429 research outputs found
Symmetry properties of Penrose type tilings
The Penrose tiling is directly related to the atomic structure of certain
decagonal quasicrystals and, despite its aperiodicity, is highly symmetric. It
is known that the numbers 1, , , , ..., where
, are scaling factors of the Penrose tiling. We show that
the set of scaling factors is much larger, and for most of them the number of
the corresponding inflation centers is infinite.Comment: Paper submitted to Phil. Mag. (for Proceedings of Quasicrystals: The
Silver Jubilee, Tel Aviv, 14-19 October, 2007
The Ammann-Beenker tilings revisited
This paper introduces two tiles whose tilings form a one-parameter family of
tilings which can all be seen as digitization of two-dimensional planes in the
four-dimensional Euclidean space. This family contains the Ammann-Beenker
tilings as the solution of a simple optimization problem.Comment: 7 pages, 4 figure
On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms
Uniquely closable skeletons of lambda terms are Motzkin-trees that
predetermine the unique closed lambda term that can be obtained by labeling
their leaves with de Bruijn indices. Likewise, uniquely typable skeletons of
closed lambda terms predetermine the unique simply-typed lambda term that can
be obtained by labeling their leaves with de Bruijn indices.
We derive, through a sequence of logic program transformations, efficient
code for their combinatorial generation and study their statistical properties.
As a result, we obtain context-free grammars describing closable and uniquely
closable skeletons of lambda terms, opening the door for their in-depth study
with tools from analytic combinatorics.
Our empirical study of the more difficult case of (uniquely) typable terms
reveals some interesting open problems about their density and asymptotic
behavior.
As a connection between the two classes of terms, we also show that uniquely
typable closed lambda term skeletons of size are in a bijection with
binary trees of size .Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
UTxO- vs Account-Based Smart Contract Blockchain Programming Paradigms
We implement two versions of a simple but illustrative smart contract: one in
Solidity on the Ethereum blockchain platform, and one in Plutus on the Cardano
platform, with annotated code excerpts and with source code attached. We get a
clearer view of the Cardano programming model in particular by introducing a
novel mathematical abstraction which we call Idealised EUTxO. For each version
of the contract, we trace how the architectures of the underlying platforms and
their mathematics affects the natural programming styles and natural classes of
errors. We prove some simple but novel results about alpha-conversion and
observational equivalence for Cardano, and explain why Ethereum does not have
them. We conclude with a wide-ranging and detailed discussion in the light of
the examples, mathematical model, and mathematical results so far
Generalized Riemann sums
The primary aim of this chapter is, commemorating the 150th anniversary of
Riemann's death, to explain how the idea of {\it Riemann sum} is linked to
other branches of mathematics. The materials I treat are more or less classical
and elementary, thus available to the "common mathematician in the streets."
However one may still see here interesting inter-connection and cohesiveness in
mathematics
An extinction rule for a class of 1D quasicrystals
We study decorated one-dimensional quasicrystal obtained by a non-standard
projection of a part of two-dimensional lattice. We focus on the impact of
varying relative positions of decorated sites. First, we give general
expression for the structure factor. Subsequently we analyze an example of
extinction rule.Comment: 5 pages, 2 figures, LaTex2e, to appear in ICQ9 Proceeding
(Philosophical Magazine
Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata
Define a certain gambler's ruin process \mathbf{X}_{j}, \mbox{ \ }j\ge 0,
such that the increments
take values and satisfy ,
all , where if , and if .
Here denote persistence parameters and with
. The process starts at and terminates when
. Denote by , , and ,
respectively, the numbers of runs, long runs, and steps in the meander portion
of the gambler's ruin process. Define and let for some . We show exists in an explicit form. We obtain a
companion theorem for the last visit portion of the gambler's ruin.Comment: Presented at 8th International Conference on Lattice Path
Combinatorics, Cal Poly Pomona, Aug., 2015. The 2nd version has been
streamlined, with references added, including reference to a companion
document with details of calculations via Mathematica. The 3rd version has 2
new figures and improved presentatio
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