scholarly journals Adelic Lefschetz formula for the action of a one-dimensional torus

2006 ◽  
pp. 31-48 ◽  
Author(s):  
S. O. Gorchinskii ◽  
A. N. Parshin
Keyword(s):  
Dimensional Torus ◽  
One Dimensional ◽  
Lefschetz Formula

Asymptotic estimate for the counting problems corresponding to the dynamical system on some decorated graphs

2017 ◽  
Vol 38 (5) ◽  
pp. 1697-1708 ◽  
Author(s):  
V. L. CHERNYSHEV ◽  
A. A. TOLCHENNIKOV
Keyword(s):  
Dynamical System ◽  
Wave Packet ◽  
General Theorem ◽  
Three Dimensional ◽  
Initial Time ◽  
Dimensional Torus ◽  
Counting Problems ◽  
One Dimensional ◽  
Moving Points ◽  
Narrow Wave

We study a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds, that is, a decorated graph. We consider the following dynamical system on decorated graphs. Suppose that, at the initial time, we have a narrow wave packet on a one-dimensional edge. It can be thought of as a point moving along the edge. When a packet arrives at the point of gluing, the expanding wavefront begins to spread on the Riemannian manifold. At the same time, there is a partial reflection of the wave packet. When the wavefront that propagates on the surface comes to another point of gluing, it generates a reflected wavefront and a wave packet on an edge. We study the number of Gaussian packets, that is, moving points on one-dimensional edges as time goes to infinity. We prove the asymptotic estimations for this number for the following decorated graphs: a cylinder with an interval, a two-dimensional torus with an interval and a three-dimensional torus with an interval. Also we prove a general theorem about a manifold with an interval and apply it to the case of a uniformly secure manifold.

scholarly journals CELLULAR AUTOMATON SUPERCOLLIDERS

2011 ◽  
Vol 22 (04) ◽  
pp. 419-439 ◽  
Author(s):  
GENARO J. MARTÍNEZ ◽  
ANDREW ADAMATZKY ◽  
CHRISTOPHER R. STEPHENS ◽  
ALEJANDRO F. HOEFLICH
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Elastic Collision ◽  
Collective Excitations ◽  
Compact Groups ◽  
Dimensional Torus ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Spatially Extended ◽  
Binary Strings

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular automaton analogous of localizations or quasi-local collective excitations traveling in a spatially extended nonlinear medium. They can be considered as binary strings or symbols traveling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyze what types of interaction occur between gliders traveling on a cellular automaton "cyclotron" and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in nonlinear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analyzed via implementation of cyclic tag systems.

scholarly journals Well-posedness of the linear heat equation with a second order memory term

2020 ◽  
Vol 34 ◽  
pp. 03011
Author(s):  
Constantin Niţă ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Unique Solution ◽  
Source Term ◽  
Memory Term ◽  
Well Posedness ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Linear Heat ◽  
The One

In this article we prove that the heat equation with a memory term on the one-dimensional torus has a unique solution and we study the smoothness properties of this solution. These properties are related with some smoothness assumptions imposed to the initial data of the problem and to the source term.

scholarly journals Large semigroups of cellular automata

2011 ◽  
Vol 32 (6) ◽  
pp. 1991-2010 ◽  
Author(s):  
YAIR HARTMAN
Keyword(s):  
Cellular Automata ◽  
Field Structure ◽  
The Other ◽  
Entire Space ◽  
Dimensional Torus ◽  
Semigroup Action ◽  
Closed Set ◽  
One Dimensional ◽  
Shift Space ◽  
The One

AbstractIn this article, we consider semigroups of transformations of cellular automata which act on a fixed shift space. In particular, we are interested in two properties of these semigroups which relate to ‘largeness’: first, a semigroup has the ID (infinite is dense) property if the only infinite invariant closed set (with respect to the semigroup action) is the entire space; the second property is maximal commutativity (MC). We shall consider two examples of semigroups: one is spanned by cellular automata transformations that represent multiplications by integers on the one-dimensional torus, and the other one consists of all the cellular automata transformations which are linear (when the symbols set is the ring ℤ/sℤ). It will be shown that these two properties of these semigroups depend on the number of symbols s. The multiplication semigroup is ID and MC if and only if s is not a power of a prime. The linear semigroup over the mentioned ring is always MC but is ID if and only if s is prime. When the symbol set is endowed with a finite field structure (when possible), the linear semigroup is both ID and MC. In addition, we associate with each semigroup which acts on a one-sided shift space a semigroup acting on a two-sided shift space, and vice versa, in a way that preserves the ID and the MC properties.

Rotation measures for homeomorphisms of the torus homotopic to a Dehn twist

1997 ◽  
Vol 17 (3) ◽  
pp. 575-591 ◽  
Author(s):  
H. ERIK DOEFF
Keyword(s):  
Fixed Point ◽  
Rotation Number ◽  
Rational Number ◽  
Dehn Twist ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Rotation Vectors ◽  
Rotation Set ◽  
Area Preserving

We extend the theory of rotation vectors to homeomorphisms of the two-dimensional torus that are homotopic to a Dehn twist. We define a one-dimensional rotation number and recreate the theory of the homotopic case to the identity case. We prove that if such a map is area preserving and has mean rotation number zero, then it must have a fixed point. We prove that the rotation set is a compact interval, and that if the rotation interval contains two distinct numbers, then for any rational number in the rotation set there exists a periodic point with that rotation number. Finally, we prove that any interval with rational endpoints can be realized as the rotation set of a map homotopic to a Dehn twist.

THE FUNDAMENTAL GROUP OF GALOIS COVER OF THE SURFACE 𝕋 × 𝕋

2008 ◽  
Vol 18 (08) ◽  
pp. 1259-1282 ◽  
Author(s):  
MEIRAV AMRAM ◽  
MINA TEICHER ◽  
UZI VISHNE
Keyword(s):  
Fundamental Group ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Generic Projection ◽  
Galois Cover ◽  
The One ◽  
First Time

This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.

THE EXPONENTIAL LOCALIZATION AND STRUCTURE OF THE SPECTRUM FOR 1D QUASI-PERIODIC DISCRETE SCHRÖDINGER OPERATORS

1991 ◽  
Vol 03 (03) ◽  
pp. 241-284 ◽  
Author(s):  
V. A. CHULAEVSKY ◽  
YA. G. SINAI
Keyword(s):  
Spectral Measure ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Discrete Schrödinger Operators ◽  
Exponential Localization

We discuss main mechanisms of the exponential localization of the eigenfunctions for one-dimensional quasi-periodic Schrödinger operators with the potential of the form V(α + nω), where V(α) is a non-degenerate C2-function on the d-dimensional torus, and ω ∈ ℝd is a typical vector with rationally incommensurate components. The exponential localization is proved so far for d ≤ 2. We emphasize the different nature of the support of the spectral measure for d = 1 and for d > 1.

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