Properties Of Walks Research Articles

Controlling and optimizing the transport (search) efficiency with local information on a class of scale-free trees.

The scale-free trees are fundamental dynamics networks with extensive applications in material and engineering fields owing to their high reliability and low power consumption characteristics. Controlling and optimizing transport (search) efficiency on scale-free trees has attracted much attention. In this paper, we first introduce degree-dependent weighted tree by assigning each edge (x,y) a weight wxy=(dxdy)θ, with dx and dy being the degree of nodes x and y, and θ being a controllable parameter. Then, we design a parameterized biased random walk strategy with the transition probability depending on the local information (the degree of neighboring nodes) and a parameter θ. Finally, we evaluate analytically the global mean first-passage time, which is an important indicator for measuring the transport (search) efficiency on the underlying networks, and find the interval for parameter θ where transport (search) efficiency can be improved on a class of scale-free trees. We also analyze the (transfinite) walk dimension for our biased random walk on the scale-free trees and find one can obtain arbitrary transfinite walk dimension in an interval by properly tuning the biased parameter θ. The results obtained here would shed light on controlling and optimizing transport (search) efficiency on objects with scale-free tree structures.

Butterfly diffusion over sparse point sets

The graph-based random walk model of fractal diffusion is limited in its use to the connected sets and does not allow for direct fractal dimension estimation based on observed data. We discuss a task of directly obtaining accurate fractal dimension estimates and propose butterfly diffusion as an alternative approach using an explicit relation between walk and fractal dimensions. The validity of the presented approach is evaluated and statistical properties of the dimension estimates are presented. Through experiments on self-similar sets, we demonstrate the effectiveness of this approach in producing unbiased dimension estimates, offering a promising tool for fractal analysis and Monte Carlo simulations. The estimate of fractal dimension can be also created from spectral dimension, but this approach is less general and less accurate.

Fractal and first-passage properties of a class of self-similar networks.

A class of self-similar networks, obtained by recursively replacing each edge of the current network with a well-designed structure (generator) and known as edge-iteration networks, has garnered considerable attention owing to its role in presenting rich network models to mimic real objects with self-similar structures. The generator dominates the structural and dynamic properties of edge-iteration networks. However, the general relationships between these networks' structural and dynamic properties and their generators remain unclear. We study the fractal and first-passage properties, such as the fractal dimension, walk dimension, resistance exponent, spectral dimension, and global mean first-passage time, which is the mean time for a walker, starting from a randomly selected node and reaching the fixed target node for the first time. We disclose the properties of the generators that dominate the fractal and first-passage properties of general edge-iteration networks. A clear relationship between the fractal and first-passage properties of the edge-iteration networks and the related properties of the generators are presented. The upper and lower bounds of these quantities are also discussed. Thus, networks can be customized to meet the requirements of fractal and dynamic properties by selecting an appropriate generator and tuning their structural parameters. The results obtained here shed light on the design and optimization of network structures.

Policy-reality gaps in Africa’s walking cities: Contextualizing institutional perspectives and residents’ lived experiences in Accra

ABSTRACT This paper confronts the current policy landscape and lived experiences of walking in African cities through the lens of policies, plans, institutional, and residents’ narratives. The paper builds on qualitative evidence drawn from content analysis and semi-structured interviews with local-level stakeholders across policy sectors concerned directly or indirectly with walking and walkability in the city. Perspectives from local government institutions are first reviewed to ascertain considerations for accessibility, safety, and pleasurability dimensions of walking and, second, contrasted with the lived walking experiences of residents of low-income settlements in Accra. Results indicate limited attention to pleasurability dimensions in both policy and municipal plans and significant incongruities with residents’ lived realities of walking. This parallel between the frameworks and plans underpinning the construction of the walking environment and the lived experiences of those affected by such decisions seeks to nuance debates about policy-reality (dis)junctures in African urbanism.

Effects of additional functional strength training on mobility in children with hemiplegic cerebral palsy

Background: Cerebral palsy (CP) affects the mobility and functional strength of children which leads to a decline in the quality of life of children. Functional strength training may considered objective-oriented to achieve some specific activity level and improve the overall performance of children. Objective: To determine the effects of additional functional strength training on mobility in Children with Hemiplegic Cerebral Palsy (HCP). Method: A single-blind randomized control trial (NCT05878756) was conducted on (n=40) children with hemiplegic cerebral palsy (HCP) at the National Institute of Rehabilitation Science, Islamabad. Children who can follow commands, not using assistive devices, and are not being treated in any other rehabilitation services were included. Children aged 4-12 years were recruited through a non-probability convenient sampling technique and a total of n=40 participants were randomly divided into Group A (n=20) which received conventional therapy along with functional strength training for 4 weeks, while Group B (n=20) only received conventional physical therapy. Both groups received intervention for 4weeks. The tools, Gross Motor Function Measure (GMFM) used for the severity of mobility and, five times sit to stand (FTSST) to assess functional lower extremity strength, transitional movement balance, and fall risk in older adults, were used at baseline and post-treatment assessment. Results: Both groups had improvements in strength and mobility but there was significant improvement in strength and mobility in Group A after 4th week (p≤ 0.05). However, no significant increase in the dimension of lying and kneeling was observed. Comparison between both groups (pre and post-treatment) for dimensions of walking, running, and jumping in both groups showed significant improvement (p=0.045). According to the result of GMFM scoring, a significant improvement in the interventional group (A) as compared to Group B (p=0.003). So, there is a considerable increase in GMFM score for Group A than Group B group. Conclusion: It is concluded that additional functional therapy has a better outcome as compared to conventional therapy on mobility and strength of children with CP. Keywords Cerebral palsy; functional strength training; mobility; Gross Motor Function Measure; Gross Motor Function Classification System

Equivalence of Besov spaces on p.c.f. self-similar sets

AbstractOn post-critically finite self-similar sets, whose walk dimensions of diffusions are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces $B^{p,q}_\sigma (K)$ and the Lipschitz–Besov spaces $\Lambda ^{p,q}_\sigma (K)$ , are identical. In particular, we provide concrete examples that $B^{p,q}_\sigma (K)=\Lambda ^{p,q}_\sigma (K)$ with $\sigma>1$ . Our method is purely analytical, and does not involve heat kernel estimate.

An elementary proof that walk dimension is greater than two for Brownian motion on Sierpiński carpets

We give an elementary self-contained proof of the fact that the walk dimension of the Brownian motion on an arbitrary generalized Sierpiński carpet is greater than 2, no proof of which in this generality had previously been available in the literature. Our proof is based solely on the self-similarity and hypercubic symmetry of the associated Dirichlet form and on several very basic pieces of functional analysis and the theory of regular symmetric Dirichlet forms. We also present an application of this fact to the singularity of the energy measures with respect to the canonical self-similar measure (uniform distribution) in this case, proved first by M. Hino in [Probab. Theory Related Fields 132 (2005), no. 2, 265–290].

On the conformal walk dimension: quasisymmetric uniformization for symmetric diffusions

We introduce the notion of conformal walk dimension, which serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that, for a given strongly local, regular symmetric Dirichlet space in which every metric ball has compact closure (MMD space), the finiteness of the conformal walk dimension characterizes the conjunction of the metric doubling property and the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension of an MMD space is defined as the infimum over all possible values of the walk dimension with which the parabolic Harnack inequality can be made to hold by a time change of the associated diffusion and by a quasisymmetric change of the metric. We show that the conformal walk dimension of any MMD space satisfying the metric doubling property and the elliptic Harnack inequality is two, and provide a necessary condition for a pair of such changes to attain the infimum defining the conformal walk dimension when it is attained by the original pair. We also prove a necessary condition for the existence of such a pair attaining the infimum in the setting of a self-similar Dirichlet form on a self-similar set, and apply it to show that the infimum fails to be attained for the Vicsek set and the N-dimensional Sierpiński gasket with Nge 3, in contrast to the attainment for the two-dimensional Sierpiński gasket due to Kigami (Math Ann 340(4):781–804, 2008).

Book review: Jennie Middleton, The Walkable City: Dimensions of Walking and Overlapping Walks of Life

Book review: Jennie Middleton, The Walkable City: Dimensions of Walking and Overlapping Walks of Life

Stochastic modelling of fractal diffusion and dimension estimation

The revision of classical methods for spectral and walk dimension estimates is the main aim of the paper. Being focused on the unbiased estimation of the walk and spectral dimensions, we aim to construct the estimates with the minimal mean square error. Accompanied simulation experiments are performed on finite substrates, spacial structures serving as a good model of both continuum and fractal sets. We compare the classical approach based on the log–log transform of asymptotic models of returning probabilities and the second moments, and we also develop a weighted approach to improve the statistical properties of dimension estimates. The other discussed aspect is whether to simulate diffusion using the classical graph diffusion model with zero probability of staying in the same vertex or to prefer the physically motivated model of diffusion over edges with the optimal value of jump probability. Finally, we present the results of simulation experiments on two-dimensional finite substrates which approximate the continuum and selected Sierpinski gaskets and carpets. The paper also summarises general suggestions based on the obtained results from the simulation experiments.

Effect of Pilates Exercises on Standing, Walking, and Balance in Children With Diplegic Cerebral Palsy.

ObjectiveTo analyze how Pilates exercises affect standing, walking, and balance in children with diplegic cerebral palsy throughout a 10-week program.MethodsWe included 40 children aged 7–9 years with diplegic cerebral palsy, and randomly allocated them into two groups of the same size: conventional therapy group (group A) and conventional therapy+Pilates group (group B). We administered the same conventional physical therapy program to both groups for 45 minutes, with group B receiving additional Pilates exercises for 45 minutes. Both groups attended the intervention program three times/week for 10 weeks. We used the Growth Motor Function Measure Scale (GMFM-88) to evaluate standing and walking (Dimensions D and E), and the Pediatric Balance Scale to evaluate balance function before and after treatment.ResultsComparison of the average values of all measured variables before and after therapy showed a statistically significant difference (p<0.05) between the two groups. All measured variables showed a significant difference between groups A and B, in favor of group B (p<0.05).ConclusionPilates exercise in addition to conventional therapy is more effective in improving balance and gross motor function in children with diplegic cerebral palsy than the conventional therapy alone.

Fractal and Knot-Free Chromosomes Facilitate Nucleoplasmic Transport

Chromosomes in the nucleus assemble into hierarchies of 3D domains that, during interphase, share essential features with a knot-free condensed polymer known as the fractal globule (FG). The FG-like chromosome likely affects macromolecular transport, yet its characteristics remain poorly understood. Using computer simulations and scaling analysis, we show that the 3D folding and macromolecular size of the chromosomes determine their transport characteristics. Large-scale subdiffusion occurs at a critical particle size where the network of accessible volumes is critically connected. Condensed chromosomes have connectivity networks akin to simple Bernoulli bond percolation clusters, regardless of the polymer models. However, even if the network structures are similar, the tracer's walk dimension varies. It turns out that the walk dimension depends on the network topology of the accessible volume and dynamic heterogeneity of the tracer's hopping rate. We find that the FG structure has a smaller walk dimension than other random geometries, suggesting that the FG-like chromosome structure accelerates macromolecular diffusion and target-search.

Scaling limits of the three-dimensional uniform spanning tree and associated random walk

We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/\beta$, where $\beta\approx 1.624\dots$ is the growth exponent of three-dimensional loop-erased random walk. Additionally, we study the random walk on the three-dimensional uniform spanning tree, deriving its walk dimension (with respect to both the intrinsic and Euclidean metric) and its spectral dimension, demonstrating the tightness of its annealed law under rescaling, and deducing heat kernel estimates for any diffusion that arises as a scaling limit.

Stability of heat kernel estimates for symmetric non-local Dirichlet forms

In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for α \alpha -stable-like processes even with α ≥ 2 \alpha \ge 2 when the underlying spaces have walk dimensions larger than 2 2 , which has been one of the major open problems in this area.

Space, time and the female body: New Delhi on foot at night

Our paper lays out an experiential account of walking the city, an otherwise mundane everyday act, as a performative action. The specific act of walking that this paper examines is part of a series of performance actions titled ‘Women Walk at Midnight’ (Delhi, 2016- present) where women collectively walk the streets of the city at night. Breaking from the conventional article format, our paper draws on reflexive feminist research and performance analysis to narrate the performative dimensions of walking that foreground how women collectively negotiate fear and vulnerability in/of the city. We draw on our own memories and experiences of walking - embodied, sensorial, even intuitive - not only as participants of the Midnight Walk but as women who often negotiate varied agressions of the city streets. The paper weaves an intricate pattern of relations between space, time and movement of female bodies in the city of New Delhi from a gendered perspective. We posit that transformations of urban atmosphere through the course of the day shape our experiences of the city, reproducing it as a gendered site. Relation of bodies to space, we argue, shifts in response to time. We are interested not only in the singular and deliberate actions carried out by individual walkers, but unpack the shared notions of risk, adventure and pleasure, experienced fleetingly and afforded only through the collective. These performative actions of female bodies at night not only challenge the rehearsed understandings of urban citizenship but also stage its emancipatory possibilities.

Effect of chronic alcohol intake on motor functions on the elderly

Alcohol use disorder (AUD) is a chronic and progressive disease influenced by genetic, psychosocial, and environmental factors. The consequences of alcohol consumption involve alterations in neural circuits of emotion and cognition, as well as in the motor planning circuit. Furthermore, during the natural aging process, several biochemical and functional alterations are also observed with neurological consequences. Thus, considering the consequences of chronic alcohol consumption on neural systems and natural aging process, we aimed to analyze the degree of motor and functional impairment in elderly with chronic alcohol consumption. Sixty elderly underwent an analysis of alcohol consumption profile (Alcohol Use Disorders Identification Test – AUDIT) that divided them into a control group (CON) and an alcohol group (ALC). The analysis of quality of life was performed using the Medical Outcomes Study 36-Item Short-Form Health Survey (SF-36), the analysis of motor function was performed using the Borg Scale, the Six-Minute Walk Test (6MWT) and the Motor Scale for Elderly (MSE). We were able to conclude that the misuse of alcohol by the elderly promotes significant physical limitations. These limitations result in a worsening of functional capacity of walking and various dimensions of motor ability: fine motor skill, global coordination, balance, body scheme, spatial organization, temporal organization, and general motor aptitude. Besides the physical limitations caused by alcohol use, the quality of life in their physical, mental, and social aspects was reduced. Thus, actions are required to help the elderly understand these losses and exercise control over alcohol misuse.

Gagliardo-Nirenberg, Trudinger-Moser and Morrey inequalities on Dirichlet spaces

With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper proves Sobolev embedding theorems in the general framework of Dirichlet spaces. Under suitable assumptions that are verified in a variety of settings, we obtain the whole family of Gagliardo-Nirenberg and Trudinger-Moser inequalities with optimal exponents. These turn out to depend not only on the Hausdorff and walk dimensions of the space but also on other invariants. In addition, we prove Morrey type inequalities and apply them to study the infimum of the exponents that ensure continuity of Sobolev functions. The results are illustrated in the case of fractals with the Vicsek set, whereas several conjectures are made for general nested fractals and the Sierpinski carpet.

Walking as a wandering ethic of (re)location: A public ‘pedagogy of hope’

This paper considers the public pedagogy of location in relation to walking. I walk and write withand from my compass orientated to the Freirean notion of a ‘pedagogy of hope’. Using an autoethnographic account of a local walk, walking is (re)presented and interpreted as a wanderingethic of (re)location. Temporal and spatial dimensions of my walking are revealed in the social,cultural and ecological context of the bushfires and the pandemic. Drawing from scholars whotheorize embodiment and the multiple natures of body~time~space, the inter and intra-actionswith/in ecologies are presenced in a sensory narrative. To consider walking as a wandering ethicof (re)location, it is argued that various temporal, spatial, material, historical and cultural dimensions are contingent within the context of change as evident in the aftermath of bushfires and thepandemic. What I examine is the inter-play in relation to what is present and otherwise absentwhilst walking that is interpreted as a ‘pedagogy of hope’ amidst the struggles and uncertaintiesof these times.

On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates

We show that, for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (sub-Gaussian estimates) imply the singularity of the energy measures with respect to the symmetric measure, verifying a conjecture by M. T. Barlow in (Contemp. Math. 338 (2003) 11–40). We also prove that in the contrary case of walk dimension two, that is, where full off-diagonal Gaussian estimates of the heat kernel hold, the symmetric measure and the energy measures are mutually absolutely continuous in the sense that a Borel subset of the state space has measure zero for the symmetric measure if and only if it has measure zero for the energy measures of all functions in the domain of the Dirichlet form.

Determining effective conductivities of fractal objects

We develop formulas to approximate the effective thermal conductivity of composite materials consisting of a matrix with nominal dimensionless conductivity Kmat=1 containing inclusions (holes) with a conductivity Khole≤1 laid out in fractal patterns: 2D Sierpinski carpets and 3D Menger sponges. Direct simulation of steady state heat conduction in finite stages of construction of these objects provide benchmark conductivity values. Recent theory on random walks in fractal media provides conductivity estimates for Sierpinski carpets for the case where Khole=0. Recognizing that the estimate in the fractal generators is related to the classic Rayleigh model, we obtain an expression for the conductivity in the more general case 0≤Khole≤1. Comparison with direct simulated values shows a relative error below 6% when the conductivity contrast is high (Khole≈0) and below 1% when the contrast is low. This significantly improves the accuracy of estimates obtained from simple resistor networks, which are interpreted as upper and lower bounds. The extension of this approach to Menger sponges gives a conductivity estimator related to the classic Maxwell conductivity model. This estimator, accounting for the effects of pattern correlations, has relative errors in the same range as the carpet estimator. A conjecture for the random walk dimension in sponges is also derived. The effective conductivities for 2D and 3D patterns, with low volume fractions of the matrix and high conductivity contrast, also represent a significant improvement over the classic formulas for uncorrelated distributions of inclusions.