175 research outputs found
Bounds on the Coefficients of Tension and Flow Polynomials
The goal of this article is to obtain bounds on the coefficients of modular
and integral flow and tension polynomials of graphs. To this end we make use of
the fact that these polynomials can be realized as Ehrhart polynomials of
inside-out polytopes. Inside-out polytopes come with an associated relative
polytopal complex and, for a wide class of inside-out polytopes, we show that
this complex has a convex ear decomposition. This leads to the desired bounds
on the coefficients of these polynomials.Comment: 16 page
A -Queens Problem. II. The Square Board
We apply to the chessboard the counting theory from Part I for
nonattacking placements of chess pieces with unbounded straight-line moves,
such as the queen. Part I showed that the number of ways to place identical
nonattacking pieces is given by a quasipolynomial function of of degree
, whose coefficients are (essentially) polynomials in that depend
cyclically on .
Here we study the periods of the quasipolynomial and its coefficients, which
are bounded by functions, not well understood, of the piece's move directions,
and we develop exact formulas for the very highest coefficients. The
coefficients of the three highest powers of do not vary with . On the
other hand, we present simple pieces for which the fourth coefficient varies
periodically. We develop detailed properties of counting quasipolynomials that
will be applied in sequels to partial queens, whose moves are subsets of those
of the queen, and the nightrider, whose moves are extended knight's moves.
We conclude with the first, though strange, formula for the classical
-Queens Problem and with several conjectures and open problems.Comment: 23 pp., 1 figure, submitted. This = second half of 1303.1879v1 with
great improvements. V2 has a new proposition, better definitions, and
corrected conjectures. V3 has results et al. renumbered to correspond with
published version, and expands dictionary's cryptic abbreviation
Laughlin's wave functions, Coulomb gases and expansions of the discriminant
In the context of the fractional quantum Hall effect, we investigate
Laughlin's celebrated ansatz for the groud state wave function at fractional
filling of the lowest Landau level. Interpreting its normalization in terms of
a one component plasma, we find the effect of an additional quadrupolar field
on the free energy, and derive estimates for the thermodynamically equivalent
spherical plasma. In a second part, we present various methods for expanding
the wave function in terms of Slater determinants, and obtain sum rules for the
coefficients. We also address the apparently simpler question of counting the
number of such Slater states using the theory of integral polytopes.Comment: 97 pages, using harvmac (with big option recommended) and epsf, 7
figures available upon request, Saclay preprint Spht 93/12
From orbital measures to Littlewood-Richardson coefficients and hive polytopes
The volume of the hive polytope (or polytope of honeycombs) associated with a
Littlewood- Richardson coefficient of SU(n), or with a given admissible triple
of highest weights, is expressed, in the generic case, in terms of the Fourier
transform of a convolution product of orbital measures. Several properties of
this function -- a function of three non-necessarily integral weights or of
three multiplets of real eigenvalues for the associated Horn problem-- are
already known. In the integral case it can be thought of as a semi-classical
approximation of Littlewood-Richardson coefficients. We prove that it may be
expressed as a local average of a finite number of such coefficients. We also
relate this function to the Littlewood-Richardson polynomials (stretching
polynomials) i.e., to the Ehrhart polynomials of the relevant hive polytopes.
Several SU(n) examples, for n=2,3,...,6, are explicitly worked out.Comment: 32 pages, 4 figures. This version (V4): a few corrected typo
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