Geometric quantization for proper moment maps

We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a non-compact symplectic manifold such that the associated moment map is proper. In particular, we give a solution to a conjecture of Michèle Vergne. To cite this article: X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Geometric realization of Adams maps

Let G/P be a homogenous space with G a compact connected Lie group and P a connected subgroup of G of equal rank. As the rational cohomology ring of G/P is concentrated in even dimensions, for an integer k we can define the Adams map of type k to be l _k : H*(G/P, ℚ) → H*(G/P, ℚ), l _k (u) = k ⁱ u, u ∈ H ²ⁱ (G/P, ℚ). We show that if k is prime to the order of the Weyl group of G, then l _k can be induced by a self map of G/P. We also obtain results which imply the condition that k is prime to the order of the Weyl group of G is necessary.     

Geometric structure in stochastic approximation

1. IntroductionLet f: Re -- R be a differelltiable fUnction. f reaChes its extremes on the setJ = {x E R"lfx(x) = 0}, (1.1)where,x(X) = (V,..., V)". (1.2)If jx can be observed exactly at any x e R", then there are various numerical methods toconstruct {xh}, xk E Re such that the distance d(xk, J) between uk and J tends to zero ask -- co. However, in many application problems jx can only be observed with noise, i.e.,the observation at time k 1 isYk 1 = fi(~k) (k 1, (1'3)where xk is …     

The runge approximation problem for holomorphic maps into grassmannians

The author was supported by an NSF grant     

Geometric quantization for proper moment maps: the Vergne conjecture

We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.     

Geometric and semigeometric approximation of spectral projections

It is proved that for a convex reflecting scatterer the support function is uniquely determined by the scattering amplitude (in particular, by the backscattering amplitude). The support function determines the position, orientation and the shape of the scatterer uniquely. Stability of the solution is discussed.     

High order approximation of implicitly defined maps

Summary Approximations for the function implicitly defined by (u)=(u, (u)) are obtained via the iterative scheme _n(u)=(u, _n–1(u)). In this paper the uniform convergence of high order derivatives of _n to the corresponding derivatives of is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Linear approximation of approximately biorthogonality preserving maps

We investigate linear properties of mappings from a bounded domain of an n-dimensional normed space into another n-dimensional normed space such that the image of some almost biorthogonal system is almost biorthogonal. In this way we generalize a result of the author on stability of orthogonality in Euclidean spaces.     

Geometric applications of Chernoff-type estimates and a ZigZag approximation for balls

In this paper we show that the euclidean ball of radius in can be approximated up to , in the Hausdorff distance, by a set defined by linear inequalities. We call this set a ZigZag set, and it is defined to be all points in space satisfying or more of the inequalities. The constant we get is , where is some universal constant. This should be compared with the result of Barron and Cheang (2000), who obtained . The main ingredient in our proof is the use of Chernoff's inequality in a geometric context. After proving the theorem, we describe several other results which can be obtained using similar methods.

     

On connections between some selection and approximation results for multivalued maps

We study the general conditions on a family of values of a multivalued map under which the existence of single-valued approximations may be derived from continuous selection results.     

Metrical diophantine approximation for continued fraction like maps of the interval

We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.

     

GENERALIZED I-NONEXPANSIVE MAPS AND INVARIANT APPROXIMATION RESULTS IN p-NORMED SPACES

We extend the concept of R-subcommuting maps due to Shahzad[17,18] to the case of non-starshaped domain and obtain a common fixed point result for this class of maps on non-starshaped domain in the setup of p-normed spaces. As applications, we establish noncommutative versions of various best approximation results for generalized I-nonexpansive maps on non-starshaped domain. Our results unify and extend that of Al-Thagafi, Dotson, Habiniak,Jungck and Sessa, Latif, Sahab, Khan and Sessa and Shahzad.     

Fourth order approximation of harmonic maps from surfaces

Let be a compact Riemannian surface and let be a compact Riemannian manifold, both without boundary, and assume that N is isometrically embedded into some ℝ ^l . We consider a sequence of critical points of the functional with uniformly bounded energy. We show that this sequence converges weakly in and strongly away from finitely many points to a smooth harmonic map. One can perform a blow-up to show that there separate at most finitely many non-trivial harmonic two-spheres at these finitely many points. Finally we prove the so called energy identity for this approximation in the case that ↪ ℝ ^l . Mathematics Subject Classification (2000) 58E20, 35J60, 53C43     

Enumeration of maps regardless of genus: Geometric approach

We use the conceptual idea of “maps on orbifolds” and the theory of the non-Euclidean crystallographic groups (NEC groups) to enumerate rooted and unrooted maps (both sensed and unsensed) on surfaces regardless of genus. As a consequence we deduce a formula for the number of chiral pairs of maps. The enumeration principle used in this paper is due to Mednykh (2006) [15], it counts the number of conjugacy classes of subgroups in NEC groups which are in one-to-one correspondence with unrooted (sensed or unsensed) maps.     

Polynomial approximation of bilinear Diffie–Hellman maps

The problem of computing bilinear Diffie–Hellman maps is considered. It is shown that the problem of computing the map is equivalent to computing a diagonal version of it. Various lower bounds on the degree of any polynomial that interpolates this diagonal version of the map are found that shows that such an interpolation will involve a polynomial of large degree, relative to the size of the set on which it interpolates.     

Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Let be a -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ). Let be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset of and let be arbitrary. Then the iteration sequence defined by , converges strongly to a fixed point of , provided that and have certain properties. If is a Hilbert space, then converges strongly to the unique fixed point of closest to .