University of Dresden

n this talk we consider a p-isometrisability property of discrete groups. If p=2 this property is equivalent to unitarisability. We prove that any group containing a non-abelian free subgroup is not p-isometrisable for any p∈(1,∞). We also discuss some open questions and possible relations of p-isometrisability with the recently introduced Littlewood exponent Lit(Γ).

To numerical analysts and other applied mathematicians Jacobians and Hessians

are matrices, i.e. rectangular arrays of numbers or algebraic expressions.

Possibly taking account of their sparsity such arrays are frequently passed

into library routines for performing various computational tasks.

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A central goal of an activity called automatic differentiation has been the

accumulation of all nonzero entries from elementary partial derivatives

according to some variant of the chainrule. The elementary partials arise

in the user-supplied procedure for evaluating the underlying vector- or

scalar-valued function at a given argument.

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We observe here that in this process a certain kind of structure that we

call "Jacobian scarcity" might be lost. This loss will make the subsequent

calculation of Jacobian vector-products unnecessarily expensive.

Instead we advocate the representation of the Jacobian as a linear computational

graph of minimal complexity. Many theoretical and practical questions remain unresolved.