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.

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.

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.

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.

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 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.

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.

The formation of gas hydrates under shock wave impact on the bubble media (two-dimensional case)

2010 ◽  
Vol 7 ◽  
pp. 90-97
Author(s):  
M.N. Galimzianov ◽  
I.A. Chiglintsev ◽  
U.O. Agisheva ◽  
V.A. Buzina
Keyword(s):  
Shock Wave ◽  
Gas Hydrates ◽  
Hydrate Formation ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Wave Impact ◽  
Bubble Liquid ◽  
One Dimensional ◽  
Bubble Media ◽  
Main Factors

Formation of gas hydrates under shock wave impact on bubble media (two-dimensional case) The dynamics of plane one-dimensional shock waves applied to the available experimental data for the water–freon media is studied on the base of the theoretical model of the bubble liquid improved with taking into account possible hydrate formation. The scheme of accounting of the bubble crushing in a shock wave that is one of the main factors in the hydrate formation intensification with increasing shock wave amplitude is proposed.

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

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