We obtain Margulis-type asymptotic estimates for the number of free homotopy classes closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces genus at least 2

We show that for each positive integer k there is a k × k matrix B with ±1 entries such that letting K1 be the symmetric convex hull of the rows of B and K2 the symmetric convex hull of √ k times the canonical unit vector basis of Rk (= √ kBk 1 ), then K1∩K2 lies between two universal multiples of the Euclidean unit ball, Bk 2 . Moreover, the probability that a random ±1 matrix satisfies the ab...

By using a complex eld with a symmetric combination of electric and magnetic elds, a rstorder covariant Lagrangian for Maxwell's equations is obtained, similar to the Lagrangian for the Dirac equation. This leads to a dual-symmetric quantum electrodynamic theory with an in nite set of local conservation laws. The dual symmetry is shown to correspond to a helical phase, conjugate to the conserve...

To every k-dimensional modular invariant vector space we associate a modular form on SL(2,Z) of weight 2k. We explore number theoretic properties of this form and found a sufficient condition for its vanishing which yields modular identities (e.g., Ramanujan-Watson’s modular identities). Furthermore, we focus on a family of modular invariant spaces coming from suitable two-dimensional spaces vi...