scholarly journals Emergence of complex patterns in a higher-dimensional phyllotactic system

2016 ◽  
Vol 85 (4) ◽  
Author(s):  
Robert M. Beyer ◽  
Jürgen Richter-Gebert
Keyword(s):  
Chaotic Behavior ◽  
Three Dimensions ◽  
Dynamical Model ◽  
Divergence Angle ◽  
Dimensional Case ◽  
Complex Structures ◽  
Two Dimensional ◽  
Higher Dimensional ◽  
Biological Problem ◽  
Space Occupation

A hypothesis commonly known as Hofmeister’s rule states that primordia appearing at the apical ring of a plant shoot in periodic time steps are formed in the position where the most space is available with respect to the space occupation of already-formed primordia. A corresponding two-dimensional dynamical model has been extensively studied by Douady and Couder, and shown to generate a variety of observable phyllotactic patterns indeed. In this study, motivated by mathematical interest in a theoretical phyllotaxis-inspired system rather than by a concrete biological problem, we generalize this model to three dimensions and present the dynamics observed in simulations, thereby illustrating the range of complex structures that phyllotactic mechanisms can give rise to. The patterns feature unexpected additional properties compared to the two-dimensional case, such as periodicity and chaotic behavior of the divergence angle.

A Graphical Exposition of the Ising Problem

1971 ◽  
Vol 12 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Higher Dimensional ◽  
The One

Ising [1] proposed the problem which now bears his name and solved it for the one-dimensional case only, leaving the higher dimensional cases as unsolved problems. The first solution to the two dimensional Ising problem was obtained by Onsager [6]. Onsager's method was subsequently explained more clearly by Kaufman [3]. More recently, Kac and Ward [2] discovered a simpler procedure involving determinants which is not logically complete.

An Intrinsically Three-Dimensional Fractal

2014 ◽  
Vol 24 (06) ◽  
pp. 1430017 ◽  
Author(s):  
M. Fernández-Guasti
Keyword(s):  
Bifurcation Diagram ◽  
Chaotic Behavior ◽  
Logistic Map ◽  
Three Dimensional ◽  
Periodic Points ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Hyperbolic Numbers ◽  
Self Similar ◽  
One To One

The quadratic iteration is mapped using a nondistributive real scator algebra in three dimensions. The bound set S has a rich fractal-like boundary. Periodic points on the scalar axis are necessarily surrounded by off axis divergent magnitude points. There is a one-to-one correspondence of this set with the bifurcation diagram of the logistic map. The three-dimensional S set exhibits self-similar 3D copies of the elementary fractal along the negative scalar axis. These 3D copies correspond to the windows amid the chaotic behavior of the logistic map. Nonetheless, the two-dimensional projection becomes identical to the nonfractal quadratic iteration produced with hyperbolic numbers. Two- and three-dimensional renderings are presented to explore some of the features of this set.

Some Helical Structures Generated by Reflexions

1985 ◽  
Vol 38 (3) ◽  
pp. 299 ◽  
Author(s):  
AC Hurley
Keyword(s):  
Dimensional Space ◽  
Helical Structure ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Helical Structures ◽  
Higher Dimensional ◽  
Translation Component ◽  
Explicit Formulae ◽  
Different Dimensions ◽  
Early Discussion

There has recently been a revival of interest in the helical structure built up as a column of face-sharing tetrahedra, because of possible applications in structural crystallography (Nelson 1983). This structure and its analogues in spaces of different dimensions are investigated here. It is shown that the only crystallographic cases are the structures in one- and two-dimensional space. For three and higher dimensional space the structures are all non-crystallographic. For the physically important case of three dimensions, this result is implicit in an early discussion by Coxeter (1969). Results obtained here include explicit formulae for the positions of all vertices of the simplexes for dimensions n = 1-4 and a demonstration that, for arbitrary n, the ratio of the translation component of the screw to the edge of the simplex is {6/ n(n+ I)(n+ 2)}1/2

scholarly journals AN ALTERNATIVE DIMENSIONAL REDUCTION PRESCRIPTION

1996 ◽  
Vol 11 (13) ◽  
pp. 1037-1045 ◽  
Author(s):  
J.D. EDELSTEIN ◽  
C. NÚÑEZ ◽  
F.A. SCHAPOSNIK ◽  
J.J. GIAMBIAGI
Keyword(s):  
Dimensional Reduction ◽  
Dimensional Space ◽  
Dimensional Case ◽  
Green Functions ◽  
Two Dimensional ◽  
Spatial Coordinate ◽  
Higher Dimensional

We propose an alternative dimensional reduction prescription which in respect with Green functions corresponds to dropping the extra spatial coordinate. From this, we construct the dimensionally reduced Lagrangians both for scalars and fermions, discussing bosonization and supersymmetry in the particular two-dimensional case. We argue that our proposal is in some situations more physical in the sense that it maintains the form of the interactions between particles thus preserving the dynamics corresponding to the higher-dimensional space.

The equation utt − Δu = |u|p for the critical value of p

1985 ◽  
Vol 101 (1-2) ◽  
pp. 31-44 ◽  
Author(s):  
Jack Schaeffer
Keyword(s):  
Compact Support ◽  
Finite Time ◽  
Blow Up ◽  
Critical Value ◽  
Cauchy Data ◽  
Three Dimensions ◽  
Dimensional Case ◽  
Small Data ◽  
Two Dimensional ◽  
Three Space

SynopsisThe equation utt − Δu = |u|p is considered in two and three space dimensions. Smooth Cauchy data of compact support are given at t = 0. For the case of three space dimensions, John has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if Glassey has shown the two dimensional case is similar. This paper shows that small data solutions blow up in finite time when p is the critical value, in three dimensions and in two.

FINDING SIMPLICES CONTAINING THE ORIGIN IN TWO AND THREE DIMENSIONS

2011 ◽  
Vol 21 (05) ◽  
pp. 495-506 ◽  
Author(s):  
KHALED ELBASSIONI ◽  
AMR ELMASRY ◽  
KAZUHISA MAKINO
Keyword(s):  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Line Segments

We show that finding the simplices containing a fixed given point among those defined on a set of n points can be done in O(n + k) time for the two-dimensional case, and in O(n2 + k) time for the three-dimensional case, where k is the number of these simplices. As a byproduct, we give an alternative (to the algorithm in Ref. 4) O(n log r) algorithm that finds the red-blue boundary for n bichromatic points on the line, where r is the size of this boundary. Another byproduct is an O(n2 + t) algorithm that finds the intersections of line segments having two red endpoints with those having two blue endpoints defined on a set of n bichromatic points in the plane, where t is the number of these intersections.

Shadowing property of geodesics in Hedlund's metric

1997 ◽  
Vol 17 (1) ◽  
pp. 187-203 ◽  
Author(s):  
MARK LEVI
Keyword(s):  
Geodesic Flow ◽  
Rotation Vector ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Shadowing Property ◽  
Mather Theory ◽  
Higher Dimensional

In this paper we show that the geodesic flow in a Hedlund-type metric on the 3-torus possesses the shadowing property. This implies, in particular, that any rotation vector is represented by a geodesic, a fact that in the two-dimensional case is given by the Aubry–Mather theory, while in the higher-dimensional case is still unknown.

scholarly journals The Velocity Tensor and the Momentum Tensor

10.14311/1356 ◽  
2011 ◽  
Vol 51 (2) ◽  
Author(s):  
T. Lanczewski
Keyword(s):  
Momentum Tensor ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Higher Dimensional ◽  
Velocity Tensor

This paper introduces a new object called the momentum tensor. Together with the velocity tensorit forms a basis for establishing the tensorial picture of classical and relativistic mechanics. Some properties of the momentum tensor are derived as well as its relation with the velocity tensor. For the sake of clarity only two-dimensional case is investigated. However, general conclusions are also valid for higher dimensional spacetimes.

scholarly journals Higher-Dimensional SUSY Quantum Mechanics

1997 ◽  
Vol 12 (08) ◽  
pp. 581-588 ◽  
Author(s):  
A. Das ◽  
S. Okubo ◽  
S. A. Pernice
Keyword(s):  
Quantum Mechanics ◽  
Spin Structure ◽  
Supersymmetric Quantum Mechanics ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Supersymmetry Algebra ◽  
General Properties ◽  
Higher Dimensional ◽  
Spatial Dimensions

Higher-dimensional supersymmetric quantum mechanics is studied. General properties of the two-dimensional case are presented. For three spatial dimensions or higher, a spin structure is shown to arise naturally from the nonrelativistic supersymmetry algebra.

Conley–Zehnder index and bifurcation of fixed points of Hamiltonian maps

2017 ◽  
Vol 38 (6) ◽  
pp. 2086-2107 ◽  
Author(s):  
YANXIA DENG ◽  
ZHIHONG XIA
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Detailed Analysis ◽  
Parameter Family ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Symplectic Diffeomorphisms ◽  
Higher Dimensional ◽  
The One

We study the bifurcations of fixed points of Hamiltonian maps and symplectic diffeomorphisms. We are particularly interested in the bifurcations where the Conley–Zehnder index of a fixed point changes. The main result is that when the Conley–Zehnder index of a fixed point increases (or decreases) by one or two, we observe that there are several bifurcation scenarios. Under some non-degeneracy conditions on the one-parameter family of maps, two, four or eight fixed points bifurcate from the original one. We give a relatively detailed analysis of the bifurcation in the two-dimensional case. We also show that higher-dimensional cases can be reduced to the two-dimensional case.

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