Affine equivalence of cubic homogeneous rotation symmetric Boolean functions

Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009. This paper studies the much more complicated cubic case for such functions. A new concept of \emph{patterns} is introduced, by means of which the structure of the smallest group G_n, whose action on the set of all such cubic functions in $n$ variables gives the affine equivalence classes for these functions under permutation of the variables, is determined. We conjecture that the equivalence classes are the same if all nonsingular affine transformations, not just permutations, are allowed. This conjecture is verified if n < 22. Our method gives much more information about the equivalence classes; for example, in this paper we give a complete description of the equivalence classes when n is a prime or a power of 3.Comment: This is the author's version of a manuscript which has been accepted by Information Sciences. The ancillary Mathematica file contains a program which computes the equivalence classes described in the paper, given n = number of variable

Caffeine Use and Associations with Sleep in Adolescents with and without Attention-Deficit/Hyperactivity Disorder (ADHD)

The objective of this study was to compare caffeine consumption in the morning, afternoon, and evening in adolescents with and without Attention Deficit/Hyperactivity Disorder (ADHD) and examine associations with sleep functioning. Participants were 302 adolescents (ages 12-14) with (n=140) and without (n=162) ADHD. Adolescents wore actigraph watches to assess total sleep time and wake after sleep onset and reported on their sleep-wake problems and the number of caffeinated beverages consumed per day in the morning, afternoon, and evening. Parents reported on adolescents’ difficulties initiating and maintaining sleep. Chi-square analyses, odds ratios, and path analyses were conducted. Analyses controlled for sex, medication status, and pubertal development. Adolescents with ADHD were 2.47 times more likely to consume caffeine in the afternoon and evening than adolescents without ADHD. Path analyses indicated significant associations between afternoon caffeine use and more self-reported sleep problems for adolescents with and without ADHD, and an association between evening caffeine use and self-reported sleep problems only in adolescents with ADHD. Afternoon caffeine use was also associated with parent-reported sleep problems in adolescents with ADHD but not in adolescents without ADHD. Caffeine use was not associated with actigraphy-assessed sleep. This is the first study to show that adolescents with ADHD consume more caffeine than their peers during later times of the day. Additionally, caffeine use is more consistently associated with poorer subjective sleep functioning in adolescents with ADHD compared to adolescents without ADHD. Pediatricians and mental health professionals should assess for caffeine use in adolescents with ADHD and co-occurring sleep problems

Balanced Symmetric Functions over GF(p)GF(p)

Under mild conditions on $n,p$, we give a lower bound on the number of $n$-variable balanced symmetric polynomials over finite fields $GF(p)$, where $p$ is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we conjecture that $X(2^t,2^{t+1}l-1)$ are the only nonlinear balanced elementary symmetric polynomials over GF(2), where $X(d,n)=\sum_{i_1<i_2<...<i_d}x_{i_1} x_{i_2}... x_{i_d}$, and we prove various results in support of this conjecture.Comment: 21 page

At the Intersection of Resident, Research and Recreation Stakeholder Interests: East Maui, Hawai‘i, as a Sustainable Tourism Destination

The experiences of contemporary protected areas indicate adaptations to challenges brought about by resource management strategies. Resident communities, protected area management, and the tourism industry stakeholders demonstrate that evolving relationships are complex webs of competing and cooperating interests. The geographic isolation of East Maui delayed the cultural disruption of traditional practices and is an area where residents simultaneously resist assimilation and re-create cultural landscapes to offer visitors a glimpse into the past and a view of an emerging future associated with the renaissance of Native Hawaiian identity. Partnerships have brought about and nurtured the perpetuation of culture and the conservation of biodiversity as stakeholders recognize shared benefits. Among the outcomes are that residents have reconstituted the identity of East Maui as a Hawaiian place with benefits to various stakeholders, including a network of protected areas. A sustainability framework suggests a reappraisal of how to nurture, not alter, East Maui’s identity