21 research outputs found
Uniform convergence for the incompressible limit of a tumor growth model
We study a model introduced by Perthame and Vauchelet that describes the
growth of a tumor governed by Brinkman's Law, which takes into account friction
between the tumor cells. We adopt the viscosity solution approach to establish
an optimal uniform convergence result of the tumor density as well as the
pressure in the incompressible limit. The system lacks standard maximum
principle, and thus modification of the usual approach is necessary
Quantitative Assessment of Robotic Swarm Coverage
This paper studies a generally applicable, sensitive, and intuitive error
metric for the assessment of robotic swarm density controller performance.
Inspired by vortex blob numerical methods, it overcomes the shortcomings of a
common strategy based on discretization, and unifies other continuous notions
of coverage. We present two benchmarks against which to compare the error
metric value of a given swarm configuration: non-trivial bounds on the error
metric, and the probability density function of the error metric when robot
positions are sampled at random from the target swarm distribution. We give
rigorous results that this probability density function of the error metric
obeys a central limit theorem, allowing for more efficient numerical
approximation. For both of these benchmarks, we present supporting theory,
computation methodology, examples, and MATLAB implementation code.Comment: Proceedings of the 15th International Conference on Informatics in
Control, Automation and Robotics (ICINCO), Porto, Portugal, 29--31 July 2018.
11 pages, 4 figure
The spiral index of knots
Two new invariants that are closely related to Milnor's curvature-torsion
invariant are introduced. The first, the spiral index of a knot, captures the
minimum number of maxima among all knot projections that are free of inflection
points. This invariant is closely related to both the bridge and braid index of
the knot. The second, the projective superbridge index, provides a method of
counting the greatest number of maxima that occur in a given knot projection.
In addition to investigating how these invariants are related to the classical
invariants, we utilize them to determine all knots with curvature-torsion
invariant equal to 6 pi.Comment: 19 pages, 14 figure
The cane toads PDE -- a model of a population with variable motility
We will describe an equation that models the invasion of cane toads in Australia. We will present results about long-time and long-range behavior of the population and connections to Hamilton-Jacobi equations.Non UBCUnreviewedAuthor affiliation: UCLAPostdoctora
Speed-up of traveling waves by negative chemotaxis
We consider the traveling wave speed for Fisher-KPP (FKPP) fronts under the influence of chemotaxis and provide an almost complete picture of its asymptotic dependence on parameters representing the strength and length-scale of chemotaxis. Our study is based on the convergence to the porous medium FKPP traveling wave and a hyperbolic FKPP-Keller-Segel traveling wave in certain asymptotic regimes. In this way, it clarifies the relationship between three equations that have each garnered intense interest on their own. Our proofs involve a variety of techniques ranging from entropy methods and decay of oscillations estimates to a general description of the qualitative behavior to the hyperbolic FKPP-Keller-Segel equation. For this latter equation, we, as a part of our limiting arguments, establish an explicit lower bound on the minimal traveling wave speed and provide a new construction of traveling waves that extends the known existence range to all parameter values
The Role of Electronic Library in the Informatization of Modern Education
В статье будет представлена роль электронной библиотеки в процессе информатизации
современного образования. Описаны результаты проведенного исследования на территории
Красноярского края относительно использования ресурсов сети Интернет школьниками и
учителями. Также описаны условия доступа к глобальной сети в школах Красноярского краяThe role of electronic library in the informatization of modern education is presented in the article.
The results of research that have been carried out on the territory of Krasnoyarsk Territory concerning
Internet sources usage by students and teachers have been described. So have been the conditions of
access to the Global Network in schools of Krasnoyarsk Territory
Speed-up of traveling waves by negative chemotaxis
We consider the traveling wave speed for Fisher-KPP (FKPP) fronts under the influence of chemotaxis and provide an almost complete picture of its asymptotic dependence on parameters representing the strength and length-scale of chemotaxis. Our study is based on the convergence to the porous medium FKPP traveling wave and a hyperbolic FKPP-Keller-Segel traveling wave in certain asymptotic regimes. In this way, it clarifies the relationship between three equations that have each garnered intense interest on their own. Our proofs involve a variety of techniques ranging from entropy methods and decay of oscillations estimates to a general description of the qualitative behavior to the hyperbolic FKPP-Keller-Segel equation. For this latter equation, we, as a part of our limiting arguments, establish an explicit lower bound on the minimal traveling wave speed and provide a new construction of traveling waves that extends the known existence range to all parameter values
A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling
As a counterpoint to classical stochastic particle methods for linear
diffusion equations, we develop a deterministic particle method for the
weighted porous medium equation (WPME) and prove its convergence on bounded
time intervals. This generalizes related work on blob methods for unweighted
porous medium equations. From a numerical analysis perspective, our method has
several advantages: it is meshfree, preserves the gradient flow structure of
the underlying PDE, converges in arbitrary dimension, and captures the correct
asymptotic behavior in simulations.
That our method succeeds in capturing the long time behavior of WPME is
significant from the perspective of related problems in quantization. Just as
the Fokker-Planck equation provides a way to quantize a probability measure
by evolving an empirical measure according to stochastic Langevin
dynamics so that the empirical measure flows toward , our particle
method provides a way to quantize according to deterministic
particle dynamics approximating WMPE. In this way, our method has natural
applications to multi-agent coverage algorithms and sampling probability
measures.
A specific case of our method corresponds exactly to the mean-field dynamics
of training a two-layer neural network for a radial basis function activation
function. From this perspective, our convergence result shows that, in the over
parametrized regime and as the variance of the radial basis functions goes to
zero, the continuum limit is given by WPME. This generalizes previous results,
which considered the case of a uniform data distribution, to the more general
inhomogeneous setting. As a consequence of our convergence result, we identify
conditions on the target function and data distribution for which convexity of
the energy landscape emerges in the continuum limit.Comment: 55 pages, 8 figure
Non-local competition slows down front acceleration during dispersal evolution
54 pages, 11 figures, Numerical appendix by Thierry DumontInternational audienceWe investigate the super-linear spreading in a reaction-diffusion model analogous to the Fisher-KPP equation, but in which the population is heterogeneous with respect to the dispersal ability of individuals, and the saturation factor is non-local with respect to one variable. We prove that the rate of acceleration is slower than the rate of acceleration predicted by the linear problem, that is, without saturation. This hindering phenomenon is the consequence of a subtle interplay between the non-local saturation and the non-trivial dynamics of some particular curves that carry the mass at the front. A careful analysis of these trajectories allows us to identify the value of the rate of acceleration. The article is complemented with numerical simulations that illustrate some behavior of the model that is beyond our analysis