CANONICALIZATION OF CONSTRAINED HAMILTONIAN EQUATIONS IN A SINGULAR SYSTEM

In this paper, the canonicalization of constrained Hamiltonian system is discussed. Because the constrained Hamiltonian equations are non-canonical, they lead to many limitations in the research. For this purpose, variable transformation is constructed that satisfies the condition of canonical equation, and the new variables can be obtained by a series of derivations. Finally, two examples are given to illustrate the applications of the result.

Adaptive dynamics of unstable cancer populations: the canonical equation

In most instances of tumour development, genetic instability plays a role in allowing cancer cell populations to respond to selection barriers, such as physical constraints or immune responses, and rapidly adapt to an always changing environment. Modelling instability is a nontrivial task, since by definition evolving changing instability leads to changes in the underlying landscape. In this paper we explore mathematically a simple version of unstable tumor progression using the formalism of Adaptive Dynamics (AD) where selection and mutation are explicitly coupled. Using a set of basic fitness landscapes, the so called canonical equation for the evolution of genetic instability on a minimal scenario associated to a population of unstable cells is derived. The implications and potential extensions of this model are discussed.

The Branching Bifurcation of Adaptive Dynamics

We unfold the bifurcation involving the loss of evolutionary stability of an equilibrium of the canonical equation of Adaptive Dynamics (AD). The equation deterministically describes the expected long-term evolution of inheritable traits — phenotypes or strategies — of coevolving populations, in the limit of rare and small mutations. In the vicinity of a stable equilibrium of the AD canonical equation, a mutant type can invade and coexist with the present — resident — types, whereas the fittest always win far from equilibrium. After coexistence, residents and mutants effectively diversify, according to the enlarged canonical equation, only if natural selection favors outer rather than intermediate traits — the equilibrium being evolutionarily unstable, rather than stable. Though the conditions for evolutionary branching — the joint effect of resident-mutant coexistence and evolutionary instability — have been known for long, the unfolding of the bifurcation has remained a missing tile of AD, the reason being related to the nonsmoothness of the mutant invasion fitness after branching. In this paper, we develop a methodology that allows the approximation of the invasion fitness after branching in terms of the expansion of the (smooth) fitness before branching. We then derive a canonical model for the branching bifurcation and perform its unfolding around the loss of evolutionary stability. We cast our analysis in the simplest (but classical) setting of asexual, unstructured populations living in an isolated, homogeneous, and constant abiotic environment; individual traits are one-dimensional; intra- as well as inter-specific ecological interactions are described in the vicinity of a stationary regime.

Canonical equation K61 for random non-Hermitian matrices A + B(U + γH)C

In this paper we apply the REFORM method for the deduction of the system of canonical equations

30 years of General Statistical Analysis and canonical equation K60 for Hermitian matrices (A + BUC)(A + BUC)*, where U is a random unitary matrix

In this paper we apply the REFORM method for the deduction of the system of canonical equations