We establish local (L, L) mapping properties for averages on curves. The exponents are sharp except for endpoints.

1 Chernoff Bounds 1.1 Bayesian Hypothesis Test A test using log-likelihood ratio statistic has the form, T (Y ) = logL(Y ) T τ. (1) Bound-1: The probability of error Pe is bounded as, Pe ≤ (π0 + π1e )eμT,0(s0)−s0τ , (2) where μT,0(s) = logE0[e ], and μ ′ T,0(s0) = τ . Bound-2: ∀ s ∈ [0, 1], Pe ≤ max(π0, π1e )eμT,0(s)−sτ . (3) Derivation of the above bound: Consider, Pe = π0P0(Γ1) + π1P1(Γ0), = ...

In this note we prove that for equilibrium states of axiom A systems the time τB(x) needed for a typical point x to enter for the first time in a typical ball B with radius r scales as τB(x) ∼ r d where d is the local dimension of the invariant measure at the center of the ball. A similar relation is proved for a full measure set of interval excanges. Some applications to Birkoff averages of un...

Hušek defines a space X to have a small diagonal if each uncountable subset of X disjoint from the diagonal, has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight ω1 which has a small diagonal, will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent tha...

We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical BorelCantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasi...

A supermanifold, Mm/n, can be caracterired by its smooth superfunctions which constitute an algebra A (Leites, Kostant). We associate canonically ~a Ia Gelfand, certain fibred manifolds on which the automorphisms (the Jordan automorphisms) of A actas diffeomorphisms. For example, the kernelsof all homomorphisms from the algebra of superfunctions onto the Grassmann algebra of dimension n form na...

The Hirsch index is a number that synthesizes a researcher’s output. It is the maximum number h such that the researcher has h papers with at least h citations each. Woeginger [Woeginger, G. J. (2008a). An axiomatic characterization of the Hirsch-index. Mathematical Social Sciences, 56(2), 224–232; Woeginger, G. J. (2008b). A symmetry axiom for scientific impact indices. Journal of Informetrics...

We provide a simple algorithmic argument to “explain” the concentration of measure phenomenon observed in the sum of independent random variables. Roughly speaking, we observe that without concentration of measure it would be possible to predict the outcome of a coin toss with probability greater than 1/2. We use this idea to derive an alternative proof of a Chernoff-Hoeffding bound for large d...

The Chernoff bounds provide very accurate information concerning the tail probabilities of most distrbhutions. in this paper we describe some relevant properties of these bounds.