1,770 research outputs found
The number of lattice paths below a cyclically shifting boundary
We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result can be viewed as an extension of well-known enumerative formulae concerning lattice paths dominated by lines of integer slope (e.g. the generalized ballot theorem). Its proof is bijective, involving a classical “reflection” argument. Moreover, a straightforward refinement of our bijection allows for the counting of paths with a specified number of corners. We also show how the result can be applied to give elegant derivations for the number of lattice walks under certain periodic boundaries. In particular, we recover known expressions concerning paths dominated by a line of half-integer slope, and some new and old formulae for paths lying under special “staircases.
Simple formulas for lattice paths avoiding certain periodic staircase boundaries
There is a strikingly simple classical formula for the number of lattice
paths avoiding the line x = ky when k is a positive integer. We show that the
natural generalization of this simple formula continues to hold when the line x
= ky is replaced by certain periodic staircase boundaries--but only under
special conditions. The simple formula fails in general, and it remains an open
question to what extent our results can be further generalized.Comment: Accepted version (JCTA); proof of Corollary 7 expanded, and 2 new
refs adde
Quantum properties of atomic-sized conductors
Using remarkably simple experimental techniques it is possible to gently
break a metallic contact and thus form conducting nanowires. During the last
stages of the pulling a neck-shaped wire connects the two electrodes, the
diameter of which is reduced to single atom upon further stretching. For some
metals it is even possible to form a chain of individual atoms in this fashion.
Although the atomic structure of contacts can be quite complicated, as soon as
the weakest point is reduced to just a single atom the complexity is removed.
The properties of the contact are then dominantly determined by the nature of
this atom. This has allowed for quantitative comparison of theory and
experiment for many properties, and atomic contacts have proven to form a rich
test-bed for concepts from mesoscopic physics. Properties investigated include
multiple Andreev reflection, shot noise, conductance quantization, conductance
fluctuations, and dynamical Coulomb blockade. In addition, pronounced quantum
effects show up in the mechanical properties of the contacts, as seen in the
force and cohesion energy of the nanowires. We review this reseach, which has
been performed mainly during the past decade, and we discuss the results in the
context of related developments.Comment: Review, 120 pages, 98 figures. In view of the file size figures have
been compressed. A higher-resolution version can be found at:
http://lions1.leidenuniv.nl/wwwhome/ruitenbe/review/QPASC-hr-ps-v2.zip (5.6MB
zip PostScript
Two-dimensional models as testing ground for principles and concepts of local quantum physics
In the past two-dimensional models of QFT have served as theoretical
laboratories for testing new concepts under mathematically controllable
condition. In more recent times low-dimensional models (e.g. chiral models,
factorizing models) often have been treated by special recipes in a way which
sometimes led to a loss of unity of QFT. In the present work I try to
counteract this apartheid tendency by reviewing past results within the setting
of the general principles of QFT. To this I add two new ideas: (1) a modular
interpretation of the chiral model Diff(S)-covariance with a close connection
to the recently formulated local covariance principle for QFT in curved
spacetime and (2) a derivation of the chiral model temperature duality from a
suitable operator formulation of the angular Wick rotation (in analogy to the
Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational
chiral theories. The SL(2,Z) modular Verlinde relation is a special case of
this thermal duality and (within the family of rational models) the matrix S
appearing in the thermal duality relation becomes identified with the
statistics character matrix S. The relevant angular Euclideanization'' is done
in the setting of the Tomita-Takesaki modular formalism of operator algebras.
I find it appropriate to dedicate this work to the memory of J. A. Swieca
with whom I shared the interest in two-dimensional models as a testing ground
for QFT for more than one decade.
This is a significantly extended version of an ``Encyclopedia of Mathematical
Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
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