548 research outputs found
Ramanujan and Extensions and Contractions of Continued Fractions
If a continued fraction is known to converge
but its limit is not easy to determine, it may be easier to use an extension of
to find the limit. By an extension of
we mean a continued fraction whose odd or even part is . One can
then possibly find the limit in one of three ways:
(i) Prove the extension converges and find its limit;
(ii) Prove the extension converges and find the limit of the other
contraction (for example, the odd part, if is the
even part);
(ii) Find the limit of the other contraction and show that the odd and even
parts of the extension tend to the same limit.
We apply these ideas to derive new proofs of certain continued fraction
identities of Ramanujan and to prove a generalization of an identity involving
the Rogers-Ramanujan continued fraction, which was conjectured by Blecksmith
and Brillhart.Comment: 16 page
The Random-bond Potts model in the large-q limit
We study the critical behavior of the q-state Potts model with random
ferromagnetic couplings. Working with the cluster representation the partition
sum of the model in the large-q limit is dominated by a single graph, the
fractal properties of which are related to the critical singularities of the
random Potts model. The optimization problem of finding the dominant graph, is
studied on the square lattice by simulated annealing and by a combinatorial
algorithm. Critical exponents of the magnetization and the correlation length
are estimated and conformal predictions are compared with numerical results.Comment: 7 pages, 6 figure
Case studies of Roots, Tubers and Bananas seed systems.
The seed systems of RTB (root, tuber, and banana) crops are unique because they are propagated from vegetative parts of the plant, not from true seed. RTB seed is thus bulkier, more perishable, and more subject to the attacks of pests and diseases than is true seed. Because of this, there is often a gap between potential and real crop yields, which seed interventions seek to narrow. Seed systems are formal or informal networks of people and organizations that produce, plant, and distribute seed. Informal systems may deliver low quality seed, but not always. This book describes 13 RTB seed system interventions, using a framework based on the concepts of seed availability, access, and quality. The 13 case studies included (1) a potato-growers’ association in Ecuador, (2) a hydroponic seed potato in Peru, (3) a yam seed technology in Nigeria, (4) a banana and plantain project in Ghana, (5) a sweetpotato seed project in Tanzania and (6) one in Rwanda, (7) a seed potato system in Kenya, (8) cassava in Nicaragua, (9) seed potato in Malawi, (10) disease-resistant cassava varieties in seven African countries, (11) a tissue culture banana project, (12) an emergency plantain and banana project in East Africa, and (13) a large cassava seed project in six African countries
Transfer matrices and partition-function zeros for antiferromagnetic Potts models. VI. Square lattice with special boundary conditions
We study, using transfer-matrix methods, the partition-function zeros of the
square-lattice q-state Potts antiferromagnet at zero temperature (=
square-lattice chromatic polynomial) for the special boundary conditions that
are obtained from an m x n grid with free boundary conditions by adjoining one
new vertex adjacent to all the sites in the leftmost column and a second new
vertex adjacent to all the sites in the rightmost column. We provide numerical
evidence that the partition-function zeros are becoming dense everywhere in the
complex q-plane outside the limiting curve B_\infty(sq) for this model with
ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the
infinite-volume free energy is perfectly analytic in this region.Comment: 114 pages (LaTeX2e). Includes tex file, three sty files, and 23
Postscript figures. Also included are Mathematica files data_Eq.m,
data_Neq.m,and data_Diff.m. Many changes from version 1, including several
proofs of previously conjectured results. Final version to be published in J.
Stat. Phy
Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice
We present exact calculations of the Potts model partition function
for arbitrary and temperature-like variable on -vertex
strip graphs of the honeycomb lattice for a variety of transverse widths
equal to vertices and for arbitrarily great length, with free
longitudinal boundary conditions and free and periodic transverse boundary
conditions. These partition functions have the form
, where
denotes the number of repeated subgraphs in the longitudinal direction. We give
general formulas for for arbitrary . We also present plots of
zeros of the partition function in the plane for various values of and
in the plane for various values of . Explicit results for partition
functions are given in the text for (free) and (cylindrical),
and plots of partition function zeros are given for up to 5 (free) and
(cylindrical). Plots of the internal energy and specific heat per site
for infinite-length strips are also presented.Comment: 39 pages, 34 eps figures, 3 sty file
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
We derive some new structural results for the transfer matrix of
square-lattice Potts models with free and cylindrical boundary conditions. In
particular, we obtain explicit closed-form expressions for the dominant (at
large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as
the solution of a special one-dimensional polymer model. We also obtain the
large-q expansion of the bulk and surface (resp. corner) free energies for the
zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47}
(resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <=
m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19
Postscript figures. Also included are Mathematica files data_CYL.m and
data_FREE.m. Many changes from version 1: new material on series expansions
and their analysis, and several proofs of previously conjectured results.
Final version to be published in J. Stat. Phy
On inversions and Doob -transforms of linear diffusions
Let be a regular linear diffusion whose state space is an open interval
. We consider a diffusion which probability law is
obtained as a Doob -transform of the law of , where is a positive
harmonic function for the infinitesimal generator of on . This is the
dual of with respect to where is the speed measure of
. Examples include the case where is conditioned to stay above
some fixed level. We provide a construction of as a deterministic
inversion of , time changed with some random clock. The study involves the
construction of some inversions which generalize the Euclidean inversions.
Brownian motion with drift and Bessel processes are considered in details.Comment: 19 page
Simulation of Potts models with real q and no critical slowing down
A Monte Carlo algorithm is proposed to simulate ferromagnetic q-state Potts
model for any real q>0. A single update is a random sequence of disordering and
deterministic moves, one for each link of the lattice. A disordering move
attributes a random value to the link, regardless of the state of the system,
while in a deterministic move this value is a state function. The relative
frequency of these moves depends on the two parameters q and beta. The
algorithm is not affected by critical slowing down and the dynamical critical
exponent z is exactly vanishing. We simulate in this way a 3D Potts model in
the range 2<q<3 for estimating the critical value q_c where the thermal
transition changes from second-order to first-order, and find q_c=2.620(5).Comment: 5 pages, 3 figures slightly extended version, to appear in Phys. Rev.
The Harris-Luck criterion for random lattices
The Harris-Luck criterion judges the relevance of (potentially) spatially
correlated, quenched disorder induced by, e.g., random bonds, randomly diluted
sites or a quasi-periodicity of the lattice, for altering the critical behavior
of a coupled matter system. We investigate the applicability of this type of
criterion to the case of spin variables coupled to random lattices. Their
aptitude to alter critical behavior depends on the degree of spatial
correlations present, which is quantified by a wandering exponent. We consider
the cases of Poissonian random graphs resulting from the Voronoi-Delaunay
construction and of planar, ``fat'' Feynman diagrams and precisely
determine their wandering exponents. The resulting predictions are compared to
various exact and numerical results for the Potts model coupled to these
quenched ensembles of random graphs.Comment: 13 pages, 9 figures, 2 tables, REVTeX 4. Version as published, one
figure added for clarification, minor re-wordings and typo cleanu
- …