Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces

The aim of this paper is twofold. On the one hand, we discuss the notions of strong chain recurrence and strong chain transitivity for flows on metric spaces, together with their characterizations in terms of rigidity properties of Lipschitz Lyapunov functions. This part extends to flows some recent results for homeomorphisms of Fathi and Pageault. On the other hand, we use these characterisations to revisit the proof of a theorem of Paternain, Polterovich and Siburg concerning the inner rigidity of a Lagrangian submanifold $\Lambda$ contained in an optical hypersurface of a cotangent bundle, under the assumption that the dynamics on $\Lambda$ is strongly chain recurrent. We also prove an outer rigidity result for such a Lagrangian submanifold $\Lambda$, under the stronger assumption that the dynamics on $\Lambda$ is strongly chain transitive.Comment: 26 pages, 2 figure

Bending rigidity of stiff polyelectrolyte chains: Single chain and a bundle of multichains

We study the bending rigidity of highly charged stiff polyelectrolytes, for both a single chain and many chains forming a bundle. A theory is developed to account for the interplay between competitive binding of counterions and charge correlations in softening the polyelectrolyte (PE) chains. The presence of even a small concentration of multivalent counterions leads to a dramatic reduction in the bending rigidity of the chains that are nominally stiffened by the repulsion between their backbone charges. The variation of the bending rigidity as a function of $f_{0}$, the fraction of charged monomers on the chain, does not exhibits simple scaling behavior; it grows with increasing $f_{0}$ below a critical value of $f_{0}$. Beyond the critical value, however, the chain becomes softer as $f_{0}$ increases. The bending rigidity also exhibits intriguing dependence on the concentration of multivalent counterion $n_{2}$; for highly charged PEs, the bending rigidity decreases as $n_2$ increases from zero, while it increases with increasing $n_{2}$ beyond a certain value of $n_{2}$. When polyelectrolyte chains form a $N$-loop condensate (e.g., a toroidal bundle formed by $N$ turns (winds) of the chain), the inter-loop coupling further softens the condensate, resulting in the bending free energy of the condensate that scales as $N$ for large $N$.Comment: 11 pages, 2 figure

Comparative analysis of rigidity across protein families

We present a comparative study in which 'pebble game' rigidity analysis is applied to multiple protein crystal structures, for each of six different protein families. We find that the main-chain rigidity of a protein structure at a given hydrogen bond energy cutoff is quite sensitive to small structural variations, and conclude that the hydrogen bond constraints in rigidity analysis should be chosen so as to form and test specific hypotheses about the rigidity of a particular protein. Our comparative approach highlights two different characteristic patterns ('sudden' or 'gradual') for protein rigidity loss as constraints are removed, in line with recent results on the rigidity transitions of glassy networks

Private Label Price Rigidity during Holiday Periods

Using weekly retail transaction scanner price data from a large U.S supermarket chain, we find significantly higher retail price rigidity for private label products than for nationally branded products during the Christmas and Thanksgiving holiday periods relative to the rest of the year. The finding cannot be explained by changes in holiday period promotional practices because we find that private label promotions appear to diminish at least as much as national brands. The increased rigidity of private label products relative to national brands is only partially accounted for by increased rigidity of wholesale prices. After ruling out other potential explanations, we suggest that the higher private label price rigidity might be due to the increased emphasis on social consumption during holiday periods, raising the customers’ value of nationally branded products relative to the private labels.Price Rigidity, Holidays, Private Label, National Brand, Social Consumption

Rigidity of Orientationally Ordered Domains of Short Chain Molecules

By molecular dynamics simulation, discovered is a strange rigid-like nature for a hexagonally packed domain of short chain molecules. In spite of the non-bonded short-range interaction potential (Lennard-Jones potential) among chain molecules, the packed domain gives rise to a resultant global moment of inertia. Accordingly, as two domains encounter obliquely, they rotate so as to be parallel to each other keeping their overall structures as if they were rigid bodies.Comment: 7 pages, 5 figures, and 2 table

Continuum model for polymers with finite thickness

We consider the continuum limit of a recently-introduced model for discretized thick polymers, or tubes. We address both analytically and numerically how the polymer thickness influences the decay of tangent-tangent correlations and find how the persistence length scales with the thickness and the torsional rigidity of the tube centerline. At variance with the worm-like chain model, the phase diagram that we obtain for a continuous tube is richer; in particular, for a given polymer thickness there exists a threshold value for the centerline torsional rigidity separating a simple exponential decay of the tangent-tangent correlation from an oscillatory one.Comment: 8 pages, 4 figures. Accepted for publication in J. Phys.

Gravity Anomalies and Flexure of the Lithosphere along the Hawaiian-Emperor Seamount Chain

Simple models for the flexure of the lithosphere caused by the load of the Hawaiian-Emperor Seamount Chain have been determined for different values of the effective flexural rigidity of the lithosphere. The gravity effect of the models have been computed and compared to observed free-air gravity anomaly profiles in the vicinity of the seamount chain. The values of the effective flexural rigidity which most satisfactorily explain both the amplitude and wavelength of the observed profiles have been determined. Computations show that if the lithosphere is modelled as a continuous elastic sheet, a single effective flexural rigidity of about 5 × 10²⁹ dyne-cm can explain profiles along the Hawaiian-Emperor Seamount Chain. If the lithosphere is modelled as a discontinuous elastic sheet an effective flexural rigidity of about 2 × 10³⁰ dyne-cm is required. Since the age of the seamount chain increases from about 3 My near Hawaii to about 70 My near the northernmost Emperor seamount these results suggest there is apparently little decrease in the effective flexural rigidity of the lithosphere with increase in the age of loading. This suggests the lithosphere is rigid enough to support the load of the seamount chain for periods of time of at least several tens of millions of years. Thus the subsidence of atolls and guyots along the chain is most likely to be regional in extent and is unlikely to be caused by an inelastic behaviour of the lithosphere beneath individual seamounts

Polymer Adsorption on Curved Surfaces: Finite chain length corrections

The structural properties of polymers adsorbed onto a surface have been widely investigated using self-consistent mean-field theories. Recently, analytical mean-field theories have been applied to study polymer adsorption on curved surfaces but all in the context of the ground state dominance approximation in which the polymer chain length (N) is essentially infinite. Using an expression for the free energy by Semenov, we determine leading order (in 1/N) corrections due to the finiteness of the polymer chain length on surface tension, spontaneous curvature, and rigidity constants.Comment: 24 pages, 7 figures. Accepted for publication in Macromolecule