Academic literature on the topic 'Sperner set systems'

AbstractFor a set L of positive integers, a set system $${\mathcal F}\subseteq 2^{[n]}$$ F ⊆ 2 [ n ] is said to be L-close Sperner, if for any pair F, G of distinct sets in $${\mathcal F}$$ F the skew distance $$sd(F,G)=\min \{|F\setminus G|,|G\setminus F|\}$$ s d ( F , G ) = min { | F \ G | , | G \ F | } belongs to L. We reprove an extremal result of Boros, Gurvich, and Milanič on the maximum size of L-close Sperner set systems for $$L=\{1\}$$ L = { 1 } , generalize it to $$|L|=1$$ | L | = 1 , and obtain slightly weaker bounds for arbitrary L. We also consider the problem when L might include 0 and reprove a theorem of Frankl, Füredi, and Pach on the size of largest set systems with all skew distances belonging to $$L=\{0,1\}$$ L = { 0 , 1 } .

Abstract Reduct of decision systems is the topic that has been attracting the interest of many researchers in data mining and machine learning for more than two decades. So far, many algorithms for finding reduct of decision systems by rough set theory have been proposed. However, most of the proposed algorithms are heuristic algorithms that find one reduct with the best classification quality. The complete study of properties of reduct of decision systems is limited. In this paper, we discover equivalence properties of reduct of consistent decision systems related to a Sperner-system. As the result, the study of the family of reducts in a consistent decision system is the study of Sperner-systems.

In multi-criteria decision making, attribute reduction has attracted the attention of researchers for more than two decades. So far, numerous scientists have proposed algorithms to construct reducts in decision tables. However, most of the suggested algorithms are heuristic which discovers a reduction based on criteria of the attribute set. In fact, studying the properties of reducts to build efficient attribute reduction models is an urgent problem. In this research, we present some properties of reducts in incomplete decision tables by the relational database theory approach. It was found that the properties of reducts in incomplete decision tables are equivalent to properties of the Sperner-systems in the theory of relational database. By studying the properties of the Sperner-systems, the efficient attribute reduction models can be built to improve the efficiency of multi-criteria decision making systems.

Purpose This paper aims to take seriously the import accorded to the interface within the digital humanities. It will probe some of the possibilities and limits of the computer interface as a reading and research tool by unpacking theoretical and practical aspects of interface design. Design/methodology/approach The authors wanted to see if they could design a tool that would meet three interrelated goals: the first was to develop a digital tool that would enable scholarship rather than mere publishing. Next, they wanted to build an interface that would acknowledge the situatedness of reading and meaning-making practices. Findings The research-oriented design approach to interface design has shown us how valuable it is to combine research and practice when thinking through issues in the digital humanities. Engaging in such a design project provides the unique opportunity to bring together theoretical concepts relating interface design with robust tools like XML mark-up and Drupal modules. Originality/value There is literature on the subject of transformation of print documents to electronic text (Hayles, 2003) and the representation of text within a computer (Sperberg-McQueen, 1991); this project attempts to build a prototype of what these theories might look like.

Abstract Mastocytosis is a hematopoietic disorder characterized by abnormal growth and accumulation of neoplastic mast cells (MC) in various organ systems. Using updated WHO criteria and the proposal of the consensus group, the disease can be divided into cutaneous mastocytosis (CM), indolent systemic mastocytosis (ISM), smouldering SM (SSM), SM with an associated hematologic neoplasm (SM-AHN), aggressive SM (ASM), and mast cell leukemia (MCL). In adult patients with skin involvement but unknown/unavailable bone marrow (BM) studies, the provisional diagnosis ´mastocytosis in the skin (MIS)´ is appropriate. Although this classification has been validated repeatedly and is of prognostic significance, additional prognostic parameters have been identified in recent years (yrs). We have established a patient-registry in the ECNM where over 1,000 cases with confirmed mastocytosis are included. The aim of this study was to identify and validate new prognostic variables predicting survival in patients with mastocytosis and to prepare a simple prognostic scoring system applicable in daily practice. Using the data set of the ECNM registry we analyzed overall survival (OS) and event-free survival (EFS; i.e. until death or progression) in 1,088 patients with mastocytosis (median age: 45.7 yrs; range: 0.1-83.3 yrs, f:m ratio, 1:0.79), including CM (n=152), MIS (n=126), ISM (n=650), SSM (n=26), SM-AHN (n=89), ASM (n=35), and MCL (n=10). The median observation period was 3.5 yrs (75-25% percentile: 1.5-6.9 yrs, maximum 34.1 yrs). In the entire cohort, the median OS was not reached. The probability to be alive after 5, 10, and 20 yrs was 89%, 83%, and 70%, respectively. As expected, the WHO classification turned out to be of utmost predictive significance (Figure 1A; p<0.005). In patients with non-advanced disease, namely CM, MIS, ISM, and SSM, the median survival was not reached, and the survival at 5 yrs was 100%, 97%, 98%, and 85%, respectively, whereas in advanced SM, namely SM-AHN, ASM, and MCL, the median survival was 2.8, 4.1, and 0.8 yrs, respectively. Patients with advanced SM were found to be older, to have higher serum tryptase- and alkaline phosphatase (aPhos) levels, higher white blood counts (WBC), lower hemoglobin (Hb) and platelet (PLT) counts, and more frequently presented with organomegaly (hepatomegaly, splenomegaly, or lymphadenopathy). Moreover, the male/female ratio was higher in advanced mastocytosis. To define the relative impact of the identified risk factors we randomly divided the total cohort (50:50) into a learning set and a validation set, performed uni- and multivariate analyses, and subsequently calculated cut off values for optimal prognostication. In these studies, the poor prognosis of patients with advanced SM was confirmed. In patients with non-advanced disease, all variables tested were significant concerning OS in univariate analyses. However, WBC, Hb, and lactate dehydrogenase had to be excluded due to variance inflation. In multivariate analyses, age >70 yrs, PLT <80 G/L (not related to SM), and aPhos ≥240 U/L were significant predictors concerning OS. Based on these parameters, we established a simple prognostic scoring system. In this score, patients with non-advanced disease (CM, MIS, ISM, SSM) without additional risk factors comprised the low risk group, those with non-advanced disease and presence of one or more risk factors (age >70 yrs, PLT <80 G/L, aPhos ≥240 U/L) the intermediate risk group, and those with advanced disease (with or without additional risk factors) the high risk group. The median OS in the low risk group was not reached, in the intermediate risk group it was 13.5 yrs, and in the high risk group 3.5 yrs (p<0.005; Figure 1B). Significant differences were also observed regarding EFS (p<0.005, Figure 1C). In conclusion, the WHO classification remains the gold-standard of prognostication in patients with mastocytosis, but additional factors, namely age, PLT, and aPhos, are powerful additional variables predicting OS in these patients. Based on in-depth analyses of the ECNM registry data-set, a simple prognostic scoring system for mastocytosis was established and is recommended to define the probability of OS and EFS in patients with mastocytosis in daily practice. Disclosures Sperr: Amgen: Honoraria, Research Funding; Novartis: Honoraria. Gotlib:Incyte Corporation: Honoraria, Membership on an entity's Board of Directors or advisory committees, Research Funding. Valent:Amgen: Honoraria; Celgene: Honoraria, Research Funding; Novartis: Honoraria, Research Funding; Deciphera Pharmaceuticals: Research Funding; Ariad: Honoraria, Research Funding.

A Sperner partition system is a system of set partitions such that any two set partitions $P$ and $Q$ in the system have the property that for all classes $A$ of $P$ and all classes $B$ of $Q$, $A \not\subseteq B$ and $B \not\subseteq A$. A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has the same cardinality $c=n/k$. In this paper, we prove a higher order generalization of Sperner's Theorem. In particular, we show that if $k$ divides $n$ the largest Sperner $k$-partition system on an $n$-set has cardinality ${n-1 \choose n/k-1}$ and is a uniform partition system. We give a bound on the cardinality of a Sperner $k$-partition system of an $n$-set for any $k$ and $n$.

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\mathcal{F}$ is a family of subsets of $X$ such that no two sets $A, B \in \mathcal{F}$ satisfy $A \subset B$ (or $B \subset A$) and $A \cap X_i=B\cap X_i$ for some $i$, then $|\mathcal{F}| \le {n \choose \lfloor n/2\rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following new result which may be of independent interest: let $\mathcal{F},\mathcal{G}$ be intersecting families of subsets of an $n$-element set that are additionally cross-Sperner, meaning that if $A \in\mathcal{F}$ and $B \in \mathcal{G}$, then $A \not\subset B$ and $B \not\subset A$. Then $|\mathcal{F}| +|\mathcal{G}| \le 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.

Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$.A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.