12,365 research outputs found
Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects
In 1981 W.L. Edge discovered and studied a pencil of highly
symmetric genus projective curves with remarkable properties. Edge's work
was based on an 1895 paper of A. Wiman. Both papers were written in the
satisfying style of 19th century algebraic geometry. In this paper and its
sequel [FL], we consider from a more modern, conceptual
perspective, whereby explicit equations are reincarnated as geometric objects.Comment: Minor revisions. Now 49 pages, 4 figures. To appear in European
Journal of Mathematics, special issue in memory of W.L. Edg
Lie Markov models with purine/pyrimidine symmetry
Continuous-time Markov chains are a standard tool in phylogenetic inference.
If homogeneity is assumed, the chain is formulated by specifying
time-independent rates of substitutions between states in the chain. In
applications, there are usually extra constraints on the rates, depending on
the situation. If a model is formulated in this way, it is possible to
generalise it and allow for an inhomogeneous process, with time-dependent rates
satisfying the same constraints. It is then useful to require that there exists
a homogeneous average of this inhomogeneous process within the same model. This
leads to the definition of "Lie Markov models", which are precisely the class
of models where such an average exists. These models form Lie algebras and
hence concepts from Lie group theory are central to their derivation. In this
paper, we concentrate on applications to phylogenetics and nucleotide
evolution, and derive the complete hierarchy of Lie Markov models that respect
the grouping of nucleotides into purines and pyrimidines -- that is, models
with purine/pyrimidine symmetry. We also discuss how to handle the subtleties
of applying Lie group methods, most naturally defined over the complex field,
to the stochastic case of a Markov process, where parameter values are
restricted to be real and positive. In particular, we explore the geometric
embedding of the cone of stochastic rate matrices within the ambient space of
the associated complex Lie algebra.
The whole list of Lie Markov models with purine/pyrimidine symmetry is
available at http://www.pagines.ma1.upc.edu/~jfernandez/LMNR.pdf.Comment: 32 page
All degree six local unitary invariants of k qudits
We give explicit index-free formulae for all the degree six (and also degree
four and two) algebraically independent local unitary invariant polynomials for
finite dimensional k-partite pure and mixed quantum states. We carry out this
by the use of graph-technical methods, which provides illustrations for this
abstract topic.Comment: 18 pages, 6 figures, extended version. Comments are welcom
Periodic-Orbit Theory of Universality in Quantum Chaos
We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory,
that full classical chaos is paralleled by quantum energy spectra with
universal spectral statistics, in agreement with random-matrix theory. For
dynamics from all three Wigner-Dyson symmetry classes, we calculate the
small-time spectral form factor as power series in the time .
Each term of that series is provided by specific families of pairs of
periodic orbits. The contributing pairs are classified in terms of close
self-encounters in phase space. The frequency of occurrence of self-encounters
is calculated by invoking ergodicity. Combinatorial rules for building pairs
involve non-trivial properties of permutations. We show our series to be
equivalent to perturbative implementations of the non-linear sigma models for
the Wigner-Dyson ensembles of random matrices and for disordered systems; our
families of orbit pairs are one-to-one with Feynman diagrams known from the
sigma model.Comment: 31 pages, 17 figure
Geometry of the Wiman Pencil, I: Algebro-Geometric Aspects
In 1981 W.L. Edge discovered and studied a pencil of highly
symmetric genus projective curves with remarkable properties. Edge's work
was based on an 1895 paper of A. Wiman. Both papers were written in the
satisfying style of 19th century algebraic geometry. In this paper and its
sequel [FL], we consider from a more modern, conceptual
perspective, whereby explicit equations are reincarnated as geometric objects.Comment: Minor revisions. Now 49 pages, 4 figures. To appear in European
Journal of Mathematics, special issue in memory of W.L. Edg
Permutation Orbifold of N=2 Supersymmetric Minimal Models
In this paper we apply the previously derived formalism of permutation
orbifold conformal field theories to N=2 supersymmetric minimal models. By
interchanging extensions and permutations of the factors we find a very
interesting structure relating various conformal field theories that seems not
to be known in literature. Moreover, unexpected exceptional simple currents
arise in the extended permuted models, coming from off-diagonal fields. In a
few situations they admit fixed points that must be resolved. We determine the
complete CFT data with all fixed point resolution matrices for all simple
currents of all Z_2-permutations orbifolds of all minimal N=2 models with k\neq
2 mod 4.Comment: 48 page
On the Black-Hole/Qubit Correspondence
The entanglement classification of four qubits is related to the extremal
black holes of the 4-dimensional STU model via a time-like reduction to three
dimensions. This correspondence is generalised to the entanglement
classification of a very special four-way entanglement of eight qubits and the
black holes of the maximally supersymmetric N = 8 and exceptional magic N = 2
supergravity theories.Comment: 32 pages, very minor changes at the start of Sec. 4.1. Version to
appear in The European Physical Journal - Plu
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