Exponentially small splitting of separatrices in a weakly hyperbolic case

We validate the Poincaré-Melnikov method in the singular case of high-frequency periodic perturbations of the Hamiltonian h₀(x,y)=(1/2)y²-x³+x⁴ under appropriate conditions, which among other things, imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one. The splitting is exponentially small.     

Exponentially small splitting of separatrices for area-preserving maps

We discuss some recent lower and upper bounds for the splitting of separatrices for close-to-identity area-preserving maps.     

Smooth Schur factorizations in the continuation of separatrices

Numerical computation of separatrices as general connecting orbits in dynamical systems is performed, and their continuation as problem parameters vary is approached via a direct application of smooth block Schur factorizations of Jacobians and monodromy matrix functions. Several numerical examples are presented to illustrate the effectiveness of the algorithms used.     

Coverings of laura algebras: The standard case

In this paper, we study the covering theory of laura algebras. We prove that if a connected laura algebra is standard (that is, has a standard connecting component), then it has Galois coverings associated to the coverings of the connecting component. As a consequence, the first Hochschild cohomology group of a standard laura algebra vanishes if and only if it has no proper Galois coverings.     

On the second adjunction mapping. The case of a 1-dimensional image

Let be a very ample line bundle on an -dimensional projective manifold , i.e., assume that for some embedding . In this article, a study is made of the meromorphic map, , associated to in the case when the Kodaira dimension of is , and has a -dimensional image. Assume for simplicity that . The first main result of the paper shows that is a morphism if either or . The second main result of this paper shows that if , then the genus, , of a fiber, , of the map induced by on hyperplane sections is . Moreover, if then , a connected component of a general fiber of is either a surface or the blowing up at one point of a surface, and . Finally the structure of the finite to one part of the Remmert-Stein factorization of is worked out.

     

Parallel calculation of a linear mapping on a computer network

We are interested in the parallel computation of a linear mapping of n real variables by a network of computers with restricted means of communication between them and without any common memory. Let $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ denote the algebra of n×n real matrices, and let G be the graph associated with a binary, reflexive and symmetric relation R over {1,2, …,n}. We define

$\text{A}{}_{\mathrm{R}}\text{= {A?M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{):a}{}_{\mathrm{ij}}\text{≠ 0}\text{implies}\text{iRj}}$

A matrix $\text{M∈M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ is said to be realizable on G if it can be expressed as a product of elements of A_R. Therefore, every matrix of $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ is realizable on G if and only if A_R generates $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$. We show that A_R generates $\text{M}{}_{\mathrm{n\times n}}\text{(}\text{R}\text{)}$ if and only if G is connected.     

The far field excited by a moving oscillating source in the resonance case

Wave fields excited by moving oscillating sources are considered. In the resonance case, the asymptotics of the far field for large values of t is found. Explicit expressions for the growing components of the field are provided.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 61–70.     

The moment mapping for unitary representations

For any unitary representation of an arbitrary Lie group I construct a moment mapping from the space of smooth vectors of the representation into the dual of the Lie algebra. This moment mapping is equivariant and smooth. For the space of analytic vectors the same construction is possible and leads to a real analytic moment mapping.     

The resonance theorem for subspaces

Under some additional assumptions on an unbounded sequence of operators and the geometry of the spaces, it is shown that, in the classical Banach-Steinhaus resonance theorem, the set of divergence contains an infinite-dimensional space, excluding zero.     

Cones with a mapping cone symmetry in the finite-dimensional case

In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamio?kowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.