9,136 research outputs found
Finite Chern-Simons matrix model - algebraic approach
We analyze the algebra of observables and the physical Fock space of the
finite Chern-Simons matrix model. We observe that the minimal algebra of
observables acting on that Fock space is identical to that of the Calogero
model. Our main result is the identification of the states in the l-th tower of
the Chern-Simons matrix model Fock space and the states of the Calogero model
with the interaction parameter nu=l+1. We describe quasiparticle and quasihole
states in the both models in terms of Schur functions, and discuss some
nontrivial consequences of our algebraic approach.Comment: 12pages, jhep cls, minor correction
Preservation of algebraicity in free probability
We show that any matrix-polynomial combination of free noncommutative random
variables each having an algebraic law has again an algebraic law. Our result
answers a question raised by a recent paper of Shlyakhtenko and Skoufranis. The
result belongs to a family of results with origins outside free probability
theory, including a result of Aomoto asserting algebraicity of the Green
function of random walk of quite general type on a free group.Comment: 41 pages, LaTeX, no figures. In v2, we added references, corrected
typos, reorganized some material, and added explanations. Main results remain
the same. In this version, v3, we added references and explanation, and
simplified the second half of the proof of the main resul
Minimal symmetric Darlington synthesis
We consider the symmetric Darlington synthesis of a p x p rational symmetric
Schur function S with the constraint that the extension is of size 2p x 2p.
Under the assumption that S is strictly contractive in at least one point of
the imaginary axis, we determine the minimal McMillan degree of the extension.
In particular, we show that it is generically given by the number of zeros of
odd multiplicity of I-SS*. A constructive characterization of all such
extensions is provided in terms of a symmetric realization of S and of the
outer spectral factor of I-SS*. The authors's motivation for the problem stems
from Surface Acoustic Wave filters where physical constraints on the
electro-acoustic scattering matrix naturally raise this mathematical issue
Towards topological quantum computer
One of the principal obstacles on the way to quantum computers is the lack of
distinguished basis in the space of unitary evolutions and thus the lack of the
commonly accepted set of basic operations (universal gates). A natural choice,
however, is at hand: it is provided by the quantum R-matrices, the entangling
deformations of non-entangling (classical) permutations, distinguished from the
points of view of group theory, integrable systems and modern theory of
non-perturbative calculations in quantum field and string theory. Observables
in this case are (square modules of) the knot polynomials, and their pronounced
integrality properties could provide a key to error correction. We suggest to
use R-matrices acting in the space of irreducible representations, which are
unitary for the real-valued couplings in Chern-Simons theory, to build a
topological version of quantum computing.Comment: 14 page
Output Feedback Invariants
The paper is concerned with the problem of determining a complete set of
invariants for output feedback. Using tools from geometric invariant theory it
is shown that there exists a quasi-projective variety whose points parameterize
the output feedback orbits in a unique way. If the McMillan degree ,
the product of number of inputs and number of outputs, then it is shown that in
the closure of every feedback orbit there is exactly one nondegenerate system.Comment: 15 page
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