## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
- Conference Publications
- AIMS Mathematics

DCDS-B

We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation
$ u_t = \nabla\cdot(u\nabla K*u) $ in $\mathbb{R}^n$,
for homogeneous potentials $K(x) = |x|^\gamma$, $\gamma>0$.
For $\gamma>2$, the aggregation happens in infinite time and
exhibits a concentration of mass along a collapsing $\delta$-ring.
We develop an asymptotic theory for the approach to this singular solution.
For $\gamma < 2$, the solution blows up in finite time and we present careful numerics of second
type similarity solutions for all $\gamma$ in this range, including
additional asymptotic behaviors in the limits $\gamma \to 0^+$ and $\gamma\to 2^-$.

DCDS-B

Studies of problems in fluid dynamics have spurred research in
many areas of mathematics, from rigorous analysis of nonlinear partial
differential equations, to numerical analysis, to modeling
and applied analysis of related physical systems.
This special issue of Discrete and Continuous Dynamical Systems Series B
is dedicated to our friend and colleague Tom Beale in recognition of
his important contributions to these areas.

For more information please click the "Full Text" above.

For more information please click the "Full Text" above.

keywords:

DCDS-B

Systems of pairwise-interacting particles model a cornucopia of physical systems, from insect swarms and bacterial colonies to nanoparticle self-assembly. We study a continuum model with densities supported on co-dimension one curves for two-species particle interaction in $\mathbb{R}^2$, and apply linear stability analysis of concentric ring steady states to characterize the steady state patterns and instabilities which form. Conditions for linear well-posedness are determined and these results are compared to simulations of the discrete particle dynamics, showing predictive power of the linear theory. Some intriguing steady state patterns are shown through numerical examples.

DCDS-B

We consider the problem of active feedback control of Rayleigh-Bénard convection via shadowgraphic measurement.
Our theoretical studies show, that when the feedback control is
positive, i.e. is
tuned to advance the onset of convection, there is a critical
threshold beyond which the system becomes linearly ill-posed
so that short-scale disturbances are greatly amplified.
Experimental observation suggests that finite size effects become
important and we develop a theory to explain these contributions.
As an efficient modelling tool for studying the dynamics of such a
controlled pattern forming system, we use a Galerkin approximation
to derive a dimension reduced model.

CPAA

We present an energy-methods-based proof of the existence and uniqueness of solutions of
a nonlocal aggregation equation with degenerate diffusion. The equation we study
is relevant to models of biological aggregation.

DCDS-B

Hotspots of crime localized in space and time are well documented. Previous mathematical models of urban crime have exhibited these hotspots but considered a static or otherwise suboptimal police response to them. We introduce a program of police response to hotspots of crime in which the police adapt dynamically to changing crime patterns. In particular, they choose their deployment to solve an optimal control problem at every time. This gives rise to a free boundary problem for the police deployment's spatial support. We present an efficient algorithm for solving this problem numerically and show that police presence can prompt surprising interactions among adjacent hotspots.

DCDS-B

This special issue is an outgrowth of a minisyposium titled ``Mathematics of Social Systems"
held at the 9th AIMS Conference
on Dynamical Systems, Differential Equations and Applications, held in Orlando, FL in July 2012.
Presenters from that session were invited to submit papers that were reviewed using the usual procedures of the DCDS journals, along with additional authors from the field.
Mathematics has already had a significant impact on basic research involving
fundamental problems in physical sciences, biological sciences, computer science and
engineering. Examples include understanding of the equations of incompressible fluid
dynamics, shock wave theory and compressible gas dynamics, ocean modeling, algorithms for
image processing and compressive sensing, and biological problems such as models for invasive
species, spread of disease, and more recently systems biology for modeling of complex
organisms and complex patterns of disease. This impact has yet to come to fruition in a
comprehensive way for complex social behavior. While computational models such as agent-based
systems and well-known statistical methods are widely used in the social sciences, applied
mathematics has not to date had a core impact in the social sciences at the level that it achieves
in the physical and life sciences. However in recent years we have seen a growth of work in this direction
and ensuing new mathematics problems that must be tackled to understand such problems.
Technical approaches include ideas from statistical physics, nonlinear partial differential equations of all types, statistics and inverse problems, and stochastic processes and social network models.
The collection of papers presented in this issue provides a backdrop of the current state of the art results in this developing new research area in applied mathematics. The body of work encompasses many of the challenges
in understanding these discrete complex systems and their related continuum approximations.

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

DCDS

We consider a class of splitting schemes for fourth order nonlinear
diffusion equations. Standard backward-time differencing requires
the solution of a higher order elliptic problem, which can be both
computationally expensive and work-intensive to code, in higher space
dimensions.
Recent papers in the literature provide computational evidence that
a biharmonic-modified, forward time-stepping method, can provide good
results for these problems.
We provide a theoretical explanation of the results.
For a basic nonlinear 'thin film' type equation we prove $H^1$
stability of the method given very simple boundedness constraints
of the numerical solution. For a more general class of long-wave
unstable problems, we prove stability and convergence, using
only constraints on the smooth solution.
Computational examples include both the model of 'thin film' type
problems and a quantitative model for electrowetting in a Hele-Shaw
cell (Lu

*et al*J. Fluid Mech. 2007). The methods considered here are related to 'convexity splitting' methods for gradient flows with nonconvex energies.
DCDS

Recently fourth order equations of the form
$u_t = -\nabla\cdot((\mathcal G(J_\sigma u)) \nabla \Delta u)$ have been proposed
for noise reduction and simplification of two dimensional images.
The operator $\mathcal G$ is a nonlinear functional involving
the gradient or Hessian of its argument, with decay in the far field.
The operator $J_\sigma$ is a standard mollifier.
Using ODE methods on Sobolev spaces,
we prove existence and uniqueness of solutions of this problem
for $H^1$ initial data.

IPI

We present a method to enhance the quality of a video sequence
captured through a turbulent atmospheric medium, and give an
estimate of the radiance of the distant scene, represented as a
``latent image,'' which is assumed to be static throughout the
video. Due to atmospheric turbulence, temporal averaging produces
a blurred version of the scene's radiance. We propose a method
combining Sobolev gradient and Laplacian to stabilize the video
sequence, and a latent image is further found utilizing the ``lucky
region" method. The video sequence is stabilized while keeping
sharp details, and the latent image shows more consistent straight
edges. We analyze the well-posedness for the stabilizing PDE and the
linear stability of the numerical scheme.

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