Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures

By establishing the intrinsic super-Poincar'e inequality,some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive.These conditions,as well as the resulting uniform upper bounds on the intrinsic heat kernels,are sharp for some concrete examples.     

ON NONLINEAR DIFFERENTIAL GALOIS THEORY

This Is an accom ofa work In course ofprogress．The aim Is the following:     

Cyclic Group Actions on 4-Manifolds

Let X be a closed, oriented Riemannian 4-manifold. Suppose that a cyclic group Z( _p (p is prime) acts on X by an orientation preserving isometry with an embedded Riemann surface as fixed point set. We study the representation of Z _p on the Spin^c-bundles and the Z _p-invariant moduli space of the solutions of the Seiberg–Witten equations for a Spin^c-structure X. When the Z _p action on the determinant bundle det L acts non-trivially on the restriction L| over the fixed point set , we consider -twisted solutions of the Seiberg-Witten equations over a Spin^c-structure ' on the quotient manifold X/Z _p X', (0,1). We relate the Z _p -invariant moduli space for the Spin^c-structure on X and the -twisted moduli space for the Spin^c-structure on X'. From this we induce a one-to-one correspondence between these moduli spaces and calculate the dimension of the -twisted moduli space. When Z _p acts trivially on L|, we prove that there is a one-to-one correspondence between the Z _p -invariant moduli space M( ^Zp and the moduli space M (") where ' is a Spin^c-structure on X' associated to the quotient bundle L/Z _p X'. vskip0pt When p = 2, we apply the above constructions to a Kahler surface X with b ₂ ⁺ (X) > 3 and H ₂(X;Z) has no 2-torsion on which an anti-holomorphic involution acts with fixed point set , a Lagrangian surface with genus greater than 0 and []2H ₂(H ;Z). If _K ^X 2 > 0 or K _X ² = 0 and the genus g()> 1, we have a vanishing theorem for Seiberg–Witten invariant of the quotient manifold X'. When K _X ² = 0 and the genus g()= 1, if there is a Z ²-equivariant Spin^c-structure on X whose virtual dimension of the Seiberg–Witten moduli space is zero then there is a Spin^c-structure " on X' such that the Seiberg-Witten invariant is ±1.     

Newton's method on Riemannian manifolds and a geometric model for the human spine

To study a geometric model of the human spine we are led tofinding a constrained minimum of a real valued function definedon a product of special orthogonal groups. To take advantgeof its Lie group structure we consider Newton's method on thismanifold. Comparisons between measured spines and computed spinesshow the pertinence of this approach.     

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then      

approximations of inertial manifolds via finite differences

We construct an inertial manifold for the evolution equation as a limit of the inertial manifolds for the difference approximations of the Trotter-Kato type and show that this limit is taken in a topology.

     

Estimation of vector ARMAX models

The asymptotic properties of maximum likelihood estimates of a vector ARMAX system are considered under general conditions, relating to the nature of the exogenous variables and the innovation sequence and to the form of the parameterization of the rational transfer functions, from exogenous variables and innovations to the output vector. The exogenous variables are assumed to be such that the sample serial covariances converge to limits. The innovations are assumed to be martingale differences and to be nondeterministic in a fairly weak sense. Stronger conditions ensure that the asymptotic distribution of the estimates has the same covariance matrix as for Gaussian innovations but these stronger conditions are somewhat implausible. With each ARMAX structure may be associated an integer (the McMillan degree) and all structures for a given value of this integer may be topologised as an analytic manifold. Other parameterizations and topologisations of spaces of structures as analytic manifolds may also be considered and the presentation is sufficiently general to cover a wide range of these. Greater generality is also achieved by allowing for general forms of constraints.     

The Singularity of Kontsevich’s Solution for QH<Superscript>*</Superscript>(CP<Superscript>2</Superscript>)

In this paper we study the nature of the singularity of the Kontsevichs solution of the WDVV equations of associativity. We prove that it corresponds to a singularity in the change of two coordinates systems of the Frobenius manifold given by the quantum cohomology of CP². Mathematics Subject Classifications (2000) 53C99, 32D99, 14N35, 88A99.