2,444 research outputs found
Bounded Independence vs. Moduli
Let k = k(n) be the largest integer such that there exists a k-wise uniform distribution over {0,1}^n that is supported on the set S_m := {x in {0,1}^n: sum_i x_i equiv 0 mod m}, where m is any integer. We show that Omega(n/m^2 log m) <= k <= 2n/m + 2. For k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over S_m. For any fixed odd m there is k ge (1 - Omega(1))n such that any k-wise uniform distribution lands in S_m with probability exponentially close to |S_m|/2^n; and this result is false for any even m
Intersection forms of toric hyperkaehler varieties
This note proves combinatorially that the intersection pairing on the middle
dimensional compactly supported cohomology of a smooth toric hyperkaehler
variety is always definite, providing a large number of non-trivial L^2
harmonic forms for toric hyperkaehler metrics on these varieties. This is
motivated by a result of Hitchin about the definiteness of the pairing of L^2
harmonic forms on complete hyperkaehler manifolds of linear growth.Comment: Latex, 7 page
A Quantum is a Complex Structure on Classical Phase Space
Duality transformations within the quantum mechanics of a finite number of
degrees of freedom can be regarded as the dependence of the notion of a
quantum, i.e., an elementary excitation of the vacuum, on the observer on
classical phase space. Under an observer we understand, as in general
relativity, a local coordinate chart. While classical mechanics can be
formulated using a symplectic structure on classical phase space, quantum
mechanics requires a complex-differentiable structure on that same space.
Complex-differentiable structures on a given real manifold are often not
unique. This article is devoted to analysing the dependence of the notion of a
quantum on the complex-differentiable structure chosen on classical phase
space
Eluding SUSY at every genus on stable closed string vacua
In closed string vacua, ergodicity of unipotent flows provide a key for
relating vacuum stability to the UV behavior of spectra and interactions.
Infrared finiteness at all genera in perturbation theory can be rephrased in
terms of cancelations involving only tree-level closed strings scattering
amplitudes. This provides quantitative results on the allowed deviations from
supersymmetry on perturbative stable vacua. From a mathematical perspective,
diagrammatic relations involving closed string amplitudes suggest a relevance
of unipotent flows dynamics for the Schottky problem and for the construction
of the superstring measure.Comment: v2, 17 pages, 8 figures, typos corrected, new figure added with 3
modular images of long horocycles,(obtained with Mathematica
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