We study lower bounds for testing membership in families of linear/affine-invariant Boolean functions over the hypercube. A family of functions P ⊆ {{0, 1} → {0, 1}} is linear/affine invariant if for any f ∈ P , it is the case that f ◦L ∈ P for any linear/affine transformation L of the domain. Motivated by the recent resurgence of attention to the permutation isomorphism problem, we first focus...

Consider a function u defined on Rn, except, perhaps, on a closed set of potential singularities S. Suppose that u solves the eikonal equation ‖Du‖ = 1 in the pointwise sense on Rn \S, where Du denotes the gradient of u and ‖ · ‖ is a norm on Rn with the dual norm ‖ · ‖∗. For a class of norms which includes the standard p-norms on Rn, 1 < p < ∞, we show that if S has Hausdorff 1-measure zero an...

Abstract: An affine Moser-Trudinger inequality, which is stronger than the Euclidean MoserTrudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L energy of gradient. The geometric inequality at the core of the affine Moser-Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies...

It has recently been shown (O. F. Pedersen T. R. Rasmussen, O. Omland, T. Sigsgaard, P. H. Quanjer. and M. R. Miller. Eur. Respir. J. 9: 828-833, 1996) that the added resistance of a mini-Wright peak flowmeter decreases peak expiratory flow (PEF) by approximately 8% compared with PEF measured by a pneumotachograph. To explore the reason for this, 10 healthy men (mean age 43 yr, range 33-58 yr) ...

Differential transcriptional modulation of duplicated fatty acid-binding protein genes by dietary fatty acids in zebrafish (Danio rerio): evidence for subfunctionalization and neofunctionalization of duplicated genes BMC Evolutionary Biology 2009, 9:219 doi:10.1186/1471-2148-9-219 Santhosh Karanth ([email protected]) Santosh P Lall ([email protected]) Eileen M Denovan-Wright (emdenova@d...

Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function f : Fp → Fp with polynomials of degree at most d ≤ p is non-negligible, while making only a constant number of queries to the function. This is an instance...

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient ...