Unbounded Spaces Research Articles

Reconstructing geographies of margins: Unbounded spaces in an immobile world

AbstractIn this paper I call for unbounding Paasi's ontological position, while urging for a research agenda that shifts from Deconstructing Regional Identity to Reconstructing geographies of margins. Reconstructing geographies of margins is an agenda for building situated knowledges from experiences of movement historically produced as ‘out of place’ and ‘out of time’ to undo fixity in spatio‐temporal categories of the state but also of social‐scientific analysis. I argue that thinking and sensing from the plural epistemological and ontological positions emerging from such movement (out of one's place and time assigned by oppressive structures of domination) is essential to emancipate the categories of the migrant/refugee/other from scientific and historical ways of knowing that produced it in the first place. This agenda cannot be fixed but must be continuously aimed, as an ongoing perpetual process of unbounding research agendas of mobility, borders and migration from settling into rigid onto‐epistemic positions.

Generic uniformly continuous mappings on unbounded hyperbolic spaces

We consider a complete, unbounded hyperbolic metric space X and a concave, nonzero and nondecreasing function ω:[0,+∞)→[0,+∞) with ω(0)=0 and study the space Cω(X) of uniformly continuous self-mappings on X whose modulus of continuity is bounded from above by ω. We endow Cω(X) with the topology of uniform convergence on bounded sets and prove that the modulus of continuity of a generic mapping in Cω(X), in the sense of Baire categories, is precisely ω. Some related results in spaces of bounded mappings and in the topology of pointwise convergence are also discussed. This note can be seen as a completion of various results due to F. Strobin, S. Reich, A.J. Zaslavski, C. Bargetz and D. Thimm.

Quantum multifractality as a probe of phase space in the Dicke model.

We study the multifractal behavior of coherent states projected in the energy eigenbasis of the spin-boson Dicke Hamiltonian, a paradigmatic model describing the collective interaction between a single bosonic mode and a set of two-level systems. By examining the linear approximation and parabolic correction to the mass exponents, we find ergodic and multifractal coherent states and show that they reflect details of the structure of the classical phase space, including chaos, regularity, and features of localization. The analysis of multifractality stands as a sensitive tool to detect changes and structures in phase space, complementary to classical tools to investigate it. We also address the difficulties involved in the multifractal analyses of systems with unbounded Hilbert spaces.

“Bardo, or the Mexican digital diaspora”

ABSTRACT Alejandro G. Iñárritu’s Bardo, falsa crónica de unas cuantas verdades/Bardo False Chronicle of a Handful of Truths (2022) raises important questions about the nature of national film culture in the age to global film commerce. The product of a collaboration between the Mexican filmmaker and Netflix, the US-based global streaming service, Bardo crystalizes long-simmering concerns about digital technologies that have shaped film production and distribution and their impact on contemporary Mexican movie-going audiences. The film’s narrative rhetoric is directed to ‘Greater Mexico’, a cultural community beyond traditional territorial borders. With the objective of calling into question the effects of digitally-driven globalization, the film focuses on the crisis of conscience of creative artist Silverio Gama (Daniel Giménez Cacho), whose symbolic cross-border journey has blurred the lines between the Mexican identity instilled in him by cultural tradition and politics and the post-national environment which, through globalized mass media flows, has since reshaped his outlook. This odyssey to his intimate past eventually leads Silverio to recognize the falseness of his assumption of his own Mexicanness and to acknowledge the progressive evolution of an emerging Mexican culture forged in the borderless, unbounded spaces formed by transnational commerce.

Estimates for negative eigenvalues of Schrödinger operators on unbounded fractal spaces

We study an asymptotic formula for the number of negative eigenvalues of Schrödinger operators on unbounded fractal spaces, which admit a cellular decomposition. We first give some sufficient conditions for Weyl-type asymptotic formula to hold. Second, we verify these conditions for the infinite blowup of Sierpiński gasket and unbounded generalized Sierpiński carpets. Finally, we demonstrate how the result can be applied to the infinite blowup of certain fractals associated with iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolution with golden ratio.

SHAPE DEPENDENT MOTION INTERPOLANTS FOR PLANAR OBJECTS.

Kinematics is most commonly about the motion of unbounded spaces. This paper deals with the kinematics of bounded shapes in a plane. This paper studies the problem of motion interpolation of a planar object with its shape taken into consideration. It applies and extends a shape dependent distance measure between two positions in the context of motion interpolation. Instead of using a fixed reference frame, a shape-dependent inertia frame of reference is used for formulating the distance between positions of a rigid object in a plane. The resulting distance function is then decomposed in two orthogonal directions and is used to formulate an interpolating function for the distance functions in these two directions. This leads to a shape dependent interpolation of translational components of a planar motion. In difference to the original concept of Kazerounian and Rastegar that comes with a shape dependent measure of the angular motion, it is assumed in this paper that the angular motion is shape independent as the angular metric is dimensionless. The resulting distance measure is not only a combination of translation and rotation parameters but also depends on the area moments of inertia of the object. It derives the explicit expressions for decomposing the shape dependent distance in two orthogonal directions, which is then used to obtain shape dependent motion interpolants in these directions. The resulting interpolants have similarities to the well-known spherical linear interpolants widely used in computer graphics in that they are defined using sinusoidal functions instead of linear interpolation in Euclidean space. The path of the interpolating motion can be adjusted by different choice of shape parameters. Examples are provided to illustrate the effect of object shapes on the resulting interpolating motions.

Rate of convergence for singular perturbations of Hamilton-Jacobi equations in unbounded spaces

We prove rate of convergence results for singular perturbations of Hamilton-Jacobi equations in unbounded spaces where the fast operator is linear, uniformly elliptic and has an Ornstein-Uhlenbeck-type drift. The slow operator is a fully nonlinear elliptic operator while the source term is assumed only locally Hölder continuous in both fast and slow variables. We obtain several rates of convergence according to the regularity of the source term.

Generic properties of nonexpansive mappings on unbounded domains

We investigate typical properties of nonexpansive mappings on unbounded complete hyperbolic metric spaces. For two families of metrics of uniform convergence on bounded sets, we show that the typical nonexpansive mapping is a Rakotch contraction on every bounded subset and that there is a bounded set which is mapped into itself by this mapping. In particular, we obtain that the typical nonexpansive mapping in this setting has a unique fixed point which can be reached by iterating the mapping. Nevertheless, it turns out that the typical mapping is not a Rakotch contraction on the whole space and that it has the maximal possible Lipschitz constant of one on a residual subset of its domain. By typical we mean that the complement of the set of mappings with this property is σ-ϕ-porous, that is, small in a metric sense. For a metric of pointwise convergence, we show that the set of Rakotch contractions is meagre.

Sometimes Size Does Not Matter

Recently Díaz, Hössjer and Marks (DHM) presented a Bayesian framework to measure cosmological tuning (either fine or coarse) that uses maximum entropy (maxent) distributions on unbounded sample spaces as priors for the parameters of the physical models ( https://doi.org/10.1088/1475-7516/2021/07/020 ). The DHM framework stands in contrast to previous attempts to measure tuning that rely on a uniform prior assumption. However, since the parameters of the models often take values in spaces of infinite size, the uniformity assumption is unwarranted. This is known as the normalization problem. In this paper we explain why and how the DHM framework not only evades the normalization problem but also circumvents other objections to the tuning measurement like the so called weak anthropic principle, the selection of a single maxent distribution and, importantly, the lack of invariance of maxent distributions with respect to data transformations. We also propose to treat fine-tuning as an emergence problem to avoid infinite loops in the prior distribution of hyperparameters (common to all Bayesian analysis), and explain that previous attempts to measure tuning using uniform priors are particular cases of the DHM framework. Finally, we prove a theorem, explaining when tuning is fine or coarse for different families of distributions. The theorem is summarized in a table for ease of reference, and the tuning of three physical parameters is analyzed using the conclusions of the theorem.

Cross-Lingual Adversarial Domain Adaptation for Novice Programming

Student modeling sits at the epicenter of adaptive learning technology. In contrast to the voluminous work on student modeling for well-defined domains such as algebra, there has been little research on student modeling in programming (SMP) due to data scarcity caused by the unbounded solution spaces of open-ended programming exercises. In this work, we focus on two essential SMP tasks: program classification and early prediction of student success and propose a Cross-Lingual Adversarial Domain Adaptation (CrossLing) framework that can leverage a large programming dataset to learn features that can improve SMP's build using a much smaller dataset in a different programming language. Our framework maintains one globally invariant latent representation across both datasets via an adversarial learning process, as well as allocating domain-specific models for each dataset to extract local latent representations that cannot and should not be united. By separating globally-shared representations from domain-specific representations, our framework outperforms existing state-of-the-art methods for both SMP tasks.

Compact and weakly compact Lipschitz operators

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat {f} : \mathcal {F}(M) \to \mathcal {F}(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat {f}$ to be compact in terms of metric conditions on $f$ . This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat {f}$ is compact if and only if it is weakly compact.

RL-QN: A Reinforcement Learning Framework for Optimal Control of Queueing Systems

With the rapid advance of information technology, network systems have become increasingly complex and hence the underlying system dynamics are often unknown or difficult to characterize. Finding a good network control policy is of significant importance to achieve desirable network performance (e.g., high throughput or low delay). In this work, we consider using model-based reinforcement learning (RL) to learn the optimal control policy for queueing networks so that the average job delay (or equivalently the average queue backlog) is minimized. Traditional approaches in RL, however, cannot handle the unbounded state spaces of the network control problem. To overcome this difficulty, we propose a new algorithm, called RL for Queueing Networks (RL-QN), which applies model-based RL methods over a finite subset of the state space while applying a known stabilizing policy for the rest of the states. We establish that the average queue backlog under RL-QN with an appropriately constructed subset can be arbitrarily close to the optimal result. We evaluate RL-QN in dynamic server allocation, routing, and switching problems. Simulation results show that RL-QN minimizes the average queue backlog effectively.

Discrete dislocation dynamics for crystal RVEs. Part 1: Periodic network kinematics

A novel implementation of the dislocation flux boundary condition in discrete dislocation dynamics is presented. The continuity of the individual dislocation loops in a periodic representative crystal volume (RVE) is enforced across the boundary of the RVE with the help of a dual topological description for representing dislocation line kinematics in two equivalent spaces representing the deforming crystal, the RVE and the unbounded crystal spaces. The former describes the motion of the dislocations in the simulated crystal RVE whereas the latter represents the motion of dislocations in an infinite space containing all replicas of the RVE. A mapping between the two spaces forms the basis of the implementation of flux boundary condition. The implementation details are discussed in the context of statistical homogeneity of bulk crystals undergoing macroscopically homogeneous plastic deformation. In this case, the boundary nodes associated with dislocation segments bear no relevance in the motion of the dislocations. Some test cases are presented and discussed to establish the proposed approach.

Identification of quantum scars via phase-space localization measures

There is no unique way to quantify the degree of delocalization of quantum states in unbounded continuous spaces. In this work, we explore a recently introduced localization measure that quantifies the portion of the classical phase space occupied by a quantum state. The measure is based on the α-moments of the Husimi function and is known as the Rényi occupation of order α. With this quantity and random pure states, we find a general expression to identify states that are maximally delocalized in phase space. Using this expression and the Dicke model, which is an interacting spin-boson model with an unbounded four-dimensional phase space, we show that the Rényi occupations with α>1 are highly effective at revealing quantum scars. Furthermore, by analyzing the high moments (α>1) of the Husimi function, we are able to identify qualitatively and quantitatively the unstable periodic orbits that scar some of the eigenstates of the model.

Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

AbstractWe consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems.In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.

Conformal Transformation of Uniform Domains Under Weights That Depend on Distance to The Boundary

Abstract The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.

Strongly elliptic operators and exponentiation of operator Lie algebras

An intriguing feature which is often present in theorems regardingthe exponentiation of Lie algebras of unbounded linear operators onBanach spaces is the assumption of hypotheses on the Laplacianoperator associated with a basis of the operator Lie algebra.The main objective of this work is to show that one can substitutethe Laplacian by an arbitrary operator in the enveloping algebra andstill obtain exponentiation, as long as its closure generates astrongly continuous one-parameter semigroup satisfying certain normestimates, which are typical in the theory of strongly ellipticoperators.

Lipschitz Algebras and Lipschitz-Free Spaces Over Unbounded Metric Spaces

Abstract We investigate a way to turn an arbitrary (usually, unbounded) metric space ${{\mathcal{M}}}$ into a bounded metric space ${{\mathcal{B}}}$ in such a way that the corresponding Lipschitz-free spaces ${{\mathcal{F}}}({{\mathcal{M}}})$ and ${{\mathcal{F}}}({{\mathcal{B}}})$ are isomorphic. The construction we provide is functorial in a weak sense and has the advantage of being explicit. Apart from its intrinsic theoretical interest, it has many applications in that it allows to transfer many arguments valid for Lipschitz-free spaces over bounded spaces to Lipschitz-free spaces over unbounded spaces. Furthermore, we show that with a slightly modified pointwise multiplication, the space ${\textrm{Lip}}_0({{\mathcal{M}}})$ of scalar-valued Lipschitz functions vanishing at zero over any (unbounded) pointed metric space is a Banach algebra with its canonical Lipschitz norm.

Quantum localization measures in phase space.

Measuring the degree of localization of quantum states in phase space is essential for the description of the dynamics and equilibration of quantum systems, but this topic is far from being understood. There is no unique way to measure localization, and individual measures can reflect different aspects of the same quantum state. Here we present a general scheme to define localization in measure spaces, which is based on what we call Rényi occupations, from which any measure of localization can be derived. We apply this scheme to the four-dimensional unbounded phase space of the interacting spin-boson Dicke model. In particular, we make a detailed comparison of two localization measures based on the Husimi function in the regime where the model is chaotic, namely, one that projects the Husimi function over the finite phase space of the spin and another that uses the Husimi function defined over classical energy shells. We elucidate the origin of their differences, showing that in unbounded spaces the definition of maximal delocalization requires a bounded reference subspace, with different selections leading to contextual answers.

Continuous functionals for unbounded convergence in Banach lattices

Recently, the different types of unbounded convergences (uo, un, uaw, uaw*) in Banach lattices were studied. In this paper, we study the continuous functionals with respect to unbounded convergences. We first characterize the continuity of linear functionals for these convergences. Then we define the corresponding unbounded dual spaces and get their exact form. Based on these results, we discuss order continuity and reflexivity of Banach lattices. Some related results are obtained as well.