Rescaling Of Space Research Articles

Options for Dynamic Similarity of Interacting Fluids, Elastic Solids and Rigid Bodies Undergoing Large Motions

Motivated by a design of a vertical axis wind turbine, we present a theory of dynamical similarity for mechanical systems consisting of interacting elastic solids, rigid bodies and incompressible fluids. Throughout, we focus on the geometrically nonlinear case. We approach the analysis by analyzing the equations of motion: we ask that a change of variables take these equations and mutual boundary conditions to themselves, while allowing a rescaling of space and time. While the disparity between the Eulerian and Lagrangian descriptions might seem to limit the possibilities, we find numerous cases that apparently have not been identified, especially for stiff nonlinear elastic materials (defined below). The results appear to be particularly adapted to structures made with origami design methods, where the tiles are allowed to deform isometrically. We collect the results in tables and discuss some particular numerical examples.

The effects of spatial configuration on relative translation gain thresholds in redirected walking

In this study, we explore how spatial configurations can be reflected in determining the threshold range of Relative Translation Gains (RTGs), a translation gain-based Redirected Walking (RDW) technique that scales the user’s movement in Virtual Reality (VR) in different ratios for width and depth. While previous works have shown that various cognitive factors influence the RDW threshold, studies investigating the impact of environmental aspects on the RDW threshold with regard to the user’s visual perception were lacking. Therefore, we examined the effect of spatial configurations on the RTG threshold by analyzing the participant’s responses and gaze distribution data in two user studies. The first study concerned the size of the virtual room and the existence of objects within it, and the second study focused on the combined impact of room size and spatial layout. Our results show that room size, the existence of objects, and spatial layout all significantly affect the RTG threshold range. Based on our findings, we propose virtual space rescaling guidelines to increase the range of adjustable movable space with RTGs for developers: placing distractors in the room, setting the perceived movable space to be larger than the adjusted movable space for an empty room, and avoiding the placement of object clusters in the center of the room. Our findings can be used to adaptively rescale VR users’ space according to the target virtual space’s configuration with a unified coordinate system that enables the utilization of physical objects in a virtual scene.

How digital citizenship regimes are rescaling European nation-states

ABSTRACT This provocation shows how five emerging digital citizenship regimes are rescaling European nation-states through a taxonomy: (i) the globalised/generalisable regime called pandemic citizenship that clarifies how post-COVID-19 datafication processes have amplified the emergence of four digital citizenship regimes in six city-regions; (ii) algorithmic citizenship (Tallinn); (iii) liquid citizenship (Barcelona/Amsterdam); (iv) metropolitan citizenship (Cardiff); and (v) stateless citizenship (Barcelona/Glasgow/Bilbao). I argue that this phenomenon should matter to us insofar as these emerging digital citizenship regimes have resulted in nation-state space rescaling, challenging its heretofore privileged position as the only natural platform for the monopoly of technopolitical and sensory power.

Atomic transport properties of liquid iron at conditions of planetary cores.

Atomic transport properties of liquid iron are important for understanding the core dynamics and magnetic field generation of terrestrial planets. Depending on the sizes of planets and their thermal histories, planetary cores may be subject to quite different pressures (P) and temperatures (T). However, previous studies on the topic mainly focus on the P-T range associated with the Earth's outer core; a systematic study covering conditions from small planets to massive exoplanets is lacking. Here, we calculate the self-diffusion coefficient D and viscosity η of liquid iron via ab initio molecular dynamics from 7.0 to 25 g/cm3 and 1800 to 25 000 K. We find that D and η are intimately related and can be fitted together using a generalized free volume model. The resulting expressions are simpler than those from previous studies where D and η were treated separately. Moreover, the new expressions are in accordance with the quasi-universal atomic excess entropy (Sex) scaling law for strongly coupled liquids, with normalized diffusivity D⋆ = 0.621 exp(0.842Sex) and viscosity η⋆ = 0.171 exp(-0.843Sex). We determine D and η along two thermal profiles of great geophysical importance: the iron melting curve and the isentropic line anchored at the ambient melting point. The variations of D and η along these thermal profiles can be explained by the atomic excess entropy scaling law, demonstrating the dynamic invariance of the system under uniform time and space rescaling. Accordingly, scale invariance may serve as an underlying mechanism to unify planetary dynamos of different sizes.

Urban investment and development corporations, new town development and China’s local state restructuring – the case of Songjiang new town, Shanghai

ABSTRACTThis article seeks to understand China’s new urban space production and associated state space rescaling through a microscopic investigation of urban investment and development corporations (UIDCs), based on a case study of Shanghai’s Songjiang New Town Development Corporation (SNTDC). It argues that the introduction of UIDCs as both developers and managers of designated urban regions is a creation of institutional reformation to accomplish customized place- and scale-specific spatial projects under the state strategy of new urban spatial development, playing an essential part in the rescaling of state space. UIDCs are economically independent of other social groups, politically bonded with the local government, and assume entrepreneurial and administrative functions within designated areas, acting as intermediary agents to enable local states greater capacity in governing new urban space production, engineering social change, and propelling economic development.

Equilibrium position of a rigid sphere in a turbulent jet: A problem of elastic reconfiguration.

The position of floating spheres trapped within an immersed turbulent water jet is investigated. Using the self-similarity properties of the jet velocity profile, the equilibrium problem is formulated in a rescaled space where the sphere is static and deformable. This approach is found to be related to a problem of elastic reconfiguration where elasticity arises here from the geometry of the flow instead of an actual deformation of a body.

Scale-invariant geometric random graphs

We introduce and analyze a class of growing geometric random graphs that are invariant under rescaling of space and time. Directed connections between nodes are drawn according to influence zones that depend on node position in space and time, mimicking the heterogeneity and increased specialization found in growing networks. Through calculations and numerical simulations we explore the consequences of scale invariance for geometric random graphs generated this way. Our analysis reveals a dichotomy between scale-free and Poisson distributions of in- and out-degree, the existence of a random number of hub nodes, high clustering, and unusual percolation behavior. These properties are similar to those of empirically observed web graphs.

Localization and limit laws of a three-state alternate quantum walk on a two-dimensional lattice

A two-dimensional discrete-time quantum walk (DTQW) can be realized by alternating a two-state DTQW in one spatial dimension followed by an evolution in the other dimension. This was shown to reproduce a probability distribution for a certain configuration of a four-state DTQW on a two-dimensional lattice. In this work we present a three-state alternate DTQW with a parameterized coin-flip operator and show that it can produce localization that is also observed for a certain other configuration of the four-state DTQW and non-reproducible using the two-state alternate DTQW. We will present two limit theorems for the three-state alternate DTQW. One of the limit theorems describes a long-time limit of a return probability, and the other presents a convergence in distribution for the position of the walker on a rescaled space by time. We will also outline the relevance of these walks in physical systems.

Coupling and Hydrodynamic Limit for the Inclusion Process

We show propagation of local equilibrium for the symmetric inclusion process (SIP) after diffusive rescaling of space and time, as well as the local equilibrium property of the non-equilibrium steady state in the boundary driven SIP. The main tool is self-duality and a coupling between \(n\) SIP particles and \(n\) independent random walkers.

Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker’s position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.

Fick’s Law in a Random Lattice Lorentz Gas

We provide a proof that the stationary macroscopic current of particles in a random lattice Lorentz gas satisfies Fick's law when connected to particles reservoirs. We consider a box on a d+1 dimensional lattice and when $d\geq7$, we show that under a diffusive rescaling of space and time, the probability to find a current different from its stationary value is exponentially small in time. Its stationary value is given by the conductivity times the difference of chemical potentials of the reservoirs. The proof is based on the fact that in high dimension, random walks have a small probability of making loops or intersecting each other when starting sufficiently far apart.

Universal flow-density relation of single-file bicycle, pedestrian and car motion

The relation between flow and density is an essential quantitative characteristic to describe the efficiency of traffic systems. We have performed experiments with single-file motion of bicycles and compare the results with previous studies for car and pedestrian motion in similar setups. In the space-time diagrams we observe three different states of motion (free flow state, jammed state and stop-and-go waves) in all these systems. Despite of their obvious differences they are described by a universal fundamental diagram after proper rescaling of space and time which takes into account the size and free velocity of the three kinds of agents. This indicates that the similarities between the systems go deeper than expected.

Les Reciproqueteurs: post-regulatory corporatism

ABSTRACTReflexive governance can be understood as an emergent encapsulated trust-building corporatism where network participants are neither state functionaries nor market entrepreneurs but network reciproqueteurs. This paper argues that such reflexive network governance results in a post-regulatory corporatism (PRC)—a more adaptable, less formalized, and flexible mode of interest intermediation, policy-making, and policy-implementation than previous modes of corporatist intermediation. Functional differentiation processes engender ‘negotiated connected contracts' in rescaled space in between inter-regional assemblages, a mode of structurally coupling new social partners in the emergent transnational knowledge-based economy. This involves the building of new social capital of network trust-building manifested in the norms of reciprocity and reflexive law constituted as a new mode of protocolism: one associated with the social learning and policy designing necessary for ecological systems' autopoeisis, resilience, and sustainability. This paper conceptualizes reflexive network governance as protocolism in constellations of PRC and discusses examples from the area of environmental policy-making. PRC is understood as a new mode of negotiated rule-making: as a recursive protocolism of multi-stakeholder social pacts constituted by frame agreements and negotiated connected network contracts.

Self-learning K-means clustering: a global optimization approach

An appropriate distance is an essential ingredient in various real-world learning tasks. Distance metric learning proposes to study a metric, which is capable of reflecting the data configuration much better in comparison with the commonly used methods. We offer an algorithm for simultaneous learning the Mahalanobis like distance and K-means clustering aiming to incorporate data rescaling and clustering so that the data separability grows iteratively in the rescaled space with its sequential clustering. At each step of the algorithm execution, a global optimization problem is resolved in order to minimize the cluster distortions resting upon the current cluster configuration. The obtained weight matrix can also be used as a cluster validation characteristic. Namely, closeness of such matrices learned during a sample process can indicate the clusters readiness; i.e. estimates the true number of clusters. Numerical experiments performed on synthetic and on real datasets verify the high reliability of the proposed method.

Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the {\epsilon}-problem converge to a solution of the limiting problem.

Designing irrigation pipes

The use of porous ducts to deliver water to agricultural fields is an old technique which helps saving water and prevents ground erosion. Designing porous duct is not as a simple task as it looks and apparently has never been the subject of mathematical research. Here the problem is addressed making use of a double rescaling of space and velocity variables, which allows the derivation of the governing equations starting from the study of the classical Navier Stokes equations in a pipe. Such equations are then solved obtaining results of practical interest in design of irrigation pipes, both for low discharge pipes (small plants) and for high discharge pipes (large plants).

A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele–Shaw cell

In this paper, we present a time and space rescaling scheme for the computation of moving interface problems. The idea is to map time–space such that the interfaces can evolve exponentially fast in the new time scale while the area/volume enclosed by the interface remains unchanged. The rescaling scheme significantly reduces the computation time (especially for slow growth), and enables one to accurately simulate the very long-time dynamics of moving interfaces. We then implement this scheme in a Hele–Shaw problem, examine the dynamics for a number of different injection fluxes, and present the largest and most pronounced viscous fingering simulations to date.

Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D

The initial value problem for the $L^{2}$ critical semilinear Schrödinger equation with periodic boundary data is considered. We show that the problem is globally well-posed in $H^{s}( T^{d} )$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in [4]. We use the "$I$-method''. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the $H^{1}(T^{d} )$ threshold. The main ingredient in the proof is a "refinement" of the Strichartz's estimates that hold true for solutions defined on the rescaled space, $T^{d}_\lambda = R^{d}/{\lambda Z^{d}}$, $d=1,2$.

Classifying individual tree species under leaf-off and leaf-on conditions using airborne lidar

In this paper, a methodology for individual tree-based species classification using high sampling density and small footprint lidar data is clarified, corrected and improved. For this purpose, a well-defined directed graph (digraph) is introduced and it plays a fundamental role in the approach. It is argued that there exists one and only one such unique digraph that describes all four pure events and resulting disjoint sets of laser points associated with a single tree in data from a two-return lidar system. However, the digraph is extendable so that it fits an n-return lidar system ( n > 2) with higher logical resolution. Furthermore, a mathematical notation for different types of groupings of the laser points is defined, and a new terminology for various types of individual tree-based concepts defined by the digraph is proposed. A novel calibration technique for estimating individual tree heights is evaluated. The approach replaces the unreliable maximum single laser point height of each tree with a more reliable prediction based on shape characteristics of a marginal height distribution of the whole first-return point cloud of each tree. The result shows a reduction of the RMSE of the tree heights of about 20% (stddev = 1.1 m reduced to stddev = 0.92 m). The method improves the species classification accuracy markedly, but it could also be used for reducing the sampling density at the time of data acquisition. Using the calibrated tree heights, a scale-invariant rescaled space for the universal set of points for each tree is defined, in which all individual tree-based geometric measurements are conducted. With the corrected and improved classification methodology the total accuracy raises from 60% to 64% for classifying three leaf-off individual tree deciduous species ( N = 200 each) in West Virginia, USA: oaks ( Quercus spp.), red maple ( Acer rubrum), and yellow poplar ( Liriodendron tuliperifera).

Nonlinear stability analysis of self-similar crystal growth: control of the Mullins–Sekerka instability

In this paper, we perform a stability analysis of 2D, noncircular self-similar crystals with isotropic surface tension growing in a supercooled melt. The existence of such self-similarly growing crystals was demonstrated recently in our previous work (J. Crystal Growth 267 (2004) 703). Here, we characterize the nonlinear morphological stability of the self-similar crystals, using a new spectrally accurate 2D boundary integral method in which a novel time and space rescaling is implemented (J. Crystal Growth 266 (2004) 552). This enables us to accurately simulate the long-time, nonlinear dynamics of evolving crystals. Our analysis and simulations reveal that self-similar shapes are stable to perturbations of the critical flux for self-similar growth. This suggests that in experiments, small oscillations in the critical flux will not change the main features of self-similar growth. Shape perturbations may either grow or decay. However, at long times there is nonlinear stabilization even though unstable growth may be significant at early times. Interestingly, this stabilization leads to the existence of universal limiting shapes. In particular, we find that the morphologies of the nonlinearly evolving crystals tend to limiting shapes that evolve self-similarly and depend on the flux. A number of limiting shapes exist for each flux (the number of possible shapes actually depends on the flux), but only one is universal in the sense that a crystal with an arbitrary initial shape will evolve to this universal shape. The universal shape can actually be retrograde. By performing a series of simulations, we construct a phase diagram that reveals the relationship between the applied flux and the achievable symmetries of the limiting shapes. Finally, we use the phase diagram to design a nonlinear protocol that might be used in a physical experiment to control the nonlinear morphological evolution of a growing crystal. Because our analysis shows that interactions among the perturbation modes are similar in both 2D and 3D, our results apply qualitatively to 3D.