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We present the development of a new theory of the pitchfork bifurcation, which removes the perspective of the third derivative and a requirement of symmetry. [ABSTRACT FROM AUTHOR]

This article presents a tribute to L. P. Shilnikov from a fan, a scientist from the next generation who owes much of his career to the results pioneered by this truly great man. The discourse is based on a mixture of personal recollections and conversations with a number of key individuals. [ABSTRACT FROM AUTHOR]

In previous work, with Bartholdi and Schick [1], the authors developed a Hodge-de Rham theory for compact metric spaces, which defined a cohomology of the space at a scale α. Here, in the case of Riemannian manifolds at a small scale, we construct explicit chain maps between the de Rham complex of differential forms and the L2 complex at scale α, which induce isomorphisms on cohomology. We also give estimates that show that on smooth functions, the Laplacian of [1], when appropriately scaled, is a good approximation of the classical Laplacian. [ABSTRACT FROM AUTHOR]

This paper attempts to give an extension of learning theory to a setting where the assumption of i.i.d. data is weakened by keeping the independence but abandoning the identical restriction. We hypothesize that a sequence of examples (xt, yt) in X × Y for t = 1, 2, 3,... is drawn from a probability distribution ρt on X × Y. The marginal probabilities on X are supposed to converge to a limit probability on X. Two main examples for this time process are discussed. The first is a stochastic one which in the special case of a finite space X is defined by a stochastic matrix and more generally by a stochastic kernel. The second is determined by an underlying discrete dynamical system on the space X. Our theoretical treatment requires that this dynamics be hyperbolic (or "Axiom A") which still permits a class of chaotic systems (with Sinai–Ruelle–Bowen attractors). Even in the case of a limit Dirac point probability, one needs the measure theory to be defined using Hölder spaces. Many implications of our work remain unexplored. These include, for example, the relation to Hidden Markov Models, as well as Markov Chain Monte Carlo methods. It seems reasonable that further work should consider the push forward of the process from X × Y by some kind of observable function to a data space. [ABSTRACT FROM AUTHOR]

In this paper, a Z2-equivariant perturbed planar Hamiltonian polynomial vector field of degree 5 is used to show how to control parameter conditions such that this system has more limit cycles and various configurations of these limit cycles. By using the method of detection functions, we prove that under different determined parameter conditions, the given system has at least 20–23 limit cycles having different configurations of the compound eyes. We give a corollary to point out that the assertion is incorrect that this system exists as 29 limit cycles in a reference. [ABSTRACT FROM AUTHOR]