546 research outputs found
Duality Relations for Potts Correlation Functions
Duality relations are obtained for correlation functions of the q-state Potts
model on any planar lattice or graph using a simple graphical analysis. For the
two-point correlation we show that the correlation length is precisely the
surface tension of the dual model, generalizing a result known to hold for the
Ising model. For the three-point correlation an explicit expression is obtained
relating the correlation function to ratios of dual partition functions under
fixed boundary conditions.Comment: REVTex with 2 ps figures, to appear in Physics Letters
Dimer statistics on the M\"obius strip and the Klein bottle
Closed-form expressions are obtained for the generating function of
close-packed dimers on a simple quartic lattice embedded on a
M\"obius strip and a Klein bottle. Finite-size corrections are also analyzed
and compared with those under cylindrical and free boundary conditions.
Particularly, it is found that, for large lattices of the same size and with a
square symmetry, the number of dimer configurations on a M\"obius strip is
70.2% of that on a cylinder. We also establish two identities relating dimer
generating functions for M\"obius strips and cylinders.Comment: 12 pages, 2 figs included, accepted by Phys. Lett.
Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces
We consider the problem of enumerating spanning trees on lattices.
Closed-form expressions are obtained for the spanning tree generating function
for a hypercubic lattice of size N_1 x N_2 x...x N_d in d dimensions under
free, periodic, and a combination of free and periodic boundary conditions.
Results are also obtained for a simple quartic net embedded on two
non-orientable surfaces, a Moebius strip and the Klein bottle. Our results are
based on the use of a formula expressing the spanning tree generating function
in terms of the eigenvalues of an associated tree matrix. An elementary
derivation of this formula is given.Comment: latex, 9 pages, no figures, to appear in Lett. Appl. Mat
Zeroes of the Jones polynomial
We study the distribution of zeroes of the Jones polynomial for a
knot . We have computed numerically the roots of the Jones polynomial for
all prime knots with crossings, and found the zeroes scattered about
the unit circle with the average distance to the circle approaching a
nonzero value as increases.
For torus knots of the type we show that all zeroes lie on the unit
circle with a uniform density in the limit of either or , a
fact confirmed by our numerical findings. We have also elucidated the relation
connecting the Jones polynomial with the Potts model, and used this relation to
derive the Jones polynomial for a repeating chain knot with crossings for
general . It is found that zeroes of its Jones polynomial lie on three
closed curves centered about the points and . In addition, there are
two isolated zeroes located one each near the points
at a distance of the order of . Closed-form expressions are
deduced for the closed curves in the limit of .Comment: 12 pages, 5 figure
Close-packed dimers on nonorientable surfaces
The problem of enumerating dimers on an M x N net embedded on non-orientable
surfaces is considered. We solve both the Moebius strip and Klein bottle
problems for all M and N with the aid of imaginary dimer weights. The use of
imaginary weights simplifies the analysis, and as a result we obtain new
compact solutions in the form of double products. The compact expressions also
permit us to establish a general reciprocity theorem.Comment: 13 pages, 1 figure, typo corrected to the version published in Phys.
Lett. A 293, 235 (2002
Interacting dimers on the honeycomb lattice: An exact solution of the five-vertex model
The problem of close-packed dimers on the honeycomb lattice was solved by
Kasteleyn in 1963. Here we extend the solution to include interactions between
neighboring dimers in two spatial lattice directions. The solution is obtained
by using the method of Bethe ansatz and by converting the dimer problem into a
five-vertex problem. The complete phase diagram is obtained and it is found
that a new frozen phase, in which the attracting dimers prevail, arises when
the interaction is attractive. For repulsive dimer interactions a new
first-order line separating two frozen phases occurs. The transitions are
continuous and the critical behavior in the disorder regime is found to be the
same as in the case of noninteracting dimers characterized by a specific heat
exponent \a=1/2.Comment: latex, 29 pages + 7 figure
Unsigned state models for the Jones polynomial
It is well a known and fundamental result that the Jones polynomial can be
expressed as Potts and vertex partition functions of signed plane graphs. Here
we consider constructions of the Jones polynomial as state models of unsigned
graphs and show that the Jones polynomial of any link can be expressed as a
vertex model of an unsigned embedded graph.
In the process of deriving this result, we show that for every diagram of a
link in the 3-sphere there exists a diagram of an alternating link in a
thickened surface (and an alternating virtual link) with the same Kauffman
bracket. We also recover two recent results in the literature relating the
Jones and Bollobas-Riordan polynomials and show they arise from two different
interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric
Recommended from our members
Effect of Second-Phase Doping on Laser Deposited Al2O3 Ceramics
Direct fabrication of engineering ceramic components by additive manufacturing (AM) is a
relatively new method for producing complex mechanical structures. This study investigates how
a second-phase doping may affect Al2O3 ceramic parts deposited by AM with a laser engineered
net shaping (LENS) system. In this study, ZrO2 and Y2O3 powders are respectively doped into
Al2O3 powders at the eutectic ratio as second-phases to improve the quality of a deposited part.
The deposited Al2O3, Al2O3/ZrO2 and Al2O3/YAG (yttrium aluminum garnet) parts are examined
for their micro-structures and micro-hardness, as well as defects. The results show that doping of
ZrO2 or Y2O3 as a second-phase performs a significant role in suppressing cracks and in refining
grains of the laser deposited parts. The micro-hardness investigation reveals that the
second-phase doping does not result in much hardness reduction in Al2O3 and the two eutectic
ceramics are both harder than 1500 Hv. The study concludes that the second-phase doping is
good for improving laser deposited ceramic parts.Mechanical Engineerin
Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings
We present exact results on the partition function of the -state Potts
model on various families of graphs in a generalized external magnetic
field that favors or disfavors spin values in a subset of
the total set of possible spin values, , where and are
temperature- and field-dependent Boltzmann variables. We remark on differences
in thermodynamic behavior between our model with a generalized external
magnetic field and the Potts model with a conventional magnetic field that
favors or disfavors a single spin value. Exact results are also given for the
interesting special case of the zero-temperature Potts antiferromagnet,
corresponding to a set-weighted chromatic polynomial that counts
the number of colorings of the vertices of subject to the condition that
colors of adjacent vertices are different, with a weighting that favors or
disfavors colors in the interval . We derive powerful new upper and lower
bounds on for the ferromagnetic case in terms of zero-field
Potts partition functions with certain transformed arguments. We also prove
general inequalities for on different families of tree graphs.
As part of our analysis, we elucidate how the field-dependent Potts partition
function and weighted-set chromatic polynomial distinguish, respectively,
between Tutte-equivalent and chromatically equivalent pairs of graphs.Comment: 39 pages, 1 figur
Analysis and design of power management scheme for an on-board solar energy storage system
This paper investigates the power management issues in a mobile solar energy storage system. A multi-converter based energy storage system is proposed, in which solar power is the primary source while the grid or the diesel generator is selected as the secondary source. The existence of the secondary source facilitates the battery state of charge detection by providing a constant battery charging current. Converter modeling, multi-converter control system design, digital implementation and experimental verification are introduced and discussed in details. The prototype experiment indicates that the converter system can provide a constant charging current during solar converter maximum power tracking operation, especially during large solar power output variation, which proves the feasibility of the proposed design
- …