We present some results about chains of monoids S₀ → S₁ → S.₂. and their associated representation rings, with particular emphasis to behavior as the index n (viz. S_n) varies. A rich supply of such chains of monoids can be found via specializations of diagrammatic algebras or variations of diagrammatic algebras, where the inclusions involve the addition of loose strands. This thesis comprises of original results along three themes associated with the above: (1) Identifying a certain polynomial property featuring operators on representation rings, and a characterization of chains of groups G₀ →G₁ → G₂ .. which satisfy this polynomial property. (2) Understanding the induced action on homology from topological actions of the chain of Temperley-Lieb monoids TL₁ → TL₂ → TL₃ ... . Making the analogy to classical representation stability. (3) Identifying chains of diagrammatic monoids S₀ → S₁ → S₂ .. on which cryptographic protocols resist linear attacks. Explicitly computes lower bounds on the dimensions of all representations of various truncations of diagrammatic monoids.

This thesis explores the relationship between the media, murderous women, and the concept of separate spheres. Murderous women challenged established gender norms. They did not conform to the societal expectations of their gender, therefore, they were not considered “normal.” As such, women like Alice Mitchell, Jane Toppan, and Amy Archer Gilligan became objects of media, medical, and public curiosity. As defined by medical science and society, newspapers policed the boundaries of “normality” by sensationalizing the lives, actions, and trials of deadly damsels. Newspaper coverage of murderous women reminded the public of the consequences of “abnormality” and non-conformity. This thesis argues that sensationalized stories of lethal ladies between 1890 and 1920 shaped public perceptions of gender, crime, mental illness, and substantiated the perceived “need” for separate spheres. Furthermore, it gives a voice to a group of historical women who existed on the fringes of society.

The purpose of the study is to enumerate the representations in high school geometry, to narrow the focus to the most common ones, and to describe how the common representations are coordinated. To achieve this purpose, I analyzed several sections of a popular geometry textbook using thematic analysis, semiotics, and a pragmatic approach to capture the variety of representations into categories and to use descriptive statistics to narrow the focus to the most common representations and coordinations. The major findings are the bringing into prominence of representations like textbook gestures, ordered pairs, written language, tables, and their uses, and at the same time understating the importance of less common representations of physical objects. Other important findings are: (1) exposing which representations are most often coordinated like written language (WL), diagrams (D), numbers (N), and ordered pairs (OP), short geometry symbols (Sy), e.g., WL to D, WL+Sy to D, and N+OP to D; (2) some of the mechanisms in that coordination like using, numbers, point names, and textbook gestures, which include color, arrows, font changes, etc. Clarifying the representations in high school geometry and narrowing the scope to the most common ones allows researchers to study various combinations of representations and their impact students. / Teaching & Learning

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In 1987, Jean-Pierre Serre gave a conjecture on the correspondence between degree 2 odd irreducible representations of the absolute Galois group of Q and modular forms. Letting M be an imaginary quadratic field, L.M. Figueiredo gave a related conjecture concerning degree 2 irreducible representations of the absolute Galois group of M and their correspondence to homology classes. He experimentally confirmed his conjecture for three representations arising from PSL(2,3)-polynomials, but only up to a sign because he did not lift them to SL(2,3)-polynomials. In this paper we compute explicit lifts and give further evidence that his conjecture is accurate.

In this paper, we examine the conjecture of Avner Ash, Darrin Doud, David Pollack, and Warren Sinnott relating Galois representations to the mod p cohomology of congruence subgroups of the general linear group of n dimensions over the integers. We present computational evidence for this conjecture (the ADPS Conjecture) for the case n = 3 by finding Galois representations which appear to correspond to cohomology eigenclasses predicted by the ADPS Conjecture for the prime p, level N, and quadratic nebentype. The examples include representations which appear to be attached to cohomology eigenclasses which arise from D8, S3, A5, and S5 extensions. Other examples include representations which are reducible as sums of characters, representations which are symmetric squares of two-dimensional representations, and representations which arise from modular forms, as predicted by Jean-Pierre Serre for n = 2.

Consider a non-Galois cubic extension K/Q ramified at a single prime p > 3. We show that if K is a subfield of an S_4-extension L/Q ramified only at p, we can determine the Artin conductor of the projective representation associated to L/Q, which is based on whether or not K/Q is totally real. We also show that the number of S_4-extensions of this type with K as a subfield is of the form 2^n - 1 for some n >= 0. If K/Q is totally real, n > 1. This proves two conjectures of Siman Wong.

In this thesis, classical computation is first introduced. Finite quantum systems are considered with D-dimensional Hilbert space, and position x and momentum p taking values in Z(D) (the integers modulo D). An analytic rep resentation of finite quantum systems that use Theta function is presented and considered. The first novel part of this thesis is contribution to study reversible classical CNOT gates and their binary inputs and outputs with reversible cir cuits. Furthermore, a reversible classical Toffoli gates are considered, as well as implementation of a Boolean expression with classical CNOT and Toffoli gates. Reversible circuits with classical CNOT and Toffoli gates are also considered. The second novel part of this thesis the study of quantum computation in terms of CNOT and Toffoli gates. Analytic representations and their zeros are considered, while zeros of the inputs and outputs for quantum CNOT and Toffoli gates are studied. Also, approximate computation of their zeros on the output are calculated. Finally, some quantum circuits are discussed. i

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