1,108 research outputs found
On plane sextics with double singular points
We compute the fundamental groups of five maximizing sextics with double
singular points only; in four cases, the groups are as expected. The approach
used would apply to other sextics as well, given their equations.Comment: A few explanations and references adde
Forbidden gap argument for phase transitions proved by means of chessboard estimates
Chessboard estimates are one of the standard tools for proving phase
coexistence in spin systems of physical interest. In this note we show that the
method not only produces a point in the phase diagram where more than one Gibbs
states coexist, but that it can also be used to rule out the existence of
shift-ergodic states that differ significantly from those proved to exist.
For models depending on a parameter (say, the temperature), this shows that
the values of the conjugate thermodynamic quantity (the energy) inside the
"transitional gap" are forbidden in all shift-ergodic Gibbs states. We point
out several models where our result provides useful additional information
concerning the set of possible thermodynamic equilibria.Comment: 26 page
Periodic GMP Matrices
We recall criteria on the spectrum of Jacobi matrices such that the
corresponding isospectral torus consists of periodic operators. Motivated by
those known results for Jacobi matrices, we define a new class of operators
called GMP matrices. They form a certain Generalization of matrices related to
the strong Moment Problem. This class allows us to give a parametrization of
almost periodic finite gap Jacobi matrices by periodic GMP matrices. Moreover,
due to their structural similarity we can carry over numerous results from the
direct and inverse spectral theory of periodic Jacobi matrices to the class of
periodic GMP matrices. In particular, we prove an analogue of the remarkable
"magic formula" for this new class
A Portable High-Quality Random Number Generator for Lattice Field Theory Simulations
The theory underlying a proposed random number generator for numerical
simulations in elementary particle physics and statistical mechanics is
discussed. The generator is based on an algorithm introduced by Marsaglia and
Zaman, with an important added feature leading to demonstrably good statistical
properties. It can be implemented exactly on any computer complying with the
IEEE--754 standard for single precision floating point arithmetic.Comment: pages 0-19, ps-file 174404 bytes, preprint DESY 93-13
Quantitative K-Theory Related to Spin Chern Numbers
We examine the various indices defined on pairs of almost commuting unitary
matrices that can detect pairs that are far from commuting pairs. We do this in
two symmetry classes, that of general unitary matrices and that of self-dual
matrices, with an emphasis on quantitative results. We determine which values
of the norm of the commutator guarantee that the indices are defined, where
they are equal, and what quantitative results on the distance to a pair with a
different index are possible. We validate a method of computing spin Chern
numbers that was developed with Hastings and only conjectured to be correct.
Specifically, the Pfaffian-Bott index can be computed by the "log method" for
commutator norms up to a specific constant
- …