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We give a new proof of Euler’s formula for the values of the Riemann zeta function at the positive even integers. The proof involves estimating a certain integral of elementary functions two different ways and using a recurrence relation for the Bernoulli polynomials evaluated at .
A frequent topic in the study of pattern avoidance is identifying when two sets of patterns are Wilf equivalent, that is, when for all . In recent work of Dokos et al. the notion of Wilf equivalence was refined to reflect when avoidance of classical patterns preserves certain statistics. We continue their work by examining des-Wilf equivalence when avoiding certain nonclassical patterns.
The involutions and the symmetric spaces associated to the family of modular groups of order are explored. We begin by analyzing the structure of the automorphism group and by establishing which automorphisms are involutions. We conclude by calculating the fixed-point group and symmetric spaces determined by each involution.
Using Fermat’s two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form can be expressed as the sum of two integer squares. We prove that is the sum of two squares for all if and only if is a square. We also prove that if , is odd, and is the sum of two squares, then is the sum of two squares for all , . Using Aurifeuillian factorization, we show that if is a prime and , then there are either zero or infinitely many odd such that is the sum of two squares. When , we define to be the least positive integer such that is the sum of two squares, and prove that if is the sum of two squares for odd, then , and both and are sums of two squares.
Representations are special functions on groups that give us a way to study abstract groups using matrices, which are often easier to understand. In particular, we are often interested in irreducible representations, which can be thought of as the building blocks of all representations. Much of the information about these representations can then be understood by instead looking at the trace of the matrices, which we call the character of the representation. This paper will address restricting characters to subgroups by shrinking the domain of the original representation to just the subgroup. In particular, we will discuss the problem of determining when such restricted characters remain irreducible for certain low-rank classical groups.
A finite graph on vertices has a prime labeling provided there is a way to label the vertices with the integers 1 through such that every pair of adjacent vertices has relatively prime labels. We extend the definition of prime labeling to infinite graphs and give a simple necessary and sufficient condition for an infinite graph to have a prime labeling. We then measure the complexity of prime labelings of infinite graphs using techniques from computability theory to verify that our condition is as simple as possible.
We study a variation of the game of best choice (also known as the secretary problem or game of googol) under an additional assumption that the ranks of interview candidates are restricted using permutation pattern-avoidance. We describe the optimal positional strategies and develop formulas for the probability of winning.
Given a graph, we can form a spanning forest by first sorting the edges in a random order, and then only keeping edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges, and so we can ask, for example, how likely is it for the process to produce a graph with trees.
We look at all graphs which can produce at most two trees in this process and determine the probabilities of having either one or two trees. From this we construct infinite families of graphs which are nonisomorphic but produce the same probabilities.
The log-concavity of the Hölder mean of two numbers, as a function of its index, is presented first. The notion of -cevian of a triangle is introduced next, for any real number . We use this property of the Hölder mean to find the smallest index such that the length of an -cevian of a triangle is less than or equal to the -Hölder mean of the lengths of the two sides of the triangle that are adjacent to that cevian.
We discuss a descriptive theory of decision making which has received much attention in recent decades: prospect theory. We specifically focus on applying the theory to problems with two attributes, assisted by different independence assumptions. We discuss a process for solving decision problems using the theory before applying it to a real life example of purchasing breakdown cover.