Maass-Shimura operators on holomorphic modular forms preserve the modularity of modular forms but not holomorphy, whereas the derivative preserves holomorphy but not modularity. Rankin-Cohen brackets are bilinear operators that preserve both and are expressed in terms of the derivatives of modular forms. We give identities relating Maass-Shimura operators and Rankin-Cohen brackets on modular forms and obtain a natural expression of the Rankin-Cohen brackets in terms of Maass-Shimura operators. We also give applications to values of L-functions and Fourier coefficients of modular forms.
This chapter considers three hands in poker. It shows how the probabilities of obtaining a Full House, Straight, and Flush level off compared to the probabilities of the other hands, and then looks at how the probabilities of these three hands shift according to how the deck has been modified. These hands, as the chapter notes, have distinct frequencies in any generalized poker game. However, the analysis here is conducted through a game of Heartless Poker, which uses a standard deck with all the hearts removed. That there are only three suits and thirty-nine cards in the deck changes the probabilities, and perhaps the rankings, of the different types of hands.
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