Using the Perron-Frobenius operator we establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give an extensive specific example using the tent map.

Almost every, essentially: Given a Lebesgue measure space (X,B, μ), a property P (x) predicated of elements of X is said to hold for almost every x ∈ X, if the set X \ {x : P (x) holds} has zero measure. Two sets A,B ∈ B are essentially disjoint if μ(A ∩B) = 0. Conservative system: Is an infinite measure preserving system such that for no set A ∈ B with positive measure are A,T−1A,T−2A, . . . p...

For a positive integer n we introduce quadratic Lie algebras trn qtr n and discrete groups Trn, QTr n naturally associated with the classical and quantum Yang-Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras trn, qtr n are Koszul, and compute their Hilbert series. We also compute the cohomology rings of these Lie algebras (which by Koszulity are...

For a positive integer n we introduce quadratic Lie algebras trn qtr n and discrete groups Trn, QTr n naturally associated with the classical and quantum Yang-Baxter equation, respectively. We prove that the universal enveloping algebras of the Lie algebras trn, qtr n are Koszul, and compute their Hilbert series. We also compute the cohomology rings of these Lie algebras (which by Koszulity are...

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer-Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann-Schwarzbach. This approach provides a definition of an L∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L∞-algebra structure as...

In this paper we introduce derivations in Krasner hyperrings and derive some basic properties of derivations. We also prove that for a strongly differential hyperring $R$ and for any strongly differential hyperideal $I$ of $R,$ the factor hyperring $R/I$ is a strongly differential hyperring. Further we prove that a map $d: R rightarrow R$ is a derivation of a hyperring $R$ if and only if the in...

A new zero-crossing edge detection method is described. Similar to Laplace of a Gaussian (LoG) algorithm, a 7x7 point spread function (PSF) is convolved with the original image to generate a smoothed image. Instead of locating edge position from the smoothed image, as in the case of LoG, we use the smoothed image as a reference to convert the original image into a three-value edge polarity map ...

We prove that a Lebesgue measure-preserving linked-twist map defined in the plane is metrically isomorphic to a Bernoulli shift (and thus strongly mixing). This is the first such result for an explicitly defined linked-twist map on a manifold other than the two-torus. Our work builds on that of Wojtkowski (1980) who established an ergodic partition for this example using an invariant cone-field...

We consider a connected symplectic manifold M acted on by a connected Lie group G in a Hamiltonian fashion. If G is compact we study the smooth function f =‖ μ ‖. We prove that if a point x ∈ M realizes a local maximum of the squared moment map ‖ μ ‖ then the orbit Gx is symplectic and Gμ(μ(x)) is G-equivariantly symplectomorphic to a product of a flag manifold and a symplectic manifold which i...

The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical observables are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest's theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the Ehrenfest group. The underlying P...