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Sturm oscillation and comparison theorems
 in ”Sturm–Liouville Theory: Past and Present”, 29–43, Birkhäuser
, 2005
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Historical Projects in Discrete Mathematics and Computer Science
"... A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itse ..."
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A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development
Minimum Area Venn Diagrams Whose Curves are Polyominoes
"... While working at the Berlin Academy, the renowned Swiss mathematician Leonard Euler was asked ..."
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While working at the Berlin Academy, the renowned Swiss mathematician Leonard Euler was asked
Early Writings on Graph Theory: Euler Circuits and The K"onigsberg Bridge Problem
"... I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude ..."
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I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude. [1, p. 30] Known today as the field of `topology, ' Leibniz's `geometry of position ' was slow to develop as a mathematical field. As C. F. Gauss (1777 1855) noted in 1833, Of the geometry of position, which Leibniz initiated and to which only two geometers, Euler and Vandermonde, have given a feeble glance, we know and possess, after a century and a half, very little more than nothing. [1, p. 30]
“Es steht schon bei Dedekind ” 1 Lecture at the DM/Algebra Seminar on 11/4/2005.
"... The content of this lecture is influenced very much by writings and translations by John Stillwell, and several books at the end of these notes which I highly recommend. The most important thing I wish to share is the realization of the similarity between several contributions by R. Dedekind to mode ..."
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The content of this lecture is influenced very much by writings and translations by John Stillwell, and several books at the end of these notes which I highly recommend. The most important thing I wish to share is the realization of the similarity between several contributions by R. Dedekind to modern mathematics: residue classes, the theory of real numbers and the theory of ideals. These notes are very rough and informal; I did not make any serious attempt to polish them. Some comments I made during the lecture may be missing: I forgot what exactly I said, and I do not believe that some of them were really important. On the other hand, several remarks that I planned to make at the lecture, and did not because of the shortage of time, are included in these notes. I do assume that most listeners had at least one course of abstract algebra or number theory, but it will be easier for those who had two. 1. When we think how we came to like mathematics, or how our tastes were formed, or which were the most memorable moments, we can make a relatively short list of these events. Thinking about this lecture I compounded about twenty five, with about five being major. To me, one of the most cherished events are those when mathematical facts which were disjoint in my mind, suddenly come together, and
Generating and drawing areaproportional . . .
, 2007
"... An Euler diagram C = {c1, c2,...,cn} is a collection of n simple closed curves (i.e., Jordan curves) that partition the plane into connected subsets, called regions, each of which is enclosed by a unique combination of curves. Typically, Euler diagrams are used to visualize the distribution of discr ..."
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An Euler diagram C = {c1, c2,...,cn} is a collection of n simple closed curves (i.e., Jordan curves) that partition the plane into connected subsets, called regions, each of which is enclosed by a unique combination of curves. Typically, Euler diagrams are used to visualize the distribution of discrete characteristics across a sample population; in this case, each curve represents a characteristic and each region represents the subpopulation possessing exactly the combination of containing curves’ properties. Venn diagrams are a subclass of Euler diagrams in which there are 2 n regions representing all possible combinations of curves (e.g., two partially overlapping circles). In this dissertation, we study the Euler Diagram Generation Problem (EDGP), which involves constructing an Euler diagram with a prescribed set of regions. We describe a graphtheoretic model of an Euler diagram’s structure and use this model to develop necessaryandsufficient existence conditions. We also use the graphtheoretic model to prove that the EDGP is NPcomplete. In addition, we study the related AreaProportional Euler Diagram Generation Problem (ωEDGP), which involves
Hints for Selected Exercises Hints For Selected Exercises of Chapter A
"... Exercise 7. Let me show that if R is transitive, then xPRy and yRz implies xPRz. Since PR ⊆ R, it is plain that xRz holds in this case. Moreover, if zRx holds as well, then yRx, for R is transitive and yRz. But this contradicts xPRy. Exercise 13. (c) Suppose c(S) = ∅ (which is possible only if S ..."
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Exercise 7. Let me show that if R is transitive, then xPRy and yRz implies xPRz. Since PR ⊆ R, it is plain that xRz holds in this case. Moreover, if zRx holds as well, then yRx, for R is transitive and yRz. But this contradicts xPRy. Exercise 13. (c) Suppose c(S) = ∅ (which is possible only if S  ≥ 3). Take any x1 ∈ S. Since c(S) = ∅, there is an x2 ∈ S\{x1} with x2 " x1. Similarly, there is an x3 ∈ S\{x1, x2} with x3 " x2. Continuing this way, I find S = {x1,..., xS} with xS  " · · · " x1. Now find a contradiction to being acyclic. Exercise 14. Apply Sziplrajn’s Theorem to the transitive closure of the relation ∗:= ∪ ({x∗} × Y). Exercise 16. (e) infA = WA and supA = W{B ∈ X: VA ⊆ B} for any class
Minimum Area Venn Diagrams Whose Curves are
"... In his lesson on categorical propositions and syllogisms, Euler used diagrams comprised of overlapping circles; these diagrams became known as Eulerian circles, or simply Euler diagrams. In an Euler diagram, a proposition's classes are represented as circles whose overlap depends on the relatio ..."
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In his lesson on categorical propositions and syllogisms, Euler used diagrams comprised of overlapping circles; these diagrams became known as Eulerian circles, or simply Euler diagrams. In an Euler diagram, a proposition's classes are represented as circles whose overlap depends on the relationship established by the proposition. For example, the propositions All arachnids are bugs Some bugs are cannibals can be represented by Fig. 1. In 1880, a Cambridge priest and mathematician named John Venn published a paper studying special instances of Euler diagrams in which the classes overlap in all possible ways [25]; although originally applied to logic reasoning, these Venn diagrams are now commonly used to teach students about set theory. For example, the Venn diagram in Fig. 2 shows all the ways in which three sets can intersect. The primary difference between Venn 1
Binary Arithmetic: From Leibniz to von Neumann
"... Before I reached the schoolclass in which logic was taught, I was deep into the historians and poets, for I began to read the historians almost as soon as I was able to read at all, and I found great pleasure and ease in verse. But as soon as I began to learn logic, I was greatly excited by the div ..."
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Before I reached the schoolclass in which logic was taught, I was deep into the historians and poets, for I began to read the historians almost as soon as I was able to read at all, and I found great pleasure and ease in verse. But as soon as I began to learn logic, I was greatly excited by the division and order in it. I immediately noticed, to the extent that a boy of 13 could, that there must be a great deal in it [5, p. 516]. His study of logic and intellectual quest for order continued throughout his life and became a basic principle to his method of inquiry. At the age of 20 he published Dissertatio de arte combinatoria (Dissertation on the Art of Combinatorics) in which he sought a characteristica generalis (general characteristic) or a lingua generalis (general language) that would serve as a universal symbolic language and reduce all debate to calculation. Leibniz maintained: