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Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k  2 and of the same level N, both eigenfunctions of the Hecke operators, and both normalized so that a1 = 1. The main result we seek is
that when E and f are congruent mod a prime p (which may be a prime ideal lying over a rational prime p > 2), the algebraic parts of the special values L(E; ; j) and L(f; ; j) satisfy congruences mod the same prime. On the way to proving the congruence result, we construct the modular symbol attached to an Eisenstein series.
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In Hilbert spaces, five classes of monotone operator of relevance to the theory of monotone operators, variational inequality problems, equilibrium problems, and differential inclusions are investigated. These are the classes of paramonotone, strictly monotone, 3-cyclic monotone, 3*-monotone (or rectangular, or *-monotone), and maximal monotone operators.
    Examples of simple operators with all possible combinations of class inclusion are given, which together with some additional results lead to an exhaustive knowledge of monotone class relationships for linear operators, linear relations, and for monotone operators in general.
Many of the example operators considered are the sum of a subdifferential with a skew linear operator (and so are Borwein-Wiersma decomposable).  Since for a single operator its Borwein-Wiersma decompositions are not unique, clean, essential, extended, and standardized decompositions are defined and the theory developed.   In particular, every Borwein-Wiersma decomposable operator has an essential decomposition, and many sufficient conditions are given for the existence of a clean decomposition.
    Various constructive methods are demonstrated together which, given any Borwein-Wiersma decomposable operator, are able to obtain a decomposition, as long as the operator has starshaped domain.  These methods are more accurate if a clean decomposition exists.  The techniques used apply a variant of Fitzpatrick's Last Function, the theory of which is developed here, where this function is shown to consist of a Riemann integration and be equivalent to Rockafellar's antiderivative when applied to subdifferentials.  Furthermore, a different saddle function representation for monotone operators is created using this function which has theoretical and numerical advantages over more classical representations.

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My thesis is dedicated to the study of various spatial stochastic processes from theoretical biology.

For finite interacting particle systems from evolutionary biology, we study two of the simple rules for the evolution of cooperation on finite graph in Ohtsuki, Hauert, Lieberman, and Nowak [Nature 441 (2006) 502-505] which were first discovered by clever, but non-rigorous, methods. We resort to the notion of voter model perturbations and give a rigorous proof, very different from the original arguments, that both of the rules of Ohtsuki et al. are valid and are sharp. Moreover, the generality of our method leads to a first-order approximation for fixation probabilities of general voter model perturbations on finite graphs in terms of the voter model fixation probabilities.

For spatial branching processes from population biology, we prove pathwise non-uniqueness in the stochastic partial differential equation (SPDE) of some one-dimensional super-Brownian motions with immigration and zero initial value. In contrast to a closely related case studied in a recent work by Mueller, Mytnik, and Perkins, the solutions of the present SPDE are assumed to be nonnegative and are unique in law. In proving possible separation of solutions, we use a novel method, called continuous decomposition, to validate natural immigrant-wise semimartingale calculations for the approximating solutions, which may be of independent interest in the study of superprocesses with immigration.

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This is followed by Niven Lecture at 2:00-3:00 pm