Let q ≥ 1 be an integer, Sq be the unit sphere embedded in Rq+1, and μq be the volume element of Sq. For x0 ∈ Sq, and α ∈ (0, π), let Sα(x0) denote the cap {ξ ∈ Sq : x0 · ξ ≥ cosα}. We prove that for any integer m ≥ 1, there exists a positive constant c = c(q,m), independent of α, with the following property. Given an arbitrary set C of points in Sα(x0), satisftying the mesh norm condition max ...

In this paper we study a new topological invariant Cat(X, ξ), where X is a finite polyhedron and ξ ∈ H(X;R) is a real cohomology class. Cat(X, ξ) is defined using open covers of X with certain geometric properties; it is a generalization of the classical Lusternik – Schnirelman category. We show that Cat(X, ξ) depends only on the homotopy type of (X, ξ). We prove that Cat(X, ξ) allows to establ...

We show that the intersection exponents for planar Brownian motions are analytic. More precisely, let B and B′ be independent planar Brownian motions started from distinct points, and define the exponent ξ(1, λ) by E [ P [ B[0, t] ∩B[0, t] = ∅ ∣∣ B[0, t] ]λ ] ≈ t, t → ∞. Then the mapping λ 7→ ξ(1, λ) is real analytic in (0,∞). The same result is proved for the exponents ξ(k, λ) where k is a pos...

We report on several studies of charm baryon production and decays by the BABAR collaboration. We confirm previous observations of the Ξ ′ 0/+ c , Ξ c (2980) + and Ξ c (3077) + baryons, measure branching ratios for Cabibbo-suppressed Λ + c decays and use baryon decays to study the properties of the light-quark baryons, Ω − and Ξ(1690) 0 .

We study the minimal number C(M, ξ) of contact charts that one needs to cover a closed connected contact manifold (M, ξ). Our basic result is C(M, ξ) ≤ dimM + 1. We also compute C(M, ξ) for all closed connected contact 3-manifolds:

Let {ξ(n)}n∈Z be a two-color random scenery, that is, a random coloring of Z in two colors, such that the ξ(i)’s are i.i.d. Bernoulli variables with parameter 1 2 . Let {S(n)}n∈N be a symmetric random walk starting at 0. Our main result shows that a.s., ξ ◦ S (the composition of ξ and S) determines ξ up to translation and reflection. In other words, by observing the scenery ξ along the random w...

We report on searches in deep inelastic ep scattering for narrow baryonic states decaying into Ξ − π − , Ξ − π + , K 0 S p and their charge conjugates, at centre-of-mass energies of 300 and 318 GeV. No signal for a new narrow baryonic state is observed in the mass ranges 1600-2100 MeV (Ξπ) and from threshold up to 1700 MeV (K 0 S p). The standard baryon Ξ(1530) 0 is observed in the decay mode Ξ...

In this note we obtain lower bounds for P(ξ ≥ 0) and P(ξ > 0) under assumptions on the moments of ξ. Here ξ is a centered real-valued random variable. For instance we consider the case where the first and p-th moment are fixed, and the case where the second and p-th moment are fixed. Such lower bounds are used in [2, 3, 5, 7] to estimate tail probabilities. It can be used to estimate P(ξ ≤ Eξ) ...

where a and W satisfy (a1) a(t) is a continuous almost periodic function of t with a(t) ≥ a0 > 0 for all t ∈ R. (W1) There is a ξ ∈ R\{0} such that W ∈ C(R\{ξ},R). (W2) limx→ξW (x) = −∞. (W3) There is a neighborhood N of ξ and U ∈ C(N\{ξ},R) such that |U(x)| → ∞ as x→ ξ and |U ′(x)|2 ≤ −W (x) for x ∈ N\{ξ}, (W4) W (x) < W (0) = 0 if x 6= 0 and W ′′(0) is negative definite. (W5) There is a const...

In the present paper we obtain a necessary and sufficient condition to prove the Riemann hypothesis in terms of certain properties of local extrema of the function Ξ(t) = ξ( 1 2 + it). First, we prove that positivity of all local maxima and negativity of all local minima of Ξ(t) form a necessary condition for the Riemann hypothesis to be true. After showing that any extremum point of Ξ(t) is a ...