7,033 research outputs found
Smooth analysis of the condition number and the least singular value
Let \a be a complex random variable with mean zero and bounded variance.
Let be the random matrix of size whose entries are iid copies of
\a and be a fixed matrix of the same size. The goal of this paper is to
give a general estimate for the condition number and least singular value of
the matrix , generalizing an earlier result of Spielman and Teng for
the case when \a is gaussian.
Our investigation reveals an interesting fact that the "core" matrix does
play a role on tail bounds for the least singular value of . This
does not occur in Spielman-Teng studies when \a is gaussian.
Consequently, our general estimate involves the norm .
In the special case when is relatively small, this estimate is nearly
optimal and extends or refines existing results.Comment: 20 pages. An erratum to the published version has been adde
Conditions for duality between fluxes and concentrations in biochemical networks
Mathematical and computational modelling of biochemical networks is often
done in terms of either the concentrations of molecular species or the fluxes
of biochemical reactions. When is mathematical modelling from either
perspective equivalent to the other? Mathematical duality translates concepts,
theorems or mathematical structures into other concepts, theorems or
structures, in a one-to-one manner. We present a novel stoichiometric condition
that is necessary and sufficient for duality between unidirectional fluxes and
concentrations. Our numerical experiments, with computational models derived
from a range of genome-scale biochemical networks, suggest that this
flux-concentration duality is a pervasive property of biochemical networks. We
also provide a combinatorial characterisation that is sufficient to ensure
flux-concentration duality. That is, for every two disjoint sets of molecular
species, there is at least one reaction complex that involves species from only
one of the two sets. When unidirectional fluxes and molecular species
concentrations are dual vectors, this implies that the behaviour of the
corresponding biochemical network can be described entirely in terms of either
concentrations or unidirectional fluxes
Ground State Spin Logic
Designing and optimizing cost functions and energy landscapes is a problem
encountered in many fields of science and engineering. These landscapes and
cost functions can be embedded and annealed in experimentally controllable spin
Hamiltonians. Using an approach based on group theory and symmetries, we
examine the embedding of Boolean logic gates into the ground state subspace of
such spin systems. We describe parameterized families of diagonal Hamiltonians
and symmetry operations which preserve the ground state subspace encoding the
truth tables of Boolean formulas. The ground state embeddings of adder circuits
are used to illustrate how gates are combined and simplified using symmetry.
Our work is relevant for experimental demonstrations of ground state embeddings
found in both classical optimization as well as adiabatic quantum optimization.Comment: 6 pages + 3 pages appendix, 7 figures, 1 tabl
Analysis and Synthesis of Digital Dyadic Sequences
We explore the space of matrix-generated (0, m, 2)-nets and (0, 2)-sequences
in base 2, also known as digital dyadic nets and sequences. In computer
graphics, they are arguably leading the competition for use in rendering. We
provide a complete characterization of the design space and count the possible
number of constructions with and without considering possible reorderings of
the point set. Based on this analysis, we then show that every digital dyadic
net can be reordered into a sequence, together with a corresponding algorithm.
Finally, we present a novel family of self-similar digital dyadic sequences, to
be named -sequences, that spans a subspace with fewer degrees of freedom.
Those -sequences are extremely efficient to sample and compute, and we
demonstrate their advantages over the classic Sobol (0, 2)-sequence.Comment: 17 pages, 11 figures. Minor improvement of exposition; references to
earlier proofs of Theorems 3.1 and 3.3 adde
Volumes of Compact Manifolds
We present a systematic calculation of the volumes of compact manifolds which
appear in physics: spheres, projective spaces, group manifolds and generalized
flag manifolds. In each case we state what we believe is the most natural scale
or normalization of the manifold, that is, the generalization of the unit
radius condition for spheres. For this aim we first describe the manifold with
some parameters, set up a metric, which induces a volume element, and perform
the integration for the adequate range of the parameters; in most cases our
manifolds will be either spheres or (twisted) products of spheres, or quotients
of spheres (homogeneous spaces).
Our results should be useful in several physical instances, as instanton
calculations, propagators in curved spaces, sigma models, geometric scattering
in homogeneous manifolds, density matrices for entangled states, etc. Some flag
manifolds have also appeared recently as exceptional holonomy manifolds; the
volumes of compact Einstein manifolds appear in String theory.Comment: 26 pages, no figures; updated addresses and bibliography. To be
published in Rep. Math. Phy
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