Can one hear the dimension of a fractal?

We consider the spectrum of the Laplacian in a bounded open domain of ⁿ with a rough boundary (i.e. with possibly non-integer dimension) and we discuss a conjecture by M. V. Berry generalizing Weyl's conjecture. Then using ideas Mark Kac developed in his famous study of the drum, we give upper and lower bounds for the second term of the expansion of the partition function. The main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on Minkowski dimensions and on Minkowski measures rather than on Haussdorff ones.Dedicated to the memory of Mark Kac     

How far can one send a photon?

The answer to the question How far can one send a photon? depends heavily on what one means by a photon and on what one intends to do with that photon. For direct quantum communication, the limit is approximately 500 km. For terrestrial quantum communication, near-future technologies based on quantum teleportation and quantum memories will soon enable quantum repeaters that will turn the development of a world-wide-quantum-web (WWQW) into a highly non-trivial engineering problem. For Device-Independent Quantum Information Processing, near-future qubit amplifiers (i.e., probabilistic heralded amplification of the probability amplitude of the presence of photonic qubits) will soon allow demonstrations over a few tens of kilometers.     

Can’t one really hear the shape of a drum?

In this paper we study the wave and the Klein-Gordon equations in frontiers with the same set of eigen values using a computational algorithm based on the finite difference method and the discrete Fourier transform. Doing this we found that although the set of eigen values in the two shapes are equal, the intensities in the spectrum are different, which means that the question, can one hear the shape of a drum? is still open.     

How much information can one store in a nonequilibrium medium?

It has recently been emphasized again that the very existence of stationary stable localized structures with short-range interactions might allow one to store information in nonequilibrium media, opening new perspectives on information storage. We show how to use generalized topological entropies to measure aspects of the quantities of storable and nonstorable information. This leads us to introduce a measure of the long-term stably storable information. As a first example to illustrate these concepts, we revisit a mechanism for the appearance of stationary stable localized structures that is related to the stabilization of fronts between structured and unstructured states (or between differently structured states).     

When can one observe good hydrodynamic Lyapunov modes?

Inspired by recent results on differences in fluctuations of finite-time Lyapunov exponents between hard-core and soft-potential systems we surmise that partial domination of the Oseledec splitting (DOS) with respect to subspaces associated with near-zero Lyapunov exponents is essential for observing good hydrodynamic Lyapunov modes (HLMs). Numerical results for coupled map lattices are presented to show the importance of DOS for observing good HLMs. This is achieved by relating splitting parameters to the maximum value of the Lyapunov mode structure factor.     

Conductance for a Quantum Wire with Weak Rashba Spin-Orbit Coupling

We theoretically study the low temperature electron transport properties of a weak Rashba spin-orbit coupling （SOC） semiconductor quantum wire connected nonadiabatically to two electrode leads without SOC. The wire and the leads are defined by a parabolic confining potential, and the influence of both the wire-lead connection and the Rashba SOC on the electron transport is treated analytically by means of scattering matrix within effective free-electron approximation. From analytical analysis and numerical examples, we find that the system shows some fractional quantum conductance behaviour, and for some particular wire width a pure spin polarized current exists. Our result may imply a simple method for the design of a spin filter without involving any magnetic materials or magnetic fields.     

Birefringent plate design for broadband spectral shaping in a Ti:sapphire regenerative amplifier

A method to design a birefringent plate(BP)for broadening spectrum in a regenerative amplifier is presented.Using this method,we design a quartz BP with thickness of 761μm inserted into a Ti:sapphire regenerative amplifier.The gain narrowing effect is reduced efficiently,and the experimental results agree with the calculation well.The bandwidth is broadened from 28 to 62 nm using the designed quartz BP and the pulses are compressed to～23 fs.     

Is plate tectonics a case of non-extensive thermodynamics?

Bird (2003) [5] proposed that the distribution of areas of the tectonic plates follows a power law and that this distribution fitted well with the concepts of a few major plates and a hierarchical self-similar organization of blocks at the boundary scale, a fractal plate distribution and a self-organized system. Here we apply the concepts of non-extensive statistical mechanics (NESM) to plate tectonics. The application of NESM is appropriate to systems such as tectonic plates where non-linearity, long-range interactions, memory effects and scaling are important. We calculate the probability density function for the areas of the tectonic plates. Our results show that three classes (small, intermediate and large) of tectonic plates can be distinguished, which is consistent with the observations of Bird. Furthermore, taking into account that for the intermediate class of tectonic plates the cumulative frequency distribution behaves as a power law with exponent 1/3, we estimate a thermodynamic q parameter of q=1.75, which supports the conclusion that the plate tectonics system is a sub-extensive one.     

Is the Kolmogorov regime the most organized hydrodynamic turbulence state? (A renormalization group approach)

The renormalization group approach to hydrodynamical turbulence is studied from the standpoint of synergetics. We have shown that the Kolmogorov regime can be regarded as a result of self-organization taking place in turbulent media. The renormalization, in turn, can be considered as an initial turbulence state mapping to the final one, which is the result of self-organization.This work was supported, in part, by the Slovak Grant agency for science; Grant 2/550/93, Physics Institute, Slovak Acad. Sci.The authors are also grateful to Drs. R. Benzi and A. Tsinober for fruitful discussions.     

Lévy walks for turbulence: A numerical study

We present a numerical study of enhanced diffusion, for which the mean-squared displacement follows asymptotically r ²(t) t , > 1. We simulate continuous time random walks with waiting-time distributions which couple the spatial and temporal parameters; this gives rise to Lévy-walks. Our results confirm the theoretically predicted long-time behavior and demonstrate its temporal regime of validity. Furthermore, the simulations document the appearance of (parameter-dependent) transitions between regular and enhanced diffusion regimes.     

What is the magnetic Weak Gravity Conjecture for axions?

The electric Weak Gravity Conjecture demands that axions with large decay constant f couple to light instantons. The resulting large instantonic corrections pose problems for natural inflation. We explore an alternative argument based on the magnetic Weak Gravity Conjecture for axions, which we try to make more precise. Roughly speaking, it demands that the minimally charged string coupled to the dual 2‐form‐field exists in the effective theory. Most naively, such large‐f strings curve space too much to exist as static solutions, thus ruling out large‐f axions. More conservatively, one might allow non‐static string solutions to play the role of the required charged objects. In this case, topological inflation would save the superplanckian axion. Furthermore, a large‐f axion may appear in the low‐energy effective theory based on two subplanckian axions in the UV. The resulting effective string is a composite object built from several elementary strings and domain walls. It may or may not satisfy the magnetic Weak Gravity Conjecture depending on how strictly the latter is interpreted and on the cosmological dynamics of this composite object, which remain to be fully understood. Finally, we recall that large‐field brane inflation is naively possible in the codimension‐one case. We show how string‐theoretic back‐reaction closes this apparent loophole of large‐f (non‐periodic) pseudo‐axions.     

Mathematics and turbulence: where do we stand?

This contribution covers the topics presented by the authors at the “Fundamental Problems of Turbulence, 50 Years after the Marseille Conference 1961” meeting that took place in Marseille in 2011. It focuses on some of the mathematical approaches to fluid dynamics and turbulence. This contribution does not pretend to cover or answer, as the reader may discover, the fundamental questions in turbulence, however, it aims toward presenting some of the most recent advances in attacking these questions using rigorous mathematical tools. Moreover, we consider that the proofs of the mathematical statements (concerning, for instance, finite time regularity, weak solutions and vanishing viscosity) may contain information as relevant, to the understanding of the underlying problem, as the statements themselves.     

How narrow can a meniscus be?

A water meniscus naturally forms in air between an atomic force microscope (AFM) tip and a substrate. This nanoscale meniscus produces a capillary force on the AFM, and also serves as a molecular transport channel in dip-pen nanolithography (DPN). A stable meniscus is a necessary condition for DPN and for the validity of the Kelvin equation commonly applied to AFM experiments. Lattice gas Monte Carlo simulations show that, due to thermal fluctuation, a stable meniscus has a lower limit in width. We find a minimum width of 5 molecular diameters (1.9 nm) when the tip becomes atomically sharp (terminated by a single atom).     

Can one define a preferred symplectic connection?

We define a variational principle for symplectic connections of the Yang-Mills type. When the symplectic manifold is a compact surface we show that the moduli space of the connections which are extremals of the functional coincides with the Teichmüller space of the surface. We indicate that the noncompact situation is very different.