For an arbitrary subset I of IR and for a function f defined on I, the number of zeros of f on I will be denoted by ZI(f) . In this paper we attempt to characterize all linear transformations T taking a linear subspace W of C(I) into functions defined on J (I, J ⊆ IR) such that ZI(f) = ZJ (Tf) for all f ∈ W .

We study the critical behavior of period doublings in N symmetrically coupled area-preserving maps for many-coupled cases withN > 3. It is found that the critical scaling behaviors depend on the range of coupling interaction. In the extreme long-range case of global coupling, in which each area-preserving map is coupled to all the other area-preserving maps with equal strength, there exist an i...

When Einstein was thinking about the theory of general relativity based on the elimination of especial relativity constraints (especially the geometric relationship of space and time), he understood the first limitation of especial relativity is ignoring changes over time. Because in especial relativity, only the curvature of the space was considered. Therefore, tensor calculations should be to...

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...

Operads are a specific tool for encoding type of algebras. For instance there are operads encoding associative algebras, commutative and associative algebras, Lie algebras, pre-Lie algebras, dendriform algebras, Poisson algebras and so on. A usual way of describing a type of algebras is by giving the generating operations and the relations among them. For instance a Lie algebra L is a vector sp...

We construct a family of integrable volume-preserving maps in R with a bidimensional heteroclinic connection of spherical shape between two fixed points of saddle-focus type. In other contexts, such structures are called Hill’s spherical vortices or spheromaks. We study the splitting of the separatrix under volume-preserving perturbations using a discrete version of the Melnikov method. Firstly...

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We consider the effect of noise on the dynamics generated by volume-preserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of various properties of the map and it...