150,352 research outputs found
Karl-Peter Hadeler: His legacy in mathematical biology
No full textKarl-Peter Hadeler is a first-generation pioneer in mathematical biology. His work inspired the contributions to this special issue. In this preface we give a brief biographical sketch of K.P. Hadelers scientific life and highlight his impact to the field
On the Convergence of the Virial Expansion
No full textThe virial expansion appears in statistical mechanics, an area where physics and mathematics intersect. Throughout this thesis we will mostly ignore the physics and mainly focus on the mathematical aspects. This is a deliberate choice, made for two reasons. Firstly, there are several books that explain statistical mechanics in a more rigorous manner than would be possible in this thesis, see for example [Rue69]. Secondly, physics is not that important for our eventual goal, which is finding the region of convergence of the virial expansion. The two main objects of this thesis are (1): deriving the cluster expansion; and (2) firstly explaining how the virial expansion came to be, and then deriving the virial expansion
The critical probability for confetti percolation equals 1/2
No full textIn the confetti percolation model, or two-coloured dead leaves model, radius one disks arrive on the plane according to a space-time Poisson process. Each disk is coloured black with probability p and white with probability 1 − p. In this paper we show that the critical probability for confetti percolation equals 1/2. That is, if p > 1/2 then a.s. there is an unbounded curve in the plane all of whose points are black; while if p ≤ 1/2 then a.s. all connected components of the set of black points are bounded. This answers a question of Benjamini and Schramm [1]. The proof builds on earlier work by Hirsch [7] and makes use of an adaptation of a sharp thresholds result of Bourgain
The hunt for canards in population dynamics: A predator-prey system
No full textEquations with periodic coefficients for singularly perturbed growth can be analysed by using fast and slow timescales which involves slow manifolds, canards and the dynamical exchanges between several slow manifolds. We extend the time-periodic P.F. Verhulst-model to predator-prey interaction where two slow manifolds are present. The fast-slow formulation enables us to obtain a detailed analysis of non-autonomous systems. In the case of sign-positive growth rate, we have the possibility of periodic solutions associated with one of the slow manifolds, also the possibility of extinction of the predator. Under certain conditions, sign-changing growth rates allow for canard periodic solutions that arise from dynamic interaction between slow manifolds
Near-integrability and recurrence in FPU cell-chains
No full textIn a neighborhood of stable equilibrium, we consider the dynamics for at least three degrees-of-freedom (dof) Hamiltonian systems (2 dof systems are not ergodic in this case). A complication is that the recurrence properties depend strongly on the resonances of the corresponding linearized system and on quasi-trapping. In contrast to the classical FPU-chain, the inhomogeneous FPU-chain shows nearly all the principal resonances. Using this fact, we construct a periodic FPU-chain of low dimension, called a FPU-cell. Such a cell can be used as a building block for a chain of FPU-cells, called a cell-chain. Recurrence phenomena depend strongly on the physical assumptions producing specific Hamiltonians; we demonstrate this for the 1:2:51:2:5 resonance, both general and for the FPU case; this resonance shows dynamics on different timescales. In addition we will study the relations and recurrence differences between several FPU-cells and a few cell-chains in the case of the classical near-integrable FPU-cell and of chaotic cells in 3:2:13:2:1 resonance
Hausdorff dimension of the arithmetic sum of self-similar sets
No full textLet β>1. We define a class of similitudes S:=(fi(x)=xβni+ai:ni∈N+,ai∈R). Taking any finite collection of similitudes (fi(x))i=1m from S, it is well known that there is a unique self-similar set K1 satisfying K1=∪i=1mfi(K1). Similarly, another self-similar set K2 can be generated via the finite contractive maps of S. We call K1+K2=(x+y:x∈K1,y∈K2) the arithmetic sum of two self-similar sets. In this paper, we prove that K1+K2 is either a self-similar set or a unique attractor of some infinite iterated function system. Using this result we can calculate the exact Hausdorff dimension of K1+K2 under some conditions, which partially provides the dimensional result of K1+K2 if the IFS's of K1 and K2 fail the irrationality assumption, see Peres and Shmerkin (2009)
Recurrence and Resonance in the Cubic Klein-Gordon Equation
No full textIn a number of models for coupled oscillators and nonlinear wave equations primary resonances dominate the phase-space phenomena. A new feature is that in a Hamiltonian framework, the interaction of primary and higher order resonances is shown to be important and can be signaled by using recurrence properties. The interaction may involve embedded double resonance. We will demonstrate these phenomena for the cubic Klein-Gordon equation on a square with Dirichlet boundary conditions using normal form techniques. The results are qualitatively and quantitatively very different from the one-dimensional spatial case
International Conference for Mathematical Modeling and Optimization in Mechanics, 6-7 March, 2014, Jyväskylä, Finland : honor of the 70th anniversary of Prof. Nikolay Banichuk
No full textThis book of abstracts presents materials of the International Conference for Mathematical Modeling
and Optimization in Mechanics (MMOM 2014) 6-7 March 2014, Jyväskylä, Finland. This event is
dedicated to Professor Nikolay Banichuk in occasion of his 70th anniversary.
It is aimed to present the latest results of leading scientists in mathematical modeling, numerical
analysis, and optimization theory and to discuss the state of the art and open problems in the field.
The book is divided in five sections:
1. Mathematical Modelling of Complex Systems
2. Stability Analysis and Vibration
3. Optimization
4. Methods of Numerical Analysis
5. Shape Optimizationunknown accessibilityei tietoa saavutettavuudest
Perturbations of Superintegrable Systems
No full textA superintegrable system has more integrals of motion than degrees d of freedom. The quasi-periodic motions then spin around tori of dimension n < d. Already under integrable perturbations almost all n-tori will break up; in the non-degenerate case the resulting d-tori have n fast and d −n slow frequencies. Such d-parameter families of d-tori do survive Hamiltonian perturbations as Cantor families of d-tori. A perturbation of a superintegrable system that admits a better approximation by a non-degenerate integrable perturbation of the superintegrable system is said to remove the degeneracy. In the minimal case d = n+1 this can be achieved by means of averaging, but the more integrals of motion the superintegrable system admits the more difficult becomes the perturbation analysis
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