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We derive precise formulas for the archimedean Euler factors occurring in certain standard Langlands 퐿-functions for unitary groups. In the 1980s, Paul Garrett, as well as Ilya Piatetski-Shapiro and Stephen Rallis (independently of Garrett), discovered integral representations of automorphic 퐿-functions that are Eulerian but, in contrast to the Rankin–Selberg and Langlands–Shahidi methods, do not require that the automorphic representations to which the 퐿-functions are associated are globally generic. Their approach, the doubling method, opened the door to a variety of applications that could not be handled by prior methods. For over three decades, though, the integrals occurring in the Euler factors at archimedean places for unitary groups eluded precise computation, except under particular simplifications (such as requiring certain representations to be one-dimensional, as Garrett did in the first major progress on this computation and only prior progress for general signatures). We compute these integrals for holomorphic discrete series of general vector weights for unitary groups of any signature. This has consequences not only for special values of 퐿-functions in the archimedean setting, but also for 푝-adic 퐿-functions, where the corresponding term had remained open. [ABSTRACT FROM AUTHOR]

We construct a p-adic Eisenstein measure with values in the space of p-adic automorphic forms on certain unitary groups. Using this measure, we p-adically interpolate certain special values of both holomorphic and non-holomorphic Eisenstein series, as both the archimedean and the p-adic weights of the Eisenstein series vary. [ABSTRACT FROM AUTHOR]

Abstract: An almost complete tripartite graph is obtained by removing an edge from the complete tripartite graph . A graph that can be decomposed into two isomorphic factors of diameter d is d-halvable. Fronček classified all 4-halvable almost complete tripartite graphs of even order in which the missing edge has its endpoints in two partite sets of odd order. In this paper, we classify 4-halvable almost complete tripartite graphs of even order for which the missing edge has an endpoint in a partite set with an even number of vertices. We also classify all 4-halvable almost complete tripartite graphs of odd order. Finally, we give a partial classification of 3- and 5-halvable almost complete tripartite graphs. [Copyright &y& Elsevier]

Two questions about the existence of structured matrices with given linesums and given zero–nonzero patterns are settled: when is there a symmetric matrix and when is there a skew-symmetric matrix? The solutions use some ideas from prior treatment of the analogous problems without structural constraints, but the current results require new conditions and new methodology. [Copyright &y& Elsevier]