Cartan models and cellular decompositions of symmetric Riemannian spaces

In this paper, by making use of the Cartan models, we will construct cellular decompositions of some symmetric Riemannian spaces such as Sp(n)/U(n), U(n)/O(n), U(2n)/Sp(n), O(2n)/U(n), SU(n)/SO(n), SU(2n)/Sp(n), SO(2n)/U(n).     

Finite linear spaces with two consecutive line degrees

Let L be a non-trivial finite linear space in which every line has n or n+1 points. We describe L completely subject to the following restrictions on n and on the number of points p: p≤n ²+n?1 and n≥3; n ²+n+2≤p≤n ²+2n?1 and n≥3; p=n ²+2n and n≥4; p=n ²+2n+2 and n≥3; p=n ²+2n+3 and n≥4.     

On some friends of the sociable numbers

Let s(n) denote the sum of the proper divisors of n. Set s ₀(n) = n, and for k > 0, put s _k (n) := s(s _k-1(n)) if s _k-1(n) > 0. Thus, perfect numbers are those n with s(n)?=?n and amicable numbers are those n with s(n) ?? n but s ₂(n)?=?n. We prove that for each fixed k ?? 1, the set of n which divide s _k (n) has density zero, and similarly for the set of n for which s _k (n) divides n. These results generalize the theorem of Erd?s that for each fixed k, the set of n for which s _k (n)?=?n has density zero.     

On the number of distinct multinomial coefficients

We study M(n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where pP(n) is the number of partitions of n into primes and p(n) is the total number of partitions of n. To use methods from commutative algebra, we encode partitions and multinomial coefficients as monomials.     

On the number of transversals in Cayley tables of cyclic groups

It is well known that if n is even, the addition table for the integers modulo n (which we denote by B_n) possesses no transversals. We show that if n is odd, then the number of transversals in B_n is at least exponential in n. Equivalently, for odd n, the number of diagonally cyclic latin squares of order n, the number of complete mappings or orthomorphisms of the cyclic group of order n, the number of magic juggling sequences of period n and the number of placements of n non-attacking semi-queens on an n×n toroidal chessboard are at least exponential in n. For all large n we show that there is a latin square of order n with at least (3.246)ⁿ transversals.We diagnose all possible sizes for the intersection of two transversals in B_n and use this result to complete the spectrum of possible sizes of homogeneous latin bitrades.We also briefly explore potential applications of our results in constructing random mutually orthogonal latin squares.     

Some designs which admit strong tactical decompositions

Whenever there exist affine planes of orders n ? 1 and n, a construction is given for a 2 ? ((n + 1)(n ? 1)², n(n ? 1), n) design admitting a strong tactical decomposition. These designs are neither symmetric nor strongly resolvable but can be embedded in symmetric 2 ? (n³ ? n + 1, n², n) designs.     

Two graph isomorphism polytopes

The convex hull ψ_n,n of certain (n!)² tensors was considered recently in connection with graph isomorphism. We consider the convex hull ψ_n of the n! diagonals among these tensors. We show: 1. The polytope ψ_n is a face of ψ_n,n. 2. Deciding if a graph G has a subgraph isomorphic to H reduces to optimization over ψ_n. 3. Optimization over ψ_n reduces to optimization over ψ_n,n. In particular, this implies that the subgraph isomorphism problem reduces to optimization over ψ_n,n.     

Primitive digraphs with smallest large exponent

A primitive digraph D on n vertices has large exponent if its exponent, γ(D), satisfies α_n?γ(D)?w_n, where α_n=⌊w_n/2⌋+2 and w_n=(n-1)²+1. It is shown that the minimum number of arcs in a primitive digraph D on n?5 vertices with exponent equal to α_n is either n+1 or n+2. Explicit constructions are given for fixed n even and odd, for a primitive digraph on n vertices with exponent α_n and n+2 arcs. These constructions extend to digraphs with some exponents between α_n and w_n. A necessary and sufficient condition is presented for the existence of a primitive digraph on n vertices with exponent α_n and n+1 arcs. Together with some number theoretic results, this gives an algorithm that determines for fixed n whether the minimum number of arcs is n+1 or n+2.     

Simplest cases of nth-degree stochastic dominance

Le p(n) be the fewest number of support points for probability distributions p and q for which p stochastically dominates q of degree n but not of any degrees less than n. Then ?(n) = n + 1 for n = 1,2,3, and ?(n) = 4 for all larger n.     

Improved book-embeddings of incomplete hypercubes

In this paper, we show that any incomplete hypercube with, at most, 2ⁿ+2ⁿ⁻¹+2ⁿ⁻² vertices can be embedded in n−1 pages for all n≥4. For the case n≥4, this result improves Fang and Lai’s result that any incomplete hypercube with, at most, 2ⁿ+2ⁿ⁻¹ vertices can be embedded in n−1 pages for all n≥2.Besides this, we show that the result can be further improved when n is large — e.g., any incomplete hypercube with at most 2ⁿ+2ⁿ⁻¹+2ⁿ⁻²+2ⁿ⁻⁷ (respectively, 2ⁿ+2ⁿ⁻¹+2ⁿ⁻²+2ⁿ⁻⁷+2ⁿ⁻²³⁰) vertices can be embedded in n−1 pages for all n≥9 (respectively, n≥232).     

Author index for volume 39

The distribution γ(c, n) of de Bruijn sequences of order n and linear complexity c is investigated. Some new results are proved on the distribution of de Bruijn sequences of low complexity, i.e., their complexity is between 2^n?1 + n and 2^n?1 + 2^n?2. It is proved that for n ? 5 and 2^n?1 + n?c<2^n?1 + 2^n?2, γ(c, n) ≡ 0 (mod 4). It is shown that for n ? 11, γ(2^n?1 + n, n) > 0. It is also proved that γ(2^n?1 + 2^n?2, n) ? 4γ(2^n?2 ? 1, n ? 2) and we give a recursive method to generate de Bruijn sequences of complexity 2^n?1 + 2^n?2.     

On a problem of rota

Let S(n, k) denote Stirling numbers of the second kind; for each n, let K_n be such that S(n, K_n) ? S(n, k) for all k. Also, let P(n) denote the lattice of partitions of an n-element set. Say that a collection of partitions from P(n) is incomparable if no two are related by refinement. Rota has asked if for all n, the largest possible incomparable collection in P(n) contains S(n, K_n) partitions. In this paper, we construct for all n sufficiently large an incomparable collection in P(n) containing strictly more than S(n, K_n) partitions. We also estimate how large n must be for this construction to work.     

The codimension of degenerate pencils

Let d_n[d_n(r)] denote the codimension of the set of pairs of n×n Hermitian [really symmetric] matrices (A, B) for which det(λI?A?xB)=p(λ,x) is a reducible polynomial. We prove that d_n(r)?n?1, d_n?_n?1 (n odd), d_n?n (n even). We conjecture that the equality holds in all three inequalities. We prove this conjecture for n=2,3.     

On the complexity of calculating factorials

It is shown that n! can be evaluated with time complexity O(log log nM (n log n)), where M(n) is the complexity of multiplying two n-digit numbers together. This is effected, in part, by writing n! in terms of its prime factors. In conjunction with a fast multiplication this yields an O(n(log n log log n)²) complexity algorithm for n!. This might be compared to computing n! by multiplying 1 times 2 times 3, etc., which is ω(n² log n) and also to computing n! by binary splitting which is O(log nM(n log n)).     

Limiting distributions of functionals of Markov chains

Let \s{X_n, n ? 0\s} and \s{Y_n, n ? 0\s} be two stochastic processes such that Y_n depends on X_n in a stationary manner, i.e. P(Y_n ? A\vbX_n) does not depend on n. Sufficient conditions are derived for Y_n to have a limiting distribution. If X_n is a Markov chain with stationary transition probabilities and Y_n = f(X_n,..., X_n+k) then Y_n depends on X_n is a stationary way. Two situations are considered: (i) \s{X_n, n ? 0\s} has a limiting distribution (ii) \s{X_n, n ? 0\s} does not have a limiting distribution and exits every finite set with probability 1. Several examples are considered including that of a non-homogeneous Poisson process with periodic rate function where we obtain the limiting distribution of the interevent times.     

On the spectrum of n quasigroups with given conjugate invariant subgroup

Let tⁿ be a vector of n positive integers that sum to 2n ? 1. Let u denote a vector of n or more positive integers that sum to n², and call u, n-universal if for every possible choice of t¹, t²,…, tⁿ, the components of the tⁱ can be arranged in the successive rows of an n-row matrix (with 0 in each unused cell) so that u is the vector of column sums.It is shown that (n,…, n)_{(n times)} is n-universal for every n. More generally, for odd n, any choice of t¹, t³,…, tⁿ can be placed in rows so that the column sums are (n, n?1,…, 2, 1); for even n, any choice of t², t⁴,…, tⁿ can be placed in rows so that the column sums are (n, n ?1,…, 2, 1). Hence, any u that can be obtained from the sum of two rows whose nonzero components are, respectively, n, n ?1,…, 2, 1 and n ?1, n ?2,…, 2, 1 (in any order, with 0's elsewhere) is n-universal.The problem examined is closely related to the graph conjecture that trees on 2, 3,…, n + 1 vertices can be superposed to yield the complete graph on n + 1 vertices.     

Weak convergence with random indices

Suppose {X_nn?-0} are random variables such that for normalizing constants a_n>0, b_n, n?0 we have Y_n(·)=(X[_{n, ·}]-b_n/a_n ? Y(·) in D(0.∞) . Then a_n and b_n must in specific ways and the process Y possesses a scaling property. If {N_n} are positive integer valued random variables we discuss when Y_Nn → Y and Y'_n=(X[_Nn]-b_n)/a_n ? Y'. Results given subsume random index limit theorems for convergence to Brownian motion, stable processes and extremal processes.     

On the decomposition of n! into prime powers

If n is a positive integer, we write n! as a product of n prime powers, each at least as large as n^δ(n). We define α(n) to be max δ(n), where the maximum is taken over all decompositions of the required type. We then show that lim_n→∞α(n) exists, and we calculate its value.     

On the symmetric hyperspace of the circle

By X(n), n?1, we denote the n-th symmetric hyperspace of a metric space X as the space of non-empty finite subsets of X with at most n elements endowed with the Hausdorff metric. In this paper we shall describe the n-th symmetric hyperspace S¹(n) as a compactification of an open cone over ΣDn−2, here Dn−2 is the higher-dimensional dunce hat introduced by Andersen, Marjanovi? and Schori (1993) [2] if n is even, and Dn−2 has the homotopy type of Sn−2 if n is odd (see Andersen et al. (1993) [2]). Then we can determine the homotopy type of S¹(n) and detect several topological properties of S¹(n).     

Lower bounds on the minimum average distance of binary codes

Let β(n,M) denote the minimum average Hamming distance of a binary code of length n and cardinality M. In this paper we consider lower bounds on β(n,M). All the known lower bounds on β(n,M) are useful when M is at least of size about 2ⁿ⁻¹/n. We derive new lower bounds which give good estimations when size of M is about n. These bounds are obtained using a linear programming approach. In particular, it is proved that lim_n→∞β(n,2n)=5/2. We also give a new recursive inequality for β(n,M).