Slow Diffeomorphisms of a Manifold with Two Dimensions Torus Action

The uniform norm of the differential of the n-th iteration of a diffeomorphism is called the growth sequence of the diffeomorphism. In this paper we show that there is no lower universal growth bound for volume preserving diffeomorphisms on manifolds with an effective two dimensions torus action by constructing a set of volume-preserving diffeomorphisms with arbitrarily slow growth.Comment: 12 p

Kick stability in groups and dynamical systems

We consider a general construction of ``kicked systems''. Let G be a group of measure preserving transformations of a probability space. Given its one-parameter/cyclic subgroup (the flow), and any sequence of elements (the kicks) we define the kicked dynamics on the space by alternately flowing with given period, then applying a kick. Our main finding is the following stability phenomenon: the kicked system often inherits recurrence properties of the original flow. We present three main examples. 1) G is the torus. We show that for generic linear flows, and any sequence of kicks, the trajectories of the kicked system are uniformly distributed for almost all periods. 2) G is a discrete subgroup of PSL(2,R) acting on the unit tangent bundle of a Riemann surface. The flow is generated by a single element of G, and we take any bounded sequence of elements of G as our kicks. We prove that the kicked system is mixing for all sufficiently large periods if and only if the generator is of infinite order and is not conjugate to its inverse in G. 3) G is the group of Hamiltonian diffeomorphisms of a closed symplectic manifold. We assume that the flow is rapidly growing in the sense of Hofer's norm, and the kicks are bounded. We prove that for a positive proportion of the periods the kicked system inherits a kind of energy conservation law and is thus superrecurrent. We use tools of geometric group theory and symplectic topology.Comment: Latex, 40 pages, revised versio

On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 1. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of n identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch-Payne-Schiffer inequality for n = 2 and show that it is strict in this cas

Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound

We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed

Importance of Verification and Validation of Data Sources in Attaining Information Superiority

Information superiority has been defined as a state that is achieved when a competitive advantage is derived from the ability to exploit a superior information position. To achieve such a superior information position enterprises and nations, alike, must not only collect and record correct, accurate, timely and useful information but also ensure that information recorded is not lost to competitors due to lack of comprehensive security and leaks. Further, enterprises that aim to attain information superiority must also ensure mechanisms of validating and verifying information to reduce the chances of mis-information. Although, research has been carried out into ways to increase the security of information and detect leaks, not enough focus has been given to the key elements of information, namely data and context. This paper outlines the importance of data in contributing to information superiority and highlights the lack of data centric approach in attaining information superiority. The paper also discusses the importance of verification and back tracking of information to ascertain the data, its source and context in validating information for its correctness, validity and accuracy. A brief list of consequences of information leaks is also provided in the document to emphasize the importance of information security in the context of data collected. Further, this paper examines the McCumber model, which outlines the various states and elements of information, to accommodate a data centric, quantitative approach. Outlining simple protocols for verification of data in the information superiority context, this paper also highlights a few steps that can be taken to verify the sources of data