(Non-)ergodicity of a degenerate diffusion modeling the fiber lay down process

We analyze the large time behavior of a stochastic model for the lay down of fibers on a moving conveyor belt in the production process of nonwovens. It is shown that under weak conditions this degenerate diffusion process has a unique invariant distribution and is even geometrically ergodic. This generalizes results from previous works [M. Grothaus and A. Klar, SIAM J. Math. Anal., 40 (2008), pp. 968–983; J. Dolbeault et al., arXiv:1201.2156] concerning the case of a stationary conveyor belt, in which the situation of a moving conveyor belt has been left open.

Estimador do Tipo Núcleo para Densidades Limites de Cadeias de Markov com Espaço de Estados Geral

In this work we studied the consistency for a class of kernel estimates of f f (.) in the Markov chains with general state space E C Rd case. This study is divided into two parts: In the first one f (.) is a stationary density of the chain, and in the second one f (x) v (dx) is the limit distribution of a geometrically ergodic chain

Ergodic capacity of the exponentially correlated slotted amplify and forward relay channel

In this paper we analyze the performance degradation of slotted amplify-and-forward protocol in wireless environments with high node density where the number of relays grows asymptotically large. Channel gains between source-destination pairs in such networks can no longer be independent. We analyze the degradation of performance in such wireless environments where channel gains are exponentially correlated by looking at the capacity per channel use. Theoretical results for eigenvalue distribution and the capacity are derived and compared with the simulation results. Both analytical and simulated results show that the capacity given by the asymptotic mutual information decreases with the network density.

Ergodic capacity of the slotted amplify and forward relay channel with finite relays

Abstract—In this paper we investigate the capacity of a general class of the slotted amplify and forward (SAF) relaying protocol where multiple, though a finite number of relays may transmit in a given cooperative slot and the relay terminals being half-duplex have a finite slot memory capacity. We derive an expression for the capacity per channel use of this generalized SAF channel assuming all source to relay, relay to destination and source to destination channel gains are independent and modeled as complex Gaussian. We show through the analysis of eigenvalue distributions that the increase in limiting capacity per channel use is marginal with the increase of relay terminals.

Ergodic capacity and outage performance of amplify-and-forward protocols

This thesis analyses the performance bounds of amplify-and-forward relay channels which are becoming increasingly popular in wireless communication applications. The statistics of cascaded Nakagami-m fading model which is a major obstacle in evaluating the outage of wireless networks is analysed using Mellin transform. Furthermore, the upper and the lower bounds for the ergodic capacity of the slotted amplify-and-forward relay channel, for finite and infinite number of relays are derived using random matrix theory. The results obtained will enable wireless network designers to optimize the network resources, benefiting the consumers.

Ergodic control of partially observed Markov chains

The ergodic or long-run average cost control problem for a partially observed finite-state Markov chain is studied via the associated fully observed separated control problem for the nonlinear filter. Dynamic programming equations for the latter are derived, leading to existence and characterization of optimal stationary policies.

A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization

While frame-invariant solutions for arbitrarily large rotational deformations have been reported through the orthogonal matrix parametrization, derivation of such solutions purely through a rotation vector parametrization, which uses only three parameters and provides a parsimonious storage of rotations, is novel and constitutes the subject of this paper. In particular, we employ interpolations of relative rotations and a new rotation vector update for a strain-objective finite element formulation in the material framework. We show that the update provides either the desired rotation vector or its complement. This rules out an additive interpolation of total rotation vectors at the nodes. Hence, interpolations of relative rotation vectors are used. Through numerical examples, we show that combining the proposed update with interpolations of relative rotations yields frame-invariant and path-independent numerical solutions. Advantages of the present approach vis-a-vis the updated Lagrangian formulation are also analyzed.

Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam

We explore an isoparametric interpolation of total quaternion for geometrically consistent, strain-objective and path-independent finite element solutions of the geometrically exact beam. This interpolation is a variant of the broader class known as slerp. The equivalence between the proposed interpolation and that of relative rotation is shown without any recourse to local bijection between quaternions and rotations. We show that, for a two-noded beam element, the use of relative rotation is not mandatory for attaining consistency cum objectivity and an appropriate interpolation of total rotation variables is sufficient. The interpolation of total quaternion, which is computationally more efficient than the one based on local rotations, converts nodal rotation vectors to quaternions and interpolates them in a manner consistent with the character of the rotation manifold. This interpolation, unlike the additive interpolation of total rotation, corresponds to a geodesic on the rotation manifold. For beam elements with more than two nodes, however, a consistent extension of the proposed quaternion interpolation is difficult. Alternatively, a quaternion-based procedure involving interpolation of relative rotations is proposed for such higher order elements. We also briefly discuss a strategy for the removal of possible singularity in the interpolation of quaternions, proposed in [I. Romero, The interpolation of rotations and its application to finite element models of geometrically exact rods, Comput. Mech. 34 (2004) 121–133]. The strain-objectivity and path-independence of solutions are justified theoretically and then demonstrated through numerical experiments. This study, being focused only on the interpolation of rotations, uses a standard finite element discretization, as adopted by Simo and Vu-Quoc [J.C. Simo, L. Vu-Quoc, A three-dimensional finite rod model part II: computational aspects, Comput. Methods Appl. Mech. Engrg. 58 (1986) 79–116]. The rotation update is achieved via quaternion multiplication followed by the extraction of the rotation vector. Nodal rotations are stored in terms of rotation vectors and no secondary storages are required.

Ordered spin-ice state in the geometrically frustrated metallic ferromagnet Sm2Mo2O7

The recent discovery of spin ice is a spectacular example of the noncoplanar spin arrangements that can arise in the pyrochlore A2B2O7 structure. We present magnetic and thermodynamic studies on the metallic ferromagnet pyrochlore Sm2Mo2O7. Our studies, carried out on oriented crystals, suggest that the Sm spins have an ordered spin-ice ground state below about T*=15 K. The temperature and field evolution of the ordered spin-ice state are governed by an antiferromagnetic coupling between the Sm and Mo spins. We propose that as a consequence of a robust feature of this coupling, the tetrahedra aligned with the external field adopt a one-in, three-out spin structure as opposed to the three-in, one-out structure in dipolar spin ices, as the field exceeds a critical value.

Geometrically simple linear weirs using circular quadrants

A detailed theoretical analysis of flow through a quadrant plate weir is made in the light of the generalized theory of proportional weirs, using a numerical optimization procedure. It is shown that the flow through the quadrant plate weir has a linear discharge-head relationship valid for certain ranges of head. It is shown that the weir is associated with a reference plane or datum from which all heads are reckoned.Further, it is shown that the measuring range of the quadrant plate weir can be considerably enhanced by extending the tangents to the quadrants at the terminals of the quadrant plate weir. The importance of this weir (when the datum of the weir lies below its crest) as an outlet weir for grit chambers is highlighted. Experiments show excellent agreement with the theory by giving a constant average coefficient of discharge.

Geometric Characterizations for Patterson-Sullivan Measures of Geometrically Finite Kleinian Groups

The main results of this thesis show that a Patterson-Sullivan measure of a non-elementary geometrically finite Kleinian group can always be characterized using geometric covering and packing constructions. This means that if the standard covering and packing constructions are modified in a suitable way, one can use either one of them to construct a geometric measure which is identical to the Patterson-Sullivan measure. The main results generalize and modify results of D. Sullivan which show that one can sometimes use the standard covering construction to construct a suitable geometric measure and sometimes the standard packing construction. Sullivan has shown also that neither or both of the standard constructions can be used to construct the geometric measure in some situations. The main modifications of the standard constructions are based on certain geometric properties of limit sets of Kleinian groups studied first by P. Tukia. These geometric properties describe how closely the limit set of a given Kleinian group resembles euclidean planes or spheres of varying dimension on small scales. The main idea is to express these geometric properties in a quantitative form which can be incorporated into the gauge functions used in the modified covering and packing constructions. Certain estimation results for general conformal measures of Kleinian groups play a crucial role in the proofs of the main results. These estimation results are generalizations and modifications of similar results considered, among others, by B. Stratmann, D. Sullivan, P. Tukia and S. Velani. The modified constructions are in general defined without reference to Kleinian groups, so they or their variants may prove useful in some other contexts in addition to that of Kleinian groups.

Geometrically Simple Logarithmic Weir

This paper discusses the design and experimental verification of a geometrically simple logarithmic weir. The weir consists of an inward trapezoidal weir of slope 1 horizontal to n vertical, or 1 in n, over two sectors of a circle of radius R and depth d, separated by a distance 2t. The weir parameters are optimized using a numerical optimization algorithm. The discharge through this weir is proportional to the logarithm of head measured above a fixed reference plane for all heads in the range 0.23R less than or equal to h less than or equal to 3.65R within a maximum deviation of +/-2% from the theoretical discharge. Experiments with two weirs show excellent agreement with the theory by giving a constant average coefficient of discharge of 0.62. The application of this weir to the field of irrigation, environmental, and chemical engineering is highlighted.

Fully Comprehensive Geometrically Non-Linear Dynamic Analysis of Multi-Body Beam Systems with Elastic Couplings

This paper is concerned with the dynamic analysis of flexible,non-linear multi-body beam systems. The focus is on problems where the strains within each elastic body (beam) remain small. Based on geometrically non-linear elasticity theory, the non-linear 3-D beam problem splits into either a linear or non-linear 2-D analysis of the beam cross-section and a non-linear 1-D analysis along the beam reference line. The splitting of the three-dimensional beam problem into two- and one-dimensional parts, called dimensional reduction,results in a tremendous savings of computational effort relative to the cost of three-dimensional finite element analysis,the only alternative for realistic beams. The analysis of beam-like structures made of laminated composite materials requires a much more complicated methodology. Hence, the analysis procedure based on Variational Asymptotic Method (VAM), a tool to carry out the dimensional reduction, is used here.The analysis methodology can be viewed as a 3-step procedure. First, the sectional properties of beams made of composite materials are determined either based on an asymptotic procedure that involves a 2-D finite element nonlinear analysis of the beam cross-section to capture trapeze effect or using strip-like beam analysis, starting from Classical Laminated Shell Theory (CLST). Second, the dynamic response of non-linear, flexible multi-body beam systems is simulated within the framework of energy-preserving and energy-decaying time integration schemes that provide unconditional stability for non-linear beam systems. Finally,local 3-D responses in the beams are recovered, based on the 1-D responses predicted in the second step. Numerical examples are presented and results from this analysis are compared with those available in the literature.

Geometrically Nonlinear Dynamics of Composite Four-Bar Mechanisms

This work intends to demonstrate the importance of geometrically nonlinear crosssectional analysis of certain composite beam-based four-bar mechanisms in predicting system dynamic characteristics. All component bars of the mechanism are made of fiber reinforced laminates and have thin rectangular cross-sections. They could, in general, be pre-twisted and/or possess initial curvature, either by design or by defect. They are linked to each other by means of revolute joints. We restrict ourselves to linear materials with small strains within each elastic body (beam). Each component of the mechanism is modeled as a beam based on geometrically nonlinear 3-D elasticity theory. The component problems are thus split into 2-D analyses of reference beam cross-sections and nonlinear 1-D analyses along the four beam reference curves. For thin rectangular cross-sections considered here, the 2-D cross-sectional nonlinearity is overwhelming. This can be perceived from the fact that such sections constitute a limiting case between thin-walled open and closed sections, thus inviting the nonlinear phenomena observed in both. The strong elastic couplings of anisotropic composite laminates complicate the model further. However, a powerful mathematical tool called the Variational Asymptotic Method (VAM) not only enables such a dimensional reduction, but also provides asymptotically correct analytical solutions to the nonlinear cross-sectional analysis. Such closed-form solutions are used here in conjunction with numerical techniques for the rest of the problem to predict multi-body dynamic responses, more quickly and accurately than would otherwise be possible. The analysis methodology can be viewed as a three-step procedure: First, the cross-sectional properties of each bar of the mechanism is determined analytically based on an asymptotic procedure, starting from Classical Laminated Shell Theory (CLST) and taking advantage of its thin strip geometry. Second, the dynamic response of the nonlinear, flexible fourbar mechanism is simulated by treating each bar as a 1-D beam, discretized using finite elements, and employing energy-preserving and -decaying time integration schemes for unconditional stability. Finally, local 3-D deformations and stresses in the entire system are recovered, based on the 1-D responses predicted in the previous step. With the model, tools and procedure in place, we shall attempt to identify and investigate a few problems where the cross-sectional nonlinearities are significant. This will be carried out by varying stacking sequences and material properties, and speculating on the dominating diagonal and coupling terms in the closed-form nonlinear beam stiffness matrix. Numerical examples will be presented and results from this analysis will be compared with those available in the literature, for linear cross-sectional analysis and isotropic materials as special cases.