Finding any given 2‐factor in sparse pseudorandom graphs efficiently

Given an $n$‐vertex pseudorandom graph $G$ and an $n$‐vertex graph $H$ with maximum degree at most two, we wish to find a copy of $H$ in $G$, that is, an embedding $\phi :V\left(H\right)\to V\left(G\right)$ so that $\phi \left(u\right)\phi \left(v\right)\in E\left(G\right)$ for all $uv\in E\left(H\right)$. Particular instances of this problem include finding a triangle‐factor and finding a Hamilton cycle in $G$. Here, we provide a deterministic polynomial time algorithm that finds a given $H$ in any suitably pseudorandom graph $G$. The pseudorandom graphs we consider are $\left(p,\lambda \right)$‐bijumbled graphs of minimum degree which is a constant proportion of the average degree, that is, $\mathrm{\Omega }\left(pn\right)$. A $\left(p,\lambda \right)$‐bijumbled graph is characterised through the discrepancy property: $|e\left(A,B\right)?p|A||B||<\lambda \sqrt{|A||B|}$ for any two sets of vertices $A$ and $B$. Our condition $\lambda =O(p^{2}n/\log n)$ on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption‐reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work, incorporating the work of Nenadov, together with additional ideas and simplifications.     

An Improved Composite Multiscale Fuzzy Entropy for Feature Extraction of MI-EEG

Motor Imagery Electroencephalography (MI-EEG) has shown good prospects in neurorehabilitation, and the entropy-based nonlinear dynamic methods have been successfully applied to feature extraction of MI-EEG. Especially based on Multiscale Fuzzy Entropy (MFE), the fuzzy entropies of the τ coarse-grained sequences in τ scale are calculated and averaged to develop the Composite MFE (CMFE) with more feature information. However, the coarse-grained process fails to match the nonstationary characteristic of MI-EEG by a mean filtering algorithm. In this paper, CMFE is improved by assigning the different weight factors to the different sample points in the coarse-grained process, i.e., using the weighted mean filters instead of the original mean filters, which is conductive to signal filtering and feature extraction, and the resulting personalized Weighted CMFE (WCMFE) is more suitable to represent the nonstationary MI-EEG for different subjects. All the WCMFEs of multi-channel MI-EEG are fused in serial to construct the feature vector, which is evaluated by a back-propagation neural network. Based on a public dataset, extensive experiments are conducted, yielding a relatively higher classification accuracy by WCMFE, and the statistical significance is examined by two-sample t-test. The results suggest that WCMFE is superior to the other entropy-based and traditional feature extraction methods.     

非线性薛定谔模型边界场算子的形式因子

讨论了可积开边界条件下的非线性薛定谔模型，给出了其贝特本征态的内积和模长.在此基础上得到了边界场算子的形式因子. 这些结果均被表达成由谱参量的函数所构成的行列式. 关键词： 可积模型 形式因子 反散射     

Fifteen consecutive integers with exactly four prime factors

We describe a successful search for a sequence of fifteen consecutive integers, each the product of exactly four prime factors. Fifteen is best possible.

     

TOF-SIMS imaging of chlorhexidine-digluconate transport in frozen hydrated biofilms of the fungus Candida albicans

The diffusion of the anti-microbial chlorhexidine digluconate (CHG) has been studied in C. albicans biofilms by time-of-flight secondary-ion mass spectrometry (TOF-SIMS). C. albicans has been shown to become resistant to common anti-microbial agents, including CHG, when growing as a biofilm. Mass transport resistance within biofilms has commonly been suggested as a resistance mechanism, but measurement of transport for most anti-microbial agents in biofilms has proven extremely difficult because of the heterogeneity of the biofilms and the difficulty in detecting these agents within an intact biofilm. In this study, TOF-SIMS has been used to study the transport of CHG and glucose in a frozen hydrated biofilm. The TOF-SIMS images reveal a progression of CHG from the top of the biofilm to its base with time. Images suggest that there are channels within the biofilm and show preferential binding of CHG to cellular components of the biofilm. Additionally, both living and dead cells can be identified in the TOF-SIMS images by the sequestration of K⁺ and the presence of cell markers. This study demonstrates that TOF-SIMS has the unique potential to simultaneously observe the presence of an antimicrobial agent, concentration of nutrients, and the viability of the cell population.     

A covering construction for packing disjoint copies of the Hoffman‐Singleton graph into K50

In this note we show how coverings induced by voltage assignments can be used to produce packings of disjoint copies of the Hoffman‐Singleton graph into K₅₀. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 408–412, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10049     

Free probability and the von Neumann algebrasof free groups

This short note summarizes the circumstances of the birth of free probability theory andsome of the recent achievements.     

On Bayes and unbiased estimators of loss

We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In 3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988, On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly, that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss. Research supported by NSF Grant DMS-97-04524.     

A new approach to obtain the non-Condon factors in closed form for two one-dimensional harmonic oscillators

A simple algebraic approach to calculate general Franck-Condon overlaps is extended to evaluate non-Condon factors for two one-dimensional harmonic oscillators. The method is based on the use of eigenstates of the harmonic oscillator annihilation operator which allows to obtain in terms of a multi-dimensional Hermite polynomial the overlap of harmonic oscillator functions associated with different Born-Oppenheimer potentials. The presented approach is self-contained, only basic concepts of quantum mechanics associated with the harmonic oscillator system are needed. The obtained expression for the Franck-Condon overlaps is similar to the Ansbacher’s formula and equivalent to the one calculated by Malkin and Man’ko. However our final expression has the advantages that only real numbers are involved and it is straightforward to get the limit case of equal frequencies. Concerning the non-Condon factors two approaches leading to different formulas are considered, both of which reduce to triple sums of products of three Hermite polynomials.     

Closed-form solutions for a rectangular layer with central crack under anti-plane shear stress

We consider the problem of determining the stress distributionin a finite rectangular elastic layer containing a Griffithcrack which is opened by internal shear stress acting alongthe length of the crack. The mode III crack is assumed to belocated in the middle plane of the rectangular layer. The followingtwo problems are considered: (A) the central crack is perpendicularto the two fixed lateral surfaces and parallel to the othertwo stress-free surfaces; (B) all the lateral surfaces of therectangular layer are clamped and the central crack is parallelto the two lateral surfaces. By using Fourier transformations,we reduce the solution of each problem to the solution of dualintegral equations with sine kernels and a weight function whichare solved exactly. Finally, we derive closed-form expressionsfor the stress intensity factor at the tip of the crack andthe numerical values for the stress intensity factor at theedges of the cracks are presented in the form of tables.