This is the second of two papers investigating for which positive integers m there exists a maximal antichain of size m in the Boolean lattice B_n (the power set of [n]:={1,2,… ,n} , ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only l-sets and (l+1) -sets for some l.

Extending a classical theorem of Sperner, we characterize the integers m such that there exists a maximal antichain of size m in the Boolean lattice Bn, that is, the power set of [n]:={1,2,… ,n} , ordered by inclusion. As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t and k, we ask which integers s have the property that there exists a family ℱ of k-sets with |ℱ|=t such that the shadow of ℱ has size s, where the shadow of ℱ is the collection of (k − 1)-sets that are contained in at least one member of ℱ . We provide a complete answer for t⩽ k+1 . Moreover, we prove that the largest integer which is not the shadow size of any family of k-sets is √(2)k^3/2+√(8)k^5/4+O(k) .

We describe in dialogue form a possible way of discovering and investigating 10-adic numbers starting from the naive question about a ‘largest natural number’. Among the topics we pursue are possibilities of extensions to transfinite 10-adic numbers, 10-adic representations of rational numbers, zero divisors, square roots and 10-adic roots of higher degree of natural numbers, and applications of 10-adic number representation in computer arithmetic. The participants of the dialogue are idealized embodiments of different philosophical attitudes towards mathematics. The article aims at illustrating how these attitudes interact, in both jarring and stimulating ways, and how they impact mathematical development.

We present and analyze the employment of the Diproche system, a natural language proof checker, within a one-semester mathematics beginners lecture with 228 participants. The system is used to check the students’ solution attempts to proving exercises in Boolean set theory and elementary number theory and to give them immediate feedback. The benefits of the employment of the system are assessed via a questionnaire at the end of the semester and via analyzing the solution attempts of a subgroup of the students. Based on our results we develop approaches for future improvements.

SummaryEverybody knows from school how to solve a quadratic equation of the form x2−px+q=0 graphically. To solve more than one equation this method can become tedious, as for each pair (p, q) a new parabola has to be drawn. Stunningly, there is one single curve that can be used to solve every quadratic equation by drawing tangent lines through a given point (p, q) to this curve.In this article we derive this method in an elementary way and generalize it to equations of the form xn−px+q=0 for arbitrary n≥2. Moreover, the number of solutions of a specific equation of this form can be seen immediately with this technique. In concluding the article, we point out connections to the duality of points and lines in the plane and to the concept of Legendre transformation.

A slogan that you find on the back of a pack of Skittles candy says ‘No two rainbows are the same. Neither are two packs of Skittles. Enjoy an odd mix.’ An online blog [1] describes how the blogger found two identical packs of Skittles, among 468 packs with a total of 27,740 Skittles. Meticulously collecting the data for this experiment was apparently triggered by some earlier calculations. More precisely, the blogger writes:

This is the second in a sequence of three papers investigating the question for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the previous paper we characterized those $m$ between $\binom{n}{\lceil n/2\rceil}-\lceil n/2\rceil^2$ and the maximum size $\binom{n}{\lceil n/2 \rceil}$ that are not sizes of maximal antichains. In this paper we show that all smaller $m$ are sizes of maximal antichains.

There are some wonderful theorems in undergraduate mathematics that everybody should know. We all have learned in school how to apply the fundamental theorem of calculus to solve an integral, say ∫...

Extending a classical theorem of Sperner, we investigate the question for which positive integers m there exists a maximal antichain of sizem in the Boolean lattice Bn, that is, the power set of [n] := {1, 2, . . . , n}, ordered by inclusion. We characterize all such integers m in the range ( n ⌈n/2⌉ ) −⌈n/2⌉2 6 m 6 ( n ⌈n/2⌉ ) . As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t and k, we ask which integers s have the property that there exists a family F of k-sets with |F| = t such that the shadow of F has size s, where the shadow of F is the collection of (k − 1)-sets that are contained in at least one member of F . We provide a complete answer for the case t 6 k + 1. Moreover, we prove that the largest integer which is not the shadow size of any family of k-sets is √ 2k3/2 + 4 √ 8k5/4 +O(k).

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"No two rainbows are the same. Neither are two packs of Skittles. Enjoy an odd mix!". Using an interpretation via spatial random walks, we quantify the probability that two randomly selected packs of Skittles candy are identical and determine the expected number of packs one has to purchase until the first match. We believe this problem to be appealing for middle and high school students as well as undergraduate students at University.

We describe in the form of a dialogue a development of various reflections on the combinatorics of set partitions; among the topics we pursue are the number of ways of partitioning a finite set into a fixed number $$d$$ of subsets of odd or even size, into $$a$$ parts of odd and $$b$$ parts of even size, and into $$d$$ parts, each of which has a size in a certain congruence class modulo some natural number $$m$$ . To this end, pattern guessing, recursion and induction, combinatorial interpretation and generating functions are employed. The participants of the dialogue represent different perspectives on and approaches to mathematics.

Ziel dieser Arbeit war die Analyse der Expression von Multidrug-Resistance-Protein (P-Glykoprotein P170) und Multidrug Resistance associated Protein 1 (MRP1) mittels FACS-Analyse und RT-PCR bei Nierenkarzinomzelllinien unterschiedlichen histologischen Typs. Auserdem wurde die Funktionalitat des P-Glykoprotein-Systems mittels des Rhodamin-Efflux-Assays in der FACS-Analyse uberpruft. Zur Untersuchung des Zusammenhangs zwischen dem Ansprechen auf Paclitaxel/Taxol und der Expression und Funktion von P-Gp bzw. MRP1 wurden Modulationsversuche sowohl im Proliferationsassay (MTT-Assay) als auch in der FACS-Analyse durchgefuhrt. Die MRP1-Expression war gering und zeigte keine Korrelation mit der Paclitaxel/Taxol-Sensitivitat. Dagegen zeigte die Expression von P-Gp eine positive Korrelation mit der Paclitaxel/Taxol-Sensitivitat, die allerdings nur fur Paclitaxel statistisch signifikant war. In den Modulationsversuchen konnte eine signifikante Beeinflussung sowohl der Palictaxel- bzw. Taxol-Sensitivitat als auch der Rhodamin-Effluxeffektivitat mit Hilfe der Multidrug-Modulatoren Verapamil und Cremophor EL erzielt werden. Die Ergebnisse dieser Arbeit sprechen dafur, dass die P-Gp-Expression von Nierenzellkarzinomen ursachlich an der Paclitaxel/Taxol-Sensitivitat beteiligt ist, dass offensichtlich aber auch noch andere Resistenzmechanismen eine Rolle spielen. Somit sind noch weitere Untersuchungen an den vorhandenen Zelllinien notig, um andere, zusatzlich wirksame Zytostatika-Resistenzmechanismen des Nierenzellkarzinoms zu identifizieren.

OF A FIELD STUDY Submitted in partial fulfillment of the requirements for the degree of Specialist in Education at the Graduate School of Eastern Illinois University . CHARLESTON, ILL.ltil.S.•1 . 1980 ~The development of a student handbook for the Middle School students of the Argenta-Oreana School District was the object of this ·particular field study. The handbook previously used by the students was merely an overview of the rules and.regulations and was definitely lacking in detail and explanation~ Our big~ school has a student handbook, but the distinct differences between the operation of the. high school and the middle school would not permit the use of the same handbook for both buildings. It was through experiences of my four years of employment as middle school principal, directives f~om the board of education, conferences with other principals, and r.equests from faculty members, that the eventual handbook was developed. The creation of this handbook was brought about mainly through the desire of the school personnel to have a more detailed explanation of how the middle school functions. My main task was to create a handbook that would be easy to read and follow and also sufficiently describe the functions of the middle school. In addition to these. activities, the purpose of this handbook was to make the transition from elementary school to middle school smooth and orderly.· Likewise, I wanted to have a handbook that would answer most of the questions that parents and students generally ask about our school and its operation. Also, we have many students who move into our district from neighboring communities. Many times our methods of operating a school are different from the manner in which the new students are famiiiar. Hopefully, this new handbook will allow these new students to understand inunediately the way our middle school will operate. and what type of behavior we expe,ct them to demonstrate. Keeping these new factor~ in mind, I developed a handbook whi_ch describes the procedures used for the proper operation of the middle school and the responsibilities necessary for the students to be successful in our school. The ~ajor reconunendation I would have is that the handbook be reviewed yearly and updated where necessary. Hopefully, a constant review of the handbook will help maintain an updated and useful instrument for both students and· parents.