28,235 research outputs found
Partial duality of hypermaps
We introduce a collection of new operations on hypermaps, partial duality,
which include the classical Euler-Poincar\'e dualities as particular cases.
These operations generalize the partial duality for maps, or ribbon graphs,
recently discovered in a connection with knot theory. Partial duality is
different from previous studied operations of S. Wilson, G. Jones, L. James,
and A. Vince. Combinatorially hypermaps may be described in one of three ways:
as three involutions on the set of flags (-model), or as three
permutations on the set of half-edges (-model in orientable case), or
as edge 3-colored graphs. We express partial duality in each of these models.Comment: 19 pages, 16 figure
Permutation Complexity via Duality between Values and Orderings
We study the permutation complexity of finite-state stationary stochastic
processes based on a duality between values and orderings between values.
First, we establish a duality between the set of all words of a fixed length
and the set of all permutations of the same length. Second, on this basis, we
give an elementary alternative proof of the equality between the permutation
entropy rate and the entropy rate for a finite-state stationary stochastic
processes first proved in [Amigo, J.M., Kennel, M. B., Kocarev, L., 2005.
Physica D 210, 77-95]. Third, we show that further information on the
relationship between the structure of values and the structure of orderings for
finite-state stationary stochastic processes beyond the entropy rate can be
obtained from the established duality. In particular, we prove that the
permutation excess entropy is equal to the excess entropy, which is a measure
of global correlation present in a stationary stochastic process, for
finite-state stationary ergodic Markov processes.Comment: 26 page
The truncated tracial moment problem
We present tracial analogs of the classical results of Curto and Fialkow on
moment matrices. A sequence of real numbers indexed by words in non-commuting
variables with values invariant under cyclic permutations of the indexes, is
called a tracial sequence. We prove that such a sequence can be represented
with tracial moments of matrices if its corresponding moment matrix is positive
semidefinite and of finite rank. A truncated tracial sequence allows for such a
representation if and only if one of its extensions admits a flat extension.
Finally, we apply the theory via duality to investigate trace-positive
polynomials in non-commuting variables.Comment: 21 page
Complex Periodic Potentials with a Finite Number of Band Gaps
We obtain several new results for the complex generalized associated Lame
potential V(x)= a(a+1)m sn^2(y,m)+ b(b+1)m sn^2(y+K(m),m) + f(f+1)m
sn^2(y+K(m)+iK'(m),m)+ g(g+1)m sn^2(y+iK'(m),m), where y = x-K(m)/2-iK'(m)/2,
sn(y,m) is a Jacobi elliptic function with modulus parameter m, and there are
four real parameters a,b,f,g. First, we derive two new duality relations which,
when coupled with a previously obtained duality relation, permit us to relate
the band edge eigenstates of the 24 potentials obtained by permutations of the
four parameters a,b,f,g. Second, we pose and answer the question: how many
independent potentials are there with a finite number "a" of band gaps when
a,b,f,g are integers? For these potentials, we clarify the nature of the band
edge eigenfunctions. We also obtain several analytic results when at least one
of the four parameters is a half-integer. As a by-product, we also obtain new
solutions of Heun's differential equation.Comment: 33 pages, 0 figure
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