Thesis (Ph. D.)--Johns Hopkins University, 1914. / Biographical note. "Reprinted from the Transactions of the American mathematical society, July, 1916."

No description available.

Thesis (M.A.)--The University of North Carolina at Greensboro, 2010. / Directed by Paul Duvall; submitted to the Dept. of Mathematics and Statistics. Title from PDF t.p. (viewed Jul. 16, 2010). Includes bibliographical references (p. 39-40).

The object of this paper is to present the properties and some of the applications of the Elliptic Integrals.

We study the appearance of a novel phenomenon for coupled identical bursters: synchronized bursts where there are changes of spike synchrony within each burst. The examples we study are for normal form elliptic bursters where there is a periodic slow passage through a Bautin (codimension two degenerate Andronov-Hopf) bifurcation. This burster has a subcritical Andronov-Hopf bifurcation at the onset of repetitive spiking while the end of burst occurs via a fold limit cycle bifurcation. We study synchronization behavior of two Bautin-type elliptic bursters for a linear direct coupling scheme as well as demonstrating its presence in an approximation of gap-junction and synaptic coupling. We also find similar behaviour in system consisted of three and four Bautin-type elliptic bursters. We note that higher order terms in the normal form that do not affect the behavior of a single burster can be responsible for changes in synchrony pattern; more precisely, we find within-burst synchrony changes associated with a turning point in the spontaneous spiking frequency (frequency transition). We also find multiple synchrony changes in similar system by incorporating multiple frequency transitions. To explain the phenomenon we considered a burst-synchronized constrained model and a bifurcation analysis of the this reduced model shows the existence of the observed within-burst synchrony states. Within-burst synchrony change is also found in the system of mutually delaycoupled two Bautin-type elliptic bursters with a constant delay. The similar phenomenon is shown to exist in the mutually-coupled conductance-based Morris-Lecar neuronal system with an additional slow variable generating elliptic bursting. We also find within-burst synchrony change in linearly coupled FitzHugh-Rinzel 2 3 elliptic bursting system where the synchrony change occurs via a period doubling bifurcation. A bifurcation analysis of a burst-synchronized constrained system identifies the periodic doubling bifurcation in this case. We show emergence of spontaneous burst synchrony cluster in the system of three Hindmarsh-Rose square-wave bursters with nonlinear coupling. The system is found to change between the available cluster states depending on the stimulus. Lyapunov exponents of the burst synchrony states are computed from the corresponding variational system to probe the stability of the states. Numerical simulation also shows existence of burst synchrony cluster in the larger network of such system.

Thesis advisor: Benjamin Howard / We study two moduli problems involving false elliptic curves with complex multiplication (CM), generalizing theorems about the arithmetic degree of certain moduli spaces of CM elliptic curves. The first moduli problem generalizes a space considered by Howard and Yang, and the formula for its arithmetic degree can be seen as a calculation of the intersection multiplicity of two CM divisors on a Shimura curve. This formula is an extension of the Gross-Zagier theorem on singular moduli to certain Shimura curves. The second moduli problem we consider deals with special endomorphisms of false elliptic curves. The formula for its arithmetic degree generalizes a theorem of Kudla, Rapoport, and Yang. / Thesis (PhD) — Boston College, 2015. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Elliptic curves have played a large role in modern cryptography. Most notably, the Elliptic Curve Digital Signature Algorithm (ECDSA) and the Elliptic Curve Di e-Hellman (ECDH) key exchange algorithm are widely used in practice today for their e ciency and small key sizes. More recently, the Supersingular Isogeny-based Di e-Hellman (SIDH) algorithm provides a method of exchanging keys which is conjectured to be secure in the post-quantum setting. For ECDSA and ECDH, e cient and secure algorithms for scalar multiplication of points are necessary for modern use of these protocols. Likewise, in SIDH it is necessary to be able to compute an isogeny from a given nite subgroup of an elliptic curve in a fast and secure fashion. We therefore nd strong motivation to study and improve the algorithms used in elliptic curve cryptography, and to develop new algorithms to be deployed within these protocols. In this thesis we design and develop d-MUL, a multidimensional scalar multiplication algorithm which is uniform in its operations and generalizes the well known 1-dimensional Montgomery ladder addition chain and the 2-dimensional addition chain due to Dan J. Bernstein. We analyze the construction and derive many optimizations, implement the algorithm in software, and prove many theoretical and practical results. In the nal chapter of the thesis we analyze the operations carried out in the construction of an isogeny from a given subgroup, as performed in SIDH. We detail how to e ciently make use of parallel processing when constructing this isogeny. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection

by Chan Chi-kin. / Includes bibliographical references / Thesis (M.Ph.)--Chinese University of Hong Kong, 1985

by Chin Pang Cheung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 38-39). / Abstracts in English and Chinese. / Chapter 1 --- Main Results and Introduction --- p.4 / Chapter 1.1 --- Preliminaries --- p.5 / Chapter 2 --- The Linearized Operator in the Weighted Sobolev Spaces --- p.8 / Chapter 2.1 --- Weighted Sobolev Space and Some Useful Properties --- p.8 / Chapter 2.2 --- The Linearized Operator --- p.10 / Chapter 3 --- Reduction to Finite Dimensions --- p.19 / Chapter 4 --- Reduced Problem --- p.27 / Chapter 4.1 --- Proof of Theorem 1.1 --- p.27 / Chapter 4.2 --- Asymptotic Behavior of uE --- p.34 / Bibliography

Yeung Sik-ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 39-42). / Chapter 1 --- Introduction and Statement of the Results --- p.3 / Chapter 2 --- Maximum Principles and Comparison Theorems --- p.12 / Chapter 3 --- Pohozaev Identity and Symmetry for p-Laplacian when 1<p<2 --- p.18 / Chapter 4 --- Singularly Perturbed p-Laplacian Equation --- p.23 / Chapter 5 --- Appendix --- p.31 / Bibliography --- p.39