back to distant past, whereas forward attractors are concerned with what happens in the distant future. Moreover, if they exist, they need not be unique. In this paper, ideas introduced in Kloeden [20] for asymptotically invariant forward attracting sets of non-autonomous ODEs are generalized and extended to study strictly invariant forward attractors of non-autonomous random dynamical systems on metric spaces. These are formulated as processes that include the effects of both noise and deterministic time variation. It is shown that the random forward attractors, when they exist, can be constructed by a pullback argument inside a positively invariant set. However, this is only a necessary condition, so additional sufficiency conditions are also given. Since forward attractors need not exist in dissipative systems, an alternative object, specifically the closed union of the forward omega limit sets for each initial time of a forward absorbing set is investigated. It is shown to be asymptotically invariant under certain conditions, so looks more and more like an invariant attractor as conventionally understood. It resembles the Vishik uniform attractor, but need not attract uniformly in the initial times and does not even require the system to be defined in the distant past. [ABSTRACT FROM AUTHOR]

q-deformations of functions and distributions have been used in the literature to explain several experimental observations. In this work, we study the dynamics of the Tinkerbell map under q -deformations. The system exhibits a rich variety of dynamical behavior as q varies, including occurrences of interior crises, paired cascades, simultaneous occurrence of Neimark-Sacker and reverse Neimark-Sacker bifurcations, and co-existence of attractors and multistability. Numerical analysis reveals the existence of 3 limit cycles occurring simultaneously in a certain parameter regime. An appropriate choice of initial conditions enables one to choose a desired attractor for the system among other co-existing ones, thus switching the system between different dynamical states. We demonstrate the possibility of secure encryption and decryption of messages with the q -deformed Tinkerbell map. The system's sensitivity to the initial conditions and to the deformation parameter makes the cryptic message secure, and decrypting the original message difficult. We propose the use of the q -deformed map as a novel method for transmission of messages securely. [ABSTRACT FROM AUTHOR]

Background  The assessment of pituitary tumor (PT) volume is important in the treatment and follow-up of patients with PT. Previously, PT volume estimation has been performed by conventional geometric equations (CGE) such as abc/2 (simplified ellipsoid volume equation) and 4πr 3 /3 (sphere), both presuming a symmetric tumor shape, which occurs uncommonly in patients with PT. In contrast, three-dimensional (3D) voxel-based software segmentation takes the irregular and asymmetric shapes that PTs often possess into account and might be a more accurate method for PT volume segmentation. The purpose of this study is twofold. (1) To compare 3D segmentation with CGE for PT volume estimation. (2) To assess inter-rater reliability in 3D segmentation of PTs. Methods  Nineteen high-resolution (1mm slice thickness) T1-weighted MRI examinations of patients with PT were independently analyzed and manually segmented, using the software ITK-SNAP, by two certified neuroradiologists. Concurrently, the volumes of the PTs were estimated with abc/2 and 4πr 3 /3 by a clinician, and the results were compared with the corresponding segmented volumes. Results  There was a significant decrease in PT volume attained from the segmentations compared with the calculations made with abc/2 (p  < 0.001, mean volume 18% higher than segmentation) and 4πr 3 /3 (p  < 0.001, mean volume 28% higher than segmentation). The intraclass correlation coefficient (ICC) for the two sets of segmented PTs was 0.99. Conclusion  CGE (abc/2 and 4πr 3 /3) significantly overestimates PT volume compared with 3D volumetric segmentation. The inter-rater agreement on manual 3D volumetric software segmentation is excellent. [ABSTRACT FROM AUTHOR]

The transition from the isotropic to nematic phase is investigated for a fluid comprised of bendable ellipsoids. When the bending amplitude is chosen such that the bodies remain convex, density functional-Onsager methods can be employed to estimate the phase transition properties. The convexity requirement places a restriction on the size of the effects; however, a trend is discernible. As the density of the fluid increases, in both the isotropic and nematic phases, the mean bending amplitude is suppressed. Furthermore, the density of the isotropic to nematic transition increases with the bending amplitude in accord with the predictions of the Khokhlov–Semenov theory. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In this paper we describe a general approach which allows the hypernetted chain (HNC) and Percus–Yevick (PY) integral equation theories to be solved numerically for fluids of hard nonspherical particles. Explicit results are given for fluids of hard ellipsoids of revolution and comparisons are made with recent Monte Carlo calculations. It is found that for dense systems of highly anisotropic ellipsoids the HNC and PY closures give significantly different results. The HNC theory is superior predicting the existance of a nematic phase in qualitative agreement with computer simulations. The PY approximation strongly and erroneously suggests that the isotropic phase is stable throughout the liquid regime. [ABSTRACT FROM AUTHOR]

Strange attractors have been extensively studied, but the same is not true for strange repellors. Some time-reversible systems have repellors that mirror their corresponding attractors and that exchange roles when time is reversed. In this paper, a conversion operator is introduced by which an easy transformation can be constructed between such a time-reversible system with an attractor/repellor pair and an irreversible one with a pair of attractors, or vice versa, thus expanding the list of such examples. [ABSTRACT FROM AUTHOR]

A finite-time version, based on implicit Lyapunov functions (ILFs), for the attractive ellipsoid method (AEM) is developed. Based on this, a robust control scheme is presented to ensure finite-time convergence of the solutions of a chain of integrators with bounded output perturbations to a minimal ellipsoidal set. The control parameters are obtained by solving a minimisation problem of the ‘size’ of the ellipsoid subject to a set of linear matrix inequalities (LMIs) constraints, and by applying the implicit function theorem. A numerical example is presented to support the implementability of these theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we consider the analysis and design of an output feedback controller for a perturbed nonlinear system in which the output is sampled and quantised. Using the attractive ellipsoid method, which is based on Lyapunov analysis techniques, together with the relaxation of a nonlinear optimisation problem, sufficient conditions for the design of a robust control law are obtained. Since the original conditions result in nonlinear matrix inequalities, a numerical algorithm to obtain the solution is presented. The obtained control ensures that the trajectories of the closed-loop system will converge to a minimal (in a sense to be made specific) ellipsoidal region. Finally, numerical examples are presented in order to illustrate the applicability of the proposed design method. [ABSTRACT FROM PUBLISHER]

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy. [ABSTRACT FROM AUTHOR]

This paper concerns the robust ‘practical’ stabilization for a class of linear controlled stochastic differential equations subject to both multiplicative and additive stochastic noises. Sufficient conditions of the stabilization are provided in two senses. In the first sense, it is proven that almost all trajectories of the stochastic model converge in a ‘mean-square sense’ to a bounded zone located in an ellipsoidal set, while the second one ensures the convergence to a zero zone in probability one. The considered control law is a linear state feedback. The stabilization problem is converted into the corresponding attractive averaged ellipsoid ‘minimization’ under some constraints of bilinear matrix inequalities (BMIs) type. Some variables permit to represent the BMIs problem in terms of linear matrix inequalities (LMIs) problem, which are resolved in a straight manner, using the conventional LMI-MATLAB toolbox. Finally, the numerical solutions of a benchmark example and a practical example are presented to show the efficiency of the proposed methodology. [ABSTRACT FROM AUTHOR]

Water level reduction with global navigation satellite systems in bathymetric surveying requires knowledge of the ellipsoidal heights of lowest astronomical tide (LAT). The traditional approach uses tidal water levels of an ocean tide model, which are subtracted from mean sea level (MSL). This approach has two major drawbacks: the modeled water levels refer to an equipotential surface, which differs from MSL, and MSL may not be known close to the coast. Here, we propose to model LAT directly relative to an equipotential surface (geoid). This is conceptually consistent with the flow equations and allows the inclusion of temporal MSL variations into the LAT definition. Numerical experiments for the North Sea show that significant differences between the traditional and the pursued approach exist if average monthly variations in MSL are included. A validation of the modeled LAT using tide gauge records reveals systematic errors, which we attribute to both the model and the tidal analysis procedure. We also show that the probability that water levels drop below LAT is high, with maximum frequency of once per week in the eastern North Sea. Therefore, we propose to reconsider the deterministic concept of LAT by a probabilistic chart datum concept, and we quantified the differences between them. [ABSTRACT FROM AUTHOR]

Controlling shapes is a challenging problem in a multi-phase fluid simulation. Bubble particles enable the details of air bubbles to be represented within a simulation based on an Euler grid. We control the target shapes of bubbles by the gradient vectors of the signed distance field and attraction forces associated with control particles. Our hybrid approach enables to simulate physically plausible movements of bubbles while preserving the details of a target shape. Furthermore, we control the paths of moving bubbles using user-defined curves and the shape of an air bubbles by drag force. An accurate model of the drag force near the fluid surface means that bubbles have realistic ellipsoidal shapes. [ABSTRACT FROM AUTHOR]

Experiments on the determination of the trajectories and orientations of ellipsoids freely floating up in a viscous fluid under the gravity action are described. [ABSTRACT FROM AUTHOR]

Abstract: We consider a general solution of the electromagnetic wave scattering problem for arbitrarily shaped homogeneous particles, whose surface can be expressed by a function of angular coordinates, using a Laplace series expansion. This can include regularly shaped particles (e.g., ellipsoids and cubes) as well as irregularly shaped particles like Gaussian spheres. For calculations of scattering properties of the particles, we use the approach based on the Sh-matrix. The Sh-matrix elements deduced from the T-matrix technique allow one to separate the shape effects from size- and refractive-index-dependent parameters. The separation also allows the corresponding surface integrals to be solved analytically for different particle shapes. In this manuscript, we give analytical expressions for the Sh-matrix elements for arbitrary shaped particles that can be presented with Laplace series. We find good agreement between results obtained comparing our and DDA calculations. [Copyright &y& Elsevier]

A complex microstructural feature has been discovered within centimeter-scale pseudotachylyte veins and is here termed “bubble collapse structure” (BCS). Three first-order features of BCSs are (1) a central quartz–dominated monocrystal or polycrystal, (2) a light-colored reaction halo, and (3) a generally radial but commonly curving pattern of opaque seams and/or elliptical quartz amygdules. Two populations of bubbles were principally responsible for forming BCSs. One was a population of millimeter-scale bubbles, and the second involved much more numerous small bubbles (mean diameter, 16 ± 8 μm). The first population likely formed from liquid water interstitial to the precursory cataclasite or from melting of hydrous minerals, whereas the second population formed by diffusional growth. BCSs began forming as the pseudotachylyte melt cooled and the pressure inside the bubbles decreased. As the large bubbles flattened into oblate ellipsoids parallel to the vein margins, melt flowed toward each collapsing bubble, deforming the smaller bubbles into radially arranged prolate shapes. Microlites nucleated in the melt on bubble margins and grew radially, and several minerals (titanite, apatite) grew on the margins of the collapsing bubbles. The nearly solidified melt formed convex-inward bulges, marking the last viscous flow. On further cooling, fluids penetrated the pseudotachylyte vein, partly along curving paths controlled by stress fields, forming pressure solutionlike seams and triggering quartz precipitation within surviving vesicles. BCSs owe their origin to changing rheology and the formation of bubbles within the pseudotachylyte melt and the subsequent transition into vesicles and amygdules. BCSs allow insights into the premelting history and nucleation of water-rich precursory fault rocks, pseudotachylyte melt formation, cooling of the melt and accompanying volume loss, and the return of fluids into the fault zone during and after cooling. [ABSTRACT FROM AUTHOR]

Abstract: A methodology for the design of sliding mode controllers for linear systems subjected to matched and unmatched perturbations is proposed. It is considered that the control signal is applied through a first-order low-pass filter. The technique is based on the existence of an attracting (invariant) ellipsoid such that the convergence to a quasi-minimal region of the origin using the suboptimal control signal is guaranteed. The design procedure is given in terms of the solution of a set of Matrix Inequalities. A benchmark example illustrating the design is given. [Copyright &y& Elsevier]

The concept of "random loose packing" (RLP) has evolved through extensive study of loose packings of spheres, which has resulted in an accepted definition as the loosest packing that can be obtained by pouring grains. We extend this consideration to packings of nonspherical grains (ellipsoids) formed by slow settling in a viscous liquid, and perform a detailed analysis of the structural properties of the resulting packings. We find that as in the case of spheres the loosest ellipsoid packings are generated for grains with high interparticle friction. However, unlike spheres, these packings cannot be considered random as they have a significant degree of orientational ordering that increases with the grain's aspect ratio. This demonstrates that applying sedimentation or pouring techniques that have become part of the commonly held definition of RLP, will not generate random packings of ellipsoids. The consequences for the accepted definition of RLP and its applicability to nonspherical grains is discussed. [ABSTRACT FROM AUTHOR]

This article presents a method for complexity reduction in reachability analysis and safety-preserving controller synthesis via Schur-based decomposition. The decomposition results in either decoupled or weakly-coupled (lower dimensional) subsystems. Reachable sets, computed independently for each subsystem, are back-projected and intersected to yield an overapproximation of the actual reachable set. Moreover, applying this technique to a class of unstable LTI systems, we show that when certain eigenvalue and state-constraint conditions are satisfied, further reduction of complexity is possible. Evaluating our method for a variety of examples we demonstrate that significant reduction in the computational costs can be achieved. This technique has considerable potential utility for use in conjunction with computationally intensive reachability tools. [ABSTRACT FROM AUTHOR]

Geodesics on the surface of an ellipsoid of revolution may be represented by canonical equations. Applying known theorems, an integral of this equation system can be found. Its exact solution is given by a definite integral (quadrature). Using power series, this integral can be calculated with arbitrary precision. As an example, a fourth order solution will be presented, especially for geodetic applications. [ABSTRACT FROM AUTHOR]

Abstract: Binaries are in vogue; many minor-planets like asteroids are being found to be binary or contact-binary systems. Even ternaries like 87 Sylvia have been discovered. The densities of these binaries are often estimated to be very low, and this, along with suspected accretionary origins, hints at a rubble interior. As in the case of fluid objects, a rubble-pile is unable to sustain all manners of spin, self-gravitation, and tidal interactions. This motivates the present study of the possible ellipsoidal shapes and mutual separations that members of a rubble-pile binary system may achieve. Conversely, knowledge of a granular binary’s shape and separation will constrain its internal structure – the ability of the binary’s members to sustain elongated shapes and/or maintain contact will hint at appreciable internal frictional strength. Because the binary’s members are allowed to be of comparable mass, the present investigation constitutes an extension of the second classical Darwin problem to granular aggregates. General equations defining the ellipsoidal rubble-pile binary system’s equilibrium are developed. These are then specialized to a pair of spin-locked, possibly unequal, prolate ellipsoidal granular aggregates aligned along their long axes. We observe that contact rubble-pile binaries can indeed exist. Further, depending on the binary’s geometry, an equilibrium contact binary’s members may, in fact, disrupt if separated. These results are applied to four suspected or known binaries: 216 Kleopatra, 25143 Itokawa, 624 Hektor and 90 Antiope. This exercise helps to bound the shapes and/or provide information about the interiors of these binaries. The binary’s interior will be modeled as a rigid-plastic, cohesionless material with a Drucker–Prager yield criterion. This rheology is a reasonable first model for rubble piles. We employ an approximate volume-averaging procedure that is based on the classical method of moments, and is an extension of the virial method (Chandrasekhar, S., 1969. Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven, CT) to granular solid bodies. The present approach also helps us present an incrementally consistent approach to investigate the equilibrium shapes of fluid binaries, while highlighting the inconsistencies and errors inherent in the popular “Roche binary approximation”. [Copyright &y& Elsevier]

We consider a system of so-called Hilbert compacts K( H) in a Hilbert space H; those Hilbert compacts admit a two-sided estimate by compact ellipsoids in H. For functionals in H, we introduce the notion of a compact extremum achieved at a certain base with respect to the imbedding in K( H). An example of the K-extremum of a variational functional in the Sobolev space W is considered. [ABSTRACT FROM AUTHOR]

This review paper deals with the dielectric and elastic characterization of composite materials constituted by dispersions of nonlinear inclusions embedded in a linear matrix. The dielectric theory deals with pseudo-oriented particles shaped as ellipsoids of revolution: it means that we are dealing with mixtures of inclusions of arbitrary aspect ratio and arbitrary non-random orientational distributions. The analysis ranges from parallel spheroidal inclusions to completely random oriented inclusions. Each ellipsoidal inclusion is made of an isotropic dielectric material described by means of the so-called Kerr nonlinear relation. On the other hand, the nonlinear elastic characterization takes into consideration a dispersion of nonlinear (spherical or cylindrical) inhomogeneities. Both phases are considered isotropic (actually it means polycrystalline or amorphous solids). Under the simplifying hypotheses of small deformation for the material body and of small volume fraction of the embedded phase, we describe a theory for obtaining the linear and nonlinear elastic properties (bulk and shear moduli and Landau coefficients) of the overall material. [ABSTRACT FROM AUTHOR]

Data from the revised Geneva-Copenhagen catalog are used to study the influence of radial migration of stars on the age dependences of parameters of the velocity ellipsoids for nearby stars in the thin disk of the Galaxy, assuming that the mean radii of the stellar orbits remain constant. It is demonstrated that precisely the radial migration of stars, together with the negative metallicity gradient in the thin disk, are responsible for the observed negative correlation between the metallicities and angular momenta of nearby stars, while the angular momenta of stars that were born at the same Galactocentric distances do not depend on either age or metallicity. The velocity components of the Sun relative to the Local Standard of Rest derived using data for stars born at the solar Galactocentric distance are ( U⊙, V⊙, W⊙) LSR = (5.1 ± 0.4, 7.9 ± 0.5, 7.7 ± 0.2) km/s. The two coordinates of the apex of the solar motion remain equal to 〈 l⊙〉 = 70° ± 7° and 〈 b⊙〉 = 41° ± 2°, within the errors. The indices for the power-law age dependences of them ajor, middle, and minor semi-axes become 0.26±0.04, 0.32±0.03, and 0.07±0.03, respectively. As a result, with age, the velocity ellipsoid for thin-disk stars born at the solar Galactocentric distance increases only in the plane of the disk, remaining virtually constant in the perpendicular direction. Its shape remains far from equilibrium, and the direction of the major axis does not change with age: the ellipsoid vertex deviation remains constant and equal to zero within the errors (〈 L〉 = 0.7° ± 0.6°, 〈 B〉 = 1.9° ± 1.1°). Such a small increase in the velocity dispersion perpendicular to the Galactic plane with age can probably be explained by “heating” of the stellar system purely by spiral density waves, without a contribution from giant molecular clouds. [ABSTRACT FROM AUTHOR]

We study the behavior of functionals of the form where ψ is a positive function, as n → ∞: The obtained results are used to establish the exact order equalities (as n → ∞) for the best n-term approximations of q-ellipsoids in metrics of the spaces S p φ: We also consider the applications of the obtained results to the determination of the exact orders of the Kolmogorov widths of octahedra in the Hilbert space. [ABSTRACT FROM AUTHOR]

Design of the control law for planar motion of the wheeled robot was studied. The aim of control lies in driving the robot to the desired smooth curvilinear trajectory and stabilizing its motion. At that, the control resource and the domain of variations of the phase variables are bounded. It was previously suggested to construct the criterion for control law stabilizability as invariant ellipsoids, that is, quadratic approximations of the attraction domains of the target trajectory. Then construction of the invariant ellipsoids came to solving a system of linear matrix inequalities and testing a scalar inequality. The paper was devoted to practical application of the previous results. Choice of the parameters of the system of linear matrix inequalities was discussed. An algorithm to construct an invariant ellipsoid in at most three iterations was developed. It also determines the maximal ellipsoid for a given maximal permissible deviation of the robot from the target trajectory. [ABSTRACT FROM AUTHOR]

The dependences of the velocity ellipsoids of F-G stars of the thin disk of the Galaxy on their ages and metallicities are analyzed based on the new version of the Geneva-Copenhagen Catalog. The age dependences of the major, middle, and minor axes of the ellipsoids, and also of the dispersion of the total residual velocity, obey power laws with indices 0.25, 0.29, 0.32, and 0.27 (with uncertainties ±0.02). Due to the presence of thick-disk objects, the analogous indices for all nearby stars are about a factor of 1.5 larger. Attempts to explain such values are usually based on modeling relaxation processes in the Galactic disk. Elimination of stars in the most numerous moving groups from the sample slightly reduces the corresponding indices (0.22, 0.26, 0.27, and 0.24). Limiting the sample to stars within 60 pc of the Sun, so that the sample can be considered to be complete, leaves both the velocity ellipsoids and their age dependences virtually unchanged. With increasing age, the velocity ellipsoid increases in size and becomes appreciablymore spherical, turns toward the direction of the Galactic center, and loses angular momentum. The shape of the velocity ellipsoid remains far from equilibrium. With increasing metallicity, the velocity ellipsoid for stars of mixed age increases in size, displays a weak tendency to become more spherical, and turns toward the direction of the Galactic center (with these changes occurring substantially more rapidly in the transition through the metallicity [Fe/H]≈−0.25). Thus, the ellipsoid changes similarly to the way it does with age; however, with decreasing metallicity, the rotational velocity about the Galactic center monotonically increases, rather than decreases (!). Moreover, the power-law indices for the age dependences of the axes depend on the metallicity, and display a maximum near [Fe/H] ≈−0.1. The age dependences of all the velocity-ellipsoid parameters for stars with equal metallicity are roughly the same. It is proposed that the appearance of a metallicity dependence of the velocity ellipsoids for thin-disk stars, recorded from the close to the Sun, is most likely due to the radial migration of stars. [ABSTRACT FROM AUTHOR]

An approach to stabilization of nonlinear oscillations in multidimensional spaces is proposed on the basis of the V.I. Zubov’s stability theory for invariant sets. As a special case, the derived controls make it possible to excite self-oscillating regimes in specified state subspaces R2 k ⊂ R2 n with simultaneous oscillation damping on Cartesian products R2 n−2 k. [ABSTRACT FROM AUTHOR]

This paper describes the redox chemistry of a tetrazene ligand on (β-diketiminato)iron complexes. Addition of 1-adamantyl azide to an iron(l) source gives the tetrazene complex L[supMe]Fe(AdNNNNAd), most likely through an imidoiron(lll) intermediate. Spectroscopic, magnetic, crystallographic, and computational investigations of the tetrazene complex show that one unpaired spin occupies a primarily ligand-based orbital, and is antiferromagnetically coupled to a high-spin iron(ll) ion to give an S = 3/2 ground state. Reversible single-electron reduction occurs at the ligand singly occupied molecular orbital (SOMO), affording a dianionic tetrazene ligand while leaving the metal oxidation state and spin state unchanged. [ABSTRACT FROM AUTHOR]

The article describes the crystal structure of a two-dimensional network in the molecular salt 2-methyllimidazolium hydrogen glutarate, and three-dimensional networks in the salts 2-methyl-imidazolium hydrogen succinate and 2-methyllimidazolium hydrogen adipate monohydrate. It shows the formation of the networks which are built from hydrogen bonded rings, with atom numbering and probability displacement ellipsoids. Also calculated are the values of the diacids and bases in an aqueous medium to identify the existence of 1:2 salt of an aliphatic diacid.

The Western Granite is the largest of several granitic bodies around the margin of the Rum Central Igneous Complex. We report palaeomagnetic and anisotropy of magnetic susceptibility (AMS) data that bear on the emplacement and deformation of the Western Granite. The collection includes samples from 27 sites throughout the Western Granite, five sites in adjacent feldspathic peridotite, and two sites in intermediate to mafic hybrid contact aureole rocks. Palaeomagnetic data from 22 of the 27 sites in the granite provide an in situ group mean D = 213.2°, I=-69.5°, α95 = 5.5° that is discordant to an early Paleocene reverse polarity expected field (about 184°, -66°, α95 = 4.3°). The discrepancy is eliminated by removing an inferred 15° of northwest-side-down tilting about a best fit horizontal tilt axis trending 040°. Data from the younger peridotite and hybrid rocks of the Rum Layered Suite provide an in situ group mean of D = 182.6°, I = -64.8°, α95 = 4.0°, which is statistically indistinguishable from an early Paleocene expected field, and imply no post-emplacement tilting of these rocks since remanence acquisition. The inferred tilt recorded in the Western Granite, which did not affect the younger Layered Suite, suggests that emplacement of the ultrabasic rocks resulted in roof uplift and associated tilt of the Western Granite to make space for mafic magma emplacement. Magnetic fabric magnitude and susceptibility parameters yield two subtle groupings in the Western Granite AMS data set. Group 1 data, defined by rocks from exposures to the east and south, have comparatively high bulk susceptibilities (Kmean, 29.51 × 10-3 in SI system), stronger anisotropies (Pj, 1.031) and oblate susceptibility ellipsoids. Group 2 data, from rocks in the west part of the pluton, have lower values of Kmean (15.89 × 10-3 SI) and Pj (1.014), and triaxial susceptibility ellipsoids. Magnetic lineations argue for emplacement of the granite as a tabular sheet from the south-southeast toward the north and west. Moderate to steeply outward-dipping magnetic foliations, together with deflection of the country rock bedding in the north, are consistent with doming accompanying magma emplacement. [ABSTRACT FROM AUTHOR]

We present analysis of lateral variations of density in the upper mantle in the area of contact of the precambrian East European Craton (EEC) and the Palaeozoic Platform (PP) in Poland, obtained by analysis of the gravimetric geoid undulations. A precise 3-D density model of the crust in the study area down to a depth of 50 km, discussed in the first part of this paper ( Majdański et al. , in press) did not explain all features of the observed geoid. This suggests that these features can be due to density inhomogeneities in the upper mantle. To estimate them, we performed inversion of a residual between the observed geoid and undulations caused by the 3-D density distribution in the crust. Basing on the assumption of local isostatic compensation and Pratt–Hayford isostasy model, the density distribution in the upper mantle was parametrized as a 40-km-thick layer located above the assumed compensation depth of 140 km and subdivided into irregular blocks. The boundaries of the blocks were defined according to boundaries of major tectonic units in the study area and position and shape of the most pronounced anomalies in the residual geoid. A series of sensitivity tests calculated for such density heterogeneities in the upper mantle showed that they can produce geoid undulations of the order of several metres. The density values in each unit were taken as model parameters for the inversion procedure, and inverse problem was solved using global optimization with constraints. The density variations in the upper mantle in the final model correlate well with the surface heat flow. This suggests that these variations can be due to diversity in mantle temperature. The Trans-European Suture Zone (TESZ), which is a major suture separating the EEC from the PP, is not observed as a distinct unit in the mantle. Instead, our study suggests continuation of the lithosphere of the EEC beneath the PP and confirms subdivision of the TESZ into terranes with distinctly different evolution. A simple calculation shows that condition of local isostatic equilibrium is almost achieved at the assumed compensation depth, with the exception of several small-scale units. However, different loads and mass deficiencies in the lithosphere are compensated at different depth levels. This proves that the lithosphere is alternation of mechanically strong and weak layers, both in the crust and upper mantle, and that different loads can be compensated at different depth. However, the total compensation of all lithospheric loads occurs in the upper mantle. [ABSTRACT FROM AUTHOR]

Using modern differential geometric methods, we study the relative equilibria for Dirichlet’s model of a self-gravitating fluid mass having at least two equal axes. We show that the only relative equilibria of this type correspond to Riemann ellipsoids for which the angular velocity and vorticity are parallel to the same principal axis of the body configuration. The two solutions found are MacLaurin and transversal spheroids. The singular reduced energy-momentum method developed in Rodríguez-Olmos (Nonlinearity 19(4):853–877, ) is applied to study their nonlinear stability and instability. We found that the transversal spheroids are nonlinearly stable for all eccentricities while for the MacLaurin spheroids, we recover the classical results. Comparisons with other existing results and methods in the literature are also made. [ABSTRACT FROM AUTHOR]

Finite element calculation is applied to assess elasticity parameters of random cellular structures with anisotropy. Cellular structure is generated using Random Sequential Addition algorithm. Anisotropy is considered by introducing elongation and orientation of ellipsoid voids. Structures are meshed using a regular mesh scheme. A uniaxial compression test is simulated under solid phase isotropy conditions. Effective Young's modulus and Poisson ratio are calculated and related to structural anisotropy parameters. Finite element results show a mesh-independence of elasticity parameters against scale and underline the large dependence of elasticity parameters to anisotropy and to the relative density of the cellular material: rigidity is increased in the anisotropy direction and mechanical instability in the other directions. Structures with large random population of ellipsoids contribute to lower such effect. [ABSTRACT FROM AUTHOR]

Suppose x̄ and s̄ lie in the interiors of a cone K and its dual K *, respectively. We seek dual ellipsoidal norms such that the product of the radii of the largest inscribed balls centered at x̄ and s̄ and inscribed in K and K *, respectively, is maximized. Here the balls are defined using the two dual norms. When the cones are symmetric, that is self-dual and homogeneous, the solution arises directly from the Nesterov–Todd primal–dual scaling. This shows a desirable geometric property of this scaling in symmetric cone programming, namely that it induces primal/dual norms that maximize the product of the distances to the boundaries of the cones. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a numerical approach to operability analysis is developed. Based on the concept of the steady-state region of attraction, which indicates the initial operating conditions that can be driven to a given setpoint, this analysis helps design engineers assess whether a process can be effectively controlled by linear control systems. Using steady-state operating data that are generated with process simulators (such as ASPEN PLUS®), the steady-state region of attraction are calculated by solving the proposed constrained nonlinear optimization problem. The objective function for the optimization problem is developed using the ellipsoid theory while the constraint functions are formulated from the complement set of the initially defined constraint set and the boundary of the available input space (AIS). Furthermore, as the solution of a nonlinear optimization problem generally depends on the initial starting point, a method of generating a set of ellipsoids in the hyperspace to start the calculation is also developed. The proposed constrained nonlinear optimization problem can be solved using numerical tools such as MATLAB®. This analysis can be performed based on process flowsheets with process simulators and thus is useful in early stages of process design. The use of the proposed numerical framework is illustrated by a case study of a methyl acetate reactive distillation column. [Copyright &y& Elsevier]

Abstract: We combine exponential sums, character sums and Fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids. [Copyright &y& Elsevier]

The distance of closest approach of particles with hard cores is a key parameter in statistical theories and computer simulations of liquid crystals and colloidal systems. In this paper, we provide a new algorithm to calculate the distance of closest approach of two ellipsoids of arbitrary shape and orientation. This algorithm is based on our previous analytic result for the distance of closest approach of 2D ellipses. The method consists of determining the intersection of the ellipsoids with the plane containing the line joining their centers and rotating the plane. The distance of closest approach of the two ellipses formed by the intersection is a periodic function of the plane orientation, whose maximum corresponds to the distance of closest approach of the two ellipsoids. [ABSTRACT FROM AUTHOR]

The ultimate state of granular material was investigated by the discrete element method. This paper discusses a series of numerical triaxial tests on samples of two kinds of ellipsoids. Samples of three different densities were loaded along various stress paths. The ellipsoids were mixed in the proportions 50:50 by weight. The longer particles have an aspect ratio (major length/minor length) of 1.5 and the other particles have an aspect ratio of 1.2. The samples were generated by deposition under gravity to simulate the air-pluviation sample preparation technique. Then they were consolidated isotropically. Fourteen stress-path controlled triaxial tests were conducted numerically for each sample. They are sheared to the ultimate state to determine the friction angle and void ratio at the ultimate state. When samples of different densities were sheared along the same loading path, a unique ultimate state was observed. The initial density does not affect the ultimate state. However, when different loading paths are applied to the same numerical specimen, they do not always arrive at the same critical state in the void-mean stress space. [ABSTRACT FROM AUTHOR]

Abstract: This paper establishes IQC-based (Integral Quadratic Constraints) conditions under which an ellipsoid is contractively invariant for a single input linear system under a saturated linear feedback law. Based on these set invariance conditions, the determination of the largest such ellipsoid, for use as an estimate of the domain of attraction, can be formulated and solved as an LMI optimization problem. Such an LMI problem can also be readily adapted for the design of the feedback gain that achieves the largest contractively invariant ellipsoid. While the advantages of the proposed IQC approach remain to be explored, it is shown in this paper that the largest contractively invariant ellipsoid determined by this approach is the same as the one determined by the existing approach based on expressing the saturated linear feedback as a linear differential inclusion (LDI), which is known to lead to non-conservative result in determining the largest contractively invariant ellipsoid for single input systems. [Copyright &y& Elsevier]

Gibbs ensemble Monte Carlo simulations are performed to obtain the vapour-liquid equilibrium of oblate-like fluids interacting through the Kihara intermolecular potential. Results confirm the validity of a perturbation theory for Kihara fluids, whose accuracy for prolate fluids was tested some years ago. As in the case of hard ellipsoids, the symmetry of the phase diagram of oblate and prolate models is analysed. An interesting relation of Boyle temperature and critical parameters with molecular volume is found for the considered models. As a particular application, this relation allows the prediction of some thermodynamic properties of a new promising biofuel 2,5dimethylfuran. [ABSTRACT FROM AUTHOR]

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of complete ellipsoidal shells of revolution with variable thickness and solid ellipsoids. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components u r , u θ, and u z in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the ellipsoidal shells of revolution and solid ellipsoids are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to three or four-digit exactness is demonstrated for the first five frequencies of the ellipsoidal shells of revolution. Numerical results are presented for a variety of ellipsoidal shells with variable thickness. Frequencies for five solid ellipsoids of different axis ratios are also given. Spherical shells and solid spheres are special cases which are included. Comparisons are also made between the frequencies from the present 3D Ritz method, a 2D Ritz method, and a 3D finite element method. The multiple degeneracies (two or more modes having the same frequency) of spherical bodies are examined by analyzing some almost-spherical ellipsoids. [ABSTRACT FROM AUTHOR]

Abstract: A new method is proposed to determine strain ellipsoids from measurements made on planar surfaces. It is similar in theory to that of Robin, P.-Y.F., [2002. Determination of fabric and strain ellipsoids from measured sectional ellipses – theory. Journal of Structural Geology 24, 531–544], but is more flexible and hence more applicable because it can incorporate different kinds of strain measurements, especially when only the pitches of the long/short axes of strain ellipses are measured on planar surfaces. It is not necessary to estimate both the relative and absolute strain ellipsoids at the same time, as Robin, P.-Y.F., [2002. Determination of fabric and strain ellipsoids from measured sectional ellipses – theory. Journal of Structural Geology 24, 531–544] and others did. The former may be estimated before the latter. The feasibility of the proposed method is demonstrated by three artificial examples and one real example. [Copyright &y& Elsevier]

We investigate the problem of minimizing the moment of inertia among convex surfaces in R3 having a specified surface area. First, we prove that a minimizing surface exists, and derive a necessary condition holding at points of positive curvature. Then we show that an equilateral triangular prism is the optimal triangular prism, that the cube is the optimal rectangular prism, and that the sphere is (locally) optimal among ellipsoids. Many examples of convex surfaces are examined, among which the lowest moment of inertia is achieved by a truncated tetrahedron. The problem of finding the global minimizing surface remains open. The analogous problem in two dimensions has been solved by Sachs and later by Hall, who showed that the equilateral triangle minimizes the moment of inertia, among all convex curves with given length. [ABSTRACT FROM AUTHOR]