67,424 research outputs found
Graham Higman's PORC theorem
Graham Higman published two important papers in 1960. In the first of these
papers he proved that for any positive integer the number of groups of
order is bounded by a polynomial in , and he formulated his famous
PORC conjecture about the form of the function giving the number of
groups of order . In the second of these two papers he proved that the
function giving the number of -class two groups of order is PORC. He
established this result as a corollary to a very general result about vector
spaces acted on by the general linear group. This theorem takes over a page to
state, and is so general that it is hard to see what is going on. Higman's
proof of this general theorem contains several new ideas and is quite hard to
follow. However in the last few years several authors have developed and
implemented algorithms for computing Higman's PORC formulae in special cases of
his general theorem. These algorithms give perspective on what are the key
points in Higman's proof, and also simplify parts of the proof.
In this note I give a proof of Higman's general theorem written in the light
of these recent developments
Exact solution of the pairing Hamiltonian by deforming the pairing algebra
The present paper makes a connection between collective bosonic states and
the exact solutions of the pairing Hamiltonian. This makes it
possible to investigate the effects of the Pauli principle on the energy
spectrum, by gradually reintroducing the Pauli principle. It also introduces an
efficient and stable numerical method to probe all the eigenstates of this
class of Hamiltonians.Comment: 18 pages, 10 figure
Frobenius theorem and invariants for Hamiltonian systems
We apply Frobenius integrability theorem in the search of invariants for
one-dimensional Hamiltonian systems with a time-dependent potential. We obtain
several classes of potential functions for which Frobenius theorem assures the
existence of a two-dimensional foliation to which the motion is constrained. In
particular, we derive a new infinite class of potentials for which the motion
is assurately restricted to a two-dimensional foliation. In some cases,
Frobenius theorem allows the explicit construction of an associated invariant.
It is proven the inverse result that, if an invariant is known, then it always
can be furnished by Frobenius theorem
Discrete Miura Opers and Solutions of the Bethe Ansatz Equations
Solutions of the Bethe ansatz equations associated to the XXX model of a
simple Lie algebra come in families called the populations. We prove that a
population is isomorphic to the flag variety of the Langlands dual Lie algebra.
The proof is based on the correspondence between the solutions of the Bethe
ansatz equations and special difference operators which we call the discrete
Miura opers. The notion of a discrete Miura oper is one of the main results of
the paper.
For a discrete Miura oper D, associated to a point of a population, we show
that all solutions of the difference equation DY=0 are rational functions, and
the solutions can be written explicitly in terms of points composing the
population.Comment: Latex, 26 page
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