Nonabelian Group Based Cryptography

From the Washington University Senior Honors Thesis Abstracts (WUSHTA), 2017. Published by the Office of Undergraduate Research. Joy Zalis Kiefer, Director of Undergraduate Research and Associate Dean in the College of Arts & Sciences; Lindsey Paunovich, Editor; Helen Human, Programs Manager and Assistant Dean in the College of Arts and Sciences Mentor: John Shareshia

Conodont biostratigraphy of the Bakken and lower Lodgepole Formations (Devonian and Mississippian), Williston Basin, North Dakota

Fourteen cores were sampled and processed, and conodonts obtained were used in the biozonation of the Bakken and basal Lodgepole Formations in the Williston Basin in North Dakota. The Bakken Formation of Devonian and Mississippian age and the Lodgepole Formation, entirely of Mississippian age, are subsurface units in North Dakota. The Bakken Formation (44 metres thick) consists of, in ascending order: a lower, dark, organic shale; a middle calcareous siltstone that is divisible into three lithic units; and an upper, dark, indurated, organic shale. The Lodgepole Formation (160 to 260 metres thick) is an argillaceous, fossiliferous limestone near the Basin center that varies to oolitic and skeletal limestone near the Basin margins. Conodonts were recovered from the three members of the Bakken, by oxidizing in household bleach, and from the basal 15.5 metres of the Lodgepole, by treatment in acetic acid, in 10 of the 14 cores sampled. The 1963 discrete elements recovered were placed in 95 species-level form-taxa assignable to 22 form-genera: 15 percent (289) from the lower member of the Bakken, less than 1 percent (3) from the middle member of the Bakken, 47 percent (930) from the upper member of the Bakken, and 37 percent (741) from the basal Lodgepole. Twenty-five Bakken Formation form-taxa and all 61 Lodgepole Formation form-taxa are reported for the first time. Three-fourths (919) of the Bakken specimens were obtained from three, separate, thin, lag zones, one in the lower member of the Bakken and two in the upper member of the Bakken. No lag zones were observed in the Lodgepole. Lags may have resulted from accumulation of coarse-grained fossil and inorganic debris under sediment-starved conditions, winnowing out of fine-grained elastics and organic material during deposition that concentrated coarse-grained fossils and inorganic material, and selective winnowing of partially lithified sediment by evanescent currents. Lags are important stratigraphically but are of little value in recognizing specific conodont biozones. At least one Famennian (Lower Palmatolepis gracilis expansa) and possibly five Kinderhookian (siphonodellid) conodont biozones were recognized in the Bakken and basal Lodgepole Formations. Thirty-eight biostratigraphically useful form-species were used for recognition of the biozones. Form-genera recovered include, in order of decreasing abundance (species-level form-taxa listed in parentheses): Siphonodella (9), Polygnathus (9), Pseudopolygnathus (5), Spathognathodus (5), Palmatolepis (3), Bispathodus (3), Dinodus (2), Branmehla (1), and Elictognathus (1). Conodonts obtained from the lower member of the Bakken suggest the presence of the palmatolepid-polygnathid biofacies indicative of an outer shelf environment. Siphonodella sulcata, from the upper member of the Bakken, suggests that this member may be as old as earliest Kinderhookian, but because of its association in lag zones with conodonts representative of younger biozones, its stratigraphic significance is diminished. Due to the paucity of conodonts in the middle member, further refinement of the Devonian-Mississippian systemic boundary was not possible. The conodont fauna supports the concept of continuous sedimentation from the Bakken Formation into the Lodgepole Formation

On cubic multisections of Eisenstein series

A systematic procedure for generating cubic multisections of Eisenstein series is given. The relevant series are determined from Fourier expansions for Eisenstein series by restricting the congruence class of the summation index modulo three. We prove that the resulting series are rational functions of η(τ) and η(3τ), where η is the Dedekind eta function. A more general treatment of cubic dissection formulas is given by describing the dissection operators in terms of linear transformations. These operators exhibit properties that mirror those of similarly defined quintic operators

Generalized reciprocal identities

Included in Ramanujan’s Notebooks are two reciprocal identities. The first identity connects the Rogers-Ramanujan continued fraction with an eta quotient. The second identity is a level thirteen analogue. These are special cases of a more general class of relations between eta quotients and modular functions defined by product generalizations of the Rogers-Ramanujan continued fraction. Each identity is shown to be a relation between generators for a certain congruence subgroup. The degree, form, and symmetry of the identities is determined from behavior at cusps of the congruence subgroup whose field of functions the parameters generate. The reciprocal identities encode information about fundamental units and class numbers for real quadratic fields

Ramanujan type congruences for quotients of level 7 Klein forms

Klein forms are used to construct generators for the graded algebra of modular forms of level 7. Dissection formulas for the series imply Ramanujan type congruences modulo powers of 7 for a family of generating functions that subsume the counting function for 7-core partitions. The broad class of arithmetic functions considered here enumerate colored partitions by weights determined by parts modulo 7. The method is a prototype for similar analysis of modular forms of level 7 and at other prime levels. As an example of the utility of the dissection method, the paper concludes with a derivation of novel congruences for the number of representations by x^2+xy+2y^2 in exactly k ways

Tidally Induced Oscillations and Orbital Decay in Compact Triple-Star Systems

We investigate the nature of tidal effects in compact triple-star systems. The hierarchical structure of a triple system produces tidal forcing at high frequencies unobtainable in binary systems, allowing for the tidal excitation of high frequency p-modes in the stellar components. The tidal forcing exists even for circular, aligned, and synchronized systems. We calculate the magnitude and frequencies of three-body tidal forcing on the central primary star for circular and coplanar orbits, and we estimate the amplitude of the tidally excited oscillation modes. We also calculate the secular orbital changes induced by the tidally excited modes, and show that they can cause significant orbital decay. During certain phases of stellar evolution, the tidal dissipation may be greatly enhanced by resonance locking. We then compare our theory to observations of HD 181068, which is a hierarchical triply eclipsing star system in the Kepler field of view. The observed oscillation frequencies in HD 181068 can be naturally explained by three-body tidal effects. We then compare the observed oscillation amplitudes and phases in HD 181068 to our predictions, finding mostly good agreement. Finally, we discuss the past and future evolution of compact triple systems like HD 181068.Comment: 19 pages, 8 figures, published in MNRA

A fragment on Euler\u27s constant in Ramanujan\u27s lost notebook

A formula for Euler’s constant found in Ramanujan’s lost notebook and also in a problem he submitted to the Journal of the Indian Mathematical Society is proved and discussed