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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]

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]

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 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]

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]

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]

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]

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.

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]

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 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]

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]

An up to date determination of a high-resolution geoid requires the use of best available databases concerning digital terrain model (DTM), bathymetry, global geopotential model and gravity field. The occasion to revisit methods to validate and merge different data sets has been created by a new project for the determination of a new European Geoid. Since the computation of the latest European geoid and quasi-geoid model (EGG97), significant new or improved data sets have become available, such as new global geopotential models from CHAMP and GRACE missions, new national and global DTMs and new or upgraded gravity data sets. In the context of the new European Gravity and Geoid Project (EGGP), within the IAG Commission 2, some data validation tests have been performed in the Italian zone. In the area 19°× 17° wide, covering Italy, three kinds of tests have been performed: comparison among different DTMs in order to choose the best one to be used; comparisons in terms of geoid computation in some coastal areas, to evaluate bathymetry effects, and the validation of the EIGEN-CG01C and EIGEN-CG03C new global models up to degree and order 360. These preliminary tests lead to the choice of SRTM DTM (integrated in no-data holes), with an added bathymetry derived by the Italian 1:25 000 official cartography near the coasts and the NOAA bathymetry in high seas. The validation of the new global models and the comparison with EGM96 model show that, in terms of geoid computation, the EGM96 yields better results. Moreover, the validation of new available land gravity data and the cross-validation of two sets of gravity data on sea have been completed. [ABSTRACT FROM AUTHOR]

Summary: We report a normal-mode method for anisotropic refinement of membrane-protein structures, based on a hypothesis that the global near-native-state disordering of membrane proteins in crystals follows low-frequency normal modes. Thus, a small set of modes is sufficient to represent the anisotropic thermal motions in X-ray crystallographic refinement. By applying the method to potassium channel KcsA at 3.2 Å, we obtained a structural model with an improved fit with the diffraction data. Moreover, the improved electron density maps allowed for large structural adjustments for 12 residues in each subunit, including the rebuilding of 3 missing side chains. Overall, the anisotropic KcsA structure at 3.2 Å was systematically closer to a 2.0 Å KcsA structure, especially in the selectivity filter. Furthermore, the anisotropic thermal ellipsoids from the refinement revealed functionally relevant structural flexibility. We expect this method to be a valuable tool for structural refinement of many membrane proteins with moderate-resolution diffraction data. [Copyright &y& Elsevier]

Abstract: Intermolecular interaction leading to formation of aggregates between gelatin, a polyampholyte, and agar, a polysaccharide was studied in the supernatant of the complex coacervate formed by these biopolymers. Electrophoresis, laser light scattering and viscometry data were used to determine the interaction and the physical structure of these intermolecular soluble complexes by modeling these to be prolate ellipsoids of revolution (rod-like structures with well defined axial ratio and Perrin''s factor). Solution ionic strength was found to reduce the axial ratio of these complexes implying the presence of screened polarization-induced electrostatic interaction between the two biopolymers. [Copyright &y& Elsevier]

A finite line of uniform charge is either ignored or handled incompletely in most textbooks. New simple and practical expressions are presented for the electric potential and electric field for this charge distribution. The equipotentials are shown to be prolate ellipsoids and the electric field lines follow hyperboloids confocal to the ellipsoids. The common foci are shown to be the endpoints of the charged line segment. These geometries are demonstrated with simple VPython programs. [ABSTRACT FROM AUTHOR]

The inertia matching ellipsoid (IME) is proposed as a new index of dynamic performance for serial-link robotic manipulators. The IME integrates the existing dynamic manipulability and manipulating-force ellipsoids to achieve an accurate measure of the dynamic torque-force transmission efficiency between the joint torque and the force applied to a load held by an end-effector. The dynamic manipulability and manipulating-force ellipsoids can both be derived from the IME as limiting forms, with respect to the weight of the load. The effectiveness of the IME is demonstrated numerically through the selection of an optimal leg posture for jumping robots and optimal active stiffness control, and experimentally through application to a pick-up task using a commercial manipulator. The index is also extended theoretically to the case of a manipulator mounted on a free-flying satellite. [ABSTRACT FROM AUTHOR]

This paper presents a sensitivity analysis of the input parameters of a program based on the discrete element method (DEM). Triaxial compression simulations were conducted on an assembly of ellipsoids with two particle shapes. We examine four input parameters including shear modulus of particles, density of particles, time step, and damping. Generally, these parameters are chosen by calibrating the result with certain known behavior of granular materials. In dynamic simulations, these input parameters are bounded by their physical attributions that should not be altered. However, in static simulations, they do not have the same physical implication. Validity of results may be questionable when input parameters are used without justification. A sensitivity analysis of the input parameters should shed light on this issue. In this paper, we will study the effect of the input values within the range of (1/10)–10 times the benchmark value. The benchmark values are commonly used by the writer. The results are presented against the benchmark simulation. The unbalanced forces in the simulations are kept below a prescribed value to enforce equilibrium. The result shows that the effect of all input parameters used in this paper is negligible as long as the small unbalanced forces in the system can be achieved. The runtimes are different. However, there are two simulations (one with low damping and the other with a large time step size) that cannot maintain the required small unbalanced force. In other words, equilibrium cannot be achieved for these simulations. [ABSTRACT FROM AUTHOR]

This study shows that a proper assessment of granule morphology is fundamental before applying any mathematical derivation or empirical formula based on the shape factor of anaerobic granular particles to determine granular characteristics. The granular images and size distribution of samples observed in this study revealed two different dimensions along two axes, which characterize these panicles as ellipsoids. In the literature. theoretical-settling velocities and panicle-size distribution of anaerobic granules have been calculated by assuming granules as spherical shaped and using the numerical correlation between the size and settling velocity. This resulted in large deviations in results reported for settling velocities and size distribution calculated by using empirical equations when compared with experimentally measured values. It is believed that, because of the non-spherical nature of these particles, errors have been made in the earlier studies (which considered these panicles as spherical), while evaluating granular physical properties. [ABSTRACT FROM AUTHOR]

The influence of particle shape on granular material response is examined by using the discrete element method. Triaxial drained and undrained tests were performed on specimens of ellipsoids of two sizes. The triaxial test boundary conditions were simulated with a recently developed boundary mechanism. Different loading paths including axial compression, axial extension, lateral compression, and true extension were employed. The specimens were composed of 1,170 ellipsoids having two types of particles. The specimen is made up of 50% by weight of Type I particles that have an aspect ratio of 1.2. The aspect ratio of the Type II particles varies between 1.5 and 2. The specimens were consolidated isotropically before shearing. Comparing with the behavior of specimens of mono-size particles, a higher friction angle and a more complex particle shape effect were observed. The friction angles from the drained tests (axial extension, true extension, and lateral compression tests) were similar and the values are higher than that of the axial compression test. All simulated results are in good agreement with laboratory observation of sands. [ABSTRACT FROM AUTHOR]

The interaction of the depth cues of binocular disparity and motion parallax could potentially be used by the visual system to recover an estimate of the viewing distance. The present study investigated whether an interaction of stereo and motion has effects that persist over time to influence the perception of shape from stereo when the motion information is removed. Static stereoscopic ellipsoids were presented following the presentation of rotating stereoscopic ellipsoids, which were located either at the same or a different viewing distance. It was predicted that shape judgements for static stimuli would be better after presentation of a rotating stimulus at the same viewing distance, than after presentation of one at a different viewing distance. No such difference was found. It was concluded that an interaction between stereo and motion depth cues does not influence the perception of subsequently presented static objects. [Copyright &y& Elsevier]

Experimental results on incompressible viscous flows in the gap between rotating spheroids are presented. Oblate ellipsoids are discussed as a connecting link between a sphere and a disk, while a prolate ellipsoid represents the geometrical intermediate stage between a sphere and a cylinder. The axes ratios of the ellipsoids are A : B = 2 : 1 and A : B = 1 : 2, respectively. The basic flow field and the occurring instabilities in form of Taylor- and cross-flow vortices have been studied by flow visualization. The common properties and their differences regarding the various flow patterns of the disks, spheres, cylinders as well as of the oblate and prolate ellipsoids are treated. The location of the generation of the different vortex systems at the various rotating bodies is determined. A comparison of the friction torque for these bodies of rotation is given by diagrams. Finally, the theoretically derived curves for the friction torque of the disks, spheres and cylinders are compared with the experimentally obtained data for the oblate and prolate ellipsoids. [ABSTRACT FROM AUTHOR]

Summary. Experimental results on incompressible viscous flows in the gap between rotating spheroids are presented. Oblate ellipsoids are discussed as a connecting link between a sphere and a disk, while a prolate ellipsoid represents the geometrical intermediate stage between a sphere and a cylinder. The axes ratios of the ellipsoids are A : B=2 : 1 and A : B=1 : 2, respectively. The basic flow field and the occurring instabilities in form of Taylor- and cross-flow vortices have been studied by flow visualization. The common properties and their differences regarding the various flow patterns of the disks, spheres, cylinders as well as of the oblate and prolate ellipsoids are treated. The location of the generation of the different vortex systems at the various rotating bodies is determined. A comparison of the friction torque for these bodies of rotation is given by diagrams. Finally, the theoretically derived curves for the friction torque of the disks, spheres and cylinders are compared with the experimentally obtained data for the oblate and prolate ellipsoids. [ABSTRACT FROM AUTHOR]

A method of postprocessing and visualizing three-dimensional vector fields, such as current density reconstruction results, is presented. This method is based on equivalent ellipsoids fitted to the vector fields. The technique has been tested with simulated data and current density reconstructions based on bioelectromagnetic data obtained from a physical thorax phantom. Three different approaches based on: 1) longest distance; 2) dominant direction; and 3) principal component analysis, for fitting the equivalent ellipsoids are proposed. Multiple foci in vector fields are extracted by multiple ellipsoids which are fitted iteratively. The method enables statistical postprocessing for the sake of comparisons of different source reconstructions algorithms or comparisons of groups of patients or volunteers. [ABSTRACT FROM AUTHOR]

Comments on the article "Unmasking Multivariate Outliers and Leverage Points," by Peter J. Rousseeuw and Bert C. van Zomeren. Use of minimum volume ellipsoids to determine the mean vector and covariance matrix of the elemental set; Reactions of the authors on the Minimum Volume Elemental Ellipse approach; Need for a graphical display that could provide insight into how outliers are distributed relative to the bulk of the data.

We study the dynamics of the Hénon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each Hénon map, whether these sets are empty or nonempty, whether they are bounded or unbounded, and whether they are equal to the unit ball or not. On a certain region of the parameter space we show that the filled Julia set is an attractor. We prove that, for infinitely many distinct Hénon maps over Q3, this attractor is infinite and supports an SRB-type measure describing the distribution of all nearby forward orbits. We include some numerical calculations which suggest the existence of such infinite attractors over Q5 and Q7 as well. On a different region of the parameter space, we show that the Hénon map is topologically conjugate on its filled Julia set to the two-sided shift map on the space of bisequences in two symbols. [ABSTRACT FROM AUTHOR]

In the title compound, C2H8N+·C12H11O5P2−, pairs of hydrogen diphenyl­diphospho­nate anions form dimers across a twofold axis, with two symmetric O⋯H⋯O hydrogen bonds [O⋯O = 2.406 (3) and 2.418 (3) Å]. The 12-membered ring thus formed has crystallographic 2 and quasi-222 symmetry. Cations on either side of the ring form N—H⋯O hydrogen bonds to the four extraannular O atoms, with N⋯O distances of 2.765 (2) and 2.748 (3) Å. [ABSTRACT FROM AUTHOR]