D oes the equation x + y + z = 29 have a solution in integers? Yes: (3,1,1), for instance. How about the equation x + y + z = 30? Again yes, although this was not known until 1999: the smallest solution is (−283059965,−2218888517, 2220422932). And how about x + y + z = 33? This is an unsolved problem. Of course, number theory does not end with the study of cubic equations in three variables: on...

We consider the generalized Newton’s least resistance problem for convex bodies: minimize functional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-integral Underscript normal upper Omega Endscripts left-parenthesis 1 plus StartAbsoluteValue nabla u x comma y right-parenthesis EndAbsoluteValue squared Superscript negative Bas...

Does the equation x+y+z = 29 have a solution in integers? Yes: (3, 1, 1), for instance. How about x+y+z = 30? Again yes, although this was not known until 1999: the smallest solution is (−283059965,−2218888517, 2220422932). And how about x+y+z = 33? This is an unsolved problem. Of course, number theory does not end with the study of cubic equations in three variables: one might ask also about x...

We give necessary and sufficient conditions for the existence of a continuous extension from a smallest-neighborhood space (Alexandrov space) X to the Khalimsky line. Using this result, we classify the subsets A ⊂ X such that every continuous function A → Z can be extended to all of X. We also consider the more general case of mappings X → Y between smallest-neighborhood spaces, and prove a dig...

The Smallest Grammar Problem – the problem of finding the smallest context-free grammar that generates exactly one given sequence – has never been successfully applied to grammatical inference. We investigate the reasons and propose an extended formulation that seeks to minimize non-recursive grammars, instead of straight-line programs. In addition, we provide very efficient algorithms that app...

We develop an abstract framework for studying the strong form of Malle’s conjecture [J. Number Theory 92 (2002), pp. 315–329; Experiment. Math. 13 (2004), 129–135] nilpotent groups G G</mml:annotatio...

The Y chromosome is one of the smallest human chromosomes with an average size of 60 million base pairs (Mb). Between X and Y chromosomes, exchange is limited to small pseudoautosomal regions (PAR) of the X-Y pair (see Figure 1). During male meiosis, recombination only takes place at the most distal short arm (PAR 1) and at the tip of the Y chromosome long arm (PAR 2). For most of its length (t...

For a permutation f of the vertex set V (G) of a connected graph G, let δf (x, y)= |d(x, y)− d(f (x), f (y))|. Define the displacement δf (G) of G with respect to f to be the sum of δf (x, y) over all unordered pairs {x, y} of distinct vertices of G. Let π(G) denote the smallest positive value of δf (G) among the n! permutations f of V (G). In this note, we determine all trees T with π(T )= 2 o...

Proof: The proof is by induction on |V (G)| + |E(G)|. The smallest graph as described in the lemma is K1,3, for which the statement holds. This gives the start of our induction. Let x = |X| and y = |Y |. If there exists a vertex v in Y of degree at least 4, then delete any edge e incident to v. The subset A of G− e guaranteed by the inductive hypothesis is adjacent in G to every vertex in Y as ...