Factors of the Gaussian coefficients

We present some simple observations on factors of the q-binomial coefficients, the q-Catalan numbers, and the q-multinomial coefficients. Writing the Gaussian coefficient with numerator n and denominator k in a form such that 2k?n by the symmetry in k, we show that this coefficient has at least k factors. Some divisibility results of Andrews, Brunetti and Del Lungo are also discussed.     

Colouring lines in projective space

Let V be a vector space of dimension v over a field of order q. The q-Kneser graph has the k-dimensional subspaces of V as its vertices, where two subspaces α and β are adjacent if and only if is the zero subspace. This paper is motivated by the problem of determining the chromatic numbers of these graphs. This problem is trivial when k=1 (and the graphs are complete) or when v<2k (and the graphs are empty). We establish some basic theory in the general case. Then specializing to the case k=2, we show that the chromatic number is q²+q when v=4 and (q^v-1-1)/(q-1) when v>4. In both cases we characterise the minimal colourings.     

Configurations in projective planes and quadrilateral-star Ramsey numbers

Some classes of configurations in projective planes with polarity are constructed. As the main result, lower bounds for the Ramsey numbers r(n)=r(C₄;K_1,n) are derived from these geometric structures, which improve some bounds due to Parsons about 30 years ago, and also yield a new class of optimal values: r(q²-2q+1)=q²-q+1 whenever q is a power of 2. Moreover, the constructions also imply a known result on C₄-K_1,n bipartite Ramsey numbers.     

Schur positivity and the <Emphasis Type="Italic">q</Emphasis>-log-convexity of the Narayana polynomials

We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity. We begin with a formula of Brändén expressing the q-Narayana numbers as a specialization of Schur functions and, by deriving several symmetric function identities, we obtain the necessary Schur-positivity results. In addition, we prove the strong q-log-concavity of the q-Narayana numbers. The q-log-concavity of the q-Narayana numbers N _q (n,k) for fixed k is a special case of a conjecture of McNamara and Sagan on the infinite q-log-concavity of the Gaussian coefficients.     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

Global stability in difference equations satisfying the generalized Yorke condition

We present several conditions sufficient for global stability of the zero solution of nonautonomous difference equation xn+1=qxn+fn(xn,…,xn−k), n∈Z, when the nonlinearities fn satisfy a sort of negative feedback condition. Moreover, for every k∈N, we indicate qk such that one of our stability conditions is sharp if q∈(0,qk]. We apply our results to discrete versions of Nicholson's blowflies equation, the Mackey-Glass equations, and the Wazewska and Lasota equation.     

On the code generated by the incidence matrix of points and k-spaces in and its dual

In this paper, we study the p-ary linear code C_k(n,q), q=p^h, p prime, h1, generated by the incidence matrix of points and k-dimensional spaces in PG(n,q). For kn/2, we link codewords of C_k(n,q)C_k(n,q) of weight smaller than 2q^k to k-blocking sets. We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets. For k<n/2, we present counterexamples to lemmas valid for kn/2. Next, we study the dual code of C_k(n,q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [H. Sachar, The F_p span of the incidence matrix of a finite projective plane, Geom. Dedicata 8 (1979) 407–415] to general dimension.     

Total palindrome complexity of finite words

The palindrome complexity function pal_w of a word w attaches to each n∈N the number of palindromes (factors equal to their mirror images) of length n contained in w. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by M_q(n) the average number of palindromes in all words of length n over an alphabet with q letters, we present an upper bound for M_q(n) and prove that the limit of M_q(n)/n is 0. A more elaborate estimation leads to .     

The Eulerian generating function of q-derangements

Let F_q denote the finite field with q elements. For nonnegative integers n,k, let d_q(n,k) denote the number of n×nF_q-matrices having k as the sum of the dimensions of the eigenspaces (of the eigenvalues lying in F_q). Let d_q(n)=d_q(n,0), i.e., d_q(n) denotes the number of n×nF_q-matrices having no eigenvalues in F_q. The Eulerian generating function of d_q(n) has been well studied in the last 20 years [Kung, The cycle structure of a linear transformation over a finite field, Linear Algebra Appl. 36 (1981) 141-155, Neumann and Praeger, Derangements and eigenvalue-free elements in finite classical groups, J. London Math. Soc. (2) 58 (1998) 564-586 and Stong, Some asymptotic results on finite vector spaces, Adv. Appl. Math. 9(2) (1988) 167-199]. The main tools have been the rational canonical form, nilpotent matrices, and a q-series identity of Euler. In this paper we take an elementary approach to this problem, based on Möbius inversion, and find the following bivariate generating function:

     

On k-regular functions with values in a universal Clifford algebra

In this paper, we mainly study properties of nullsolutions of the operator Dk (k∈N^∗=N?{0}), so-called k-regular functions. Firstly, we study the set of all homogeneous polynomials of degree p in x₁,…,xn which are k-regular in the whole Rn, clearly is a right module over C(Vn,n), we construct a basis for the right module . Secondly, we study the k-regular and analytic functions, and we give the Taylor expansions for these functions. At last, the corresponding Taylor expansions for k-regular functions are given since each k-regular function is a real analytic function.     

A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration

In this paper, we consider the modified q-Bernstein polynomials for functions of several variables on q-Volkenborn integral and investigate some new interesting properties of these polynomials related to q-Stirling numbers, Hermite polynomials and Carlitz’s type q-Bernoulli numbers.     

The locally 2-arc transitive graphs admitting an almost simple group of Suzuki type

In this paper, seven families of vertex-intransitive locally (G,2)-arc transitive graphs are constructed, where Sz(q)?G?Aut(Sz(q)), q=²2k+1 for some k∈N. It is then shown that for any graph Γ in one of these families, Sz(q)?Aut(Γ)?Aut(Sz(q)) and that the only locally 2-arc transitive graphs admitting an almost simple group of Suzuki type whose vertices all have valency at least three are (i) graphs in these seven families, (ii) (vertex transitive) 2-arc transitive graphs admitting an almost simple group of Suzuki type, or (iii) double covers of the graphs in (ii). Since the graphs in (ii) have been classified by Fang and Praeger (1999) [6], this completes the classification of locally 2-arc transitive graphs admitting a Suzuki simple group     

The weight distribution of irreducible cyclic codes with block lengths n1((q1−1)N)

We study the weight distribution of irreducible cyclic (n, k) codeswith block lengths n = n₁((q¹ ? 1)/N), where N|q ? 1, gcd(n₁,N) = 1, and gcd(l,N) = 1. We present the weight enumerator polynomial, A(z), when k = n₁l, k = (n₁ ? 1)l, and k = 2l. We also show how to find A(z) in general by studying the generator matrix of an (n₁, m) linear code, $\text{V}{}^1{}_{\mathrm{d}}$ over GF(q^d) where d = gcd (ord_n1(q), l). Specifically we study A(z) when $\text{V}{}^1{}_{\mathrm{d}}$ is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers A_μ which completely determine the weight distributionof any irreducible cyclic (n₁(2¹ ? 1), k) code over GF(2) for all n₁ ? 17.     

On the distribution of points in orbits of PGL(2,q) acting on GF(q)

We prove results on the distribution of points in an orbit of PGL(2,q) acting on an element of GF(qⁿ). These results support a conjecture of Klapper. More precisely, we show that the points in an orbit are uniformly distributed if n is small with respect to q.     

Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG₁(n,q), where n?3, among all non-symmetric 2-(v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasi-symmetric designs PGn−2(n,q), where n?4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and all intersection numbers at least qn−4+?+q+1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG₁(n,q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.     

Embedding of products Q(k) × B(τ) in absolute A-sets

Theorems about closed embeddings in absolute A-sets of the products Q(k) × B(τ), Q(k) ×N, and Q(k) × C are proved. These are generalizations to the nonseparable case of theorems of Saint-Raymond, van Mill, and van Engelen about closed embeddings in separable absolute Borel sets of the products Q × N and Q × C, where Q is the space of rational numbers, C is the Cantor perfect set, and N is the space of irrational numbers.     

A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table

Let G be a graph consisting of powers of disjoint cycles and let A be an intersecting family of independent r-sets of vertices. Provided that G satisfies a further condition related to the clique numbers of the powers of the cycles, then |A| will be as large as possible if it consists of all independent r-sets containing one vertex from a specified cycle. Here r can take any value, 1?r?α(G), where α(G) is the independence number of G. This generalizes a theorem of Talbot dealing with the case when G consists of a cycle of order n raised to the power k. Talbot showed that .     

The sharp Markov-Nikol’skii inequality for algebraic polynomials in the spaces L q and L 0 on a closed interval

In this paper, an inequality between the L _q -mean of the kth derivative of an algebraic polynomial of degree n ≥ 1 and the L ₀-mean of the polynomial on a closed interval is obtained. Earlier, the author obtained the best constant in this inequality for k = 0, q ∈ [0,∞] and 1 ≤ k ≤ n, q ∈ {0} ∪ [1,∞]. Here a newmethod for finding the best constant for all 0 ≤ k ≤ n, q ∈ [0,∞], and, in particular, for the case 1 ≤ k ≤ n, q ∈ (0, 1), which has not been studied before is proposed. We find the order of growth of the best constant with respect to n as n → ∞ for fixed k and q.     

On the twisted q-Euler numbers and polynomials associated with basic q-l-functions

One of the purposes of this paper is to construct the twisted q-Euler numbers by using p-adic invariant integral on Zp in the fermionic sense. Moreover, we consider the twisted Euler q-zeta functions and q-l-functions which interpolate the twisted q-Euler numbers and polynomials at a negative integer.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.