Edges in models of shear flow

2013 ◽  
Vol 721 ◽  
pp. 386-402 ◽  
Author(s):  
Norman Lebovitz ◽  
Giulio Mariotti
Keyword(s):  
Shear Flow ◽  
Characteristic Feature ◽  
Stable Manifold ◽  
Geometric Features ◽  
Higher Dimension ◽  
Lower Branch ◽  
Codimension One ◽  
Onset Of Turbulence ◽  
Low Dimensional ◽  
Laminar State

AbstractA characteristic feature of the onset of turbulence in shear flows is the appearance of an ‘edge’, a codimension-one invariant manifold that separates ‘lower’ orbits, which decay directly to the laminar state, from ‘upper’ orbits, which decay more slowly and less directly. The object of this paper is to elucidate the structure of the edge that makes this behaviour possible. To this end we consider a succession of low-dimensional models. In doing this we isolate geometric features that are robust under increase of dimension and are therefore candidates for explaining analogous features in higher dimension. We find that the edge, which is the stable manifold of a ‘lower-branch’ state, winds endlessly around an ‘upper-branch’ state in such a way that upper orbits are able to circumnavigate the edge and return to the laminar state.

scholarly journals Flags of holomorphic foliations

2011 ◽  
Vol 83 (3) ◽  
pp. 775-786 ◽  
Author(s):  
Rogério S. Mol
Keyword(s):  
Finite Number ◽  
Complex Manifold ◽  
Necessary Conditions ◽  
Holomorphic Foliations ◽  
Lower Dimension ◽  
Higher Dimension ◽  
Basic Properties ◽  
Codimension One ◽  
Tangent Sheaf ◽  
Polar Classes

A flag of holomorphic foliations on a complex manifold M is an object consisting of a finite number of singular holomorphic foliations on M of growing dimensions such that the tangent sheaf of a fixed foliation is a subsheaf of the tangent sheaf of any of the foliations of higher dimension. We study some basic properties oft hese objects and, in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" />, n > 3, we establish some necessary conditions for a foliation, we find bounds of lower dimension to leave invariant foliations of codimension one. Finally, still in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" /> involving the degrees of polar classes of foliations in a flag.

scholarly journals Studying edge geometry in transiently turbulent shear flows

2014 ◽  
Vol 747 ◽  
pp. 506-517 ◽  
Author(s):  
Matthew Chantry ◽  
Tobias M. Schneider
Keyword(s):  
Shear Flows ◽  
Feedback Loops ◽  
Turbulent Shear ◽  
Dynamical Structure ◽  
Plane Couette Flow ◽  
Edge Of Chaos ◽  
Edge Geometry ◽  
Codimension One ◽  
Moderate Reynolds Number ◽  
Low Dimensional

AbstractIn linearly stable shear flows at moderate Reynolds number, turbulence spontaneously decays despite the existence of a codimension-one manifold, termed the edge, which separates decaying perturbations from those triggering turbulence. We statistically analyse the decay in plane Couette flow, quantify the breaking of self-sustaining feedback loops and demonstrate the existence of a whole continuum of possible decay paths. Drawing parallels with low-dimensional models and monitoring the location of the edge relative to decaying trajectories, we provide evidence that the edge of chaos does not separate state space globally. It is instead wrapped around the turbulence generating structures and not an independent dynamical structure but part of the chaotic saddle. Thereby, decaying trajectories need not cross the edge, but circumnavigate it while unwrapping from the turbulent saddle.

scholarly journals Decoherence-Free Subspaces for a Quantum Register Interacting with a Spin Environment

2013 ◽  
Vol 20 (04) ◽  
pp. 1350014 ◽  
Author(s):  
Paweł Należyty ◽  
Dariusz Chruściński
Keyword(s):  
Necessary Conditions ◽  
General Setting ◽  
Quantum Spin ◽  
High Dimensional ◽  
Higher Dimension ◽  
Quantum Register ◽  
Measurement Limit ◽  
Sufficient And Necessary Conditions ◽  
Low Dimensional ◽  
Decoherence Free Subspaces

We study a model of a quantum spin register interacting with an environment of spin particles in quantum-measurement limit. In the limit of collective decoherence we obtain the form of state vectors that constitute high-dimensional decoherence-free subspaces (DFS). In a more general setting we present sufficient and necessary conditions for the existence of low-dimensional DFSs that can be used to construct subspaces of higher dimension.

scholarly journals Universal continuous transition to turbulence in a planar shear flow

2017 ◽  
Vol 824 ◽  
Author(s):  
Matthew Chantry ◽  
Laurette S. Tuckerman ◽  
Dwight Barkley
Keyword(s):  
Shear Flow ◽  
Universality Class ◽  
Transition To Turbulence ◽  
Free Boundaries ◽  
Turbulent Structures ◽  
Continuous Transition ◽  
Normal Representation ◽  
Onset Of Turbulence ◽  
Planar Shear ◽  
Order Of Magnitude

We examine the onset of turbulence in Waleffe flow – the planar shear flow between stress-free boundaries driven by a sinusoidal body force. By truncating the wall-normal representation to four modes, we are able to simulate system sizes an order of magnitude larger than any previously simulated, and thereby to attack the question of universality for a planar shear flow. We demonstrate that the equilibrium turbulence fraction increases continuously from zero above a critical Reynolds number and that statistics of the turbulent structures exhibit the power-law scalings of the (2 + 1)-D directed-percolation universality class.

The Structure of the Polytope of Hereditary Information

2019 ◽  
Vol 8 (2) ◽  
pp. 7-22
Author(s):  
Gennadiy Vladimirovich Zhizhin
Keyword(s):  
Nucleic Acid ◽  
Hydrogen Bonds ◽  
Phosphoric Acid ◽  
High Intensity ◽  
Information Exchange ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Nitrogen Bases ◽  
Higher Dimensional ◽  
Low Dimensional

The representations of the sugar molecule and the residue of phosphoric acid in the form of polytopes of higher dimension are used. Based on these ideas and their simplified three-dimensional images, a three-dimensional image of nucleic acids is constructed. The geometry of the neighborhood of the compound of two nucleic acid helices with nitrogen bases has been investigated in detail. It is proved that this neighborhood is a cross-polytope of dimension 13 (polytope of hereditary information), in the coordinate planes of which there are complementary hydrogen bonds of nitrogenous bases. The structure of this polytope is defined, and its image is given. The total incident flows from the low-dimensional elements to the higher-dimensional elements and vice versa of the hereditary information polytope are calculated equal to each other. High values of these flows indicate a high intensity of information exchange in the polytope of hereditary information that ensures the transfer of this information.

Polytopes Dual to Polytopic Prismahedrons

2018 ◽  
pp. 141-160
Keyword(s):  
Higher Dimension ◽  
New Class ◽  
Low Dimensional

The polytopes are dual to polytypic prismahedrons. In particular, polytopes dual to the product of two canons. It is shown that these polytopes form a new class of polytopes with different values of the incidence of elements of low-dimensional polytopes to polytopes of higher dimension entering the polytope. If the polygons in their product have equal sides, then the dual polytope to the product consists of tetrahedrons, and the degree of incidence of the edge of the dual polytope is determined by the number of sides of the polygon. The existence of a previously unknown polytope consisting of one hundred tetrahedrons is established. Its election is constructed, all its constituent tetrahedrons are listed.

scholarly journals Superstable manifolds of invariant circles and codimension-one Böttcher functions

2013 ◽  
Vol 35 (1) ◽  
pp. 152-175 ◽  
Author(s):  
SCOTT R. KASCHNER ◽  
ROLAND K. W. ROEDER
Keyword(s):  
Complex Manifold ◽  
Stable Manifold ◽  
Invariant Circle ◽  
Codimension One ◽  
Invariant Circles ◽  
Local Stable Manifold ◽  
Real Analytic

AbstractLet $f: X~\dashrightarrow ~X$ be a dominant meromorphic self-map, where $X$ is a compact, connected complex manifold of dimension $n\gt 1$. Suppose that there is an embedded copy of ${ \mathbb{P} }^{1} $ that is invariant under $f$, with $f$ holomorphic and transversally superattracting with degree $a$ in some neighborhood. Suppose that $f$ restricted to this line is given by $z\mapsto {z}^{b} $, with resulting invariant circle $S$. We prove that if $a\geq b$, then the local stable manifold ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition $a\geq b$ cannot be relaxed without adding additional hypotheses by presenting two examples with $a\lt b$ for which ${ \mathcal{W} }_{\mathrm{loc} }^{s} (S)$ is not real analytic in the neighborhood of any point.

scholarly journals Birational boundedness of low-dimensional elliptic Calabi–Yau varieties with a section

2021 ◽  
Vol 157 (8) ◽  
pp. 1766-1806
Author(s):  
Gabriele Di Cerbo ◽  
Roberto Svaldi
Keyword(s):  
Product Type ◽  
Codimension One ◽  
Low Dimensional ◽  
Calabi Yau Manifolds

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi–Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim (Y)\leq 5$ and $Y$ is not of product type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of Kawamata log terminal pairs $(X, \Delta )$ with $K_X+\Delta$ numerically trivial and not of product type, in dimension at most four.

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