299,449 research outputs found
Contact Geometry of Hyperbolic Equations of Generic Type
We study the contact geometry of scalar second order hyperbolic equations in
the plane of generic type. Following a derivation of parametrized
contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7)
and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence
method to study the generic case. An intriguing feature of this class of
equations is that every generic hyperbolic equation admits at most a
nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp:
normal forms for the contact-equivalence classes of these maximally symmetric
generic hyperbolic equations are derived and explicit symmetry algebras are
presented. Moreover, these maximally symmetric equations are Darboux
integrable. An enumeration of several submaximally symmetric (eight and
seven-dimensional) generic hyperbolic structures is also given.Comment: This is a contribution to the Special Issue "Elie Cartan and
Differential Geometry", published in SIGMA (Symmetry, Integrability and
Geometry: Methods and Applications) at http://www.emis.de/journals/SIGM
Explicit lower bounds on the modular degree of an elliptic curve
We derive an explicit zero-free region for symmetric square L-functions of
elliptic curves, and use this to derive an explicit lower bound for the modular
degree of rational elliptic curves. The techniques are similar to those used in
the classical derivation of zero-free regions for Dirichlet L-functions, but
here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no
Siegel zeros, which leads to a strengthened result
Lower Bounds on the Communication Complexity of Shifting
We study the communication complexity of the SHIFT (equivalently, SUM-INDEX) function in a 3-party simultaneous message model. Alice and Bob share an n-bit string x and Alice holds an index i and Bob an index j. They must send messages to a referee who knows only n, i and j, enabling him to determine x[(i+j) mod n]. Surprisingly, it is possible to achieve nontrivial savings even with such a strong restriction: Bob can now make do with only ceil(n/2) bits. Here we show that this bound is completely tight, for all n. This is an exact lower bound, with no asymptotics involved
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