We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex critical eigenvalues together with either an eigenvalue −1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher order terms and these are investigated here. We (re)derive the normal forms and reduce the...

Free coherent states for a system with two degrees of freedom is defined. A linear map of the space of free coherent states to the space of distributions on 2-adic disc is constructed. 1 Construction of free (or Boltzmannian) coherent states Free (or Boltzmannian) Fock space has been considered in some recent works on quantum chromodynamics [1], [2], [3] and noncommutative probability [4], [5]....

The details and functional significance of the intrinsic horizontal connections between neurons in the motor cortex (MCx) remain to be clarified. To further elucidate the nature of this intracortical connectivity pattern, experiments were done on the MCx of three cats. The anterograde tracer biocytin was ejected iontophoretically in layers II, III, and V. Some 30-50 neurons within a radius of a...

In 1973/74 Bennett and (independently) Carl proved that for 1 ≤ u ≤ 2 the identity map id: `u ↪→ `2 is absolutely (u, 1)-summing, i. e., for every unconditionally summable sequence (xn) in `u the scalar sequence (‖xn‖`2 ) is contained in `u, which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a 2-concave symmetric Banach s...

We consider the Dirichlet-to-Neumann map associated to the Schrö– dinger equation with a potential on a bounded domain Ω ⊂ Rn, n ≥ 3. We show that the integral of the potential over a two-plane Π is determined by the values of the integral kernel of the Dirichlet-to-Neumann map on any open subset U ⊂ ∂Ω which contains Π ∩ ∂Ω. 0 Introduction For Ω a bounded domain in R with Lipschitz boundary, ∂...

For projectivizations of rational maps Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rationalmaps of projective spaces (Theorem6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps w...

Certain simple models of structural glasses ref. [1, 2] map onto random matrix models. These random matrix models have gaps in their eigenvalue distribution. It turns out that matrix models with gaps in their eigenvalue distributions have the unusual property of multiple solutions or minimas of the free energy at the same point in phase space. I present evidence for the presence of multiple sol...

We formulate and study the set of coupled nonlinear differential equations which define a series of shape invariant potentials which repeats after a cycle of p iterations. These cyclic shape invariant potentials enlarge the limited reservoir of known analytically solvable quantum mechanical eigenvalue problems. At large values of x, cyclic superpotentials are found to have a linear harmonic osc...

A theorem of J. Hersch (1970) states that for any smooth metric on S2, with total area equal to 4π, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard S1-action on S2, one can restrict the Laplace operator to the subspace of S1-invariant functions and c...

We present two theoretical results for the linear response eigenvalue problem. The first result is a minimization principle for the sum of the smallest eigenvalues with the positive sign. The second result is Cauchy-like interlacing inequalities. Although the linear response eigenvalue problem is a nonsymmetric eigenvalue problem, these results mirror the well-known trace minimization principle...