Hankel determinants and orthogonal polynomials

We develop a general context for the computation of the determinant of a Hankel matrix H_n = (α_i+j)0i,jn, assuming some suitable conditions for the exponential (or ordinary) generating function of the sequence (α_n)_n0. Several well-known particular cases are thus derived in a unified way.     

On the chromatic number of random graphs

In this paper we consider the classical Erdős–Rényi model of random graphs G_n,p. We show that for p=p(n)n^−3/4−δ, for any fixed δ>0, the chromatic number χ(G_n,p) is a.a.s. ℓ, ℓ+1, or ℓ+2, where ℓ is the maximum integer satisfying 2(ℓ−1)log(ℓ−1)p(n−1).     

Double piling structure of matrix monotone functions and of matrix convex functions

There are basic equivalent assertions known for operator monotone functions and operator convex functions in two papers by Hansen and Pedersen. In this note we consider their results as correlation problem between two sequences of matrix n-monotone functions and matrix n-convex functions, and we focus the following three assertions at each label n among them:

(i) f(0)0 and f is n-convex in [0,α),

(ii) For each matrix a with its spectrum in [0,α) and a contraction c in the matrix algebra M_n,

f(cac)cf(a)c,

(iii) The function is n-monotone in (0,α).

We show that for any nN two conditions (ii) and (iii) are equivalent. The assertion that f is n-convex with f(0)0 implies that g(t) is (n-1)-monotone holds. The implication from (iii) to (i) does not hold even for n=1. We also show in a limited case that the condition (i) implies (ii).     

Rook-by-rook rook theory: Bijective proofs of rook and hit equivalences

Suppose μ and ν are integer partitions of n, and N>n. It is well known that the Ferrers boards associated to μ and ν are rook-equivalent iff the multisets [μ_i+i:1iN] and [ν_i+i:1iN] are equal. We use the Garsia–Milne involution principle to produce a bijective proof of this theorem in which non-attacking rook placements for μ are explicitly matched with corresponding placements for ν. One byproduct is a direct combinatorial proof that the matrix of Stirling numbers of the first kind is the inverse of the matrix of Stirling numbers of the second kind. We also prove q-analogues and p,q-analogues of these results. We also use the Garsia–Milne involution principle to show that for any two rook boards B and B^′, if B and B^′ are bijectively rook-equivalent, then B and B^′ are bijectively hit-equivalent.     

The maximum spectral radius of -free graphs of given order and size

Suppose that G is a graph with n vertices and m edges, and let μ be the spectral radius of its adjacency matrix.Recently we showed that if G has no 4-cycle, then μ²-μn-1, with equality if and only if G is the friendship graph.Here we prove that if m9 and G has no 4-cycle, then μ²m, with equality if G is a star. For 4m8 this assertion fails.     

All hypoenergetic graphs with maximum degree at most 3

The energy E(G) of a graph G is defined as the sum of the absolute values of its eigenvalues. A connected graph G of order n is said to be hypoenergetic if E(G)<n. All connected hypoenergetic graphs with maximum degree Δ3 have been characterized. In addition to the four (earlier known) hypoenergetic trees, we now show that complete bipartite graph K_2,3 is the only hypoenergetic cycle-containing hypoenergetic graph. By this, the validity of a conjecture by Majstorović et al. has been confirmed.     

The de Schipper formula and squares of Riesz spaces

In this paper we introduce and study the square mean and the geometric mean in Riesz spaces. We prove that every geometric mean closed Riesz space is square mean closed and give a counterexample to the converse. We define for positive a, b in a square mean closed Riesz space E an addition via the formulaab=sup {(cos x)a + (sin x)b: 0 x 2π},which goes back to a formula by de Schipper. In case that E is geometric mean closed this turns the mldeflying set of the positive cone of E into a lattice ordered semigroup, which in turn is the positive cone ofa Riesz space E^□. We prove, under the additional condition that E is geometric mean closed, that E^□ is Riesz isomorphic to the square of E as introduced earlier by Buskes and van Rooij.     

Universal Polynomial Majorants on Convex Bodies

Let K be a convex body in ^d (d2), and denote by B_n(K) the set of all polynomials p_n in ^d of total degree n such that |p_n|1 on K. In this paper we consider the following question: does there exist a p*_nB_n(K) which majorates every element of B_n(K) outside of K? In other words can we find a minimal γ1 and p*_nB_n(K) so that |p_n(x)|γ |p*_n(x)| for every p_nB_n(K) and x ^d\K? We discuss the magnitude of γ and construct the universal majorants p*_n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K.     

Existence of HSOLSSOMs of type

In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2ⁿu¹ (HSOLSSOM(2ⁿu¹)). For u2, necessary conditions for existence of such an HSOLSSOM are that u must be even and n3u/2+1. Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n9 and n3u/2+1 or (2) n263 and n2(u-2). In this paper we show that in (1) the condition n9 can be extended to n30 and that in (2), the condition n263 can be improved to n4, except possibly for 19 pairs (n,u), the largest of which is (53,28).     

On groups with conjugacy classes of distinct sizes

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following:

1. , for 1a5, a≠2.

2. A₈.

3. PSL(3,4)^e, for 1e10.

4. A₅×PSL(3,4)^e, for 1e10.

Based on this result, we virtually show that if G is an ah-group with π(G) 2,3,5,7 , then F(G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S₃, then the non-abelian socle of G is either trivial or isomorphic to one of the following:

1. , for 3a5.

2. PSL(3,4)^e, for 1e10.

Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z₂S_n. These results are of independent interest and are located in appendices for greater autonomy.     

Group actions and invariants in algebras of generic matrices

We show that the fixed elements for the natural GL_m-action on the universal division algebra UD(m,n) of m generic n×n-matrices form a division subalgebra of degree n, assuming n3 and 2mn²−2. This allows us to describe the asymptotic behavior of the dimension of the space of SL_m-invariant homogeneous central polynomials p(X₁,…,X_m) for n×n-matrices. Here the base field is assumed to be of characteristic zero.     

Radon, Cosine and Sine Transforms on Real Hyperbolic Space

New pointwise inversion formulae are obtained for the d-dimensional totally geodesic Radon transform on the n-dimensional real hyperbolic space, 1dn−1, in terms of polynomials of the Laplace–Beltrami operator and intertwining fractional integrals. Similar results are established for hyperbolic cosine and sine transforms.     

An anisotropic, superconvergent nonconforming plate finite element

The classical finite element convergence analysis relies on the following regularity condition: there exists a constant c independent of the element K and the mesh such that h_K/ρ_Kc, where h_K and ρ_K are diameters of K and the biggest ball contained in K, respectively. In this paper, we construct a new, nonconforming rectangular plate element by the double set parameter method. We prove the convergence of this element without the above regularity condition. The key in our proof is to obtain the O(h²) consistency error. We also prove the superconvergence of this element for narrow rectangular meshes. Results of our numerical tests agree well with our analysis.     

On the Positivity of the Fundamental Polynomials for Generalized Hermite–Fejér Interpolation on the Chebyshev Nodes

It is shown that the fundamental polynomials for (0, 1, …, 2m+1) Hermite–Fejér interpolation on the zeros of the Chebyshev polynomials of the first kind are non-negative for −1x1, thereby generalising a well-known property of the original Hermite–Fejér interpolation method. As an application of the result, Korovkin's 10theorem on monotone operators is used to present a new proof that the (0, 1, …, 2m+1) Hermite–Fejér interpolation polynomials offC[−1, 1], based onnChebyshev nodes, converge uniformly tofasn→∞.     

On a problem of H. Shapiro

Let μ be a real measure on the line such that its Poisson integral M(z) converges and satisfies|M(x+iy)|Ae^−cyα, y→+∞,for some constants A,c>0 and 0<α1. We show that for 1/2<α1 the measure μ must have many sign changes on both positive and negative rays. For 0<α1/2 this is true for at least one of the rays, and not always true for both rays. Asymptotical bounds for the number of sign changes are given which are sharp in some sense.     

A Nonsymmetric Correlation Inequality for Gaussian Measure

Letμbe a Gaussian measure (say, onRⁿ) and letK,LRⁿbe such thatKis convex,Lis a “layer” (i.e.,L={x: ax, ub} for somea, bRanduRⁿ), and the centers of mass (with respect toμ) ofKandLcoincide. Thenμ(K∩L)μ(K)·μ(L). This is motivated by the well-known “positive correlation conjecture” for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses the estimateΦ(x)> 1−((8/π)^1/2/(3x+(x²+8)^1/2))e^−x2/2,x>−1, for the (standard) Gaussian cumulative distribution function, which is sharper than the classical inequality of Komatsu.     

-Linked planar graphs

A graph G is (m,n)-linked if for any two disjoint subsets R,BV(G) with |R|m and |B|n, G has two disjoint connected subgraphs containing R and B, respectively. We shall prove that a planar graph with at least six vertices is (3,3)-linked if and only if G is 4-connected and maximal.     

On the pullback equation

We discuss the existence of a diffeomorphism such that

φ^*(g)=f