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The purpose of this paper is to apply particle methods to the numerical solution of the
EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry
momentum so that wavefront interactions represent collisions in which momentum is
exchanged. This behavior allows for the description of many rich physical applications,
but also introduces difficult numerical challenges. We present a particle method for the
EPDiff equation that...
Despite recent advances, treatment of patients with aggressive Non-Hodgkin's
lymphoma (NHL2) has yet to be optimally designed. Notwithstanding the contribution of
molecular treatments, intensification of chemotherapeutic regimens may still be beneficial.
Hoping to aid in the design of intensified chemotherapy, we put forward a mathematical
and computational model that analyses the effect of Doxorubicin on NHL over a wide
range of patho-physiological conditions. The model represents tumour growth...
Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensional
superdiffusive Brusselator model are analyzed. The superdiffusive Brusselator model
differs from its regular counterpart in that the Laplacian operator of the regular model
is replaced by ∂α/∂|ξ|α, 1 < α
< 2, an integro-differential operator that reflects the nonlocal behavior of
superdiffusion. The order of the operator, α, is a measure of the rate of
...
We study the asymptotics of the even moments of solutions to a stochastic wave equation in spatial dimension 3 with linear multiplicative spatially homogeneous gaussian noise that is white in time. Our main theorem states that these moments grow more quickly than one might expect. This phenomenon is well known for parabolic stochastic partial differential equations, under the name of intermittency. Our results seem to be the first example of this phenomenon for hyperbolic equations. For comparison,...
We show how the well-known concept of external rays in polynomial dynamics may be extended throughout the Julia set of certain rational maps. These new types of rays, which we call internal rays, meet the Julia set in a Cantor set of points, and each of these rays crosses infinitely many other internal rays at many points. We then use this construction to show that there are infinitely many disjoint copies of the Mandelbrot set in the parameter planes for these maps.
It has been known for a long time that for birth-and-death processes started in zero the first passage time of a given level is distributed as a sum of independent exponentially distributed random variables, the parameters of which are the negatives of the eigenvalues of the stopped process. Recently, Diaconis and Miclo have given a probabilistic proof of this fact by constructing a coupling between a general birth-and-death process and a process whose birth rates are the negatives of the eigenvalues,...
When invading the tissue, malignant tumour cells (i.e. cancer cells) need to detach from
neighbouring cells, degrade the basement membrane, and migrate through the extracellular
matrix. These processes require loss of cell-cell adhesion and enhancement of cell-matrix
adhesion. In this paper we present a mathematical model of an intracellular pathway for
the interactions between a cancer cell and the extracellular matrix. Cancer cells use
similar...
The theory of iterated monodromy groups was developed by Nekrashevych [9]. It is a wonderful example of application of group theory in dynamical systems and, in particular, in holomorphic dynamics. Iterated monodromy groups encode in a computationally efficient way combinatorial information about any dynamical system induced by a post-critically finite branched covering. Their power was illustrated by a solution of the Hubbard Twisted Rabbit Problem given by Bartholdi and Nekrashevych [2].These...
We show that every global viscosity solution of the Hamilton-Jacobi equation associated with a convex and superlinear Hamiltonian on the cotangent bundle of a closed manifold is necessarily invariant under the identity component of the group of symmetries of the Hamiltonian (we prove that this group is a compact Lie group). In particular, every Lagrangian section invariant under the Hamiltonian flow is also invariant under this group.
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