On the completion of partial latin squares

In this paper a certain condition on partial latin squares is shown to be sufficient to guarantee that the partial square can be completed, namely, that it have fewer than n entries, and that at most $\text{[}\text{(n + 1)}\text{2}\text{]}$ of these lie off some line, where n is the order of the square. This is applied to establish that the Evans conjecture is true for n ? 8; i.e., that given a partial latin square of order n with fewer than n entries, n ? 8, the square can be completed. Finally, the results are viewed in a conjugate way to establish different conditions sufficient for the completion of a partial latin square.     

On a set puncturing problem

It is established that the maximum number of points required to puncture 3n sets of probability $\text{2}\text{3n}$ is 2n, as had been conjectured. In fact, for 1 ≤ m ≤ n, a family of m sets of probability $\text{2}\text{n}$ can be punctured using no more than $\text{min}\text{(m,[}\text{(n + m)}\text{3}\text{])}$ points. The more general statement that (k + 1)n sets of probability $\text{k}\text{(k + 1)n}$ require at most 2n points for puncturing is shown to be false already for k = 3.     

Bounds for sorting by prefix reversal

For a permutation σ of the integers from 1 to n, let ?(σ) be the smallest number of prefix reversals that will transform σ to the identity permutation, and let ?(n) be the largest such ?(σ) for all σ in (the symmetric group) S_n. We show that $\text{?(n)?}\text{(5n+5)}\text{3}$, and that $\text{?(n)?}\text{17n}\text{16}$ for n a multiple of 16. If, furthermore, each integer is required to participate in an even number of reversed prefixes, the corresponding function g(n) is shown to obey $\text{3n}\text{2?1}\text{?g(n)?2n+3}$.     

On finite differences,permutations, and Quadratic residues

Let p be an odd prime and n an integer relatively prime to p. In this work three criteria which give the value of the Legendre symbol ($\text{n}\text{p}$) are developed. The first uses two adjacent rows of Pascal's triangle which depend only on p to express ($\text{n}\text{p}$) explicitly in terms of the numerically least residues (mod p) of the numbers n, 2n, …, [$\text{(p + 1)}\text{4}\text{]n}$ or of the numbers $\text{[}\text{(p + 1)}\text{4}\text{]n,…, [}\text{(p ? 1)}\text{2}\text{]n}$. The second, analogous to a theorem of Zolotareff and valid only if p ≡ 1 (mod 4), expresses ($\text{n}\text{p}$) in terms of the parity of the permutation of the set {$\text{1,2,…, (}\text{(p? 1)}\text{2}$} defined by the absolute values of the numerically least residues of $\text{n, 2n,…,[}\text{(p? 1}\text{2}\text{]n}$. The third is a result dual to Gauss' lemma which can be derived directly without Euler's criterion. The applications of the dual include a proof of Gauss' lemma free of Euler's criterion and a proof of the Quadratic Reciprocity Law.     

The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy

Let M be the n-dimensional Minkowski space, n ? 3. One consequence of [1] is that the null space of the equation $\text{{(n ? 2k + 2)d}{}^1\text{d + (n ? 2k ? 2)dd}{}^1\text{} Φ = 0}$ on differential k-forms Φ in M is conformally covariant. The same is true of a nonlinear equation obtained by adding to the above a term homogeneous of degree $\text{(n + 2)}\text{(n ? 2)}$. This generalizes the well-known conformal covariance properties of the wave equation and the equations $\text{φ ± φ}{}^{\mathrm{<mtext>(n\; +\; 2)</mtext><mtext>(n\; ?\; 2)</mtext>}}\text{= 0}$ when k = 0, and of Maxwell's equations on a vector potential when $\text{k =}\text{(n ± 2)}\text{2}$ (and n is even). We define a natural (conformally invariant) symplectic structure for the new equations, and use it to calculate the $\text{(n + 1)(n + 2)}\text{2}$ conserved quantities corresponding to the standard conformal group generators.     

On weighted lattice paths

A weighted lattice path from (1, 1) to (n, m) is a path consisting of unit vertical, horizontal, and diagonal steps of weight w. Let f(0), f(1), f(2), … be a nondecreasing sequence of positive integers; the path connecting the points of the set $\text{{(n, m) ¦ f(n ? 1) ? m ? f(n), n = 1, 2, …}}$ will be called the roof determined by f. We determine the number of weighted lattice paths from (1, 1) to (n + 1, f(n)) which do not cross the roof determined by f. We also determine the polynomials that must be placed in each cell below the roof such that if a 1 is placed in each cell whose lower left-hand corner is a point of the roof, every k × k square subarray comprised of adjacent rows and columns and containing at least one 1 will have determinant $\text{x(}\text{k}\text{2}\text{)}$.     

Spanning subgraphs of k-connected digraphs

It is shown that every k-edge-connected digraph with m edges and n vertices contains a spanning connected subgraph having at most $\text{2m + 6(k ?1)(n ? 1))}\text{(5k ? 3)}$ edges. When k = 2 the bound is improved to $\text{(3m + 8(n ? 1))}\text{10}$, which implies that a 2-edge-connected digraph is connected by less than 70% of its edges. Examples are given which require almost two-thirds of the edges to connect all vertices.     

On the number of combinations without unit separation

Let f(n, k) denote the number of ways of selecting k objects from n objects arrayed in a line with no two selected having unit separation (i.e., having exactly one object between them). Then, if $\text{n ? 2(k ? 1), f(n,k)=}\text{∑}\text{i=0}\text{κ}\text{(}\text{n?k+I?2i}\text{k?2i}\text{)}$ (where $\text{κ = [}\text{k}\text{2}\text{]}$). If n < 2(k ? 1), then f(n, k) = 0. In addition, f(n, k) satisfies the recurrence relation f(n, k) = f(n ? 1, k) + f(n ? 3, k ? 1) + f(n ? 4, k ? 2). If the objects are arrayed in a circle, and the corresponding number is denoted by g(n, k), then for n > 3, g(n, k) = f(n ? 2, k) + 2f(n ? 5, k ? 1) + 3f(n ? 6, k ? 2). In particular, if n ? 2k + 1 then $\text{(n,k)=(}\text{n?k}\text{k}\text{)+(}\text{n?k?1}\text{k?1}\text{)}$.     

Spheres tangent to all the faces of a simplex

There are at most 2ⁿ spheres tangent to all n + 1 faces of an n-simplex. It has been shown that the minimum number of such spheres is $\text{2}{}^{\mathrm{n}}\text{? c(n,}\text{1}\text{2}\text{(n + 1))}$ if n is odd and $\text{2}{}^{\mathrm{n}}\text{? c(n,}\text{1}\text{2}\text{(n + 1))}$ if n is even. The object of this note is to show that this result is a consequence of a theorem in graph theory.     

On a conjecture about the exponent set of primitive matrices

M. Lewin and Y. Vitek conjecture [7] that every integer $\text{?[}\text{(n>}{}^2\text{?2n+2)}\text{2}\text{]+1}$ is an exponent of some n×n primitive matrix. In this paper, we prove three results related to Lewin and Vitek's conjecture: (1) Every integer $\text{?[}\text{(n}{}^2\text{?2n+2)}\text{4}\text{]+1}$ is an exponent of some n×n primitive matrix. (2) The conjecture is true when n is sufficiently large. (3) We give a counterexample to show that the conjecture is not true in the case when n=11.     

Minimally k-connected graphs of low order and maximal size

Let G be a minimally k-connected graph of order n and size e(G).Mader [4] proved that (i) $\text{e(G)?kn?(}\text{k+1}\text{2}\text{)}$; (ii) e(G)?k(n?k) if n?3k?2, and the complete bipartite graph K_k,n?k is the only minimally k-connected graph of order; n and size k(n?k) when k?2 and n?3k?1.The purpose of the present paper is to determine all minimally k-connected graphs of low order and maximal size. For each n such that k+1?n?3k?2 we prove $\text{e(G)??}\text{(n+k)}{}^2\text{8}\text{?}$ and characterize all minimally k-connected graphs of order n and size $\text{?(}\text{(n+k)}{}^2\text{8}\text{?}$.     

A property of the colored complete graph

If the lines of the complete graph K_n are colored so that no point is on more than $\text{1}\text{7}\text{(n?1)}$ lines of the same color or so that each point lies on more than $\text{1}\text{7}\text{(5n+8)}$ lines of different colors, then K_n contains a cycle of length n with adjacent lines having different colors.     

Nonnegative factorization of completely positive matrices

Let A be a real symmetric n × n matrix of rank k, and suppose that A = BB′ for some real n × m matrix B with nonnegative entries (for some m). (Such an A is called completely positive.) It is shown that such a B exists with $\text{m?}\text{1}\text{2}\text{k(k+1)?N}$, where 2N is the maximal number of (off-diagonal) entries which equal zero in a nonsingular principal submatrix of A. An example is given where the least m which works is $\text{(k+1)}{}^2\text{4}$ (k odd),$\text{k(k+2)}\text{4}$ (k even).     

Enumeration of arrays of a given size

Enumeration of arrays whose row and column sums are specified have been studied by a number of people. It has been determined that the function that enumerates square arrays of dimension n, whose rows and columns sum to a fixed non-negative integer r, is a polynomial in r of degree (n ? 1)².In this paper we consider rectangular arrays whose rows sum to a fixed non-negative integer r and whose columns sum to a fixed non-negative integer s, determined by ns = mr. in particular, we show that the functions which enumerate 2 × n and 3 × n arrays with fixed row sums $\text{nr}\text{(2, n)}$ and $\text{nr}\text{(3, n)}$, where the symbol (a, b) denotes the greatest common divisor of a and b, and fixed column sums, are polynomials in r of degrees (n ? 1) and 2(n ? 1) respectively. We have found simple formulas to evaluate these polynomials for negative values, - r, and we show that for certain small negative integers our polynomials will always be zero. We also considered the generating functions of these polynomials and show that they are rational functions of degrees less than zero, whose denominators are of the forms (1 ? y)ⁿ and (1 ? y)²ⁿ?1 respectively and whose numerators are polynomials in y whose coefficients satisfy certain properties. In the last section we list the actual polynomials and generating functions in the 2 × n and 3 × n cases for small specific values of n.     

On some covering designs

Let n ? k ? t be positive integers, and let Ω be a set of n elements. Let C(n, k, t) denote the number of k-tuples of Ω in a minimal system of k-tuples such that every t-tuple is contained in at least one k-tuple of the system. C(n, k, t) has been determined in all cases for which $\text{C(n, k, t) ?}\text{3(t + 1)}\text{2}$ [W. H. Mills, Ars Combinatoria8 (1979), 199–315]. C(n, k, t) is determined in the case $\text{3(t + 1)}\text{2}\text{< C(n, k, t) ?}\text{3(t + 2)}\text{2}$.     

Probabilistic analysis of solving the assignment problem for the traveling salesman problem

This paper deals with probabilistic analysis of optimal solutions of the asymmetric traveling salesman problem. The exact distribution for the number of required next-best solutions of the assignment problem with random data in order to find an optimal tour is given. For every n-city asymmetric problem, there exists an algorithm such that (i) with probability 1 ? s, s?(0,1) the algorithm produces an optimal tour, (ii) it runs in time $\text{O}\text{(}\text{n}{}^4\text{3}\text{)}$, and (iii) it requires less than w((w + n ? 1)$\text{log}\text{(w + n ? 1) + w + 1) +}\text{1}\text{6}\text{w(n}{}^3\text{+ 3n}{}^2\text{+ 2n ? 6)}$ computational steps, where w = log(s)/log(1 ? E_n); E_n ?(0,1) is given by a simple mathematical formula. Additionally, the polynomial of (iii) gives the exact (deterministic) execution time to find w =1 ,2…. next-best solutions of the assignment problem.     

Recurrence relations based on minimization and maximization

Explicit and asymptotic solutions are presented to the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + min_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) for the cases (1) α + β < 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and g(n) = δ_nI. (2) α + β > 1, min(α, β) > 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and (a) g(n) = δ_n1, (b) g(n) = 1. The general form of this recurrence was studied extensively by Fredman and Knuth [J. Math. Anal. Appl.48 (1974), 534–559], who showed, without actually solving the recurrence, that in the above cases $\text{M(n) = Ω(n}{}^{\mathrm{<mtext>1\; +\; </mtext><mtext>1</mtext><mtext>\gamma </mtext>}}\text{)}$, where γ is defined by α^?γ + β^?γ = 1, and that $\text{lim}{}_{\mathrm{n\; \to \; \infty }}\text{M(n)}\text{n}{}^{\mathrm{1\; +\; \gamma }}$ does not exist. Using similar techniques, the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + max_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) is also investigated for the special case α = β < 1 and g(n) = 1 if n is odd = 0 if n is even.     

The multiplier for the ball and radial functions

It is shown that the multiplier for the ball is restricted weak type on radial functions in L^p($\text{R}$ⁿ) when $\text{p =}\text{2n}\text{(n + 1)}$. Interpolation then yields a theorem of Herz.     

On the minimum order of graphs with given semigroup

Denote by M(n) the smallest positive integer such that for every n-element monoid M there is a graph G with at most M(n) vertices such that End(G) is isomorphic to M. It is proved that $\text{2}\text{(1 + o(1))n}\text{log}{}_2\text{n}\text{≤M(n)≤n ·}\text{2n}\text{+ O(n)}$. Moreover, for almost all n-element monoids M there is a graph G with at most 12 · n · log₂n + n vertices such that End(G) is isomorphic to M.