Riemann‐Hilbert Problems for the Shapes Formed by Bodies Dissolving,Melting, and Eroding in Fluid Flows

The classical Stefan problem involves the motion of boundaries during phase transition, but this process can be greatly complicated by the presence of a fluid flow. Here we consider a body undergoing material loss due to either dissolution (from molecular diffusion), melting (from thermodynamic phase change), or erosion (from fluid‐mechanical stresses) in a fast‐flowing fluid. In each case, the task of finding the shape formed by the shrinking body can be posed as a singular Riemann‐Hilbert problem. A class of exact solutions captures the rounded surfaces formed during dissolution/melting, as well as the angular features formed during erosion, thus unifying these different physical processes under a common framework. This study, which merges boundary‐layer theory, separated‐flow theory, and Riemann‐Hilbert analysis, represents a rare instance of an exactly solvable model for high‐speed fluid flows with free boundaries.© 2017 Wiley Periodicals, Inc.     

A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains

A highly accurate new solver is developed to deal with the Dirichlet problems for the 2D Laplace equation in the doubly connected domains. We introduce two circular artificial boundaries determined uniquely by the physical problem domain, and derive a Dirichlet to Dirichlet mapping on these two circles, which are exact boundary conditions described by the first kind Fredholm integral equations. As a direct result, we obtain a modified Trefftz method equipped with two characteristic length factors, ensuring that the new solver is stable because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the unknown coefficients. The new method possesses several advantages: mesh‐free, singularity‐free, non‐illposedness, semi‐analyticity of solution, efficiency, accuracy, and stability. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Technological integration and hyperconnectivity: Tools for promoting extreme human lifespans

Artificial, neurobiological, and social networks are three distinct complex adaptive systems (CASs), each containing discrete processing units (nodes, neurons, and humans, respectively). Despite the apparent differences, these three networks are bound by common underlying principles which describe the behavior of the system in terms of the connections of its components, and its emergent properties. The longevity (long‐term retention and functionality) of the components of each of these systems is also defined by common principles. Here, I will examine some properties of the longevity and function of the components of artificial and neurobiological systems, and generalize these to the longevity and function of the components of social CAS. In other words, I will show that principles governing the long‐term functionality of computer nodes and of neurons, may be extrapolated to the study of the long‐term functionality of humans (or more precisely, of the noemes, an abstract combination of “existence” and “digital fame”). The study of these phenomena can provide useful insights regarding practical ways that can be used to maximize human longevity. The basic law governing these behaviors is the “Law of Requisite Usefulness,” which states that the length of retention of an agent within a CAS is proportional to the agent's contribution to the overall adaptability of the system. © 2014 Wiley Periodicals, Inc. Complexity 20: 15–24, 2015     

Fractal response of physiological signals to stress conditions,environmental changes,and neurodegenerative diseases

In the past two decades the biomedical community has witnessed several applications of nonlinear system theory to the analysis of biomedical time series and the development of nonlinear dynamic models. The development of this area of medicine can best be described as nonlinear and fractal physiology. These studies have been intended to develop more reliable methodologies for understanding how biological systems respond to peculiar altered conditions induced by internal stress, environment stress, and/or disease. Herein, we summarize the theory and some of our results showing the fractal dependency on different conditions of physiological signals such as inter‐breath intervals, heart inter‐beat intervals, and human stride intervals. © 2007 Wiley Periodicals, Inc. Complexity 12: 12–17, 2007     

BML revisited: Statistical physics,computer simulation,and probability

Statistical physics, computer simulation, and discrete mathematics are intimately related through the study of shared lattice models. These models lie at the foundation of all three fields, are studied extensively, and can be highly influential. Yet new computational and mathematical tools may challenge even well‐established beliefs. Consider the BML model, which is a paradigm for modeling self‐organized patterns of traffic flow and first‐order jamming transitions. Recent findings, on the existence of intermediate states, bring into question the standard understanding of the jamming transition. We review the results and show that the onset of full‐jamming can be considerably delayed based on the geometry of the system. We also introduce an asynchronous version of BML, which lacks the self‐organizing properties of BML, has none of the puzzling intermediate states, but has a sharp, discontinuous, transition to full jamming. We believe this asynchronous version will be more amenable to rigorous mathematical analysis than standard BML. We discuss additional models, such as bootstrap percolation, the honey‐comb dimer model and the rotor‐router, all of which exemplify the interplay between the three fields, while also providing cautionary tales. Finally, we synthesize implications for how results from one field may relate to the other, and also implications specific to computer implementations. © 2006 Wiley Periodicals, Inc. Complexity, 12, 30–39, 2006     

An algorithm for detecting community structure of social networks based on prior knowledge and modularity

An algorithm is proposed to detect community structure in social network. The algorithm begins with a community division based on prior knowledge of the degrees of the nodes, and then combines the communities until a clear partition is obtained. In applications such as a computer‐generated network, Ucinet networks, and Chinese rural‐urban migrants' social networks, the algorithm can achieve higher modularity and greater speed than others in the recent literature. © 2007 Wiley Periodicals, Inc. Complexity 12: 53–60, 2007     

Global controllability between steady supercritical flows in channel networks

We consider a tree‐like network of open channels with outflow at the root. Controls are exerted at the boundary nodes of the network except for the root. In each channel, the flow is modelled by the de St. Venant equations. The node conditions require the conservation of mass and the conservation of energy. We show that the states of the system can be controlled within the entire network in finite time from a stationary supercritical initial state to a given supercritical terminal state with the same orientation. During this transition, the states stay in the class of C1‐functions, so no shocks occur. Copyright 2004 John Wiley & Sons, Ltd.     

Task‐performing dynamics in irregular,biomimetic networks

Understanding self‐organized collective dynamics—especially in sparsely connected, noisy, and imperfect networks—has important implications for designing and optimizing task‐performing technological systems as well as for deciphering biological structures and functions. We note that stomatal arrays on plant leaves might provide an ideal example of task‐performance in this context. Guided by observations of stomatal networks, we examined a simple model of task‐performing, collective dynamics that included state noise, spatial rule heterogeneity, dynamic modules, and network rewiring. Our results indicate that task‐performance in such networks can actually be enhanced by various kinds of spatial and temporal irregularity. © 2007 Wiley Periodicals, Inc. Complexity 12: 14–21, 2007     

Searching for mobile intruders in circular corridors by two 1-searchers

We consider the problem of searching for a mobile intruder in a circular corridor by two mobile searchers, who hold one flashlight. A circular corridor is a polygon with one polygonal hole such that its outer and inner boundaries are mutually weakly visible. Both 1-searchers always direct their flashlights at the inner boundary. The objective is to decide whether there exists a search schedule for two 1-searchers to detect the intruder, no matter how fast he moves, and if so, generate a search schedule. We give a characterization of the circular corridors, which are searchable by two 1-searchers. Based on our characterization, an O(nlogn) time algorithm is then presented to determine the searchability of a circular corridor, where n denotes the total number of vertices of the outer and inner boundaries. Moreover, a search schedule can be reported in time linear in its size, if it exists.     

Repeating earthquakes,episodic tremor and slip: Emerging patterns in complex earthquake cycles?

The lack of regularity in earthquake cycles continues to be a confounding issue in earthquake science. Lately, observations of episodic nonvolcanic tremor and slip (ETS) along a few well‐instrumented tectonic plate boundaries are intriguing: these features recur together with predictable time intervals. Data now trace recurring ETS back to 1990 and no significant earthquake ever followed an ETS episode. This observation and the fact that stress drops associated with episodic slips are low, only on the order of 0.01 MPa, suggests that repeated ETS has little cumulative effects in priming the fault for the next large earthquake. Another known regularity in seismic activity is the so‐called repeating earthquakes that rupture the same patch of fault repetitively. Current hypotheses for repeating earthquakes point to the interaction of continual, aseismic fault slip with locked, seismogenic patches of the fault. Interestingly, ETS also recur near where the transition between brittle faulting and plastic flow is expected, although it is not clear how and why regularities in space and time are interconnected. New data show that ETS recurs throughout the entire length of the Cascadia subduction zone, thus ruling out any special, local factors as necessary conditions for ETS. Recurrence intervals of ETS vary along the Cascadia. Such variations are not governed by the rate of plate motion that ultimately drives the earthquake process, but they do coincide with variations in the geology of the overriding plate that can influence the rheology along the plate interface. To this end, we call attention to the Portevin‐Le Chatelier effect (PLC, or jerky flow) as a potential analog to the earthquake process. The dynamics of the PLC has been extensively studied and shows many intriguing features as the system goes from chaotic to self‐organized critical regimes as strain rate increases. In particular, the PLC exhibits not only stick‐slip behavior (stress serration) over time but also spatial interactions over extended regions—features that are necessary to account for complex spatio‐temporal variations associated with earthquake activities. © 2007 Wiley Periodicals, Inc. Complexity 12: 33–43, 2007     

The evolution of cooperation in spatial groups

Much of human cooperation remains an evolutionary riddle. There is evidence that individuals are often organized into groups in many social situations. Inspired by this observation, we propose a simple model of evolutionary public goods games in which individuals are organized into networked groups. Here, nodes in the network represent groups; the edges, connecting the nodes, refer to the interactions between the groups. Individuals establish public goods games with partners in the same group and migrate among neighboring groups depending on their payoffs and expectations. We show that the paradigmatic public goods social dilemma can be resolved and high cooperation levels are attained in structured groups, even in relatively harsh conditions for cooperation. Further, by means of numerical simulations and mean-field analysis, we arrive at the result: larger average group size and milder cooperation environment would lead to lower cooperation level but higher average payoffs of the entire population. Altogether, these results emphasize that our understanding of cooperation can be enhanced by investigations of how spatial groups of individuals affect the evolution dynamics, which might help in explaining the emergence and evolution of cooperation.     

Clifford algebra method for network expression,computation, and algorithm construction

Clifford algebra is introduced as a theoretical foundation for network topology expression and algorithm construction. Network nodes are coded with basis vectors in a vector space , and the edges and k‐walk routes can be expressed by 2‐blades and k‐blades, respectively, in the Clifford algebra Cl(n,0). The topologies among nodes, edges, and routes of networks can be directly calculated, and the network routes can be extended and traversed with oriented join products. The network algorithm construction processes based on Clifford algebra are instantiated by the single source shortest path algorithm. The experimental results on different scale random networks suggest that Clifford algebra is suited for network expression and relation computation. The Clifford algebra‐based shortest path algorithm is vivid and clear in geometric meaning and has great advantage on temporal and spatial complexity. Copyright © 2013 John Wiley & Sons, Ltd.     

Gauss‐Green theorem for weakly differentiable vector fields,sets of finite perimeter,and balance laws

We analyze a class of weakly differentiable vector fields F : ?ⁿ → ?ⁿ with the property that F ∈ L^∞ and div F is a (signed) Radon measure. These fields are called bounded divergence‐measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence‐measure field F over the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss‐Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure μ that is absolutely continuous with respect to ??^{N ? 1} on ?^N, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure‐theoretic interior of the set with respect to the measure ||μ||, the total variation measure. We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter E as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary so that the Gauss‐Green theorem for F holds on E. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N ? 1)‐dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure‐valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc.     

Topological properties of the one dimensional exponential random geometric graph

In this article we study the one‐dimensional random geometric (random interval) graph when the location of the nodes are independent and exponentially distributed. We derive exact results and limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Network design through forests with degree- and role-constrained minimum spanning trees

Finding the degree-constrained minimum spanning tree (DCMST) of a graph is a widely studied NP-hard problem. One of its most important applications is network design. Here we deal with a new variant of the DCMST problem, which consists of finding not only the degree- but also the role-constrained minimum spanning tree (DRCMST), i.e., we add constraints to restrict the role of the nodes in the tree to root, intermediate or leaf node. Furthermore, we do not limit the number of root nodes to one, thereby, generally, building a forest of DRCMSTs. The modeling of network design problems can benefit from the possibility of generating more than one tree and determining the role of the nodes in the network. We propose a novel permutation-based representation to encode these forests. In this new representation, one permutation simultaneously encodes all the trees to be built. We simulate a wide variety of DRCMST problem instances which we optimize using different evolutionary computation algorithms encoding individuals of the population using the proposed representation. To illustrate the applicability of our approach, we formulate the trans-European transport network as a DRCMST problem. In this network design, we simultaneously optimize nine transport corridors and show that it is straightforward using the proposed representation to add constraints depending on the specific characteristics of the network.     

Perturbation analysis in finite LD‐QBD processes and applications to epidemic models

In this paper, our interest is in the perturbation analysis of level‐dependent quasi‐birth‐and‐death (LD‐QBD) processes, which constitute a wide class of structured Markov chains. An LD‐QBD process has the special feature that its space of states can be structured by levels (groups of states), so that a tridiagonal‐by‐blocks structure is obtained for its infinitesimal generator. For these processes, a number of algorithmic procedures exist in the literature in order to compute several performance measures while exploiting the underlying matrix structure; among others, these measures are related to first‐passage times to a certain level L(0) and hitting probabilities at this level, the maximum level visited by the process before reaching states of level L(0), and the stationary distribution. For the case of a finite number of states, our aim here is to develop analogous algorithms to the ones analyzing these measures, for their perturbation analysis. This approach uses matrix calculus and exploits the specific structure of the infinitesimal generator, which allows us to obtain additional information during the perturbation analysis of the LD‐QBD process by dealing with specific matrices carrying probabilistic insights of the dynamics of the process. We illustrate the approach by means of applying multitype versions of the susceptible‐infective (SI) and susceptible‐infective‐susceptible (SIS) epidemic models to the spread of antibiotic‐sensitive and antibiotic‐resistant bacterial strains in a hospital ward.     

Conditions for the discovery of solution horizons

We present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable. These problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network. We provide general forward algorithms with stopping rules for their solution. The key condition required is that of weak reachability, which roughly requires that for any sequence of nodes or states, it must be possible from optimal states to reach states close in cost to states along this sequence. Moreover the costs to reach these states must converge to zero. Applications are considered in optimal search, undiscounted Markov decision processes, and deterministic infinite horizon optimization.This work was supported in part by NSF Grant ECS-8700836 to The University of Michigan.     

A one-sweep method in the solution of finite-element equations: An application to the vibration of structures in heavy fluids

A one-sweep method for the numerical solution of finite-element equations is presented. This procedure is especially efficient in computing time and storage when the solution is required at only a few nodes of the finite-element mesh. Furthermore, the method is particularly useful in dealing with problems on infinite or semi-infinite domains. Artificial boundaries must be introduced in such cases, and the one-sweep method affords an extremely efficient algorithm by which the dependence of the solution on the location of these boundaries can be assessed. An application of the method to the vibration of a half-submerged circular cylinder in a heavy fluid is presented.The second author wishes to express his thanks to Professor J. L. Sackman for reviewing this work.(deceased).     

A local radial basis function collocation method to solve the variable‐order time fractional diffusion equation in a two‐dimensional irregular domain

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.