Consider two finite-dimensional normed spaces X and Y , a fixed convex cone K ⊂ X, and a linear mapping A : X → Y . We call A well-posed if AK = Y . In particular, in the purely linear case K = X, well-posedness coincides with surjectivity. Our interest is in the “distance to ill-posedness”: that is, we seek the smallest structured linear perturbation ∆A : X → Y such that the perturbed mapping ...

We consider the simply typed-calculus with primitive re-cursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization and ground (base-type) connuence is proved for the full calculus; full connuence is proved for the calcul...

We extend the classical inverse and implicit function theorems, the implicit function theorems of Lyusternik and Graves, and the results of Clarke and Pourciau to the situation when the given function is not smooth, but it has a convex strict prederivative whose measure of noncompactness is smaller than its measure of surjectivity. The proof of the main results requires Banach’s open mapping th...

The Besicovitch pseudodistance measures the relative size of the set of points where two functions take different values; the quotient space modulo the induced equivalence relation is endowed with a natural metric. We study the behaviour of cellular automata in the new topology and show that, under suitable additional hypotheses, they retain certain properties possessed in the usual product top...

The notions of relaxed submonotone and relaxed monotone mappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hy...

The Besicovitch pseudodistance measures the relative size of the set of points where two functions take different values; the quotient space modulo the induced equivalence relation is endowed with a natural metric. We study the behaviour of cellular automata in the new topology and show that, under suitable additional hypotheses, they retain certain properties possessed in the usual product top...

For a quartic primitive CM field K, we say that a rational prime p is evil if at least one of the abelian varieties with CM by K reduces modulo a prime ideal p|p to a product of supersingular elliptic curves with the product polarization. We call such primes evil primes for K. In [GL], we showed that for fixed K, such primes are bounded by a quantity related to the discriminant of the field K. ...