Harold M. Stark

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We discuss zeta functions of finite irregular undirected connected graphs (which may be weighted) and apply them to obtain, for example an analog of the prime number theorem for cycles in graphs. We consider 3 types of zeta functions of graphs: vertex, edge, and path. Analogs of the Riemann hypothesis are also introduced.
In Articles 303 and 304 of his 1801 Disquisitiones Arithmeticae [Gau86], Gauss put forward several conjectures that continue to occupy us to this day. Gauss stated his conjectures in the language of binary quadratic forms (of even discriminant only, a complication that was later dispensed with). Since Dedekind’s time, these conjectures have been phrased in(More)
We investigate spectra of Cayley graphs for the Heisenberg group over Þnite rings Z/pZ, where p is a prime. Emphasis is on graphs of degree four. We show that for odd p there is only one such connected graph up to isomorphism. When p = 2, there are at most two isomorphism classes. We study the spectra using two main tools representations of the Heisenberg(More)
OF THE DISSERTATION Quadratic Forms and Relative Quadratic Extensions by Michael William Mastropietro Doctor of Philosophy in Mathematics University of California San Diego, 2000 Professor Harold M. Stark, Chair Let k be a real quadratic eld of class number one, and K be a totally complex extension of k. We investigate the correspondence between ideal(More)
It is well known that every Euclidean ring is a principal ideal ring. It is also known for a very long time that the converse is not valid. Counterexamples exist under the rings R of integral algebraic numbers in quadratic complex fields Q [√−D ], for D = 19, 43, 67, and 163. In conection with these counterexamples several results were published in an(More)