Compatibility of arithmetic and algebraic local constants (the case )

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$. We deduce new cases of the $p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

A LOWER BOUND FOR THE NUMBER OF FORBIDDEN MOVES TO UNKNOT A LONG VIRTUAL KNOT

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fn-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fn-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fn-moves needed to transform K to K′.

DIVISIBILITY OF DEDEKIND FINITE SETS

A Dedekind-finite set is said to be divisible by a natural number n if it can be partitioned into pieces of size n. We study several aspects of this notion, as well as the stronger notion of being partitionable into n pieces of equal size. Among our results are that the divisors of a Dedekind-finite set can consistently be any set of natural numbers (containing 1 but not 0), that a Dedekind-finite power of 2 cannot be divisible by 3, and that a Dedekind-finite set can be congruent modulo 3, to all of 0, 1, and 2 simultaneously. (In these results, 2 and 3 serve as typical examples; the full results are more general.)

Wiener Number of Polyphenyls and Phenylenes

It is shown that the values which the Wiener numbers of isomeric polyphenyls may assume are all congruent modulo 36. Similarly, the Wiener numbers of isomeric phenylens are always congruent modulo 18. These findings provide a rationalization for the known fact that the isomer-discriminating power of the Wiener number is rather weak. Extensions of the present results to more general classes of graphs are also pointed out.

Probability and the elementary symmetric functions

Let p be a prime, and suppose that x1,…,xN are independent random variables which take the values 0, 1,…,p − 1 with probabilities s0, sl…,sp−1 where s0+…+sp−1 = 1 and 0 < sk < 1 for each k. PN(n) denotes the probability that the elementary symmetric function σr(x1,…,xN) = ∑x1…,xr of the rth degree in the variables x1,…,xN is congruent, modulo p, to a prescribed integer n.

Corrigendum: Block idempotents and the Brauer correspondence

Professor William F. Reynolds has discovered a mistake in the main theorem of [1]. The problem occurs with the first display on page 339 in that is only congruent modulo radZ(G) to . The theorem still holds if θ(radZ(G)) ⊆ radZ(H), but this is not the case in general.