A two-dimensional version of the folklore theorem

1995 ◽  
pp. 89-105
Author(s):  
Michael Jakobson ◽  
Sheldon Newhouse
Keyword(s):  
Two Dimensional ◽  
Dimensional Version

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?

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.

Analysis on arithmetic schemes. II

2010 ◽  
Vol 5 (3) ◽  
pp. 437-557 ◽  
Author(s):  
Ivan Fesenko
Keyword(s):  
Zeta Function ◽  
Logarithmic Derivative ◽  
Boundary Function ◽  
Boundary Integral ◽  
Integration Theory ◽  
Meromorphic Continuation ◽  
Two Dimensional ◽  
Fundamental Properties ◽  
Arithmetic Surfaces ◽  
Dimensional Version

AbstractWe construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K2-delic and K1×K1-delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.

Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves

2002 ◽  
Vol 450 ◽  
pp. 201-205 ◽  
Author(s):  
ELIEZER KIT ◽  
LEV SHEMER
Keyword(s):  
Gravity Waves ◽  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Velocity Potential ◽  
Surface Elevation ◽  
Two Dimensional ◽  
Zakharov Equation ◽  
Dimensional Version

A spatial two-dimensional version of the Zakharov equation describing the evolution of deep-water gravity waves is used to derive two fourth-order evolution equations, for the amplitudes of the surface elevation and of the velocity potential. The scaled form of the equations is presented.

A global two-dimensional version of Smale’s cancellation theorem via spectral sequences

2015 ◽  
Vol 36 (6) ◽  
pp. 1795-1838 ◽  
Author(s):  
M. A. BERTOLIM ◽  
D. V. S. LIMA ◽  
M. P. MELLO ◽  
K. A. DE REZENDE ◽  
M. R. DA SILVEIRA
Keyword(s):  
Sequence Analysis ◽  
Spectral Sequence ◽  
Matrix Theory ◽  
Chain Complex ◽  
Connection Matrix ◽  
Local Version ◽  
Two Dimensional ◽  
Dimensional Version ◽  
Cancellation Theorem ◽  
Flows On Surfaces

In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$. The local version of this theorem relates differentials $d^{r}$ of the $r$th page $E^{r}$ to Smale’s theorem on cancellation of critical points.

scholarly journals Global Attractors of Sixth Order PDEs Describing the Faceting of Growing Surfaces

2015 ◽  
Vol 28 (1) ◽  
pp. 49-67 ◽  
Author(s):  
M. D. Korzec ◽  
P. Nayar ◽  
P. Rybka
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Initial Conditions ◽  
Periodic Boundary Conditions ◽  
Global Attractors ◽  
Sixth Order ◽  
Two Dimensional ◽  
One Dimensional ◽  
Crystalline Surface ◽  
Dimensional Version

Abstract A spatially two-dimensional sixth order PDE describing the evolution of a growing crystalline surface h(x, y, t) that undergoes faceting is considered with periodic boundary conditions, as well as its reduced one-dimensional version. These equations are expressed in terms of the slopes $$u_1=h_{x}$$ u 1 = h x and $$u_2=h_y$$ u 2 = h y to establish the existence of global, connected attractors for both equations. Since unique solutions are guaranteed for initial conditions in $$\dot{H}^2_{per}$$ H ˙ p e r 2 , we consider the solution operator $$S(t): \dot{H}^2_{per} \rightarrow \dot{H}^2_{per}$$ S ( t ) : H ˙ p e r 2 → H ˙ p e r 2 , to gain our results. We prove the necessary continuity, dissipation and compactness properties.

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