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Unsolved Haiku, Scott W. Williams
Unsolved Haiku, Scott W. Williams
Journal of Humanistic Mathematics
This poem describes the still unsolved 1937 conjecture of Lloyd Collatz: Do repeated applications of the algorithm described yield the number 1?
The Genesis Of A Theorem, Osvaldo Marrero
The Genesis Of A Theorem, Osvaldo Marrero
Journal of Humanistic Mathematics
We present the story of a theorem's conception and birth. The tale begins with the circumstances in which the idea sprouted; then is the question's origin; next comes the preliminary investigation, which led to the conjecture and the proof; finally, we state the theorem. Our discussion is accessible to anyone who knows mathematical induction. Therefore, this material can be used for instruction in a variety of courses. In particular, this story may be used in undergraduate courses as an example of how mathematicians do research. As a bonus, the proof by induction is not of the simplest kind, because it …
Lattice Extensions And Zeros Of Multilinear Polynomials, Maxwell Forst
Lattice Extensions And Zeros Of Multilinear Polynomials, Maxwell Forst
CGU Theses & Dissertations
We treat several problems related to the existence of lattice extensions preserving certain geometric properties and small-height zeros of various multilinear polynomials. An extension of a Euclidean lattice $L_1$ is a lattice $L_2$ of higher rank containing $L_1$ so that the intersection of $L_2$ with the subspace spanned by $L_1$ is equal to $L_1$. Our first result provides a counting estimate on the number of ways a primitive collection of vectors in a lattice can be extended to a basis for this lattice. Next, we discuss the existence of lattice extensions with controlled determinant, successive minima and covering radius. In …
Squate, Tom Blackford
Squate, Tom Blackford
Journal of Humanistic Mathematics
This is the story of a middle school student who befriends an irrational number, the square root of eight.
On The Characterization Of Prime Sets Of Polynomials By Congruence Conditions, Arvind Suresh
On The Characterization Of Prime Sets Of Polynomials By Congruence Conditions, Arvind Suresh
CMC Senior Theses
This project is concerned with the set of primes modulo which some monic, irreducible polynomial over the integers has a root, called the Prime Set of the polynomial. We completely characterise these sets for degree 2 polynomials, and develop sufficient machinery from algebraic number theory to show that if the Galois group of a monic, irreducible polynomial over the integers is abelian, then its Prime Set can be written as the union of primes in some congruence classes modulo some integer.
A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, Gizem Karaali
A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Prove It!, Kenny W. Moran
Prove It!, Kenny W. Moran
Journal of Humanistic Mathematics
A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.
Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger
Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger
CMC Senior Theses
Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.
Effective Structure Theorems For Quadratic Spaces Via Height, Lenny Fukshansky
Effective Structure Theorems For Quadratic Spaces Via Height, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at the Second International Conference on The Algebraic and Arithmetic Theory of Quadratic Forms, December 2007.