Confluentes Mathematici

We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of some algebraic group H defined over R. In the case when R = ℝ, this result was claimed in [5], but a mistake in the proof was recently found, and the new proof we obtained has the advantage of being valid over an arbitrary real closed field. We also extend the result to not necessarily connected affine Nash groups over arbitrary real closed fields.

We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in the group is a union of a family of quantifier free definable sets, where the parameter of the family runs over a set definable (with quantifiers) in a sort which carries the structure of an ordered set with some additional unary predicates.

As a corollary, we find that all definable functions in ordered abelian groups are piecewise linear on finitely many definable pieces.

We consider the gradient flow of a one-homogeneous functional, whose dual involves the derivative of a constrained scalar function. We show in this case that the gradient flow is related to a weak, generalized formulation of a Hele–Shaw flow. The equivalence follows from a variational representation, which is a variant of well-known variational representations for the Hele–Shaw problem. As a consequence we get existence and uniqueness of a weak solution to the Hele–Shaw flow. We also obtain an explicit representation for the Total Variation flow in dimension 1, and easily deduce basic qualitative properties, concerning in particular the "staircasing effect".

We show that ℓ_p norms are characterized as the unique norms which are both invariant under coordinate permutation and multiplicative with respect to tensor products. Similarly, the L_p norms are the unique rearrangement-invariant norms on a probability space such that ‖XY‖ = ‖X‖ ⋅ ‖Y‖ for every pair X, Y of independent random variables. Our proof combines the tensor power trick and Cramér's large deviation theorem.