On the Homology of Elementary Abelian Groups as Modules over the Steenrod Algebra

We examine the dual of the so-called "hit problem", the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as module over the Steenrod Algebra $\mathcal{A}$ at the prime 2. The dual problem is to determine the set of $\mathcal {A}$-annihilated elements in homology. The set of $\mathcal{A}$-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each $k \geq 0$, the set of {\it $k$ partially $\mathcal{A}$-annihilateds}, the set of elements that are annihilated by $Sq^i$ for each $i\leq 2^k$, itself forms a free associative algebra.Comment: 6 pages + reference

The Good, the Bad, and the Ugly: Student Writing in College Algebra

Students often protest when they are expected to write to a professional standard in courses beyond English or literature. In this panel discussion, a group of professors from diverse disciplines will describe methods for integrating specific writing projects into teaching in subjects including college algebra, biology, geosciences, and library and information science

Introducing Abstract Mathematics through Digit Sums and Cyclic Patterns

Using simple concepts that middle and high school students should be able to grasp, including “clock face arithmetic,” the standard multiplication table, and adding the digits of a number together, more abstract concepts such as modular arithmetic and cyclic groups may be introduced at an early stage in the students’ mathematical career. We find this approach to be organic and appealing to most students, encouraging them to think in different ways about familiar objects, and we encourage educators to test the concepts in their own classrooms

Analytic Geometry and Calculus I

This Grants Collection for Analytic Geometry and Calculus I was created under a Round Two ALG Textbook Transformation Grant. Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process. Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials: Linked Syllabus Initial Proposal Final Reporthttps://oer.galileo.usg.edu/mathematics-collections/1008/thumbnail.jp

Statistics of Random Permutations and the Cryptanalysis Of Periodic Block Ciphers

A block cipher is intended to be computationally indistinguishable from a random permutation of appropriate domain and range. But what are the properties of a random permutation? By the aid of exponential and ordinary generating functions, we derive a series of collolaries of interest to the cryptographic community. These follow from the Strong Cycle Structure Theorem of permutations, and are useful in rendering rigorous two attacks on Keeloq, a block cipher in wide-spread use. These attacks formerly had heuristic approximations of their probability of success. Moreover, we delineate an attack against the (roughly) millionth-fold iteration of a random permutation. In particular, we create a distinguishing attack, whereby the iteration of a cipher a number of times equal to a particularly chosen highly-composite number is breakable, but merely one fewer round is considerably more secure. We then extend this to a key-recovery attack in a "Triple-DES" style construction, but using AES-256 and iterating the middle cipher (roughly) a million-fold. It is hoped that these results will showcase the utility of exponential and ordinary generating functions and will encourage their use in cryptanalytic research.Comment: 20 page