scholarly journals Non-Repetitive Tilings

10.37236/1644 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
James D. Currie ◽  
Jamie Simpson
Keyword(s):  
Higher Dimensions ◽  
Two Dimensional ◽  
Dimensional Version ◽  
New Type ◽  
Free Word

In 1906 Axel Thue showed how to construct an infinite non-repetitive (or square-free) word on an alphabet of size 3. Since then this result has been rediscovered many times and extended in many ways. We present a two-dimensional version of this result. We show how to construct a rectangular tiling of the plane using 5 symbols which has the property that lines of tiles which are horizontal, vertical or have slope +1 or $-1$ contain no repetitions. As part of the construction we introduce a new type of word, one that is non-repetitive up to mod k, which is of interest in itself. We also indicate how our results might be extended to higher dimensions.

Ferroelectric Liquid Crystal Light Valve Using SiO2 / A-Si:H Photodiode

1991 ◽  
Vol 219 ◽  
Author(s):  
T. Horikawa ◽  
S. Tahata ◽  
S. Kaho ◽  
T. Masumi ◽  
N. Mikami ◽  
...  
Keyword(s):  
Liquid Crystal ◽  
Dark Conductivity ◽  
Ferroelectric Liquid Crystal ◽  
Two Dimensional ◽  
Metal Insulator ◽  
Resolution Capability ◽  
New Type ◽  
Light Valve ◽  
Mis Diode ◽  
High Photosensitivity

ABSTRACTA new type of ferroelectric liquid crystal light valve (FLCLV) is presented. The design of the FLCLV is based upon the linear equivalent circuit analyses. A photosensor in the FLCLV consists of a metal-insulator-semiconductor (MIS) photodiode. A-Si:H doped with boron and nitrogen [a-Si:(:N:B)] is used in the MIS diode. The a-Si:H(:B:N) film has a dark-conductivity of less than 1×1012 S/cm and a high photosensitivity.Consequently, the writing characteristics of the FLCLV for a two dimensional (2D) image are evaluated. Using writing light of 630 nm and 1 mW/cm2, a high resolution capability of 120∼140 1p/mm is obtained.

Two-dimensional material-based functional aerogels for treating hazards in environment: Synthesis, functional tailoring, applications, and sustainability analysis

2022 ◽  
Author(s):  
Hao Kong ◽  
Yun Chen ◽  
Guozheng Yang ◽  
Bin Liu ◽  
Lei Guo ◽  
...  
Keyword(s):  
Environmental Pollution ◽  
Human Health ◽  
Two Dimensional ◽  
Ecological Balance ◽  
Global Problem ◽  
Sustainability Analysis ◽  
New Type

Environmental pollution is a global problem that endangers human health and ecological balance. As a new type of nanomaterial, two-dimensional material (2DM)-based aerogel is one of the most promising candidates...

scholarly journals Geometry of the 3D Pythagoras' Theorem

2016 ◽  
Vol 8 (6) ◽  
pp. 78 ◽  
Author(s):  
Luis Teia
Keyword(s):  
Three Dimensional ◽  
Transformation Process ◽  
Two Dimensional ◽  
Natural Evolution ◽  
Dimensional Version ◽  
Pythagoras Theorem

This paper explains step-by-step how to construct the 3D Pythagoras' theorem by geometric manipulation of the two dimensional version. In it is shown how $x+y=z$ (1D Pythagoras' theorem) transforms into $x^2+y^2=z^2$ (2D Pythagoras' theorem) via two steps: a 90-degree rotation, and a perpendicular extrusion. Similarly, the 2D Pythagoras' theorem transforms into 3D using the same steps. Octahedrons emerge naturally during this transformation process. Hence, each of the two dimensional elements has a direct three dimensional equivalent. Just like squares govern the 2D, octahedrons are the basic elements that govern the geometry of the 3D Pythagoras' theorem. As a conclusion, the geometry of the 3D Pythagoras' theorem is a natural evolution of the 1D and 2D. This interdimensional evolution begs the question -- Is there a bigger theorem at play that encompasses all three?

Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime

1995 ◽  
Vol 291 ◽  
pp. 57-81 ◽  
Author(s):  
S. M. Churilov ◽  
I. G. Shukhman
Keyword(s):  
Travelling Waves ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Critical Layer ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
New Type ◽  
Two Stages

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

Solutions to the Tipping Point Problem

2012 ◽  
Vol 106 (1) ◽  
pp. 60-63
Keyword(s):  
Ice Cream ◽  
Tipping Point ◽  
Rectangular Prism ◽  
Point Problem ◽  
Two Dimensional ◽  
Dimensional Version ◽  
Fixed Area

The problem posed in MT August 2011 (vol. 105, no. 1, pp. 62-66) asked readers to consider the two-dimensional version of tipping a bowl (assumed to be a rectangular prism) to spoon out the last little bit of melted ice cream. Here is the essence of the problem: Given a fluid region of fixed area A contained in a rectangle whose width is W, find a formula for the fluid depth D when the container is tilted through a known angle T that is measured from horizontal.

ON TWO-DIMENSIONAL DYNAMICAL SYSTEMS ASSOCIATED WITH MEMBRANE KINETICS UNDERLYING CARDIAC ARRHYTHMIAS

2009 ◽  
Vol 19 (05) ◽  
pp. 1709-1732 ◽  
Author(s):  
B. M. BAKER ◽  
M. E. KIDWELL ◽  
R. P. KLINE ◽  
I. POPOVICI
Keyword(s):  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Two Dimensional ◽  
Stimulus Period ◽  
Smooth Maps ◽  
Periodic Stimulation ◽  
Piecewise Smooth Maps ◽  
Dimensional Version ◽  
Bifurcation Law ◽  
The One

We study the orbits, stability and coexistence of orbits in the two-dimensional dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation — as a function of (a) kinetic parameters (two amplitudes, two rate constants) and (b) stimulus period. The original paper focused mostly on the one-dimensional version of this model (one amplitude, one rate constant), whose orbits, stability properties, and bifurcations were analyzed via the theory of skew-tent (hence unimodal) maps; the principal family of orbits were so-called "n-escalators", with n a positive integer. The two-dimensional analog (motivated by experimental results) has led to the current study of continuous, piecewise smooth maps of a polygonal planar region into itself, whose dynamical behavior includes the coexistence of stable orbits. Our principal results show (1) how the amplitude parameters control which escalators can come into existence, (2) escalator bifurcation behavior as the stimulus period is lowered — leading to a "1/n bifurcation law", and (3) the existence of basins of attraction via the coexistence of three orbits (two of them stable, one unstable) at the first (largest stimulus period) bifurcation. We consider the latter result our most important, as it is conjectured to be connected with arrhythmia.

Microstrip Gas Counters

2014 ◽  
pp. 31-51
Keyword(s):  
Glass Surface ◽  
Two Dimensional ◽  
Large Area ◽  
Proportional Chamber ◽  
Microelectronic Technology ◽  
Multiwire Proportional Chamber ◽  
Dimensional Version ◽  
A Chain ◽  
New Generation ◽  
First Time

In this chapter, the first micropattern gaseous detector, the microstrip gas counter, invented in 1988 by A. Oed, is presented. It consists of alternating anode and cathode strips with a pitch of less than 1 mm created on a glass surface. It can be considered a two-dimensional version of a multiwire proportional chamber. This was the first time microelectronic technology was applied to manufacturing of gaseous detectors. This pioneering work offers new possibilities for large area planar detectors with small gaps between the anode and the cathode electrodes (less than 0.1 mm). Initially, this detector suffered from several serious problems, such as charging up of the substrate, discharges which destroyed the thin anode strips, etc. However, by efforts of the international RD28 collaboration hosted by CERN, most of them were solved. Although nowadays this detector has very limited applications, its importance was that it triggered a chain of similar developments made by various groups, and these collective efforts finally led to the creation of a new generation of gaseous detectors-micropattern detectors.

scholarly journals The In2SeS/g-C3N4 heterostructure: a new two-dimensional material for photocatalytic water splitting

2020 ◽  
Vol 8 (20) ◽  
pp. 6923-6930 ◽  
Author(s):  
C. He ◽  
F. S. Han ◽  
J. H. Zhang ◽  
W. X. Zhang
Keyword(s):  
Visible Light ◽  
Water Splitting ◽  
Semiconductor Material ◽  
Two Dimensional ◽  
Photocatalytic Water Splitting ◽  
New Type

The In2SeS/g-C3N4 heterostructure is a new type of semiconductor material that uses visible light to split water.

An Energy-Based Similarity Subgrid-Scale Model in Two-Dimensional Planar Shear Flow

2013 ◽  
Vol 694-697 ◽  
pp. 594-600
Author(s):  
Yu Xuan Zhang ◽  
Song Ping Wu
Keyword(s):  
Shear Flow ◽  
Scale Model ◽  
Two Dimensional ◽  
Subgrid Scale ◽  
Planar Shear ◽  
New Type ◽  
Subgrid Scale Model ◽  
Layer Flow ◽  
Refined Grid ◽  
Dissipative Scale

A new type of similarity subgrid-scale (SGS) model which based on energy and dissipative scale isotropy assumption is presented. This model combines the advantages of traditional Smagorinsky SGS model with similarity SGS model. And a two-dimensional shear layer flow is simulated using refined grid result as a standard and comparing witch LES method including multiple SGS models. The results indicate that the result of SIM model much approximates to refined grid result than other SGS models.

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