结霜发生随机性的实验研究

结霜是影响制冷设备工作的不利因素之一,对结霜发生的研究可以指导改进除霜方案。本文通过显微摄像的手段研究了结霜发生过程。对于不同的冷壁面温度和水蒸汽分压力,有两类结霜发生过程:一是经过气液固相变形成霜,二是经过气固相变形成霜。实验观察到第一类结霜的转变时间具有随机性,并对此现象进行了分析。     

From Heisenberg to Gödel via Chaitin

In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.” Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the converse implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a “formal uncertainty principle” which implies Chaitin’s information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. In fact, the formal uncertainty principle applies to all systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies algorithmic randomness not only in mathematics, but also in physics.     

On extractors and exposure‐resilient functions for sublogarithmic entropy

We study resilient functions and exposure‐resilient functions in the low‐entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit‐fixing sources) maps any distribution on n ‐bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure‐resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n ‐ k input bits. In this paper, we focus on the case that k is sublogarithmic in n. We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space‐bounded streaming algorithms. Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure‐resilient function with high probability even if k is as small as a constant (resp. loglog n). No explicit exposure‐resilient functions achieving these parameters are known. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

A minimal pair of -degrees

We construct a minimal pair of -degrees. We do this by showing the existence of an unbounded nondecreasing function which forces -triviality in the sense that is -trivial if and only if for all , .

     

Weak quasi‐randomness for uniform hypergraphs

We study quasi‐random properties of k‐uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung‐Graham‐Wilson theorem for quasi‐random graphs. Moreover, let K_k be the complete graph on k vertices and M(k) the line graph of the graph of the k‐dimensional hypercube. We will show that the pair of graphs (K_k,M(k)) has the property that if the number of copies of both K_k and M(k) in another graph G are as expected in the random graph of density d, then G is quasi‐random (in the sense of the Chung‐Graham‐Wilson theorem) with density close to d. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

A Langevin approach to the Log–Gauss–Pareto composite statistical structure

The distribution of wealth in human populations displays a Log–Gauss–Pareto composite statistical structure: its density is Log–Gauss in its central body, and follows power-law decay in its tails. This composite statistical structure is further observed in other complex systems, and on a logarithmic scale it displays a Gauss-Exponential structure: its density is Gauss in its central body, and follows exponential decay in its tails. In this paper we establish an equilibrium Langevin explanation for this statistical phenomenon, and show that: (i) the stationary distributions of Langevin dynamics with sigmoidal force functions display a Gauss-Exponential composite statistical structure; (ii) the stationary distributions of geometric Langevin dynamics with sigmoidal force functions display a Log–Gauss–Pareto composite statistical structure. This equilibrium Langevin explanation is universal — as it is invariant with respect to the specific details of the sigmoidal force functions applied, and as it is invariant with respect to the specific statistics of the underlying noise.     

Cluster approaches to random alloys: An appraisal

Some of the cluster extensions of the coherent potential approximation (CPA) based on the effective medium theory have been critically studied with respect to the decoupling schemes involved in them. Their computational tractability has been examined and it has been found that theself-consistent calculations in three-dimensional systems are immensely difficult to perform. A self-consistent calculation has been reported for simple cubic lattices with diagonal and off-diagonal disorder using a pair-CPA method. A significant finding of the paper is that it has been shown thatnon-analyticities are a general feature of extensions of CPA within multiple scattering framework. The non-analyticities were reported several times but a general proof of their existence was not noticed. It was also believed that the so-called molecular—CPA is analytic, this has been shown to be wrong here. The density of states results with off-diagonal randomness have been qualitatively understood to yield some information about the influence of off-diagonal randomness on Anderson localisation of an electron.     

Probabilistic bootstrap percolation

In bootstrap percolation, sites are occupied with probabilityp, but those with less thanm occupied first neighbors are removed. This culling process is repeated until a stable configuration (all occupied sites have at leastm occupied first neighbors or the whole lattice is empty) is achieved. Formm ₁ the transition is first order, while form<m ₁ it is second order, withm-dependent exponents. In probabilistic bootstrap percolation, sites have probabilityr or (1–r) of beingm- orm-sites, respectively (m-sites are those which need at leastm occupied first neighbors to remain occupied). We have studied the model on Bethe lattices, where an exact solution is available. Form=2 andm=3, the transition changes from second to first order atr ₁=1/2, and the exponent is different forr<1/2,r=1/2, andr>1/2. The same qualitative behavior is found form=1 andm=3. On the other hand, form=1 andm=2 the transition is always second order, with the same exponents ofm=1, for any value ofr>0. We found, form=z–1 andm=z, wherez is the coordination number of the lattice, thatp _c=1 for a value ofr which depends onz, but is always above zero. Finally, we argue that, for bootstrap percolation on real lattices, the exponents and form=2 andm=1 are equal, for dimensions below 6.On leave from Universidade Federal de Santa Catarina, Depto. de Fisica, 88049, Florianópolis, SC, Brazil     

Uncertainty Propagation in Layered Unsaturated Soils

This study analyzes the wetting front migration in layered unsaturated soils which have uncertain hydraulic properties. A Monte Carlo scheme was used to propagate the uncertainty of hydraulic parameters. RANUF, a computer program, was developed to solve the one-dimensional, pressure-based form of Richards' equation and to implement the Monte Carlo scheme.Uncertainty propagation was investigated for two-layered soils of various alternating fine over coarse or coarse over fine layer configurations and of various nonrandomized and/or randomized layer arrangements. The effects of changing initial and boundary conditions were also investigated. Randomness was introduced via the saturated hydraulic conductivity, K _s, which was assumed to be distributed lognormally with a coefficient of variation of about 10 percent.It was found that in layered soils the mean profiles (i.e., water content and pressure head) remained essentially unchanged regardless of which layer (or layers) was (or were) randomized; however, the variance profiles were affected. Also, higher uniform initial water content tended to inhibit uncertainty, but higher supply rates did not show any characteristic trend for uncertainty behavior.