Efficient indexing of necklaces and irreducible polynomials over finite fields

We study the problem of indexing irreducible polynomials over finite fields, and give the first efficient algorithm for this problem. Specifically, we show the existence of poly(n, log q)-size circuits that compute a bijection between {1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials of degree n over a finite field F_q. This has applications in pseudorandomness, and answers an open question of Alon, Goldreich, H{\aa}stad and Peralta[AGHP]. Our approach uses a connection between irreducible polynomials and necklaces ( equivalence classes of strings under cyclic rotation). Along the way, we give the first efficient algorithm for indexing necklaces of a given length over a given alphabet, which may be of independent interest

Normal, Abby Normal, Prefix Normal

A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present results about the number $pnw(n)$ of prefix normal words of length $n$, showing that $pnw(n) =\Omega\left(2^{n - c\sqrt{n\ln n}}\right)$ for some $c$ and $pnw(n) = O \left(\frac{2^n (\ln n)^2}{n}\right)$. We introduce efficient algorithms for testing the prefix normal property and a "mechanical algorithm" for computing prefix normal forms. We also include games which can be played with prefix normal words. In these games Alice wishes to stay normal but Bob wants to drive her "abnormal" -- we discuss which parameter settings allow Alice to succeed.Comment: Accepted at FUN '1

Patterns within Patterns within the Smart Living Experience

Modern technology is increasingly being employed to create a “smart” living experience. These “smart” technology entities are producing copious of amounts data, which in turn rely on increased storage, distribution and computation capacity to manage the data. Depending on the scenario, the diversity of piecemeal solutions almost reflects the diversity of problems they address. But some solutions can be reapplied. In the field of computing, design patterns can provide a general, reusable solution to commonly recurring problems within a given context through software design. This work seeks to determine the core elements of a technology-independent design pattern format and an open software framework can be developed to capture, share and redeploy existing successful and reusable strategies for commonly encountered smart environment use cases. Applying in areas such as assistive technology, energy management and environmental monitoring. The underpinning notion of this paper is to introduce “how, where and why” a rule set based in “design pattern” format could contribute to describe a general “understanding” of given cases in the smart environment domain, as well as allow different processes to collaborate with each other

Topological self-similarity on the random binary-tree model

Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We introduce a transformation of a binary tree, and derive a recursive equation about branch orders. As an application of the analysis, topological self-similarity and its generalization is proved in an asymptotic sense. Also, some important examples are presented

Genome-wide association study of REM sleep behavior disorder identifies polygenic risk and brain expression effects

Rapid-eye movement (REM) sleep behavior disorder (RBD), enactment of dreams during REM sleep, is an early clinical symptom of alpha-synucleinopathies and defines a more severe subtype. The genetic background of RBD and its underlying mechanisms are not well understood. Here, we perform a genome-wide association study of RBD, identifying five RBD risk loci near SNCA, GBA, TMEM175, INPP5F, and SCARB2. Expression analyses highlight SNCA-AS1 and potentially SCARB2 differential expression in different brain regions in RBD, with SNCA-AS1 further supported by colocalization analyses. Polygenic risk score, pathway analysis, and genetic correlations provide further insights into RBD genetics, highlighting RBD as a unique alpha-synucleinopathy subpopulation that will allow future early intervention

GBA variants in REM sleep behavior disorder: a multicenter study

To study the role of GBA variants in the risk for isolated rapid-eye-movement (REM)-sleep behavior disorder (iRBD) and conversion to overt neurodegeneration

Investigation of autosomal genetic sex differences in Parkinson's disease

Objective: Parkinson's disease (PD) is a complex neurodegenerative disorder. Men are on average similar to 1.5 times more likely to develop PD compared to women with European ancestry. Over the years, genomewide association studies (GWAS) have identified numerous genetic risk factors for PD, however, it is unclear whether genetics contribute to disease etiology in a sex-specific manner.Methods: In an effort to study sex-specific genetic factors associated with PD, we explored 2 large genetic datasets from the International Parkinson's Disease Genomics Consortium and the UK Biobank consisting of 13,020 male PD cases, 7,936 paternal proxy cases, 89,660 male controls, 7,947 female PD cases, 5,473 maternal proxy cases, and 90,662 female controls. We performed GWAS meta-analyses to identify distinct patterns of genetic risk contributing to disease in male versus female PD cases.Results: In total, 19 genomewide significant regions were identified and no sex-specific effects were observed. A high genetic correlation between the male and female PD GWAS were identified (rg = 0.877) and heritability estimates were identical between male and female PD cases (similar to 20%).Interpretation: We did not detect any significant genetic differences between male or female PD cases. Our study does not support the notion that common genetic variation on the autosomes could explain the difference in prevalence of PD between males and females cases at least when considering the current sample size under study. Further studies are warranted to investigate the genetic architecture of PD explained by X and Y chromosomes and further evaluate environmental effects that could potentially contribute to PD etiology in male versus female patients.Neurological Motor Disorder