The Direction of Time: Seven Arrows for One Stream? A Program

I review the role of entropy in the Second Law of Thermodynamics in providing time with a direction and explaining irreversibility. I then list seven seemingly different features, each characterizing an additional arrow. In one case, connected to Black Hole Physics, the arrow has been successfully merged with the thermodynamical and may serve as model; in another case, that of the Evolutionary drive, the adequate function has been identified as complexity. We define the aims of the program and also provide information for an alternative, geometrical (effective) approach.     

Leptogenesis from supersymmetry breaking

We show that soft supersymmetry breaking terms involving the heavy sneutrinos can lead to sneutrino-antisneutrino mixing and to new sources of CP violation, which are present even if a single generation is considered. These terms are naturally present in supersymmetric versions of leptogenesis scenarios, and they induce indirect CP violation in the decays of the heavy sneutrinos, eventually generating a baryon asymmetry. This new contribution can be comparable to or even dominate over the asymmetry produced in traditional leptogenesis scenarios.     

Two-channel Kondo effect in a modified single electron transistor

We suggest a simple system of two electron droplets which should display two-channel Kondo behavior at experimentally accessible temperatures. Stabilization of the two-channel Kondo fixed point requires fine control of the electrochemical potential in each droplet, which can be achieved by adjusting voltages on nearby gate electrodes. We study the conditions for obtaining this type of two-channel Kondo behavior, discuss the experimentally observable consequences, and explore the generalization to the multichannel Kondo case.     

New Coins from Old, Smoothly

Given a (known) function f:[0,1]→(0,1), we consider the problem of simulating a coin with probability of heads f(p) by tossing a coin with unknown heads probability p, as well as a fair coin, N times each, where N may be random. The work of Keane and O’Brien (ACM Trans. Model. Comput. Simul. 4(2):213–219, 1994) implies that such a simulation scheme with the probability ℙ _p (N<∞) equal to 1 exists if and only if f is continuous. Nacu and Peres (Ann. Appl. Probab. 15(1A):93–115, 2005) proved that f is real analytic in an open set S⊂(0,1) if and only if such a simulation scheme exists with the probability ℙ _p (N>n) decaying exponentially in n for every p∈S. We prove that for α>0 noninteger, f is in the space C ^α [0,1] if and only if a simulation scheme as above exists with ℙ _p (N>n)≤C(Δ _n (p)) ^α , where \varDelta _n(x):=max{?{x(1-x)/n},1/n}\varDelta _{n}(x):=\max\{\sqrt{x(1-x)/n},1/n\}. The key to the proof is a new result in approximation theory: Let B⁺_n\mathcal{B}^{+}_{n} be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree n. We show that a function f:[0,1]→(0,1) is in C ^α [0,1] if and only if f has a series representation ?_n=1^￥F_n\sum_{n=1}^{\infty}F_{n} with F_n ? B⁺_nF_{n}\in \mathcal{B}^{+}_{n} and ∑ _k>n F _k (x)≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1. We also provide a counterexample to a theorem stated without proof by Lorentz (Math. Ann. 151:239–251, 1963), who claimed that if some j_n ? B⁺_n\varphi_{n}\in\mathcal{B}^{+}_{n} satisfy |f(x)−φ _n (x)|≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1, then f∈C ^α [0,1].     

On the sum of the L1 influences of bounded functions

Let f: {-1, 1}ⁿ → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L₁ influence of f can also be bounded as a function of d. Ba?kurs and Bavarian answered this question in the affirmative, providing a bound of O(d³) for general functions and O(d²) for homogeneous functions. We improve on their results by providing a bound of d² for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.     

On the Hausdorff dimension of fibres

It is well known that there are planar sets of Hausdorff dimension greater than 1 which are graphs of functions, i.e., all their vertical fibres consist of 1 point. We show this phenomenon does not occur for sets constructed in a certain “regular” fashion. Specifically, we consider sets obtained by partitioning a square into 4 subsquares, discarding 1 of them and repeating this on each of the 3 remaining squares, etc.; then almost all vertical fibres of a set so obtained have Hausdorff dimension at least 1/2. Sharp bounds on the dimensions of sets of exceptional fibres are presented. Partially supported by a grant from the Landau Centre for Mathematical Analysis.     

Effect of hot acid etching on the mechanical strength of ground YAG laser elements

High-power pumped Nd:YAG elements may exceed their tensile strength under high thermally induced stress. Providing extra strength to such rods is essential for their employment in high-power lasers. The tensile strength of YAG elements was increased by chemical etching in concentrated phosphoric acid. The highest tensile strength was achieved by etching of fine-ground YAG components: an average for slabs, and for rods, which are 3.6 times and 5 times higher than those of non-etched elements, respectively. The measurements were carried out by four-point flexure strength test. We have established a dependency among the micro-roughness of YAG elements, the surface morphology obtained by etching, and the tensile strength: the tensile strength of the etched element improves for finer after-etch surface texture, which is obtained for finer initial micro-roughness.

To assure the withstanding of Nd:YAG rods under high thermal gradients, a new approach was employed, namely, increasing the pump-power applied to the Nd:YAG rod till fracture. Our results show an increase by more than 2.7 times in tensile strength of etched Nd:YAG rods as compared to standard commercial rods, which corresponds to a thermal loading of excess of 434 W/cm.     

Mixing of the upper triangular matrix walk

We study a natural random walk over the upper triangular matrices, with entries in the field ${\mathbb{Z}_2}$ , generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n ²) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields ${\mathbb{Z}_q}$ for q prime.     

Points of increase for random walks

Say that a sequenceS ₀, ..., S_n has a (global) point of increase atk ifS _k is maximal amongS ₀, ..., S_k and minimal amongS _k, ..., S_n. We give an elementary proof that ann-step symmetric random walk on the line has a (global) point of increase with probability comparable to 1/logn. (No moment assumptions are needed.) This implies the classical fact, due to Dvoretzky, Erdős and Kakutani (1961), that Brownian motion has no points of increase. Research partially supported by NSF grant # DMS-9404391.