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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]

This paper completes the construction of p-adic L-functions for unitary groups. More precisely, in Harris, Li and Skinner ['p-adic L-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure', Doc. Math. Extra Vol. (2006), 393-464 (electronic)], three of the authors proposed an approach to constructing such p-adic L-functions (Part I). Building on more recent results, including the first named author's construction of Eisenstein measures and p-adic differential operators [Eischen, 'A p-adic Eisenstein measure for unitary groups', J. Reine Angew. Math. 699 (2015), 111-142; 'p-adic differential operators on automorphic forms on unitary groups', Ann. Inst. Fourier (Grenoble) 62(1) (2012), 177-243], Part II of the present paper provides the calculations of local ξ-integrals occurring in the Euler product (including at p). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method. [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]

Clustering is a challenging problem in unsupervised learning. In lieu of a gold standard, stability has become a valuable surrogate to performance and robustness. In this work, we propose a non-parametric bootstrapping approach to estimating the stability of a clustering method, which also captures stability of the individual clusters and observations. This flexible framework enables different types of comparisons between clusterings and can be used in connection with two possible bootstrap approaches for stability. The first approach, scheme 1, can be used to assess confidence (stability) around clustering from the original dataset based on bootstrap replications. A second approach, scheme 2, searches over the bootstrap clusterings for an optimally stable partitioning of the data. The two schemes accommodate different model assumptions that can be motivated by an investigator's trust (or lack thereof) in the original data and additional computational considerations. We propose a hierarchical visualization extrapolated from the stability profiles that give insights into the separation of groups, and projected visualizations for the inspection of the stability of individual operations. Our approaches show good performance in simulation and on real data. These approaches can be implemented using the R package bootcluster that is available on the Comprehensive R Archive Network (CRAN). [ABSTRACT FROM AUTHOR]