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Register a new userA discrete Faa di Brunos formula
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We derive some formulas that rule the behaviour of finite differences under composition of functions with vector values and arguments.
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About 160 years ago, the Italian mathematician Fa`a di Bruno published two notes dealing about the now eponymous formula giving the derivative of any order of a composition of two functions. We reproduce here the two original notes, Fa`a di Bruno (1855, 1857), written respectively in Italian and in French, and propose a translation in English.
A closed linkage mechanism in three-dimensional space is an object comprising rigid bodies connected with hinges in a circular form like a rosary. Such linkages include Bricard6R and Bennett4R. To design such a closed linkage, it is necessary to solve a high-degree algebraic equation, which is generally difficult. In this lecture, the author proposes a new family of closed linkage mechanisms with an arbitrary number of hinges as an extension of a certain Bricard6R. They have singular properties, such as one-dimensional degree of freedom (1-DOF), and certain energies taking a constant value regardless of the state. These linkage mechanisms can be regarded as discrete Mobius strips and may be of interest in the context of pure mathematics as well. However, many of the properties described here have been confirmed only numerically, with no rigorous mathematical proof, and should be interpreted with caution.
We show that an apparently overlooked result of Euler from cite{E421} is essentially equivalent to the general multiplication formula for the $Gamma$-function that was proven by Gauss in cite{Ga28}.
The Hardy-Ramanujan formula for the number of integer partitions of $n$ is one of the most popular results in partition theory. While the unabridged final formula has been celebrated as reflecting the genius of its authors, it has become all too common to attribute either some simplified version of the formula which is not as ingenious, or an alternative more elegant version which was expanded on afterwards by other authors. We attempt to provide a clear and compelling justification for distinguishing between the various formulas and simplifications, with a summarizing list of key take-aways in the final section.
We present a representation formula for discrete indefinite affine spheres via loop group factorizations. This formula is derived from the Birkhoff decomposition of loop groups associated with discrete indefinite affine spheres. In particular we show that a discrete indefinite improper affine sphere can be constructed from two discrete plane curves.
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