Discrete Logarithms in Cryptography
There are many cryptosystems whose security is reliant on the difficulty of solving the discrete logarithm problem. Due to this difficulty of solving the discrete logarithm problem, many algorithms have been discovered for reducing this difficulty. This paper will explore the Diffie-Hellman public key exchange and its relation to the El Gamal cryptosystem. As well, algorithms such as Baby-step Giant-step, Pohlig-Helmman, and Pollard's Rho which reduce the time complexity of solving the discrete logarithm problem will be analyzed in terms of their run times. All of these algorithms have been implemented using java programming to allow testing of the effectiveness of the algorithms mentioned above.
Who's Got the Power? An Introduction to Power Indices
The idea of voting power has many real world applications. Whether it is through purchasing shares of ownership of a corporation, or voting in a national election through the electoral college, individual power is a key part of voting scenarios. But how many votes does one really need to significantly increase their power? Power indices are highly useful tools for calculating voting power of both individuals and groups. Two such indices that will be examined in this paper are the Banzhaf Power Index and the Hapley-Shubik Power Index, both of which use the concepts of swing votes and winning coalitions to determine individual voting power. Indices like these can be constructed through brute force computations, by using generating functions, or through other methods that will be discussed in this paper. We will also consider an example of the simple legislative system in the United States, made up of a President, House of Representatives, and the Senate. Furthermore, the relationship between these three bodies and their proportional voting powers will be examined to determine how the size of a group affects an individual’s voting power.
Penrose Tiling and Nonperiodic Islamic Architectural Ornament
This paper introduces the idea of a tiling of the plane. Nonperiodic tilings have been a topic of interest to mathematicians since the early 1960's. The Penrose tilings are examples of such a tiling. We present the basic ideas in the proof that any Penrose tiling must be nonperiodic. Finally, we will connect the concepts of Penrose tiles to the underlying structures present in medieval Islamic architectural ornament.
Continuous Dependence on Modeling for the Backward Heat Equation
Given the one-dimensional backward heat equation, we first show how to derive the general form of solutions, uc (x; t), and show that solutions do not depend continuously on initial data to demonstrate ill-posedness. We then propose a well-posed approximate problem with general solution given by vc (x; t). Finally, we prove continuous dependence results for the difference between solutions of the ill-posed backward heat equation, and the well-posed model problem.
Classifying Compact, Connected Surfaces
A surface is a second countable Hausdorff space which is locally Euclidean of dimension 2. A well-known result in algebraic topology states that any compact, connected surface is homeomorphic to the sphere, a connected sum of finitely many tori, or a connected sum of finitely many projective planes. After giving a careful definition of a surface and some examples, we prove that any surface is of one of these three forms. We then develop the theory behind the fundamental group of a topological space, which is necessary to prove that the surfaces in this classification are pairwise distinct up to homeomorphism.
A Study in Ring Multiplication Tables
In 2012, Sunil Chebolu proved that all 1s in the multiplication table of Zn are located on the main diagonal if and only if n is a divisor of 24. This paper is concerned with finding the n for which certain proportions of the 1s in the multiplication tables of Zn lie on the diagonal. In particular, we prove that for a prime p, there exists an n such that 1/p of the 1's in the multiplication table for Zn lie on the diagonal if and only if p is a Sophie Germain prime.
Mathematical Modeling of the Sleeping Brain:
Understanding the Dynamics of Early Child Sleep
The human brain is highly active during sleep. Neuronal pathways in the brain are developed and strengthened during sleep, particularly in early child development. The current project utilizes a differential equations based model, in combination with sleep data from preschool-aged children, to investigate age-related changes in sleep. The aim of the project is three-fold: (1) identify sleep/wake patterns in children aged 2.5 to 6.5 years and compare these with adult-like sleep/wake patterns; (2) fit the model to these patterns to predict physiological mechanisms (e.g. strengthening of specific neuronal pathways) that may underlie the changes observed in sleep during development; and (3) analyze model dynamics using numerically-simulated discrete maps.
Statistical analyses of electroencephalography (EEG) and actigraphy data were used to characterize the general sleep architecture of participants' sleep. Single factor analyses of variance revealed a significant decrease of rapid eye movement (REM) sleep, both during day time naps and night time sleep in the preschool participants. Overall, the total amount of sleep that participants had significantly decreased as a function of age as well. Consequently, we identified a parameter regime in which the model generates bi-phasic sleep that accurately reflected the distinct nap and night time patterns of observed child sleep. In addition, discrete numerical simulations of the model revealed valuable insight on the effects of the circadian parameters on sleep/wake patterns, both in adults and children. By using circle maps, we identified fixed points of the system, as well as their stability. There was a single stable fixed point in the adult case. While the child case (for ages 2.5-3.5) did not show fixed points, it showed stable period-2 points.
Using Euclid's Theorems to Minimize Skin Grafts in the Operating Room
Cosmetic surgery may be one of the most well-known types of plastic surgery, but it is by far not the only type. Microsurgery, hand surgery and reconstructive surgery are just a few other important types of plastic surgery. Within these types, plastic surgeons use skin grafts and aps to fix an injury. Inspired by a paper written by Margaret Symington, we look into the best angle to cut a skin graft in order to cover a wound without wasting additional skin from the donor site and minimizing scaring and pain for the patient. By using Euclid's Theorems and GeoGebra, we determine the best cut angle. We also examine how large a wound we would be able to cover if we restricted the size of the lens.