The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^{-3/2}, plus an error term of order n^{-3/2-\epsilon} with \epsilon >0. This is a strong statement that the critical exponent delta is given by delta =2.Comment: 56 pages, 3 Postscript figures, in AMS-LaTeX, with graphicx, epic, and xr package

Optimization of the Asymptotic Property of Mutual Learning Involving an Integration Mechanism of Ensemble Learning

We propose an optimization method of mutual learning which converges into the identical state of optimum ensemble learning within the framework of on-line learning, and have analyzed its asymptotic property through the statistical mechanics method.The proposed model consists of two learning steps: two students independently learn from a teacher, and then the students learn from each other through the mutual learning. In mutual learning, students learn from each other and the generalization error is improved even if the teacher has not taken part in the mutual learning. However, in the case of different initial overlaps(direction cosine) between teacher and students, a student with a larger initial overlap tends to have a larger generalization error than that of before the mutual learning. To overcome this problem, our proposed optimization method of mutual learning optimizes the step sizes of two students to minimize the asymptotic property of the generalization error. Consequently, the optimized mutual learning converges to a generalization error identical to that of the optimal ensemble learning. In addition, we show the relationship between the optimum step size of the mutual learning and the integration mechanism of the ensemble learning.Comment: 13 pages, 3 figures, submitted to Journal of Physical Society of Japa

Conductance of a single molecule anchored by an isocyanide substituent to gold electrodes

The effect of anchoring group on the electrical conductance of a single molecule bridging two Au electrodes was studied using di-substituted (isocyanide (CN-), thiol (S-) or cyanide (NC-)) benzene. The conductance of a single Au/1,4-diisocyanobenzene/Au junction anchored by isocyanide via a C atom (junction with the Au-CN bond) was $3 \times 10 ^{-3} G_{0}$ ($2e^{2}/h$). The value was comparable to $4 \times 10 ^{-3} G_{0}$ of a single Au/1,4-benzenedithiol/Au junction with the Au-S bond. The Au/1,4-dicyanobenzene/Au molecular junction with the Au-NC bond did not show well-defined conductance values. The metal-molecule bond strength was estimated by the distance over which the molecular junction was stretched before breakdown. The stretched length of the molecular junction with the Au-CN bond was comparable to that of the Au junction, indicating that the Au-CN bond was stronger than the Au-Au bond.Comment: 3 figures, to be appear in Appl. Phys. Let

Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Renyi random graphs. In this updated version we incorporate an erratum to be published in a forthcoming issue of Probab. Theory Relat. Fields. This results in a modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming issue of Probab. Theory Relat. Field

On the Formation Age of the First Planetary System

Recently, it has been observed the extreme metal-poor stars in the Galactic halo, which must be formed just after Pop III objects. On the other hand, the first gas clouds of mass $\sim 10^6 M_{\odot}$ are supposed to be formed at $z \sim$ 10, 20, and 30 for the $1\sigma$, $2\sigma$ and $3\sigma$, where the density perturbations are assumed of the standard $\Lambda$CDM cosmology. If we could apply this gaussian distribution to the extreme small probability, the gas clouds would be formed at $z \sim$40, 60, and 80 for the $4\sigma$, $6\sigma$, and $8\sigma$. The first gas clouds within our galaxy must be formed around $z\sim 40$. Even if the gas cloud is metal poor, there is a lot of possibility to form the planets around such stars. The first planetary systems could be formed within $\sim 6\times 10^7$ years after the Big Bang in the universe. Even in our galaxies, it could be formed within $\sim 1.7\times 10^8$ years. It is interesting to wait the observations of planets around metal-poor stars. For the panspermia theory, the origin of life could be expected in such systems.Comment: 5 pages,Proceedings IAU Symposium No. 249, 2007, Exoplanets:Y-S. Sun, S. Ferraz-Mello and J.-L, Zhou, eds. (p325

Vibration Alert Bracelet for Notification of the Visually and Hearing Impaired

This paper presents the prototype of an electronic vibration bracelet designed to help the visually and hearing impaired to receive and send emergency alerts. The bracelet has two basic functions. The first function is to receive a wireless signal and respond with a vibration to alert the user. The second function is implemented by pushing one button of the bracelet to send an emergency signal. We report testing on a prototype system formed by a mobile application and two bracelets. The bracelets and the application form a complete system intended to be used in retirement apartment communities. However, the system is flexible and could be expanded to add new features or to serve as a research platform for gait analysis and location services. The medical and professional potential of the proposed system is that it offers a simple, modular, and cost-effective alternative to all the existing medical devices with similar functionality currently on the market. The proposed system has an educational potential as well: it can be used as a starting point for capstone projects and demonstration purposes in schools to attract students to STEM disciplines

New Lower Bounds on the Self-Avoiding-Walk Connective Constant

We give an elementary new method for obtaining rigorous lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice $Z^d$. The method is based on loop erasure and restoration, and does not require exact enumeration data. Our bounds are best for high $d$, and in fact agree with the first four terms of the $1/d$ expansion for the connective constant. The bounds are the best to date for dimensions $d \geq 3$, but do not produce good results in two dimensions. For $d=3,4,5,6$, respectively, our lower bound is within 2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.Comment: 35 pages, 388480 bytes Postscript, NYU-TH-93/02/0