scholarly journals Continuity of the Peierls barrier and robustness of laminations

2014 ◽  
Vol 35 (4) ◽  
pp. 1263-1288 ◽  
Author(s):  
BLAŽ MRAMOR ◽  
BOB RINK
Keyword(s):  
Broad Class ◽  
Variational Problems ◽  
Similar Estimate ◽  
Solid State Physics ◽  
Research Workshop ◽  
Rotation Numbers ◽  
Peierls Barrier ◽  
Twist Maps ◽  
The Difference ◽  
Math Phys

AbstractWe study the Peierls barrier$P_{\omega }(\xi )$for a broad class of monotone variational problems. These problems arise naturally in solid state physics and from Hamiltonian twist maps. We start by deriving an estimate for the difference$\vert P_{\omega }(\xi ) - P_{q/p}(\xi ) \vert $of the Peierls barriers of rotation numbers$\omega \in {{\mathbb{R}}}$and$q/p\in {\mathbb{Q}}$. A similar estimate was obtained by Mather [Modulus of continuity for Peierls’s barrier.Proc. NATO Advanced Research Workshop on Periodic Solutions of Hamiltonian Systems and Related Topics (Il Ciocco, Italy, 13–18 October 1986) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 209).Eds. P. H. Rabinowitz, A. Ambrosetti and I. Eckeland. D. Reidel, Dordrecht, 1987, pp. 177–202] in the context of twist maps, but our proof is different and applies more generally. It follows from the estimate that$\omega \mapsto P_{\omega }(\xi )$is continuous at irrational points. Moreover, we show that the Peierls barrier depends continuously on parameters and hence that the property that a monotone variational problem admits a lamination of minimizers of rotation number$\omega \in {{\mathbb{R}}}\delimiter "026E30F {\mathbb{Q}}$is open in the$C^1$-topology.

scholarly journals LOW ENTROPY OUTPUT STATES FOR PRODUCTS OF RANDOM UNITARY CHANNELS

2013 ◽  
Vol 02 (01) ◽  
pp. 1250018 ◽  
Author(s):  
BENOÎT COLLINS ◽  
MOTOHISA FUKUDA ◽  
ION NECHITA
Keyword(s):  
Free Probability ◽  
Bell State ◽  
Quantum Channels ◽  
Unitary Operators ◽  
Graphical Calculus ◽  
Additivity Problems ◽  
Communication Capacity ◽  
The Difference ◽  
Low Entropy ◽  
Math Phys

In this paper, we study the behavior of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings [Superadditivity of communication capacity using entangled inputs, Nat. Phys.5 (2009) 255–257] and Hayden–Winter [Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1, Comm. Math. Phys.284(1) (2008) 263–280] to the additivity problems. In particular, we study in depth the difference of behavior between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in [B. Collins and I. Nechita, Random quantum channels I: Graphical calculus and the Bell state phenomenon, Comm. Math. Phys.297(2) (2010) 345–370].

scholarly journals Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

1986 ◽  
Vol 6 (2) ◽  
pp. 205-239 ◽  
Author(s):  
Kevin Hockett ◽  
Philip Holmes
Keyword(s):  
Symbolic Dynamics ◽  
Homoclinic Orbits ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Irrational Rotation Number ◽  
Rotation Set ◽  
Area Preserving

AbstractWe investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

Propriétés des attracteurs de Birkhoff

1988 ◽  
Vol 8 (2) ◽  
pp. 241-310 ◽  
Author(s):  
P. Le Calvez
Keyword(s):  
Periodic Point ◽  
Infinite Number ◽  
Twist Map ◽  
Rotation Numbers ◽  
Twist Maps

AbstractWe study dissipative twist maps of the annulus, following the ideas of G. D. Birkhoff explained in an article of 1932.In the first part, we give complete and rigorous proofs of the results of this article. We define the Birkhoff attractor of a dissipative twist map which has an attracting bounded annulus, we give its main properties and we define its upper and lower rotation numbers.In the second part we give further results on these sets, thus we show that they often coincide with the closure of a hyperbolic periodic point and that they can contain an infinite number of sinks. We also show that the Birkhoff attractors don't depend on a continuous way on the maps.

scholarly journals Tracking Control Based on Recurrent Neural Networks for Nonlinear Systems with Multiple Inputs and Unknown Deadzone

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
J. Humberto Pérez-Cruz ◽  
José de Jesús Rubio ◽  
E. Ruiz-Velázquez ◽  
G. Solís-Perales
Keyword(s):  
Neural Network ◽  
Nonlinear Systems ◽  
Exponential Convergence ◽  
Broad Class ◽  
Tracking Error ◽  
The State ◽  
Control Law ◽  
Reference Trajectory ◽  
Uncertain Dynamics ◽  
The Difference

This paper deals with the problem of trajectory tracking for a broad class of uncertain nonlinear systems with multiple inputs each one subject to an unknown symmetric deadzone. On the basis of a model of the deadzone as a combination of a linear term and a disturbance-like term, a continuous-time recurrent neural network is directly employed in order to identify the uncertain dynamics. By using a Lyapunov analysis, the exponential convergence of the identification error to a bounded zone is demonstrated. Subsequently, by a proper control law, the state of the neural network is compelled to follow a bounded reference trajectory. This control law is designed in such a way that the singularity problem is conveniently avoided and the exponential convergence to a bounded zone of the difference between the state of the neural identifier and the reference trajectory can be proven. Thus, the exponential convergence of the tracking error to a bounded zone and the boundedness of all closed-loop signals can be guaranteed. One of the main advantages of the proposed strategy is that the controller can work satisfactorily without any specific knowledge of an upper bound for the unmodeled dynamics and/or the disturbance term.

scholarly journals A more accurate Parameterization based on cosmic Age (MAPAge)

2021 ◽  
Vol 21 (11) ◽  
pp. 277
Author(s):  
Lu Huang ◽  
Zhi-Qi Huang ◽  
Zhuo-Yang Li ◽  
Huan Zhou
Keyword(s):  
Cold Dark Matter ◽  
Broad Class ◽  
Angular Diameter ◽  
Recent Letter ◽  
Λcdm Model ◽  
Forecast Data ◽  
Theoretical Predictions ◽  
The Difference ◽  
Percent Accuracy ◽  
Better Than

Abstract Recently, several statistically significant tensions between different cosmological datasets have raised doubts about the standard Lambda cold dark matter (ΛCDM) model. A recent letter (Huang 2020) suggests to use “Parameterization based on cosmic Age” (PAge) to approximate a broad class of beyond-ΛCDM models, with a typical accuracy ∼1% in angular diameter distances at z ≲ 10. In this work, we extend PAge to a More Accurate Parameterization based on cosmic Age (MAPAge) by adding a new degree of freedom η 2. The parameter η 2 describes the difference between physically motivated models and their phenomenological PAge approximations. The accuracy of MAPAge, typically of order 10−3 in angular diameter distances at z ≲ 10, is significantly better than PAge. We compare PAge and MAPAge with current observational data and forecast data. The conjecture in Huang (2020), that PAge approximation is sufficiently good for current observations, is quantitatively confirmed in this work. We also show that the extension from PAge to MAPAge is important for future observations, which typically require sub-percent accuracy in theoretical predictions.

scholarly journals Strategies for identifying dynamic regions in protein complexes: flexibility changes accompany methylation in chemotaxis receptor signaling states

2020 ◽  
Author(s):  
Nikita Malik ◽  
Katherine A Wahlbeck ◽  
Lynmarie K Thompson
Keyword(s):  
Histidine Kinase ◽  
Protein Complexes ◽  
Broad Class ◽  
Signal Propagation ◽  
List Type ◽  
Inverse Dynamic ◽  
Multiple Timescales ◽  
External Domain ◽  
Dynamic Regions ◽  
The Difference

AbstractBacterial chemoreceptors are organized in arrays composed of helical receptors arranged as trimers of dimers, coupled to a histidine kinase CheA and a coupling protein CheW. Ligand binding to the external domain inhibits the kinase activity, leading to a change in the swimming behavior. Adaptation to an ongoing stimulus involves reversible methylation and demethylation of specific glutamate residues. However, the exact mechanism of signal propagation through the helical receptor to the histidine kinase remains elusive. Dynamics of the receptor cytoplasmic domain is thought to play an important role in the signal transduction, and current models propose inverse dynamic changes in different regions of the receptor. We hypothesize that the adaptational modification (methylation) controls the dynamics by stabilizing a partially ordered domain, which in turn modulates the binding of the kinase, CheA. We investigated the difference in dynamics between the methylated and unmethylated states of the chemoreceptor using solid-state NMR. The unmethylated receptor (CF4E) shows increased flexibility relative to the methylation mimic (CF4Q). Methylation helix 1 (MH1) has been shown to be flexible in the methylated receptor. Our analysis indicates that in addition to MH1, methylation helix 2 also becomes flexible in the unmethylated receptor. In addition, we have demonstrated that both states of the receptor have a rigid region and segments with intermediate dynamics. The strategies used in the study for identifying dynamic regions are applicable to a broad class of proteins and protein complexes with intrinsic disorder and dynamics spanning multiple timescales.Graphical AbstractHighlightsReceptors exhibit greater ns timescale dynamics in unmethylated vs methylated stateMethylation helix 2 likely involved in increased flexibility of unmethylated stateDynamics occur on multiple timescales in both states of the receptor

scholarly journals Regular dependence of invariant curves and Aubry–Mather sets of twist maps of an annulus

1988 ◽  
Vol 8 (4) ◽  
pp. 555-584 ◽  
Author(s):  
Raphaël Douady
Keyword(s):  
Hamiltonian Systems ◽  
Rotation Number ◽  
Positive Measure ◽  
Invariant Curves ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Area Preserving Maps ◽  
Area Preserving

AbstractWe prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets.Then we show that invariant curves of the same map with different rotation numbers ω and ω′ cannot approach each other at a distance less than cst. |ω−ω′|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure.Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Pöschel (for the second one), in the case of Hamiltonian systems and area preserving maps.

Courts, Constitutions, and Labor Politics in England and America: A Study of the Constitutive Power of Law

1991 ◽  
Vol 16 (01) ◽  
pp. 1-34 ◽  
Author(s):  
William E. Forbath
Keyword(s):  
United States ◽  
Broad Class ◽  
The United States ◽  
Turn Of The Century ◽  
Labor Politics ◽  
Trade Unionism ◽  
Judicial Supremacy ◽  
The Difference ◽  
Parliamentary Supremacy ◽  
The 19Th Century

For most of the 19th century, the labor movements of England and America seemed to be developing along similar lines. Then, in the decades around the turn of the century, both movements were embroiled in a common battle over the political soul of trade unionism. In England, the champions of broad, class-based social and industrial reforms prevailed. In the United States, they lost, and the winners were the voluntarists, who held that labor should steer clear of politics as much as possible. This article suggests that the key reasons for the divergence lie not in the sociology of the working class or labor movement, so much as in the character of the state and polity and the lessons trade unionists drew from experiences in those arenas. The difference between judicial supremacy in the United States and parliamentary supremacy in England combined with other differences in the two nations’ forms of government to produce sharply contrasting lessons about the value of state-based reforms.

A method for proving that monotone twist maps have no invariant circles

1991 ◽  
Vol 11 (1) ◽  
pp. 79-84 ◽  
Author(s):  
Irwin Jungreis
Keyword(s):  
Existence Theorem ◽  
Computer Assisted ◽  
Twist Map ◽  
Standard Map ◽  
Rotation Numbers ◽  
Invariant Circles ◽  
Twist Maps ◽  
Parameter Values

AbstractWe present an existence theorem for certain kinds of orbits of a monotone twist map and use it to obtain a criterion for proving that there are no invariant circles with a certain range of rotation numbers. We have used this criterion to prove (computer assisted) that the standard map has no invariant circles for several parameter values includingk= 0.9718.

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