W. Edwin Clark

The domination numbers of the 5 x n and 6 x n grid graphs

The k × n grid graph is the product Pk × Pn of a path of length k − 1 and a path of length n − 1. We prove here formulas found by E. O. Hare for the domination numbers of P5 × Pn and P6 × Pn.

INEQUALITIES INVOLVING GAMMA AND PSI FUNCTIONS

We prove that certain functions involving the gamma and q-gamma function are monotone. We also prove that (xmψ(x))(m+1) is completely monotonic. We conjecture that -(xmψ(m)(x))(m) is completely monotonic for m ≥ 2; and we prove it, with help from Maple, for 2 ≤ m ≤ 16. We give a very useful Maple procedure to verify this for higher values of m. A stronger result is also formulated where we conjecture that the power series coefficients of a certain function are all positive.

Enumeration of finite commutative chain rings

Abstract A chain ring is an associative, commutative ring with an identity whose ideals form a chain. We associate with each finite chain ring five invariants (integers) and determine (in certain cases) the number of isomorphism classes of rings with given invariants. These results yield immediate corollaries for Pappian Hjelmslev planes which are coordinatized by such rings.

Quandle colorings of knots and applications

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q1 → Q0. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramins list. Links to the data computed and various programs in C, GAP and Maple are provided.

Binomial andQ-Binomial Coefficient Inequalities Related to the Hamiltonicity of the Kneser Graphs and TheirQ-Analogues

The Kneser graphK(n, k) has as vertices all thek-subsets of a fixedn-set and has as edges the pairs {A, B} of vertices such thatAandBare disjoint. It is known that these graphs are Hamiltonian ifformula]forn?2k+1. We determine asymptotically for fixedkthe minimum valuen=e(k) for which this inequality holds. In addition we give an asymptotic formula for the solution ofk?(n)?(n?2k+1)=?2(n?k+1) forn?2k+1, ask?∞, whennandkare not restricted to take integer values. We also show that for all prime powersqandn?2k,k?1, theq-analoguesKq(n, k) are Hamiltonian by consideration of the analogous inequality forq-binomial coefficients.

Connected quandles associated with pointed abelian groups

A quandle is a self-distributive algebraic structure that appears in quasigroup and knot theories. For each abelian group A and c2 A, we define a quandle G. A; c/ on Z3 A. These quandles are generalizations of a class of nonmedial Latin quandles defined by V. M. Galkin, so we call them Galkin quandles. Each G. A; c/ is connected but not Latin unless A has odd order. G. A; c/ is nonmedial unless 3 AD 0. We classify their isomorphism classes in terms of pointed abelian groups and study their various properties. A family of symmetric connected quandles is constructed from Galkin quandles, and some aspects of knot colorings by Galkin quandles are also discussed.

Sum-free sets in vector spaces over GF(2)

Abstract A subset S of an abelian group G is said to be sum-free if whenever a, b ∈ S, then a + b ∉ S. A maximal sum-free (msf) set S in G is a sum-free set which is not properly contained in another sum-free subset of G. We consider only the case where G is the vector space (V(n) of dimension n over GF(2). We are concerned with the problem of determining all msf sets in V(n). It is well known that if S is a msf set then |S| ⩽ 2n − 1. We prove that there are no msf sets S in V(n) with 5 × 2n − 4 2 ⩽ s ⩽ [ (n − t) 2 ] . These methods suffice to construct msf sets of all possible cardinalities for n ⩽ 6. We also present some of the results of our computer searches for msf sets in V(n). Up to equivalence we found all msf-sets for n ⩽ 6. For n > 6 our searches used random sampling and, in this case, we find many more msf sets than our present methods of construction can account for.

The automorphism class group of the category of rings

This paper is motivated by the observation that the property of having a left identity cannot be described “categorically” in the category of rings, since the functor ( )OP which takes a ring into its opposite ring is a category automorphism but does not preserve this property. In general, a property of objects, morphisms, etc., in a category V can be characterized category- theoretically if and only if it is invariant under the automorphisms (self- equivalences) of ‘27. This makes it desirable to know whether ( )OP is the only nontrivial automorphism of the category of rings up to equivalence of func- tors. We shall show that the answer is yes, and that in the case of commutative rings, there are no nontrivial automorphisms. More generally, let

q-Analogue of a binomial coefficient congruence

We establish a q-analogue of the congruence (papb)≡(ab)   (modp2) where p is a prime and a and b are positive integers.

Quandle coloring and cocycle invariants of composite knots and abelian extensions

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.