Daniel J. Velleman

Pascal's matrices

(1993). Pascals Matrices. The American Mathematical Monthly: Vol. 100, No. 4, pp. 372-376.

Permutations and Combination Locks

Consider a combination lock with n buttons, numbered 1 through n. A valid combination consists of a sequence of button-pushes, in which each button is pushed exactly once. If the buttons must be pushed one at a time, then clearly there will be n! possible combinations. But what if we are allowed to push butt6ns simultaneously? We can represent a valid combination for such a lock as a sequence of disjoint, nonempty subsets of the set B = {1,2,. . ., n} whose union is B. Each set in the sequence specifies a collection of buttons to be pushed simultaneously. For example, if n = 3 then we have the following possible combinations:

-morasses, and a weak form of Martin’s axiom provable in

We prove, in ZFC, that simplified gap-i morasses of height w exist. By earlier work on the relationship between morasses and forcing it immediately follows that a certain Martins axiom-type forcing axiom is provable in ZFC. We show that this forcing axiom can be thought of as a weak form of MA,,1 and give some applications.

ω-Morasses, and a Weak form of Martin's Axiom Provable in ZFC

We prove, in ZFC, that simplified gap-i morasses of height w exist. By earlier work on the relationship between morasses and forcing it immediately follows that a certain Martins axiom-type forcing axiom is provable in ZFC. We show that this forcing axiom can be thought of as a weak form of MA,,1 and give some applications.

Variable declarations in natural deduction

Abstract We propose the use of variable declarations in natural deduction. A variable declaration is a line in a derivation that introduces a new variable into the derivation. Semantically, it can be regarded as declaring that the variable denotes an element of the universe of discourse. Undeclared variables, in contrast, do not denote anything, and may not occur free in any formula in the derivation. Although most natural deduction systems in use today do not have variable declarations, the idea can be traced back to one of the first papers on natural deduction. We show how the use of variable declarations in natural deduction leads to a formal system that has a number of desirable features: It is simple, easy to use and understand, and corresponds closely to ordinary informal reasoning. Soundness and completeness of the system are easily proven. Furthermore, the system clarifies the role of the existential instantiation rule in natural deduction.

On a Combinatorial Principle of Hajnal and Komjath

In their paper [3], Hajnal and Komjath define the following combinatorial principle: Definition 1.1. Suppose κ is an infinite cardinal and n ω . Then H n ( κ ) is the statement: There is a function F : [ κ ] n → [[ κ ] ω ] ≤ ω such that (a) ∀ A ∈[ κ ] n ∀ Y ∈ F ( A )( Y ⊆ min ( A )), and (b) . H n ( κ ) is related to a more general principle introduced by Hajnal and Nagy in [4]. For applications of these principles to free sets for set mappings and Ramsey games we refer the reader to [3] and [4]. In [3] Hajnal and Komjath prove the consistency of ZFC + GCH + ∀ n ∈ ω ( H n + 1 ( ω n + 1 )), relative to an ω -Mahlo cardinal. They conjecture that L is a model of this theory, and suggest that the proof might require higher gap morasses. The first few cases of this conjecture are known to be true; it is easy to see that if CH holds then H 1 ( ω 1 ) is true, and Laver proved that V = L implies H 2 ( ω 2 ). In this paper we go one step further and prove V = L → H 3 ( ω 3 ). Unfortunately our methods do not appear to give H n ( ω n ) for n ≥ 4. Most of our notation is standard. If X is any set and κ is a cardinal number then [ X ] κ is the set of subsets of X with cardinality κ , and [ X ] ≤ κ is the set of subsets of X with cardinality ≤ κ . If X is a set of ordinals then tp( X ) is the order type of X .

What to Expect in a Game of Memory

Abstract The game of memory is played with a deck of n pairs of cards. The cards in each pair are identical. The deck is shuffled and the cards laid face down. A move consists of flipping over first one card and then another. The cards are removed from play if they match. Otherwise, they are flipped back over and the next move commences. A game ends when all pairs have been matched. We determine that, when the game is played optimally, as n → ∞: • The expected number of moves is (3 − 2 ln 2)n + 7/8 − 2 ln 2 ≈ 1.61n.• The expected number of times two matching cards are unwittingly flipped over is ln 2.• The expected number of flips until two matching cards have been seen is

Partitioning Pairs of Countable Sets of Ordinals

In [3], Todorcevic showed that ω 1 ⇸ [ ω 1 ] ω 1 2 . In this paper we use similar methods to prove an analogous partition theorem for P ω 1( λ ), for certain uncountable cardinals λ . Recall that ω 1 → [ ω 1 ] ω 1 2 , means that for every function f : [ ω 1 ] 2 → ω 1 there is a set A ∈ [ ω 1 ] ω 1 such that f “[ A ] 2 ≠ ω 1 , and of course ω 1 ⇸ [ ω 1 ] ω 1 2 , is the negation of this statement. For partition relations on P ω 1 ( → ) it is customary to partition only those pairs of sets in which the first set is a subset of the second. Thus for A ⊆ P ω 1 ( λ ) we define We will write P ω 1 ( λ ) → [unbdd] λ 2 to mean that for every function f : [ P ω 1 ( λ )] ⊂ 2 → λ there is an unbounded set A ⊆ P ω 1 ( λ ) such that f “[ A ] ⊂ 2 ≠ λ , and again P ω 1 ( λ ) ⇸ [unbdd] λ 2 is the negation of this statement.

On Gauss's First Proof of the Fundamental Theorem of Algebra

Abstract Carl Friedrich Gauss is often given credit for providing the first correct proof of the fundamental theorem of algebra in his 1799 doctoral dissertation. However, Gausss proof contained a significant gap. In this paper, we give an elementary way of filling the gap in Gausss proof.

Simpson Symmetrized and Surpassed

Simpson’s rule is a well-known numerical method for approximating definite integrals. It is named after Thomas Simpson, who published it in 1743, although it was known already more than a century before that. Bonaventura Cavalieri gave a geometric version of Simpson’s rule in 1639, and James Gregory published the rule in 1668. Others who published not only Simpson’s rule but also more general formulas before Simpson’s publication in 1743 include Isaac Newton, Roger Cotes, and James Stirling [4, p. 77]. Many calculus textbooks state Simpson’s rule in its composite form, which says that ∫ b