On the Exponential Bound of the Cutoff Resolvent
A simpler proof of a result of Burq [1] is presented.
On the finite blocking property
A planar polygonal billiard is said to have the finite blocking property if for every pair of points in there exists a finite number of “blocking” points such that every billiard trajectory from to meets one of the ’s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces....
On the forcing relation for surface homeomorphisms
On the formal first cocycle equation for iteration groups of type II
Let x be an indeterminate over ℂ. We investigate solutions αn : ℂ → ℂ, n ≥ 0, of the first cocycle equation in ℂ [[x]], the ring of formal power series over ℂ, where (F(s,x))s ∈ ℂ is an iteration group of type II, i.e. it is a solution of the translation equation of the form F(s,x) ≡ x + ck(s)xk mod xk+1, where k ≥ 2 and ck ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions αn(s) of are polynomials in ck(s).It is possible to replace...
On the fractality of the biological tree-like structures.
On the -entropy and its Hudetz correction
The Hudetz correction of the fuzzy entropy is applied to the -entropy. The new invariant is expressed by the Hudetz correction of fuzzy entropy.
On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory
A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N-wave equation are reviewed. New results on explicit construction of the wave functions for radial...
On the generalization of the Stein-Weiss theorem for the ergodic Hilbert transform
The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized to the ergodic Hilbert transform.
On the generalized Maxwell-Bloch equations.
On the genesis of symbolic dynamics as we know it
We trace the beginning of symbolic dynamics-the study of the shift dynamical system-as it arose from the use of coding to study recurrence and transitivity of geodesics. It is our assertion that neither Hadamard's 1898 paper, nor the Morse-Hedlund papers of 1938 and 1940, which are normally cited as the first instances of symbolic dynamics, truly present the abstract point of view associated with the subject today. Based in part on the evidence of a 1941 letter from Hedlund to Morse, we place the...
On the geodesics flow of tori without conjugate points.
On the geometric structure of the set of solutions of Einstein equations [Book]
On the global asymptotic stability problem and the Jacobian conjecture
On the global maximum of the solution to a stochastic heat equation with compact-support initial data
Consider a stochastic heat equation ∂tu=κ ∂xx2u+σ(u)ẇ for a space–time white noise ẇ and a constant κ>0. Under some suitable conditions on the initial function u0 and σ, we show that the quantities lim sup t→∞t−1sup x∈Rln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup x∈R|ut(x)|2) are equal, as well as bounded away from zero and infinity by explicit multiples of 1/κ. Our proof works by demonstrating quantitatively that the peaks of the stochastic process x↦ut(x) are highly concentrated...
On the global stable manifold
We give an alternative proof of the stable manifold theorem as an application of the (right and left) inverse mapping theorem on a space of sequences. We investigate the diffeomorphism class of the global stable manifold, a problem which in the general Banach setting gives rise to subtle questions about the possibility of extending germs of diffeomorphisms.
On the group of isometries on a locally compact metric space.
On the group of real analytic diffeomorphisms
The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the -dimensional torus, its identity component is a simple group. For fibered manifolds, for manifolds admitting special semi-free actions and for 2- or 3-dimensional manifolds with nontrivial actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.
On the growth of averaged Weyl sums for rigid rotations
On the growth of Sobolev norms for the cubic Szegő equation
We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed...
On the Hausdorff dimension of a family of self-similar sets with complicated overlaps
We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) {γx,λx,λx+1}. Since two maps have the same fixed point, there are very complicated overlaps, and it is not possible to directly apply known techniques. We give a formula for the Hausdorff dimension of the attractor for Lebesgue almost all parameters (γ,λ), γ < λ. This result only holds for almost all parameters: we find a dense set of parameters (γ,λ) for which the Hausdorff dimension...