On Fuglede’s conjecture and the existence of universal spectra

Recent methods developed by, Too [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in R(5) Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed all example of a nonspectral tile. In search of lower dimensional nonspectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang [14]. In particular, we prove here that the USC and the "tile --> spectral " direction of Fuglede's conjecture are equivalent in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szabo [13] for the existence of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples of tiles which have no spectra. We overcome these difficulties by invoking some ideas of Revesz and Farkas [2], and obtain nonspectral tiles in R(3)

Machine learning based session drop prediction in LTE networks and its SON aspects

Abnormal bearer session release (i.e. bearer session drop) in cellular telecommunication networks may seriously impact the quality of experience of mobile users. The latest mobile technologies enable high granularity real-time reporting of all conditions of individual sessions, which gives rise to use data analytics methods to process and monetize this data for network optimization. One such example for analytics is Machine Learning (ML) to predict session drops well before the end of session. In this paper a novel ML method is presented that is able to predict session drops with higher accuracy than using traditional models. The method is applied and tested on live LTE data offline. The high accuracy predictor can be part of a SON function in order to eliminate the session drops or mitigate their effects. © 2015 IEEE

Triangle percolation in mean field random graphs -- with PDE

We apply a PDE-based method to deduce the critical time and the size of the giant component of the ``triangle percolation'' on the Erd\H{o}s-R\'enyi random graph process investigated by Palla, Der\'enyi and VicsekComment: Summary of the changes made: We have changed a remark about k-clique percolation in the first paragraph. Two new paragraphs are inserted after equation (4.4) with two applications of the equation. We have changed the names of some variables in our formula

Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata

Define a certain gambler's ruin process \mathbf{X}_{j}, \mbox{ \ }j\ge 0, such that the increments $\varepsilon_{j}:=\mathbf{X}_{j}-\mathbf{X}_{j-1}$ take values $\pm1$ and satisfy $P(\varepsilon_{j+1}=1|\varepsilon_{j}=1, |\mathbf{X}_{j}|=k)=P(\varepsilon_{j+1}=-1|\varepsilon_{j}=-1,|\mathbf{X}_{j}|=k)=a_k$, all $j\ge 1$, where $a_k=a$ if $0\le k\le f-1$, and $a_k=b$ if $f\le k<N$. Here $0<a, b <1$ denote persistence parameters and $f ,N\in \mathbb{N}$ with $f<N$. The process starts at $\mathbf{X}_0=m\in (-N,N)$ and terminates when $|\mathbf{X}_j|=N$. Denote by ${\cal R}'_N$, ${\cal U}'_N$, and ${\cal L}'_N$, respectively, the numbers of runs, long runs, and steps in the meander portion of the gambler's ruin process. Define $X_N:=\left ({\cal L}'_N-\frac{1-a-b}{(1-a)(1-b)}{\cal R}'_N-\frac{1}{(1-a)(1-b)}{\cal U}'_N\right )/N$ and let $f\sim\eta N$ for some $0<\eta <1$. We show $\lim_{N\to\infty} E\{e^{itX_N}\}=\hat{\varphi}(t)$ exists in an explicit form. We obtain a companion theorem for the last visit portion of the gambler's ruin.Comment: Presented at 8th International Conference on Lattice Path Combinatorics, Cal Poly Pomona, Aug., 2015. The 2nd version has been streamlined, with references added, including reference to a companion document with details of calculations via Mathematica. The 3rd version has 2 new figures and improved presentatio

Notes on the identity of the male paralectotype of Thecla heodes and description of a new species : Strymon cryptodes sp. nov. from northern Peru (Lepidoptera: Lycaenidae)

A new species, Strymon cryptodes sp. nov. is described from puna grasslands of northern Peru. Its phenotype corresponds with the male paralectotype of a congeneric species originally described as Thecla heodes Druce, 1909, which, in consequence, turns out to be a mixture of two biological species. Males of S. heodes have conspicuous scent patches, absent in S. cryptodes sp. nov. The new species is known from four individuals recently collected in the vicinity of the city of Cajamarca and two historical specimens from other localities in the department of Cajamarca. Based on wing colour patterns and other morphological characters S. cryptodes sp. nov. is placed in the Strymon istapa species-group defined by Robbins & Nicolay (2002), although the presence or absence of male androconial patch and genitalia brush organ are demonstrated not to be a valid diagnostic infrageneric character

The spectrum of the random environment and localization of noise

We consider random walk on a mildly random environment on finite transitive d- regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized.Comment: 18 pages, 1 figur