In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the m(52) knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least n different Legendrian representatives with maximal Thurston–Bennequin number of the twist knot K−2n with crossing number 2n+1. In this paper we give a complete classification of Legendrian and transverse representatives o...

Let E/K be an elliptic curve defined over a number field, let ĥ be the canonical height on E, and let K/K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E/K) > 0 so that every nontorsion point P ∈ E(K) satisfies ĥ(P ) > C(E/K).

Let T be a tree on n vertices and L(T ) be its Laplacian matrix. The eigenvalues and eigenvectors of T are respectively referred to those of L(T ). With respect to a given eigenvector Y of T , a vertex u of T is called a characteristic vertex if Y [u] = 0 and there is a vertex w adjacent to u with Y [w] 6= 0; an edge e = (u, w) of T is called a characteristic edge if Y [u]Y [w] < 0. C(T, Y ) de...

For a countably generated group K and a function on K taking values that are either nonnega-tive integers or 1, we determine in Theorem 8 equivalent conditions for there to exist a Dedekind domain D with class group K such that, for each k in K, exactly (k) maximal ideals of D belong to k.

This study investigated whether fatiguing dynamic exercise depresses maximal in vitro Na(+)-K(+)-ATPase activity and whether any depression is attenuated with chronic training. Eight untrained (UT), eight resistance-trained (RT), and eight endurance-trained (ET) subjects performed a quadriceps fatigue test, comprising 50 maximal isokinetic contractions (180 degrees /s, 0.5 Hz). Muscle biopsies ...

We approximately solve, by reduction to Maximum Clique, the graph k-coloring NP-hard problem in a binary weights Hoppeld net special case. This network was used earlier to approximately solve Maximum Clique and some other NP-hard problems by reduction to Maximum Clique. We determine k-coloring approximation performance on random graphs and on one other distribution of \harder" graphs. We compar...

A combinatorial characterization of resolvable Steiner 2-(v, k, 1) designs embeddable as maximal arcs in a projective plane of order (v − k)/(k − 1) is proved, and a generalization of a conjecture by Andries Brouwer [9] is formulated.

We investigate what conditions on the minimum or average degree of a graph ensure that it contains a k-block , a (maximal) set of at least k vertices no two of which can be separated by fewer than k vertices.

We deene the VC-dimension of a set of permutations A S n to be the maximal k such that there exist distinct i 1 ; :::; i k 2 f1; :::; ng that appear in A in all possible linear orders, that is, every linear order of fi 1 ; :::; i k g is equivalent to the standard order of f(i 1); :::; (i k)g for at least one permutation 2 A. In other words, the VC-dimension of A is the maximal k such that for s...