Low-dimensional Objects Research Articles

On the occurrence of lumped forces at corners in classical plate theories: a physically based interpretation

The paradigmatic example of a twisted square plate is here considered. An equivalent partition of the plate in a grid of beams a la Grashof is found such that, as the number of beams tends to infinity, the grid exhibits the same deflection of the plate. This scheme is used to interpret, through the distinction between Euler‐Bernoulli and Timoshenko beam theories, the different types of natural boundary conditions that can arise in the Kirchhoff‐Love and Mindlin‐Reissner theories of plates. A physically based interpretation for the occurrence of lumped forces at the plate corners through the formation of a boundary layer is provided. It is well known that the solution of the biharmonic equation governing the bending of plates in Kirchhoff‐ Love theory is compatible with only two distinct conditions at each boundary point, whereas in general three boundary data can be independently assigned on an unconstrained border. This contradiction for the order of the equation is a two-hundred-year-old problem. The paradox arose when the three-boundarydata statement by Poisson [1829] was criticized by Kirchhoff [1850], who obtained only two natural conditions at the border within a variational framework, using a static equivalence sometimes referred to as the “Kirchhoff transformation” [Vasil’ev 2012]. This result arose from the first variation of the energy functional, but it was not corroborated by any physically based interpretation. A long discussion ensued among the most eminent scientists of the period with the purpose of reconciling the Poisson and Kirchhoff theories. The dispute culminated with the elementary interpretation by Thomson and Tait [1883], who showed how to reduce the torque per unit length on the contour to a shear transverse force. Friedrichs and Dressler [1961] and Gol’Denveiser and Kolos [1965] have independently shown that the plate theory is the leading term of the expansion solution (in a small thickness parameter) for the linear elastostatics of thin, flat, isotropic bodies. As expected, this leading term alone is unable to satisfy arbitrarily prescribed edge conditions. There has been a renewed interest during the last years in the fundamental problem of understanding the relationship between the three-dimensional elasticity theory and theories for lower-dimensional objects (plates, shells, rods). Due to the availability of sophisticated methods of variational convergence [Ciarlet 1997], important achievements have been obtained by showing that various theories of plates arise as a rigorous variational limit (or 0-limit) of the equations of three-dimensional elasticity as the

Regularity of densities in relaxed and penalized average distance problem

The average distance problem finds application in data parameterization, which involves``representing'' the data using lower dimensional objects. Froma computational point of view it is often convenient to restrictthe unknown to the family of parameterized curves.The original formulation of the average distance problem exhibits several undesirable properties. In this paper we propose an alternative variant: we minimize the functional \begin{equation*}\int_{{\mathbb{R}}^d\times \Gamma_\gamma} |x-y|^p {\,{d}}\Pi(x,y)+\lambda L_\gamma +\varepsilon\alpha(\nu)+\varepsilon' \eta(\gamma)+\varepsilon''\|\gamma'\|_{TV},\end{equation*}where $\gamma$ varies among the family of parametrized curves, $\nu$ among probability measures on $\gamma$,and $\Pi$ among transport plans between $\mu$ and $\nu$. Here $\lambda,\varepsilon,\varepsilon',\varepsilon''$ are given parameters,$\alpha$ is a penalization term on $\mu$, $\Gamma_\gamma$ (resp. $L_\gamma$) denotes the graph (resp. length)of $\gamma$, and $\|\cdot\|_{TV}$ denotes the total variation semi-norm.We will use techniques from optimal transport theory and calculus of variations. The main aim is to prove essential boundedness, and Lipschitz continuity for Radon-Nikodym derivative of $\nu$, when $(\gamma,\nu,\Pi)$is a minimizer.

Dimensional reduction of BPS attractors in AdS gauged supergravities

We relate across dimensions BPS attractors of black strings and black holes of various topology in gauged supergravities with nontrivial scalar potential. The attractors are of the form AdS$_{2, 3} \times \Sigma^{2, 3}$ in 4, 5, and 6 dimensions, and can be generalized to some higher dimensional analogs. Even though the attractor geometries admit standard Kaluza-Klein and Scherk-Schwarz reductions, their asymptotic AdS spaces in general do not. The resulting lower dimensional objects are black holes with runaway asymptotics in supergravity theories with no maximally symmetric vacua. Such classes of solutions are already known to exist in literature, and results here suggest an interpretation in terms of their higher-dimensional origin that often has a full string theory embedding. In a particular relevant example, the relation between 5d Benini-Bobev black strings arXiv:1302.4451 and a class of 4d Cacciatori-Klemm black holes arXiv:0911.4926 is worked out in full detail, providing a type IIB and dual field theory description of the latter solutions. As a consistency check, the Cardy formula for the field theory is shown to match the Bekenstein-Hawking entropy for horizon topology of any genus.

Simulations of electromechanical shape transformations of Au nanoparticles

Metallic nanocrystals can exhibit intriguing properties due to large surfaces specific for low-dimensional objects. Recent experimental observation showed an interesting shape transformation of Au nanoparticle during simultaneous mechanical and electrical load. By applying finite element methods extended to include nanosize effects, we study the mechanisms of such transformations. We utilize fully coupled electro-thermal calculations with the nanoscale correction to the electric and thermal conductivities. The mechanical response of the material is simulated using the elastoplastic material model while the nanoscale mechanical interaction between gold and tungsten particles is simulated using adhesive contact modelling. We observe that Joule heating due to high electric currents increases the temperature of the nanoparticle to the value close to the melting point. In combination with the mechanical stress, this causes significant plastic deformations within the nanoparticle, which can explain the observed shape modification of the latter.

Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems.

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima-Zwanzig-Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection-reaction problems.

Invariant manifolds for nonsmooth systems with sliding mode

Invariant manifolds play an important role in the study of Dynamical Systems, since they help to reduce the essential dynamics to lower dimensional objects. In that way, a bifurcation analysis can easily be performed. In the classical approach, the reduction to invariant manifolds requires smoothness of the system which is typically not given for nonsmooth systems. For that reason, techniques have been developed to extend such a reduction procedure to nonsmooth systems. In the present paper, we present such an approach for systems involving sliding motion. In addition, an analysis of the reduced equation shows that the generation of periodic orbits through nonlinear perturbations which is usually related to Hopf bifurcation follows a different type of bifurcation if nonsmooth elements are present, since generically symmetry is broken by the nonsmooth terms.

Slender-body theory for viscous flow via dimensional reduction and hyperviscous regularization

A new slender-body theory for viscous flow, based on the concepts of dimensional reduction and hyperviscous regularization, is presented. The geometry of flat, elongated, or point-like rigid bodies immersed in a viscous fluid is approximated by lower-dimensional objects, and a hyperviscous term is added to the flow equation. The hyperviscosity is given by the product of the ordinary viscosity with the square of a length that is shown to play the role of effective thickness of any lower-dimensional object. Explicit solutions of simple problems illustrate how the proposed method is able to represent with good approximation both the velocity field and the drag forces generated by rigid motions of the immersed bodies, in analogy with classical slender-body theories. This approach has the potential to open up the way to more effective computational techniques, since geometrical complexities can be significantly reduced. This, however, is achieved at the expense of involving higher-order derivatives of the velocity field. Importantly, both the dimensional reduction and the hyperviscous regularization, combined with suitable numerical schemes, can be used also in situations where inertia is not negligible.

Functional architectures and structured flows on manifolds: a dynamical framework for motor behavior.

We outline a dynamical framework for sequential sensorimotor behavior based on the sequential composition of basic behavioral units. Basic units are conceptualized as temporarily existing low-dimensional dynamical objects, or structured flows, emerging from a high-dimensional system, referred to as structured flows on manifolds. Theorems from dynamical system theory allow for the unambiguous classification of behaviors as represented by structured flows, and thus provide a means to define and identify basic units. The ensemble of structured flows available to an individual defines his or her dynamical repertoire. We briefly review experimental evidence that has identified a few basic elements likely to contribute to each individual's repertoire. Complex behavior requires the involvement of a (typically high-dimensional) dynamics operating at a time scale slower than that of the elements in the dynamical repertoire. At any given time, in the competition between units of the repertoire, the slow dynamics temporarily favor the dominance of one element over others in a sequential fashion, binding together the units and generating complex behavior. The time scale separation between the elements of the repertoire and the slow dynamics define a time scale hierarchy, and their ensemble defines a functional architecture. We illustrate the approach with a functional architecture for handwriting as proof of concept and discuss the implications of the framework for motor control.

From lines and circles to hyperspheres: a problem and its generalization

In this article we illustrate, by way of a particular example, that simple problems in two or three dimensions may possess elegant generalizations to higher dimensions. Such problems provide a natural setting for undergraduates in which to consider n- dimensional generalizations of familiar low-dimensional objects, and to develop the advanced mathematical ideas and techniques necessary to work in dimensions lying beyond the realm of everyday experience. Exact and asymptotic results are derived, and educational aspects are discussed.

Polarity effects in unsupported polar nanoribbons

We analyze the characteristics of polarity in unsupported nanoribbons with zigzag edges, by a combination of analytic models, semiempirical Hartree-Fock simulations, and first-principles approach. We consider two materials with widely different ionic-covalent character, MgO and MoS${}_{2}$, and two polarity healing mechanisms: the so-called electronic compensation in ribbons with pristine edges, and ionic compensation in ribbons with an adequately chosen density of missing edge ions. The general expression of compensating charges, the edge metallization and spin polarization in the electronic mechanism, and the efficiency of the ionic mechanism are similar to those known in thin films and at polar surfaces. Differences, however, exist and are related to the low dimensionality, the atomic structure, and the strong undercoordination of edge atoms in nanoribbons. Polarity signatures are specified and a discussion of the possible origins of metallic edge states in these low dimensional objects is provided.

ASPECT RATIO DEPENDENCE OF THE FREE-FALL TIME FOR NON-SPHERICAL SYMMETRIES

We investigate the collapse of non-spherical substructures, such as sheets and filaments, which are ubiquitous in molecular clouds. Such non-spherical substructures collapse homologously in their interiors but are influenced by an edge effect that causes their edges to be preferentially accelerated. We analytically compute the homologous collapse timescales of the interiors of uniform-density, self-gravitating filaments and find that the homologous collapse timescale scales linearly with the aspect ratio. The characteristic timescale for an edge driven collapse mode in a filament, however, is shown to have a square root dependence on the aspect ratio. For both filaments and circular sheets, we find that selective edge acceleration becomes more important with increasing aspect ratio. In general, we find that lower dimensional objects and objects with larger aspect ratios have longer collapse timescales. We show that estimates for star formation rates, based upon gas densities, can be overestimated by an order of magnitude if the geometry of a cloud is not taken into account.

MODES OF STAR FORMATION IN FINITE MOLECULAR CLOUDS

We analytically investigate the modes of gravity-induced star formation possible in idealized finite molecular clouds where global collapse competes against both local Jeans instabilities and discontinuous edge instabilities. We examine these timescales for collapse in spheres, discs, and cylinders, with emphasis on the structure, size, and degree of internal perturbations required in order for local collapse to occur before global collapse. We find that internal, local collapse is more effective for the lower dimensional objects. Spheres and discs, if unsupported against global collapse, must either contain strong perturbations or must be unrealistically large in order for small density perturbations to collapse significantly faster than the entire cloud. We find, on the other hand, that filamentary geometry is the most favorable situation for the smallest perturbations to grow before global collapse overwhelms them and that filaments containing only a few Jeans masses and weak density perturbations can readily fragment. These idealized solutions are compared with simulations of star-forming regions in an attempt to delineate the role of global, local, and edge instabilities in determining the fragmentation properties of molecular clouds. The combined results are also discussed in the context of recent observations of Galactic molecular clouds.

Time Scale Hierarchies in the Functional Organization of Complex Behaviors

Traditional approaches to cognitive modelling generally portray cognitive events in terms of ‘discrete’ states (point attractor dynamics) rather than in terms of processes, thereby neglecting the time structure of cognition. In contrast, more recent approaches explicitly address this temporal dimension, but typically provide no entry points into cognitive categorization of events and experiences. With the aim to incorporate both these aspects, we propose a framework for functional architectures. Our approach is grounded in the notion that arbitrary complex (human) behaviour is decomposable into functional modes (elementary units), which we conceptualize as low-dimensional dynamical objects (structured flows on manifolds). The ensemble of modes at an agent’s disposal constitutes his/her functional repertoire. The modes may be subjected to additional dynamics (termed operational signals), in particular, instantaneous inputs, and a mechanism that sequentially selects a mode so that it temporarily dominates the functional dynamics. The inputs and selection mechanisms act on faster and slower time scales then that inherent to the modes, respectively. The dynamics across the three time scales are coupled via feedback, rendering the entire architecture autonomous. We illustrate the functional architecture in the context of serial behaviour, namely cursive handwriting. Subsequently, we investigate the possibility of recovering the contributions of functional modes and operational signals from the output, which appears to be possible only when examining the output phase flow (i.e., not from trajectories in phase space or time).

A hierarchical feature fusion framework for adaptive visual tracking

A Hierarchical Model Fusion (HMF) framework for object tracking in video sequences is presented. The Bayesian tracking equations are extended to account for multiple object models. With these equations as a basis a particle filter algorithm is developed to efficiently cope with the multi-modal distributions emerging from cluttered scenes. The update of each object model takes place hierarchically so that the lower dimensional object models, which are updated first, guide the search in the parameter space of the subsequent object models to relevant regions thus reducing the computational complexity. A method for object model adaptation is also developed. We apply the proposed framework by fusing salient points, blobs, and edges as features and verify experimentally its effectiveness in challenging conditions.

Homogenization of many‐body structures in fully nonlinear elasticity: A first approach with some applied flavour

AbstractWe expose and discuss a model for certain “cord‐belt like” many‐body structures in a geometrically and constitutively nonlinear regime that accounts for noninterpenetration of matter in the sense of Ciarlet and Nečas. As the structures compose of more and simultaneously smaller bodies, we state homogenization limits for the respective models based on a Γ‐convergence analysis. Depending on the structures' geometries, the homogenization process may result in different mechanical effects, ranging from anisotropy to completely new kinematic constraints. The homogenization results pose moreover as first rigorous evidence that appropriately arranged and laminated structures composed of mechanically low dimensional objects can have the same properties as if they were composed of higher dimensional objects (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Scene Representation Based on Dynamic Field Theory: From Human to Machine

Event Abstract Back to Event Scene Representation Based on Dynamic Field Theory: From Human to Machine Stephan K. Zibner1*, Christian Faubel1, Ioannis Iossifidis1 and Gregor Schöner1 1 Ruhr-Universität Bochum, Institut für Neuroinformatik, Germany A typical human-robot cooperation task puts an autonomous robot in the role of an assistant for the human user. Fulfilling requests of the human user like “hand me the red screwdriver” demands an internal representation of the spatial layout of the scene as well as an understanding of labels and identifiers like “red”, associated with the spatial information. In addition, the representation of the scene must be updated according to ongoing changes in the outside world.Studies trying to comprehend the nature and extent of the internal representation of humans [1] hint that we keep a limited, non-pictorial representation of our perception in memory. Experimental studies on human visual working memory using a change detection paradigm [2] define space as being the key to bind multiple features of an object. Dynamic Field Theory, which is a theory of embodied cognition[4], provides the process models to explain such results. The models are based on Dynamic Neural Fields, which are a model of neural activity in the human cortex related to metric spaces.Applying the theory in the paradigm of autonomous robots provides a guideline for designing a neurally plausible robotic scene representation.We approach the problems of building up, maintaining, and updating a robotic scene representation with an architecture built from a set of Dynamic Neural Fields [5]. Core of this architecture are three-dimensional Dynamic Neural Fields, which, inspired by [2], dynamically associate two-dimensional object locations in an allocentric reference frame with extracted low-dimensional object features like color. The unit of information representing an association in these fields is localized supra-threshold activity, called a peak, that both gives a detection decision on input and an estimate of continuous parameters like space or color hue represented by the field’s dimensions. Through the architectural connections, the three-dimensional fields are sequentially filled up with associative peaks as soon as the robot perceives a scene. Using the stability characteristics of fields and a steady coupling to current camera input, changes in object positions are tracked even for multiple moving objects, objects are memorized when they get out of view, and associations are removed automatically once an object is removed from the scene.The resulting application is tested on the robotic platform CoRA (Cooperative Robotic Assistant).The resulting peaks of an autonomous scanning sequence can be successfully held in the three-dimensional associative Dynamic Neural Field, even when objects get out of view. Here, the application shows a capacity limit of four to five objects that can be kept concurrently in the scene representation. If multiple objects are moved within the robot’s field of view, a smaller amount of objects can be tracked simultaneously. The capacity limit is reduced to three objects for multiobject tracking, bearing in mind that not only the spatial position of the objects is updated correctly, but the stored feature associations are carried along as well. The decrease of capacity between static and moving objects is also observable in the human counterpart [3].

A dual latent class unfolding model for two-way two-mode preference rating data

In unfolding for two-way two-mode preference ratings data, the categorization of the set of individuals while the categories are represented in a low dimensional space may be an advisable procedure to facilitate their understanding. In addition to considering groups of individuals of a similar preference pattern, homogeneous groups of objects are also considered, such that within each group there are clustered objects perceived to have similar attributes. A dual latent class model is proposed for a matrix of preference ratings data, which will partition the individuals and the objects into classes, and simultaneously represent the cluster centers in a low dimensional space, while individuals and objects retain their preference relationship. Both the categories achieved and the unfolding configuration are estimated to be simultaneously optimal, by means of a conditional maximum likelihood estimation procedure, in a simulated annealing framework that enables us to take a statistical decision about the parameters of the model. The adjusted BIC statistic is employed to test the number of mixture components, and the dimensionality of the representation. Real and artificial data sets are analyzed to illustrate the model’s performance.

Lasing in low-dimensional ZnO objects: Interrelation between the crystallite morphology and feedback formation mechanisms

Specific features of UV lasing in low-dimensional zinc oxide materials have been investigated. Two main types of lasers, differing in the feedback formation mechanism-classical random lasers and microlasers—have been selected in active disordered zinc oxide materials. In the first case, feedback is provided by strong light backscattering, and a large number of scattering particles are involved in the cavity formation. In the second case, each crystallite operates as an individual microlaser. The main lasing spectral parameters are reported for the distinguished types of lasers.

Dynamic Semiparametric Factor Models in Risk Neutral Density Estimation

Dimension reduction techniques for functional data analysis model and approximate smooth random functions by lower dimensional objects. In many applications the focus of interest lies not only in dimension reduction but also in the dynamic behaviour of the lower dimensional objects. The most prominent dimension reduction technique - functional principal components analysis - however, does not model time dependences embedded in functional data. In this paper we use dynamic semiparametric factor models (DSFM) to reduce dimensionality and analyse the dynamic structure of unknown random functions by means of inference based on their lower dimensional representation. We apply DSFM to estimate the dynamic structure of risk neutral densities implied by prices of option on the DAX stock index.

Elastic light scattering by semiconductor quantum dots of arbitrary shape

Elastic light scattering by low-dimensional semiconductor objects is investigated theoretically. The differential cross section of resonant light scattering on excitons in quantum dots is calculated. The polarization and angular distribution of scattered light do not depend on the quantum-dot form, sizes and potential configuration if light wave lengths exceed considerably the quantum-dot size. In this case the magnitude of the total light scattering cross section does not depend on quantum-dot sizes. The resonant total light scattering cross section is about a square of light wave length if the exciton radiative broadening exceeds the nonradiative broadening. Radiative broadenings are calculated.