A slight improvement to Korenblum's constant

Let A²(D) be the Bergman space over the open unit disk D in the complex plane. Korenblum conjectured that there is an absolute constant c∈(0,1) such that whenever |f(z)|?|g(z)| in the annulus c<|z|<1, then ‖f(z)‖?‖g(z)‖. This conjecture had been solved by Hayman [W.K. Hayman, On a conjecture of Korenblum, Analysis (Munich) 19 (1999) 195-205. [1]], but the constant c in that paper is not optimal. Since then, there are many papers dealing with improving the upper and lower bounds for the best constant c. For example, in 2004 C. Wang gave an upper bound on c, that is, c<0.67795, and in 2006 A. Schuster gave a lower bound, c>0.21. In this paper we slightly improve the upper bound for c.     

A note on Faber operator

For a rectifable Jordan curve Γ with complementary domainsD and D,Anderson conjectured that the Faber operator is a bounded isomorphism between the Besov spaces Bp(1 p ∞) of analytic functions in the unit disk and in the inner domain D,respectively.We point out that the conjecture is not true in the special case p=2,and show that in this case the Faber operator is a bounded isomorphism if and only if Γ is a quasi-circle.     

Upper bounds for permanents of (1, − 1)-matrices

Let Ω _n denote the set of alln×n (1, ? 1)-matrices. In 1974 E. T. H. Wang posed the following problems: Is there a decent upper bound for |perA| whenAσΩ _n is nonsingular? We recently conjectured that the best possible bound is the permanent of the matrix with exactlyn?1 negative entries in the main diagonal, and affirmed that conjecture by the study of a large class of matrices in Ω _n . Here we prove that this conjecture also holds for another large class of (1, ?1)-matrices which are all nonsingular. We also give an upper bound for the permanents of a class of matrices in Ω _n which are not all regular.     

Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 2, let p(1, n) real, and let H₁^p (M) be the standard Sobolev space of order p. By the Sobolev embedding theorem, H₁^p(M) ⊂ L^p* (M) where p^* = np/(n - p). Classically, this leads to some Sobolev inequality (I_p¹), and then to some Sobolev inequality (I_p^p) where each term in (I_p¹) is elevated to the power p. Long standing questions were to know if the optimal versions with respect to the first constant of (I_p¹) and (I_p^p) do hold. Such questions received an affirmative answer by Hebey-Vaugon for p = 2, and on what concerns (I_p¹), by Aubin for two-dimensional manifolds and for manifolds of constant sectional curvature. Recently, Druet proved that for p > 2, and p² < n, the optimal version of (I_p^p) is false if the scalar curvature of g is positive somewhere, while for p > 1, the optimal version of (I_p^p) does hold on flat torii and compact hyperbolic spaces. We prove here that the optimal version of (I_p^p), p > 1, does hold for compact manifolds of nonpositive sectional curvature in any dimension where the Cartan-Hadamard conjecture is true. In particular, since the Cartan-Hadmard conjecture is true in dimensions 2, 3, and 4, the optimal version of (I_p^p) does hold on any compact manifold of nonpositive sectional curvature of dimension 2, 3, or 4.     

On mockenhoupt’s conjecture in the Hardy-Littlewood majorant problem

The Hardy-Littlewood majorant problem has a positive answer only for even integer exponents p, while there are counterexamples for all p {ie91-1} 2?. Montgomery conjectured that even among the idempotent polynomials there must exist counterexamples, i.e. there exist a finite set of characters and some ± signs with which the signed character sum has larger p ^th norm than the idempotent obtained with all the signs chosen + in the character sum. That conjecture was proved recently by Mockenhaupt and Schlag. However, Mockenhaupt conjectured that even the classical {ie91-2} three-term character sums, used for p = 3 and k = 1 already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. In our previous work we proved this conjecture for k = 0, 1, 2, i.e. in the range 0 < p < 6, p {ie91-3} 2?. Continuing this work here we demonstrate that even k = 3, 4 cases hold true. Several refinement in the technical features of our approach include improved fourth order quadrature formulae, finite estimation of G′²/G (with G being the absolute value square function of an idempotent), valid even at a zero of G, and detailed error estimates of approximations of various derivatives in subintervals, chosen to have accelerated convergence due to smaller radius of the Taylor approximation     

On a conjecture by le Veque and Trefethen related to the kreiss matrix theorem

In the Kreiss matrix theorem the power boundedness ofN ×N matrices is related to a resolvent condition on these matrices. LeVeque and Trefethen proved that the ratio of the constants in these two conditions can be bounded by 2eN. They conjectured that this bound can be improved toeN.In this note the conjecture is proved to be true. The proof relies on a lemma which provides an upper bound for the arc length of the image of the unit circle in the complex plane under a rational function. This lemma may be of independent interest.     

Betti numbers and degree bounds for some linked zero-schemes

In [J. Herzog, H. Srinivasan, Bounds for multiplicities, Trans. Amer. Math. Soc. 350 (1998) 2879-2902], Herzog and Srinivasan study the relationship between the graded Betti numbers of a homogeneous ideal I in a polynomial ring R and the degree of I. For certain classes of ideals, they prove a bound on the degree in terms of the largest and smallest Betti numbers, generalizing results of Huneke and Miller in [C. Huneke, M. Miller, A note on the multiplicity of Cohen-Macaulay algebras with pure resolutions, Canad. J. Math. 37 (1985) 1149-1162]. The bound is conjectured to hold in general; we study this using linkage. If R/I is Cohen-Macaulay, we may reduce to the case where I defines a zero-dimensional subscheme Y. If Y is residual to a zero-scheme Z of a certain type (low degree or points in special position), then we show that the conjecture is true for I_Y.     

Direct and inverse asymptotic scattering problems for Dirac-Krein systems

The asymptotic scattering matrix s _ε(λ) for a Dirac-Krein system with signature matrix J = diag{ I _p ,-I _p }, integrable potential, and the boundary condition u ₁(0, λ) = u ₂(0, λ)ε(λ) with a coefficient ε(λ) that belongs to the Schur class of holomorphic contractive p × p matrix-valued functions in the open upper half-plane is defined. The inverse asymptotic scattering problem for a given s _ε is analyzed by Krein’s method. Earlier studies by Krein and others focused on the case in which ε = I _p (or a constant unitary matrix).     

Relationship between strictly collapsible and perfect contingency tables

Whittemore (1978) conjectured that an N-dimensional contingency table p is strictly collapsible over each factor with respect to the set of remaining factors if and only if p has a certain factorization. I prove this conjecture for N = 3 and show by counterexamples that it is false for N > 3.     

Monotonicity of permanents of certain doubly stochastic circulant matrices

Let p_k(A), k=2,…,n, denote the sum of the permanents of all k×k submatrices of the n×n matrix A. A conjecture of Ðokovi?, which is stronger than the famed van der Waerden permanent conjecture, asserts that the functions p_k((1?θ)J_n+;θA), k=2,…, n, are strictly increasing in the interval 0?θ?1 for every doubly stochastic matrix A. Here J_n is the n×n matrix all whose entries are equal $\text{1}\text{n}$. In the present paper it is proved that the conjecture holds true for the circulant matrices A=αI_n+ βP_n, α, β?0, α+;β=1, and $\text{A=}\text{(nJ}{}_{\mathrm{n}}\text{?I}{}_{\mathrm{n}}\text{?P}{}_{\mathrm{n}}\text{)}\text{(n?2)}$, where I_n and P_n are respectively the n×n identify matrix and the n×n permutation matrix with 1's in positions (1,2), (2,3),…, (n?1, n), (n, 1).     

On a generalized James constant

We introduce a generalized James constant J(a,X) for a Banach space X, and prove that, if J(a,X)<(3+a)/2 for some a∈[0,1], then X has uniform normal structure. The class of spaces X with J(1,X)<2 is proved to contain all u-spaces and their generalizations. For the James constant J(X) itself, we show that X has uniform normal structure provided that , improving the previous known upper bound at 3/2. Finally, we establish the stability of uniform normal structure of Banach spaces.     

On the Cameron-Praeger conjecture

This paper takes a significant step towards confirming a long-standing and far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They conjectured in 1993 that there are no non-trivial block-transitive 6-designs. We prove that the Cameron-Praeger conjecture is true for the important case of non-trivial Steiner 6-designs, i.e. for 6-(v,k,λ) designs with λ=1, except possibly when the group is PΓL(2,pe) with p=2 or 3, and e is an odd prime power.     

Matrices and set intersections

The incidence matrix of a (υ, k, λ)-design is a (0, 1)-matrix A of order υ that satisfies the matrix equation AA^T=(k?λ)I+λJ, where A^T denotes the transpose of the matrix A, I is the identity matrix of order υ, J is the matrix of 1's of order υ, and υ, k, λ are integers such that 0<λ<k<υ?1. This matrix equation along with various modifications and generalizations has been extensively studied over many years. The theory presents an intriguing joining together of combinatorics, number theory, and matrix theory. We survey a portion of the recent literature. We discuss such varied topics as integral solutions, completion theorems, and λ-designs. We also discuss related topics such as Hadamard matrices and finite projective planes. Throughout the discussion we mention a number of basic problems that remain unsolved.     

Unavoidable patterns

Let Fk denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollobás conjectured that for every ?>0 and positive integer k there is n(k,?) such that every 2-edge-coloring of the complete graph of order n?n(k,?) which has at least edges in each color contains a member of Fk. This conjecture was proved by Cutler and Montágh, who showed that n(k,?)<⁴k/?. We give a much simpler proof of this conjecture which in addition shows that n(k,?)<?−ck for some constant c. This bound is tight up to the constant factor in the exponent for all k and ?. We also discuss similar results for tournaments and hypergraphs.     

Note on bounds for multiplicities

Let S=K[x₁,…,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.     

Order of the smallest counterexample to Gallai’s conjecture

In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Zamfirescu conjectured that the smallest counterexample to Gallai’s conjecture is a graph on 12 vertices. We prove that Gallai’s conjecture is true for every connected graph G with α′(G) ≤ 5, which implies that Zamfirescu’s conjecture is true.     

Doubly stochastic matrices with equal subpermanents

It has been conjectured that if A is a doubly stochastic n>× n matrix such that per A(i, j)≥perA for all i, j, then either A = J_n, then n × n matrix with each entry equal to $\text{1}\text{n}$, or, up to permutations of rows and columns, $\text{A =}\text{1}\text{2}\text{(I}{}_{\mathrm{n}}\text{+ P}{}_{\mathrm{n}}\text{)}$, where P_n is a full cycle permutation matrix. This conjecture is proved.     

Packing and Covering Triangles in Planar Graphs

Tuza conjectured that if a simple graph G does not contain more than k pairwise edge-disjoint triangles, then there exists a set of at most 2k edges that meets all triangles in G. It has been shown that this conjecture is true for planar graphs and the bound is sharp. In this paper, we characterize the set of extremal planar graphs.     

On the Courant-Fischer theory for Krein spaces

Let J=I_r⊕-I_n-r,0<r<n. An n×n complex matrix A is said to be J-Hermitian if JA=A^∗J. An extension of the classical theory of Courant and Fischer on the Rayleigh ratio of Hermitian matrices is stated for J-Hermitian matrices. Applications to the theory of small oscilations of a mechanical system are presented.