Maximal antichains of minimum size

Let $n\geqslant 4$ be a natural number, and let $K$ be a set $K\subseteq [n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a maximal family $\mathcal{A}$ of subsets of $[n]$ such that $\mathcal{A}$ contains only sets whose size is in $K$, and $A\not\subseteq B$ for all ${A,B}\subseteq\mathcal{A}$, i.e. $\mathcal{A}$ is an antichain. We present a general construction of such antichains for sets $K$ containing 2, but not 1. If $3\in K$ our construction asymptotically yields the smallest possible size of such a family, up to an $o(n^2)$ error. We conjecture our construction to be asymptotically optimal also for $3\not\in K$, and we prove a weaker bound for the case $K={2,4}$. Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory which is interesting in its own right.Comment: fixed faulty argument in Section 2, added reference

Preventing Incomplete/Hidden Requirements: Reflections on Survey Data from Austria and Brazil

Many software projects fail due to problems in requirements engineering (RE). The goal of this paper is analyzing a specific and relevant RE problem in detail: incomplete/hidden requirements. We replicated a global family of RE surveys with representatives of software organizations in Austria and Brazil. We used the data to (a) characterize the criticality of the selected RE problem, and to (b) analyze the reported main causes and mitigation actions. Based on the analysis, we discuss how to prevent the problem. The survey includes 14 different organizations in Austria and 74 in Brazil, including small, medium and large sized companies, conducting both, plan-driven and agile development processes. Respondents from both countries cited the incomplete/hidden requirements problem as one of the most critical RE problems. We identified and graphically represented the main causes and documented solution options to address these causes. Further, we compiled a list of reported mitigation actions. From a practical point of view, this paper provides further insights into common causes of incomplete/hidden requirements and on how to prevent this problem.Comment: in Proceedings of the Software Quality Days, 201

Minimizing the regularity of maximal regular antichains of 2- and 3-sets

Let $n\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\{1,2,...,n\}$ with the following properties: (1) $\mathcal{A}$ is an antichain, i.e. no member of $\mathcal A$ is a subset of any other member of $\mathcal A$, (2) $\mathcal A$ is maximal, i.e. for every $X\in 2^{[n]}\setminus\mathcal A$ there is an $A\in\mathcal A$ with $X\subseteq A$ or $A\subseteq X$, and (3) $\mathcal A$ is $r$-regular, i.e. every point $x\in[n]$ is contained in exactly $r$ members of $\mathcal A$. We prove lower bounds on $r$, and we describe constructions for regular maximal antichains with small regularity.Comment: 7 pages, updated reference

Narzędziowe wsparcie procesu scoutingu – ocena wyników prac B+R z wykorzystaniem metodyki Quicklook oraz wpływ oceny na procesy komercjalizacji

Publikacja współfinansowana w ramach projektu: „SCOUTING – aktywny system monitoringu i oceny potencjału rynkowego prac badawczych kluczem do współpracy nauki i przedsiębiorców” współfinansowanego ze środków Unii Europejskiej w ramach Europejskiego Funduszu Społeczneg

CP violation at one loop in the polarization-independent chargino production in e+e- collisions

Recently Osland and Vereshagin noticed, based on sample calculations of some box diagrams, that in unpolarised e+e- collisions CP-odd effects in the non-diagonal chargino-pair production process are generated at one-loop. Here we perform a full one-loop analysis of these effects and point out that in some cases the neglected vertex and self-energy contributions may play a dominant role. We also show that CP asymmetries in chargino production are sensitive not only to the phase of mu parameter in the chargino sector but also to the phase of stop trilinear coupling A_t.Comment: 14 pages, 5 figure

Minimum Weight Flat Antichains of Subsets

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal F=\mathcal{A}\cup\mathcal{B}$, where $\mathcal{A}$ contains only $k$-subsets, while $\mathcal{B}$ contains only $(k-1)$-subsets. Moreover, we assume $\mathcal{A}$ consists of the first $m$ $k$-subsets in squashed (colexicographic) order, while $\mathcal{B}$ consists of all $(k-1)$-subsets not contained in the subsets in $\mathcal{A}$. Given reals $\alpha,\beta>0$, we say the weight of $\mathcal F$ is $\alpha\cdot|\mathcal{A}|+\beta\cdot|\mathcal{B}|$. We characterize the minimum weight antichains $\mathcal F$ for any given $n,k,\alpha,\beta$, and we do the same when in addition $\mathcal F$ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function

Evaluating the Barrier Effect of a Major Highway on Movement and Gene Flow of the Northern Flying Squirrel

Roads are pervasive sources of habitat fragmentation around the world, affecting an estimated 19 percent of the land area of the coterminous United States (Forman 2000). The barrier effect of roads has been demonstrated for species from multiple taxa. Still, information DNA was extracted from cheek cells of 41 individuals and genotyped at 12 microsatellite loci to characterize patterns of population structure. Seven of 16 monitored squirrels crossed the highway at least once during their nightly movements. Randomization tests of the movement data do not indicate significant avoidance of crossing the highway corridor. Movement does not necessarily equate to gene flow, however, and forthcoming analysis of microsatellite data will help elucidate whether current rates of movement are sufficient to maintain genetic connectivity across the highway. regarding the response of the vast majority of species to roads is lacking. We examine the effects of a major roadway on the movement and population genetics of Northern flying squirrels (Glaucomys sabrinus) in the Cascade Mountains of Washington, USA. During 2009 and 2010, flying squirrels (n = 16) were trapped and radio-tracked to gather data on movement within their home ranges and to detect movement across the roadway. Additionally

Observing the Effects of Inbreeding and Local Adaptation on Fitness in Westslope Cutthroat Trout Populations in a Common Garden

Montana Westslope cutthroat trout (Oncorhynchus clarkii lewisi, WCT) populations, particularly those east of the continental divide, are predominantly small and isolated from one other. Small population size inevitably leads to a more inbred population and can lead to lowered fitness (inbreeding depression). Isolated populations may experience local adaptation, which increases the fitness of a population within its native habitat. If inbreeding is the greatest threat to a population, introducing individuals from another population might be the best management decision. However, if local adaptation has also occurred, introducing new individuals might lower the fitness of the population. Our goal was to evaluate the relative importance of inbreeding and local adaptation on fitness in several WCT populations. This study combines eggs from several populations of WCT into several different natural habitats. Remote-site incubators were used to introduce eggs to six sites over four years in the Cherry Creek drainage. Each year a colder and warmer site was selected to test for the potential of local adaptation to stream temperature. Electrofishing was used to sample above and below introduction sites one, two, and three years after eggs were introduced. DNA sequencing of microsatellite loci in parents and offspring were then used to determine the population of origin of 511 offspring sampled in Cherry Creek in 2008 and will ultimately be used for over 750 offspring sampled in 2007 and 2009. Preliminary results suggest that a colder common habitat produces larger differences in relative fitness than a warmer habitat