780 research outputs found
Enumerative Galois theory for cubics and quartics
We show that there are monic, cubic
polynomials with integer coefficients bounded by in absolute value whose
Galois group is . We also show that the order of magnitude for
quartics is , and that the respective counts for , ,
are , , . Our work establishes
that irreducible non- cubic polynomials are less numerous than reducible
ones, and similarly in the quartic setting: these are the first two solved
cases of a 1936 conjecture made by van der Waerden
Open Strings and Electric Fields in Compact Spaces
We analyse open strings with background electric fields in the internal
space, T-dual to branes moving with constant velocities in the internal space.
We find that the direction of the electric fields inside a two torus, dual to
the D brane velocities, has to be quantised such that the corresponding
direction is compact. This implies that D-brane motion in the internal torus is
periodic, with a periodicity that can be parametrically large in terms of the
internal radii. By S-duality, this is mapped into an internal magnetic field in
a three torus, a quantum mechanical analysis of which yields a similar result,
i.e. the parallel direction to the magnetic field has to be compact.
Furthermore, for the magnetic case, we find the Landau level degeneracy as
being given by the greatest common divisor of the flux numbers. We carry on the
string quantisation and derive the relevant partition functions for these
models. Our analysis includes also the case of oblique electric fields which
can arise when several stacks of branes are present. Compact dimensions and/or
oblique sectors influence the energy loss of the system through pair-creation
and thus can be relevant for inflationary scenarios with branes. Finally, we
show that the compact energy loss is always larger than the non-compact one.Comment: 44 pages, 4 figures; v2 corrections in section 8, one reference adde
The Simulator: Understanding Adaptive Sampling in the Moderate-Confidence Regime
We propose a novel technique for analyzing adaptive sampling called the {\em
Simulator}. Our approach differs from the existing methods by considering not
how much information could be gathered by any fixed sampling strategy, but how
difficult it is to distinguish a good sampling strategy from a bad one given
the limited amount of data collected up to any given time. This change of
perspective allows us to match the strength of both Fano and change-of-measure
techniques, without succumbing to the limitations of either method. For
concreteness, we apply our techniques to a structured multi-arm bandit problem
in the fixed-confidence pure exploration setting, where we show that the
constraints on the means imply a substantial gap between the
moderate-confidence sample complexity, and the asymptotic sample complexity as
found in the literature. We also prove the first instance-based
lower bounds for the top-k problem which incorporate the appropriate
log-factors. Moreover, our lower bounds zero-in on the number of times each
\emph{individual} arm needs to be pulled, uncovering new phenomena which are
drowned out in the aggregate sample complexity. Our new analysis inspires a
simple and near-optimal algorithm for the best-arm and top-k identification,
the first {\em practical} algorithm of its kind for the latter problem which
removes extraneous log factors, and outperforms the state-of-the-art in
experiments
On the hardness of the shortest vector problem
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 77-84).An n-dimensional lattice is the set of all integral linear combinations of n linearly independent vectors in Rm. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest non-zero vector in it. We prove that the shortest vector problem is NP-hard (for randomized reductions) to approximate within some constant factor greater than 1 in any 1, norm (p >\=1). In particular, we prove the NP-hardness of approximating SVP in the Euclidean norm 12 within any factor less than [square root of]2. The same NP-hardness results hold for deterministic non-uniform reductions. A deterministic uniform reduction is also given under a reasonable number theoretic conjecture concerning the distribution of smooth numbers. In proving the NP-hardness of SVP we develop a number of technical tools that might be of independent interest. In particular, a lattice packing is constructed with the property that the number of unit spheres contained in an n-dimensional ball of radius greater than 1 + [square root of] 2 grows exponentially in n, and a new constructive version of Sauer's lemma (a combinatorial result somehow related to the notion of VC-dimension) is presented, considerably simplifying all previously known constructions.by Daniele Micciancio.Ph.D
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