Richard Douglas Chatham

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• Doctor of Philosophy
• Professor (Full) at Morehead State University

We present placements of mutually non-attacking chess pieces of mixed type that occupy more than half of the squares of an m × n board. If both white and black pawns are allowed as separate types, there are arrangements, which we also present, that occupy at least two-thirds of the board squares.

We define the queens (resp., rooks) diameter-separation number to be the minimum number of pawns for which some placement of those pawns on an n × n board produces a board with a queens graph (resp., rooks graph) with a desired diameter d . We determine these numbers for some small values of d .

A dragon king is a shogi piece that moves any number of squares vertically or horizontally or one square diagonally but does not move through or jump over other pieces. We construct infinite families of solutions to the n + k dragon kings problem of placing k pawns and n + k mutually nonattacking dragon kings on an n × n board, including solutions...

Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, whi...

For a chessboard graph and a given graph parameter π, a π separation number is the minimum number of pawns for which some arrangement of those pawns on the board will produce a board where π has some desired value. We extend previous results on independence and domination separation. We also consider separation of other domination-related parameter...

The classic n-queens problem asks for placements of just n mutually non-attacking queens on an n × n board. By adding enough pawns, we can arrange to fill roughly one-quarter of the board with mutually non-attacking queens. How many pawns do we need? We discuss that question for square boards as well as rectangular m × n boards.

Chessboard separation problems are modifications to classic chessboard problems, such as the N queens problem, in which obstacles are placed on the chessboard. The N + k queens problem requires placements of k pawns and N + k mutually non-attacking queens on an N -by-N chessboard. Here we examine centrosymmetric (half-turn symmetric) and doubly cen...

Abstract Chessboard separation problems are modiflcations to classic chess- board problems, such as the N Queens Problem, in which obstacles are placed on the chessboard. This paper focuses on a variation known as the N + k Queens Problem, in which k Pawns and N + k mutually non-attacking Queens are to be placed on an N-by-N chess- board. Results a...

A legal placement of Queens is any placement of Queens on an order N chessboard in which any two attacking Queens can be separated by a Pawn. The Queens independence separation number is the minimum number of Pawns which can be placed on an N × N board to result in a separated board on which a maximum of m independent Queens can be placed. We prove...

If O ≤ n ≤ ∞ and R⊆T are (commutative unital) rings, then (R, T) is called an n-dimensional pair if each intermediate ring A (that is, each ring A such that R ⊆ A ⊆ T) has Krull dimension n. If 1 ≤ n ≤ ∞, examples are given of n-dimensional pairs that are not integral extensions, including an infinite family of n-dimensional pairs that are neither...

We define a legal placement of Queens to be any placement in which any two attacking Queens can be separated by a Pawn. The Queens separation number is defined to be equal to the minimum number of Pawns which can separate some legal placement of m Queens on an order n chess board. We prove that n + 1 Queens can be separated by 1 Pawn and conjecture...

If R ⊆ T is an integral extension of (commutative) rings such that R is an open ring, R[a, b] is a going-down ring for each a, b ∈ T and T is semiquasilocal, then each ring contained between R and T is an open ring. An example is given to show that the "semiquasilocal" hypothesis cannot be deleted. If R ⊆ T are rings such that R[a, b] is an open ri...

This note shows that if T is a commutative ring with prime subring R, then all subrings of T have Krull dimension at most m if and only if the supremum over all minimal prime ideals of T of the transcendence degrees of T/P over R/(P∩R) is less than or equal to m minus the dimension of R. More generally, this note shows that, given a commutative rin...

It is proved that if R ⊂ T are going-down domains such that Spec(R) = Spec(T) as sets (for instance, a proper field extension) and M denotes the common maximal ideal of R and T, then each ring between R and T is a going-down domain if and only if the transcendence degree of T/M over R/M is at most 1. As a consequence, transcendence degree is used t...

D'après D. E. Dobbs, Houston J. Math. 23 (1997), 1–11, nous disons que l'anneau (commutatif)A est un anneau-“going-down” siA/P est un domaine-“going-down” pour chaque idéal premier deA. Etant donné une extension,R⊆T, nous disons que (R, T) est une paire d'anneaux-“going-down” (respectivement, une paire “going-down”) siS est un anneau-“going-down” p...