Hierarchy of Hamilton equations on Banach Lie-Poisson spaces related to restricted Grassmannian

We consider the Banach Lie-Poisson space and its complexification , where the first one of them contains the restricted Grassmannian Grres as a symplectic leaf. Using the Magri method we define an involutive family of Hamiltonians on these Banach Lie-Poisson spaces. The hierarchy of Hamilton equations given by these Hamiltonians is investigated. The operator equations of Ricatti-type are included in this hierarchy. For a few particular cases we give the explicit solutions.     

Induced and coinduced Banach Lie–Poisson spaces and integrability

The Poisson induction and coinduction procedures are used to construct Banach Lie–Poisson spaces as well as related systems of integrals in involution. This general method applied to the Banach Lie–Poisson space of trace class operators leads to infinite Hamiltonian systems of k-diagonal trace class operators which have infinitely many integrals. The bidiagonal case is investigated in detail.     

Matrix coefficients and coadjoint orbits of compact Lie groups

Let be a compact Lie group. We use Weyl functional calculus (Anderson, 1969) and symplectic convexity theorems to determine the support and singular support of the operator-valued Fourier transform of the product of the -function and the pull-back of an arbitrary unitary irreducible representation of to the Lie algebra, strengthening and generalizing the results of Cazzaniga, 1992. We obtain as a consequence a new demonstration of the Kirillov correspondence for compact Lie groups.

     

On the classification of the coadjoint orbits of the Sobolev Bott-Virasoro group

The purpose of this paper is to give the classification of the Bott-Virasoro coadjoint orbits, with nonzero central charge, in the functional analytic setting of smooth Hilbert manifolds. The central object of the paper is thus the completion of the Bott-Virasoro group with respect to a Sobolev topology, giving rise to a smooth Hilbert manifold and topological group, called the Sobolev Bott-Virasoro group. As a consequence of this approach, analytic and geometric properties of the coadjoint orbits are studied.     

A bijection between the adjoint and coadjoint orbits of a semidirect product

We prove that there exists a geometric bijection between the sets of adjoint and coadjoint orbits of a semidirect product, provided a similar bijection holds for particular subgroups. We also show that under certain conditions the homotopy types of any two orbits in bijection with each other are the same. We apply our theory to the examples of the affine group and the Poincaré group, and discuss the limitations and extent of this result to other groups.     

Deformations of vector fields and canonical coordinates on coadjoint orbits

We establish the concurrence of the existence of a canonical linear embedding, enabling us to pass to the Darboux coordinates on the coadjoint orbits, and the existence of a polarization of a certain linear functional. In a corollary to the main theorem we prove that every polarization is normal, which means that it satisfies the Pukansky condition.     

The existence of Silnikov''''s orbits in one coupled Duffing equation

The existence of Silnikov's orbits in one coupled Duffing equation is discussed by using the fiber structure of invariant manifold and high-dimensional Melnikov's method. Example and numerical simulation results are also given to demonstrate the theoretical analysis.     

The Deodhar decomposition of the Grassmannian and the regularity of KP solitons

Given a point A$A$ in the real Grassmannian, it is well-known that one can construct a soliton solution _uA(x,y,t)$u_A(x,y,t)$ to the KP equation. The contour plot   of such a solution provides a tropical approximation to the solution when the variables x$x$, y$y$, and t$t$ are considered on a large scale and the time t$t$ is fixed. In this paper we use several decompositions of the Grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions. First we use the positroid stratification   of the real Grassmannian in order to characterize the unbounded line-solitons in the contour plots at y?0$y?0$ and y?0$y?0$. Next we use the Deodhar decomposition   of the Grassmannian–a refinement of the positroid stratification–to study contour plots at t?0$t?0$. More specifically, we index the components of the Deodhar decomposition of the Grassmannian by certain tableaux which we call Go-diagrams  , and then use these Go-diagrams to characterize the contour plots of solitons solutions when t?0$t?0$. Finally we use these results to show that a soliton solution _uA(x,y,t)$u_A(x,y,t)$ is regular for all times t$t$ if and only if A$A$ comes from the totally non-negative part of the Grassmannian.     

Hyperkähler structures and infinite-dimensional Grassmannians

In this paper, we describe an example of a hyperkähler quotient of a Banach manifold by a Banach Lie group. Although the initial manifold is not diffeomorphic to a Hilbert manifold (not even to a manifold modelled on a reflexive Banach space), the quotient space obtained is a Hilbert manifold, which can be furthermore identified either with the cotangent space of a connected component (j∈Z), of the restricted Grassmannian or with a natural complexification of this connected component, thus proving that these two manifolds are isomorphic hyperkähler manifolds. Moreover, Kähler potentials associated with the natural complex structure of the cotangent space of and with the natural complex structure of the complexification of are computed using Kostant-Souriau's theory of prequantization.     

Matching polytopes,toric geometry,and the totally non-negative Grassmannian

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Gr _k,n )_≥0. This is a cell complex whose cells Δ _G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Δ _G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We use the data of P(G) to define an associated toric variety X _G . We use our technology to prove that the cell decomposition of (Gr _k,n )_≥0 is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr _k,n )_≥0 is 1. Alexander Postnikov was supported in part by NSF CAREER Award DMS-0504629. David Speyer was supported by a research fellowship from the Clay Mathematics Institute. Lauren Williams was supported in part by the NSF.     

An implicit function theorem for Banach spaces and some applications

We prove a generalized implicit function theorem for Banach spaces, without the usual assumption that the subspaces involved being complemented. Then we apply it to the problem of parametrization of fibers of differentiable maps, the Lie subgroup problem for Banach–Lie groups, as well as Weil’s local rigidity for homomorphisms from finitely generated groups to Banach–Lie groups.      

Herz spaces and restricted summability of Fourier transforms and Fourier series

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1,1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that over a cone-like set a.e. for all f∈L₁(Rd). Moreover, converges to f(x) over a cone-like set at each Lebesgue point of f∈L₁(Rd) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.     

Complementably universal Banach spaces, II

The two main results are:

A.

If a Banach space X is complementably universal for all subspaces of c₀ which have the bounded approximation property, then X^∗ is non-separable (and hence X does not embed into c₀).

B.

There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.

Theorem B solves a problem that dates from the 1970s.     

Metric generalized inverse for linear manifolds and extremal solutions of linear inclusion in Banach spaces

Let X,Y be Banach spaces and M a linear manifold in X×Y={{x,y}∣x∈X,y∈Y}. The central problem which motivates many of the concepts and results of this paper is the problem of characterization and construction of all extremal solutions of a linear inclusion y∈M(x). First of all, concept of metric operator parts and metric generalized inverses for linear manifolds are introduced and investigated, and then, characterizations of the set of all extremal or least extremal solutions in terms of metric operator parts and metric generalized inverses of linear manifolds are given by the methods of geometry of Banach spaces. The principal tool in this paper is the generalized orthogonal decomposition theorem in Banach spaces.     

Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {U_t: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {U_tn(μ₁ * μ₂*…*μ_n)*δ_xn}, where {μ_n}Q, {x_n}X and measures U_tnμ_j (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes L_m( ) as follows: L₀( )= ( (X); ), L_m( )= (L_m−1( ); ) for m=1, 2,… and L_∞( )=_m=0^∞L_m( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from L_m( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.     

Toughness,binding number and restricted matching extension in a graph

A connected graph $G$ with at least $2m+2n+2$ vertices is said to satisfy the property $E(m,n)$ if $G$ contains a perfect matching and for any two sets of independent edges $M$ and $N$ with $\left|M\right|=m$ and $\left|N\right|=n$ with $M\cap N=?$, there is a perfect matching $F$ in $G$ such that $M?F$ and $N\cap F=?$. In particular, if $G$ is $E(m,0)$, we say that $G$ is $m$-extendable. One of the authors has proved that every $m$-tough graph of even order at least $2m+2$ is $m$-extendable (Plummer, 1988). Chen (1995) and Robertshaw and Woodall (2002) gave sufficient conditions on binding number for $m$-extendability. In this paper, we extend these results and give lower bounds on toughness and binding number which guarantee $E(m,n)$.     

The polynomial degree of the Grassmannian G(1,n,q) of lines in finite projective space PG(n, q)

Let G: = G(1,n,q) denote the Grassmannian of lines in PG(n,q), embedded as a point-set in PG(N, q) with For n = 2 or 3 the characteristic function of the complement of G is contained in the linear code generated by characteristic functions of complements of n-flats in PG(N, q). In this paper we prove this to be true for all cases (n, q) with q = 2 and we conjecture this to be true for all remaining cases (n, q). We show that the exact polynomial degree of is for δ: = δ(n, q) = 0 or 1, and that the possibility δ = 1 is ruled out if the above conjecture is true. The result deg( for the binary cases (n,2) can be used to construct quantum codes by intersecting G with subspaces of dimension at least