A short proof of Meyniel's theorem

Meyniel's theorem states that a strict diconnected digraph has a directed Hamilton cycle if d(u) + d(v) ? 2n ? 1 for every pair u, v of nonadjacent vertices. We give short proof of this theorem.     

On systems of finite sets with constraints on their unions and intersections

Let F be a family of subsets of an n-element set. F is said to be of type (n, r, s) if A ∈ F, B ∈ F implies that |A ∪ B| ? n ? r, and |A ∩ B| ? s. Let f(n, r, s) = max {|F| : F is of type (n, r, s)}. We prove that f(n, r, s) ? f(n ? 1, r ? 1, s) + f(n ? 1, r + 1, s) if r > 0, n > s. And this result is used to give simple and unified proofs of Katona's and Frankl's results on f(n, r, s) when s = 0 and s = 1.     

Some mutually disjoint steiner systems

A t-design λ; t-d-n is a system of subsets of size d (called blocks) from an n-set S, such that each t-subset from S is contained in precisely λ blocks. A Steiner system S(l, m, n) is a t-design with parameters 1; l-m-n. Two Steiner systems (or t-designs) are disjoint if they share no blocks. A search has been conducted which resulted in discovering 9 mutually disjoint S(5, 8, 24)'s, 24 mutually disjoint S(4, 7, 23)'s, 60 mutually disjoint S(3, 6, 22)'s, and 197 mutually disjoint S(2, 5, 21)'s. Taking unions of several mutually disjoint Steiner systems will then produce t-designs (with varying λ's) on 21, 22, 23, and 24 points.     

Gray codes in graphs of subsets

Let G(n,k) be a graph whose vertices are the k-element subsets of an n-set represented as n-tuples of “O's” and “1's” with k “1's”. Two such subsets are adjacent if one can be obtained from the other by switching a “O” and a “1” which are in adjacent positions, where the first and nth positions are also considered adjacent. The problem of finding hamiltonian cycles in G(n,k) is discussed. This may be considered a problem of finding “Gray codes” of the k-element subsets of an n-set. It is shown that no such cycle exists if n and k are both even or if k=2 and n?7 and that such a cycle does exist in all other cases where k?5.     

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.     

On the separation power and the completion of partial latin squares

Let n and m be natural numbers, n ? m. The separation power of order n and degree m is the largest integer k = k(n, m) such that for every (0, 1)-matrix A of order n with constant linesums equal to m and any set of k 1's in A there exist (disjoint) permutation matrices P₁,…, P_m such that A = P₁ + … + P_m and each of the k 1's lies in a different P_i. Almost immediately we have 1 ? k(n, m) ? m ? 1, yet in all cases where the value of k(n, m) is actually known it equals m ? 1 (except under the somewhat trivial circumstances of k(n, m) = 1). This leads to a conjecture about the separation power, namely that k(n, m) = m ? 1 if $\text{m ? [}\text{n}\text{2}\text{] + 1}$. We obtain the bound $\text{k(n, m) ? m ? [}\text{n}\text{2}\text{] + 2}$, so that this conjecture holds for n ? 7. We then move on to latin squares, describing several equivalent formulations of the concept. After establishing a sufficient condition for the completion of a partial latin square in terms of the separation power, we can show that the Evans conjecture follows from this conjecture about the separation power. Finally the lower bound on k(n, m) allows us to show, after some calculations, that the Evans conjecture is true for orders n ? 11.     

A second-order nonlinear difference equation: Oscillation and asymptotic behavior

Discrete analogues are investigated for well-known results on oscillation, growth, and asymptotic behavior of solutions of y″ + q(t) y^γ = 0, for q(t) ? 0 and for q(t) ? 0. The analogue of Atkinson's oscillation criterion is shown to be true for Δ²y_{n ? 1} + q_ny_n^γ = 0, but the analogue for Atkinson's nonoscillation criterion is shown to be false.     

On cographic regular matroids

The presentation of alternating permutatioas via labelled binary trees is used to define polynomials H_2n?1(x) as enumerating polynomials for the height of peaks in alternating permutations of length 2n?1. A divisibility property of the coefficients of these polynomials is proved, which generalizes and explains combinatirially a well-known property of the tangent numbers. Furthermore, a version of the exponential generating function for the H_2n?1(x) is given, leading to a new combinatorial interpretation of Dumont's modified Ghandi-polynomials.     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

On large sets of disjoint Steiner triple systems II

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that if n is an odd number, there exist 12 mutually orthogonal Latin squares of order n and D(1 + 2n) = 2n ? 1, then D(1 + 12n) = 12n ? 1.     

Under-recurrence in the Khintchine recurrence theorem

The Khintchine recurrence theorem asserts that in a measure preserving system, for every set A and ε > 0, we have μ(A ∩ T^?nA) ≥ μ(A)² ? ε for infinitely many n ∈ N. We show that there are systems having underrecurrent sets A, in the sense that the inequality μ(A ∩ T^?nA) < μ(A)² holds for every n ∈ N. In particular, all ergodic systems of positive entropy have under-recurrent sets. On the other hand, answering a question of V. Bergelson, we show that not all mixing systems have under-recurrent sets. We also study variants of these problems where the previous strict inequality is reversed, and deduce that under-recurrence is a much more rare phenomenon than over-recurrence. Finally, we study related problems pertaining to multiple recurrence and derive some interesting combinatorial consequences.     

Intersections among Steiner systems

A Steiner system S(l, m, n) is a system of subsets of size m (called blocks) from an n-set S, such that each d-subset from S is contained in precisely one block. Two Steiner systems have intersection k if they share exactly k blocks. The possible intersections among S(5, 6, 12)'s, among S(4, 5, 11)'s, among S(3, 4, 10)'s, and among S(2, 3, 9)'s are determined, together with associated orbits under the action of the automorphism group of an initial Steiner system. The following are results: (i) the maximal number of mutually disjoint S(5, 6, 12)'s is two and any two such pairs are isomorphic; (ii) the maximal number of mutually disjoint S(4, 5, 11)'s is two and any two such pairs are isomorphic; (iii) the maximal number of mutually disjoint S(3, 4, 10)'s is five and any two such sets of five are isomorphic; (iv) a result due to Bays in 1917 that there are exactly two non-isomorphic ways to partition all 3-subsets of a 9-set into seven mutually disjoint S(2, 3, 9)'s.     

Hypergeometric series well-poised in SU(n) and a generalization of Biedenharn's G-functions

We prove that Holman's hypergeometric series well-poised in SU(n) satisfy a general difference equation. We make use of the “path sum” function developed by Biedenharn and this equation to show that a special class of these series, multiplied by simple products, may be regarded as a U(n) generalization of Biedenharn and Louck's G(Δ; X) functions for U(3). The fact that these generalized G-functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations, we give an elementary proof of Holman's U(n) generalization of the ₅F₄(1) summation theorem.     

Block designs and graph imbeddings

Heffter first observed that certain imbeddings of complete graphs give rise to BIBD's with k = 3 and λ = 2 (and sometimes λ = 1); Alpert established a one-to-one correspondence between BIBD's with k = 3 and λ = 2 and triangulation systems for complete graphs. In this paper we extend this correspondence to PBIBD's on two association classes with k = 3, λ₁ = 0 and λ₂ = 2, and triangulation systems for strongly regular graphs. The group divisible designs of Hanani are used to construct triangulations for the graphs K_n(m), in each case permitted by the euler formula. Conversely, triangular imbeddings of K_n(m) are constructed which lead to new group divisible designs. A process is developed for “doubling” a given PBIBD of an appropriate form. Various extensions of these ideas are discussed, as is an application to the construction of quasigroups.     

Further gossip problems

n people have distinct bits of information. They can communicate via k-party conference calls. How many such calls are needed to inform everyone of everyone else's information? Let f(n,k) be this minimum number. Then we give a simple proof that f(n,k)= [(n?k)(k?1)]+[nk] for 1?n?k², f(n,k)=2[(n?k)(k?1)] for n>k².In the 2-party case we consider the case in which certain of the calls may permit information flow in only one direction. We show that any 2n-4 call scheme that conveys everone's information to all must contain a 4-cycle, each of whose calls is “two way”, along with some other results. The method follows that of Bumby who first proved the 4-cycle conjecture.     

On symplectic polar spaces over non-perfect fields of characteristic 2

Given a field 𝕂 of characteristic 2 and an integer n ≥ 2, let W(2n ? 1, 𝕂) be the symplectic polar space defined in PG(2n ? 1, 𝕂) by a non-degenerate alternating form of V(2n, 𝕂) and let Q(2n, 𝕂) be the quadric of PG(2n, 𝕂) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W(2n ? 1, 𝕂) ? Q(2n, 𝕂). This is true when 𝕂 is perfect, but false otherwise. In this article, we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W(2n ? 1, 𝕂) is indeed isomorphic to a non-singular quadric Q, but when 𝕂 is non-perfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, 𝕂) is a proper subgeometry of W(2n ? 1, 𝕂). We show that, in spite of this fact, W(2n ? 1, 𝕂) can be embedded in Q(2n, 𝕂) as a subgeometry and that this embedding induces a full embedding of the dual DW(2n ? 1, 𝕂) of W(2n ? 1, 𝕂) into the dual DQ(2n, 𝕂) of Q(2n, 𝕂).     

Fast solution of toeplitz systems of equations and computation of Padé approximants

We present two new algorithms, ADT and MDT, for solving order-n Toeplitz systems of linear equations Tz = b in time O(n log²n) and space O(n). The fastest algorithms previously known, such as Trench's algorithm, require time $\text{Ω(n}{}^2\text{)}$ and require that all principal submatrices of T be nonsingular. Our algorithm ADT requires only that T be nonsingular. Both our algorithms for Toeplitz systems are derived from algorithms for computing entries in the Padé table for a given power series. We prove that entries in the Padé table can be computed by the Extended Euclidean Algorithm. We describe an algorithm EMGCD (Extended Middle Greatest Common Divisor) which is faster than the algorithm HGCD of Aho, Hopcroft and Ullman, although both require time O(n log²n), and we generalize EMGCD to produce PRSDC (Polynomial Remainder Sequence Divide and Conquer) which produces any iterate in the PRS, not just the middle term, in time O(n log²n). Applying PRSDC to the polynomials U₀(x) = x²ⁿ⁺¹ and U₁(x) = a₀ + a₁x + … + a_2nx²ⁿ gives algorithm AD (Anti-Diagonal), which computes any (m, p) entry along the antidiagonal m + p = 2n of the Padé table for U₁ in time O(n log²n). Our other algorithm, MD (Main-Diagonal), computes any diagonal entry (n, n) in the Padé table for a normal power series, also in time O(n log²n). MD is related to Schönhage's fast continued fraction algorithm. A Toeplitz matrix T is naturally associated with U₁, and the (n, n) Padé approximation to U₁ gives the first column of T^?1. We show how a formula due to Trench can be used to compute the solution z of Tz = b in time O(n log n) from the first row and column of T^?1. Thus, the Padé table algorithms AD and MD give O(n log²n) Toeplitz algorithms ADT and MDT. Trench's formula breaks down in certain degenerate cases, but in such cases a companion formula, the discrete analog of the Christoffel-Darboux formula, is valid and may be used to compute z in time O(n log²n) via the fast computation (by algorithm AD) of at most four Padé approximants. We also apply our results to obtain new complexity bounds for the solution of banded Toeplitz systems and for BCH decoding via Berlekamp's algorithm.     

Regular orientable imbeddings of complete graphs

This paper classifies the regular imbeddings of the complete graphs K_n in orientable surfaces. Biggs showed that these exist if and only if n is a prime power p^e, his examples being Cayley maps based on the finite field F = GF(n). We show that these are the only examples, and that there are $\text{φ(n ? 1)}\text{e}$ isomorphism classes of such maps (where φ is Euler's function), each corresponding to a conjugacy class of primitive elements of F, or equivalently to an irreducible factor of the cyclotomic polynomial Φ_{n ? 1}(z) over GF(p). We show that these maps are all equivalent under Wilson's map-operations H_i, and we determined for which n they are reflexible or self-dual.     

An umbral calculus for polynomials characterizing U(n) tensor operators

After the change of variables Δ_i = γ_i ? δ_i and x_{i,i + 1} = δ_i ? δ_{i + 1} we show that the invariant polynomials _μG⁽ⁿ⁾_q(, Δ_i, ; , x_i,i+1,) characterizing U(n) tensor operators 〈p, q,…, q, 0,…, 0〉 become an integral linear combination of Schur functions S_λ(γ ? δ) in the symbol γ ? δ, where γ ? δ denotes the difference of the two sets of variables {γ₁ ,…, γ_n} and {δ₁ ,…, δ_n}. We obtain a similar result for the yet more general bisymmetric polynomials ^m_μG⁽ⁿ⁾_q(γ₁ ,…, γ_n; δ₁ ,…, δ_m). Making use of properties of skew Schur functions $\text{S}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\rho </mtext>}}$ and S_λ(γ ? δ) we put together an umbral calculus for ^m_μG⁽ⁿ⁾_q(γ; δ). That is, working entirely with polynomials, we uniquely determine ^m_μG⁽ⁿ⁾_q(γ; δ) from ^m_μG⁽ⁿ⁾_{q ? 1}(γ; δ) and combinatorial rules involving Ferrers diagrams (i.e., partitions), provided that n ≥ (μ + 1)q. (This restriction does not interfere with writing the general case of ^m_μG⁽ⁿ⁾_q(γ; δ) as a linear combination of S_λ(γ ? δ).) As an application we deduce “conjugation” symmetry for ⁿ_μG⁽ⁿ⁾_q(γ; δ) from “transposition” symmetry by showing that these two symmetries are equivalent.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.