A discontinuous solution for an evolution compressible stokes system in a bounded domain

An evolution compressible Stokes system is studied in a bounded cylindrical region . The initial datum of pressure is assumed to have a jump at a specified curve C₀ in Ω. As predicted by the Rankine-Hugoniot conditions, the pressure and velocity derivatives have jump discontinuities along the characteristic plane of the curve C₀ directed by an ambient velocity vector. An explicit formula for the jump discontinuity is presented. The jump decays exponentially in time, more rapidly for smaller viscosities. Under suitable conditions of the data, a regularity of the solution is established in a compact subregion of Q away from the jump plane.     

Deffuant model with general opinion distributions: First impression and critical confidence bound

In the Deffuant model for social influence, pairs of adjacent agents interact at a constant rate and mix up their opinions (represented by continuous variables) only if the distance between opinions is short according to a threshold. We derive a critical threshold for the Deffuant model on , above which the opinions converge toward the average value of the initial opinion distribution with probability one, provided the initial distribution has a finite second order moment. We demonstrate our theoretical results by performing extensive numerical simulations on some continuous probability distributions including uniform, Beta, power‐law and normal distributions. Noticed is a clear differentiation of convergence rate that unimodal opinions (regardless of being biased or not) achieve consensus much faster than even or polarized opinions. Hereby, the emergence of a single mainstream view is a prominent feature giving rise to fast consensus in public opinion formation and social contagious behavior. Finally, we discuss the Deffuant model on an infinite Cayley tree, through which general network architectures might be factored in. © 2013 Wiley Periodicals, Inc. Complexity 19: 38–49, 2013     

Global dissipativity of mixed time-varying delayed neural networks with discontinuous activations

In this paper, the problems of global dissipativity are investigated for neural networks with mixed time-varying delays and discontinuous activations. Some new criteria for checking the global dissipativity of the addressed neural networks are established by constructing appropriate Lyapunov functionals and employing the theory of Filippov systems and M-matrix properties. Finally, two numerical examples with simulations are presented to demonstrate the effectiveness of the theoretical results.     

Bernstein polynomial and discontinuous functions

We obtain new estimates of the rate of convergence of Bernstein operators for discontinuous functions on [0,1]$[0,1]$ which can be used to derive known results for continuous functions and functions of bounded variation.     

Consensus formation of two-level opinion dynamics

Opinion dynamics have received significant attention in recent years. This paper proposes a bounded confidence opinion model for a group of agents with two different confidence levels. Each agent in the population is endowed with a confidence interval around her opinion with radius αd or (1-α)d, where α ∈ (0,1/2] represents the differentiation of confidence levels. We analytically derived the critical confidence bound d_c = 1/(4α) for the two-level opinion dynamics on ?. A single opinion cluster is formed with probability 1 above this critical value regardless of the ratio p of agents with high/low confidence. Extensive numerical simulations are performed to illustrate our theoretical results. Noticed is a clear impact of p on the collective behavior: more agents with high confidence lead to harder agreement. It is also experimentally revealed that the sharpness of the threshold d_c increases with α but does not depend on p.     

Nyström-Clenshaw-Curtis quadrature for integral equations with discontinuous kernels

A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis quadrature for Fredholm integral equations of the second kind

whose kernel is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when is infinitely differentiable away from the diagonal . Relation to the singular value decomposition is indicated. Application to integro-differential Schrödinger equations with nonlocal potentials is given.

     

Analyzing social media data having discontinuous underlying dynamics

This paper develops a real time algorithm which identifies times of emotional discontinuity as reflected in social media. The paper formulates the optimization problem to solve, develops an algorithm to solve it using dynamic programming, and illustrates the new method by analyzing mood shifts reflected in 380,000 Twitter messages related to one of the world’s most popular soccer teams, Manchester United, during their 2011–12 season.     

Generalized solutions of parabolic initial value problems with discontinuous nonlinearities

We proved recently that parabolic initial value problems with discontinuous nonlinearities have no unique weak solution in general, but have a unique generalized solution in the sense of Colombeau. In this paper we study the relationship between generalized solutions and weak solutions.     

Multiplicity result for a discrete eigenvalue problem with discontinuous nonlinearities

Using a three critical points theorem for nondifferentiable functionals, we investigate a class of second order difference equation with discontinuous nonlinearities. A new multiplicity result is obtained.     

Spatial and spectral superconvergence of discontinuous Galerkin method for hyperbolic problems

In this paper, we analyze the spatial and spectral superconvergence properties of one-dimensional hyperbolic conservation law by a discontinuous Galerkin (DG) method. The analyses combine classical mathematical arguments with MATLAB experiments. Some properties of the DG schemes are discovered using discrete Fourier analyses: superconvergence of the numerical wave numbers, Radau structure of the X spatial error.     

Asymptotic behaviour of three-dimensional singularly perturbed convection-diffusion problems with discontinuous data

We consider three singularly perturbed convection-diffusion problems defined in three-dimensional domains: (i) a parabolic problem −?(uxx+uyy)+ut+v₁ux+v₂uy=0 in an octant, (ii) an elliptic problem −?(uxx+uyy+uzz)+v₁ux+v₂uy+v₃uz=0 in an octant and (iii) the same elliptic problem in a half-space. We consider for all of these problems discontinuous boundary conditions at certain regions of the boundaries of the domains. For each problem, an asymptotic approximation of the solution is obtained from an integral representation when the singular parameter ?→⁰⁺. The solution is approximated by a product of two error functions, and this approximation characterizes the effect of the discontinuities on the small ?− behaviour of the solution and its derivatives in the boundary layers or the internal layers.     

Mayer and optimal stopping stochastic control problems with discontinuous cost

We study two classes of stochastic control problems with semicontinuous cost: the Mayer problem and optimal stopping for controlled diffusions. The value functions are introduced via linear optimization problems on appropriate sets of probability measures. These sets of constraints are described deterministically with respect to the coefficient functions. Both the lower and upper semicontinuous cases are considered. The value function is shown to be a generalized viscosity solution of the associated HJB system, respectively, of some variational inequality. Dual formulations are given, as well as the relations between the primal and dual value functions. Under classical convexity assumptions, we prove the equivalence between the linearized Mayer problem and the standard weak control formulation. Counter-examples are given for the general framework.     

Limit cycle bifurcations in discontinuous planar systems with multiple lines

In this paper, the limit cycle bifurcation problem is investigated for a class of planar discontinuous perturbed systems with $n$ parallel switch lines. Under the assumption that the unperturbed system has a family of periodic orbits crossing all of the lines, an explicit expression of the first order Melnikov function along the periodic orbits is presented, which plays an important role in studying the problem of limit cycle bifurcations. As an application of the established method, the maximal number of limit cycles of a discontinuous system is considered.     

Diffusion dynamics in small-world networks with heterogeneous consumers

Diffusions of new products and technologies through social networks can be formalized as spreading of infectious diseases. However, while epidemiological models describe infection in terms of transmissibility, we propose a diffusion model that explicitly includes consumer decision-making affected by social influences and word-of-mouth processes. In our agent-based model consumers’ probability of adoption depends on the external marketing effort and on the internal influence that each consumer perceives in his/her personal networks. Maintaining a given marketing effort and assuming its effect on the probability of adoption as linear, we can study how social processes affect diffusion dynamics and how the speed of the diffusion depends on the network structure and on consumer heterogeneity. First, we show that the speed of diffusion changes with the degree of randomness in the network. In markets with high social influence and in which consumers have a sufficiently large local network, the speed is low in regular networks, it increases in small-world networks and, contrarily to what epidemic models suggest, it becomes very low again in random networks. Second, we show that heterogeneity helps the diffusion. Ceteris paribus and varying the degree of heterogeneity in the population of agents simulation results show that the more heterogeneous the population, the faster the speed of the diffusion. These results can contribute to the development of marketing strategies for the launch and the dissemination of new products and technologies, especially in turbulent and fashionable markets. This paper won the best student paper award at the North American Association for Computational Social and Organizational Science (NAACSOS) Conference 2005, University of Notre Dame, South Bend, Indiana, USA. Preceding versions of this paper have been presented to the Conference of the North American Association for Computational Social and Organizational Science (NAACSOS), 2005, University of Notre Dame, South Bend, USA and to the Conference of the European Social Simulation Association (ESSA), 2005, Koblenz, Germany. Sebastiano Alessio Delre received his Master Degree in Communication Science at the University of Salerno. After one year collaboration at the Institute of Science and Technologies of Cognition (ISTC, Rome, Italy), now he is a PhD student at the faculty of economics, University of Groningen, the Netherlands. His work focus on how different network structures affect market dynamics. His current application domain concerns Agent-Based Simulation Models for social and economic phenomena like innovation diffusion, fashions and turbulent market. Wander Jager is an associate professor of marketing at the University of Groningen. He studied social psychology and obtained his PhD in the behavioral and social sciences, based on a dissertation about the computer modeling of consumer behaviors in situations of common resource use. His present research is about consumer decision making, innovation diffusion, market dynamics, crowd behavior, stock-market dynamics and opinion dynamics. In his work he combines methods of computer simulation and empirical surveys. He is involved in the management committee of the European Social Simulation Association (ESSA). Marco Janssen is an assistant professor in the School of Human Evolution and Social Change and in the Department of Computer Science and Engineering at Arizona State University. He got his degrees in Operations Research and Applied Mathematics. During the last 15 years, he uses computational tools to study social phenomena, especially human-environmental interactions. His present research focuses on diffusion dynamics, institutional innovation and robustness of social-ecological systems. He combined computational studies with laboratory and field experiments, case study analysis and archeological data. He is an associate editor-in-chief of the journal Ecology and Society.     

WHEN WILL THEY EVER MAKE UP THEIR MINDS? THE SOCIAL STRUCTURE OF UNSTABLE DECISION MAKING

French (1977) French, J. R. 1977. “A formal theory of social power”. In Social networks: A developing paradigm, Edited by: Leinhardt, S. pp. 35–48. New York: Academic Press.  [Google Scholar], Harary (1959) Harary, F. 1959. “A criterion for unanimity in French's theory of social power”. In Studies in social power, Edited by: Cartwright, D. pp. 168–182. Ann Arbor: Institute for Social Research.  [Google Scholar], and Abelson (1964) Abelson, R. P. 1964. “Mathematical models of the distribution of attitudes under controversy”. In Contributions to mathematical psychology, Edited by: Frederiksen, N. and Gulliksen, H. pp. 142–160. New York: Holt, Rinehart &; Winston.  [Google Scholar] initiated a prominent line of social influence models to explain social norms or collective decisions from the structure of influence networks. These models fail to generate unstable decision dynamics, a phenomenon that can be observed in collective decision-making. To capture instability, we assume that decision-makers raise their level of salience to reduce expected losses from decision-outcomes. Our model generates persistently unstable outcome patterns under conditions related to the social network and to intolerance for expected losses. A 6-actor example reveals stable outcomes for low intolerance, complex oscillations for intermediate levels of intolerance, and simple and regular oscillation for high intolerance. We discuss implications for the predictability of collective decision-making.