1 Motivations In machine learning, the standard measure of accuracy for models is the prediction error (PE), i.e. the expected loss on future examples. We consider here the i.i.d. regression or classification setups, where future examples are assumed to be independently sampled from the distribution that generated the training set. When the data distribution is unknown, PE cannot be computed. T...

The first enzyme of the pathway for uridine diphosphate N-acetyl-D-glucosamine (UDPAG) biosynthesis i.e. L-glutamine: D-fructose 6-P amidotransferase (E.C. 2.6.1.16) was purified 52-fold from human placenta using methanol fractionation and column chromatography on DEAE-Sephadex A-50. The enzyme showed optimal activity in a broad range of pH from 5.8 to 7.8 in both phosphate and cacodylate ...

A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, . . . , x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by χk(G). We determine the exact value of this parameter when G is a web or an antiweb. Our r...

Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p = 1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang’s argument requires only 2-fold (...

A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, . . . , x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by χk(G). We determine the exact value of this parameter when G is a web or an antiweb. Our r...