On Exchangeability in Network Models

We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size.Comment: Dedicated to the memory of Steve Fienber

The orbit method solution for the deformed three coupled scalar fields

In this work, we present a deformed solutions starting from systems of three coupled scalar fields with super-potential $W(\phi_1, \phi_2, \phi_3)$ by orbit method. First, we deform the corresponding super-potential and obtain defect solutions. It is shown that how to construct new models altogether with its defect solutions in terms of the non-deformed model. Therefore, we draw the graph of super-potential and different fields in terms of $x.$ So we observe that the graphs for deformed and non - deformed cases are changed by the scale.Comment: 9 pages, 5 figure

Influence of clamp-widening on the quality factor of nanomechanical silicon nitride resonators

Nanomechanical resonators based on strained silicon nitride (Si$_3$N$_4$) have received a large amount of attention in fields such as sensing and quantum optomechanics due to their exceptionally high quality factors ($Q$s). Room-temperature $Q$s approaching 1 billion are now in reach by means of phononic crystals (soft-clamping) and strain engineering. Despite great progress in enhancing $Q$s, difficulties in fabrication of soft-clamped samples limits their implementation into actual devices. An alternative means of achieving ultra-high $Q$s was shown using trampoline resonators with engineered clamps, which serves to localize the stress to the center of the resonator, while minimizing stress at the clamping. The effectiveness of this approach has since come into question from recent studies employing string resonators with clamp-tapering. Here, we investigate this idea using nanomechanical string resonators with engineered clampings similar to those presented for trampolines. Importantly, the effect of orienting the strings diagonally or perpendicularly with respect to the silicon frame is investigated. It is found that increasing the clamp width for diagonal strings slightly increases the $Q$s of the fundamental out-of-plane mode at small radii, while perpendicular strings only deteriorate with increasing clamp width. Measured $Q$s agree well with finite element method simulations even for higher-order resonances. The small increase cannot account for previously reported $Q$s of trampoline resonators. Instead, we propose the effect to be intrinsic and related to surface and radiation losses.Comment: 7 pages, 4 figure

Time-dependent backgrounds of two dimensional string theory from the c=1c=1 matrix model

The aim of this paper is to use correspondence between solutions in the $c=1$ matrix model collective field theory and coupled dilaton-gravity to a massless scalar field. First, we obtain the incoming and outgoing fluctuations for the time-dependent backgrounds with the lightlike and spacelike boundaries. In the case of spacelike boundaries, we have done here for the first time. Then by using the leg-pole transformations we find corresponding tachyon field in two dimensional string theory for lightlikes and spacelikes boundary.Comment: 10 page

The main transition in the Pink membrane model: finite-size scaling and the influence of surface roughness

We consider the main transition in single-component membranes using computer simulations of the Pink model [D. Pink {\it et al.}, Biochemistry {\bf 19}, 349 (1980)]. We first show that the accepted parameters of the Pink model yield a main transition temperature that is systematically below experimental values. This resolves an issue that was first pointed out by Corvera and co-workers [Phys. Rev. E {\bf 47}, 696 (1993)]. In order to yield the correct transition temperature, the strength of the van der Waals coupling in the Pink model must be increased; by using finite-size scaling, a set of optimal values is proposed. We also provide finite-size scaling evidence that the Pink model belongs to the universality class of the two-dimensional Ising model. This finding holds irrespective of the number of conformational states. Finally, we address the main transition in the presence of quenched disorder, which may arise in situations where the membrane is deposited on a rough support. In this case, we observe a stable multi-domain structure of gel and fluid domains, and the absence of a sharp transition in the thermodynamic limit.Comment: submitted to PR