Fermat's theorem

A. Defining Reaction Mechanisms and Catalyst Structure The methane-to-methanol reactions that we aimed to evaluate consist of four critical steps starting from the initial reactants (CH4, NH4+, oxo): C-H activation followed by a hydrogen atom abstraction (HAA) from CH4 to form some combination of the hydrogenated complex, ammonium or ammonia, and a methyl radical; a radical rebound (RR) to form a methanol adduct and ammonium; methanol dissociation from the metal; and catalysis regeneration via oxygen atom transfer. We modeled these steps as stationary points throughout the reaction coordinate, each point consisting of a sum of the reactants, intermediates, or products of each step. Our research mainly focuses on the energy barrier of the C-H activation of methane (∆G‡), as it is the proposed rate determining step [12]. In detail, this step involves the breaking of a C-H methane bond and the subsequent transfer of the hydrogen to the oxo, forming a hydroxyl group. We characterized this energy barrier by building a plausible transition state guess that lay between our reactants and HAA intermediates, performing a DFT optimization, and then calculating ∆G‡ by evaluating the resulting output molecule. One of the most crucial aspects of this study involves the minimization of ∆G‡, the relative free energy of transition states. We proposed variations of a catalyst structure inspired by a cofactor of EDBH in the hopes of replicating its efficient hydrocarbon oxidation. The original…

was noticed. Although Germain had a challenging start to her career, she was persistent, ambitious, and determined; these qualities, as well as her work in mathematics, made her hard to ignore. Sophie Germain began her career by attempting to solve Fermat’s Last Theorem which states that “xn + yn = zn has no positive integer solutions for x, y, z when n > 2” (Riddle). Germain worked relentlessly on proving this theorem and eventually believed had she solved it. Her conclusion was this: “for…

“Sophie Germain’s Theorem can be applied for many prime exponents, by producing a valid auxiliary prime, to eliminate the existence of solutions to the Fermat equation involving numbers not divisible by the exponent p” (Laubenbacher and Pengelley, 9). Legendre later used Germain’s findings in a supplement to the second edition of his book, Essai sur le Théorie des Nombres (O’Connor and Robertson). The Institut de France awarded Germain a medal for her work on Fermat’s Last Theorem and Germain…

under the alias of Le Blanc, affianced in correspondent with the author. Sophie Germain was no ivory-tower mathematician, which later became evident in 1807, when French troops were occupying Hanover. Recalling Archimedes’ destiny and dreading for Gausss’s safety in the story she once read, she addressed an inquiry to the French commanding officer, General Pernety, who was a family friend of the Germains. One of Sophie Germain’s theorems is connected to the baffling and still unanswered problem…

standard practices during the time period because he prefers pleasure from sex from unconventional methods over loyal monogamy. Septimus is academically tutoring Thomasina, but is also teaching her sex in a unique manner considering their time period. This introduces Thomasina’s desire for knowledge, but also how sex and mathematics can be a societal battle. Septimus then immediate contrasts sex to math in order to portray how even though they seem like binary oppositions, they are immensely…

Mathematical proof is a method used in both primary and secondary schools at varying levels. Children should be able to justify and give reasons for a conjecture they have produced in order to validate a mathematical proof. There are various techniques that can be used to illustrate the nature of proof in schools. Mathematical proof is defined as a thorough and believable argument to support the truth of a prediction in maths, which turns the assumptions into valid conclusions (Haylock and…

As we delve deeper into our theories on interpersonal communication, we begin to learn more about ourselves and how to interact with the people around us. Whether they are in our lives on a personal, professional or combined capacity. This week I have decided to look at the theories of Interactional View and Genderlect Styles (Griffin, 2015). When we study Interactional View, a theory developed by Paul Watzlawick, we can see how communication has shaped us into the people we are today.…

that one of the central goals of school mathematics, notably at the upper secondary level, is the development of proof concepts (Coe & Ruthven, 1994). The incorporation of algebraic proofs in Victorian Mathematics education also refutes the criticism that the focus on algebraic proofs theme is sporadic at the secondary school level (Pedemonte, 2008). Many different methods of proof are stated in both the VCE Mathematics Study Design 2016-2018 and some VCE Mathematics textbooks: Specialist…

are people that will disagree and say that the reason of evolution is because with time our species became more observant and gained knowledge and just knew that these things would work then they created them. I believe that this is completely wrong I think that there needs to be a theory then experiments, that is the only way for evolution to work. People believe that the basis of math is based off of assumptions. And I completely agree because assumptions and theories to me are synonyms.…

These things don't exist in math. Math is full of "rules" that don't have exceptions. Things such as the Pythagorean Theorem will always work for right triangles. There will never be a right triangle where a^2 + b^2 does not = c^2. This is what I like about math. Math is a subject that I think would never Let me down ; It has no contradictions. It's something that's reliable and useful. I've learned that without math, life would be a cycle of events without reason. If I ever wonder why when I…