Some Best Constants in the Landau Inequality on a Finite Interval

An additive form of the Landau inequality forf∈Wⁿ_∞[−1, 1],is proved for 0<c?(cos(π/2n))⁻², 1?m?n−1, with equality forf(x)=T_n(1+(x−1)/c), 1?c?(cos(π/2n))⁻², whereT_nis the Chebyshev polynomial. From this follows a sharp multiplicative inequality,for ‖f⁽ⁿ⁾‖?σ ‖f‖, 2ⁿ⁻¹n! cos²ⁿ(π/2n)?σ?2ⁿ⁻¹n!, 1?m?n−1. For these values ofσ, the result confirms Karlin's conjecture on the Landau inequality for intermediate derivatives on a finite interval. For the proof of the additive inequality a Duffin and Schaeffer-type inequality for polynomials is shown.     

On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces

Given a bounded operator A on a Banach space X with Drazin inverse AD and index r, we study the class of group invertible bounded operators B such that I+AD(B−A) is invertible and R(B)∩N(Ar)={0}. We show that they can be written with respect to the decomposition X=R(Ar)⊕N(Ar) as a matrix operator, , where B₁ and are invertible. Several characterizations of the perturbed operators are established, extending matrix results. We analyze the perturbation of the Drazin inverse and we provide explicit upper bounds of ‖B^?−AD‖ and ‖BB^?−ADA‖. We obtain a result on the continuity of the group inverse for operators on Banach spaces.     

On a property of plane curves

Let γ:[0,1]→²[0,1] be a continuous curve such that γ(0)=(0,0), γ(1)=(1,1), and γ(t)∈²(0,1) for all t∈(0,1). We prove that, for each n∈N, there exists a sequence of points Ai, 0?i?n+1, on γ such that A₀=(0,0), An+1=(1,1), and the sequences and , 0?i?n, are positive and the same up to order, where π₁, π₂ are projections on the axes.     

André-Quillen homology of algebra retracts

Given a homomorphism of commutative noetherian rings ?:R→S, Daniel Quillen conjectured in 1970 that if the André-Quillen homology functors D_n(S∣R;−) vanish for all n?0, then they vanish for all n?3. We prove the conjecture under the additional hypothesis that there exists a homomorphism of rings ψ:S→R such that ?°ψ=id_S. More precisely, in this case we show that ψ is a complete intersection at for every prime ideal  of S. Using these results, we describe all algebra retracts S→R→S for which the algebra Tor_•^R(S,S) is finitely generated over Tor₀^R(S,S)=S.     

Transition matrices for well-conditioned Markov chains

Let T∈R^n×n be an irreducible stochastic matrix with stationary distribution vector π. Set A = I − T, and define the quantity , where A_j, j = 1, … , n, are the (n − 1) × (n − 1) principal submatrices of A obtained by deleting the jth row and column of A. Results of Cho and Meyer, and of Kirkland show that κ₃ provides a sensitive measure of the conditioning of π under perturbation of T. Moreover, it is known that .In this paper, we investigate the class of irreducible stochastic matrices T of order n such that , for such matrices correspond to Markov chains with desirable conditioning properties. We identify some restrictions on the zero-nonzero patterns of such matrices, and construct several infinite classes of matrices for which κ₃ is as small as possible.     

Blaschke products and the rank of backward shift invariant subspaces over the bidisk

Let H²(D²) be the Hardy space over the bidisk. For sequences of Blaschke products {φn(z):−∞<n<∞} and {ψn(w):−∞<n<∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H²(D²)?M for the pair of operators and .     

The evaluation subgroup of a fibre inclusion

Let be a fibration of simply connected CW complexes of finite type with classifying map . We study the evaluation subgroup Gn(E,X;j) of the fibre inclusion as an invariant of the fibre-homotopy type of ξ. For spherical fibrations, we show the evaluation subgroup may be expressed as an extension of the Gottlieb group of the fibre sphere provided the classifying map h induces the trivial map on homotopy groups. We extend this result after rationalization: We show that the decomposition G_∗(E,X;j)⊗Q=(G_∗(X)⊗Q)⊕(π_∗(B)⊗Q) is equivalent to the condition Q(h_?)=0.     

Bounds and conjectures for the signless Laplacian index of graphs

Using the AutoGraphiX system, we obtain conjectures of the form l(n)?q₁⊕i(G)?u(n) where q₁ denotes the signless Laplacian index of graph is one the four operations is another invariant chosen among minimum, average and maximum degree, average distance, diameter, radius, girth, proximity, remoteness, vertex, edge and algebraic connectivities, independence number, domination number, clique number, chromatic number and matching number, Randi? index, l(n) and u(n) are best possible lower and upper bounds function of the order n of G. Algebraic conjectures are obtained in 120 cases out of 152 and structural conjectures in 12 of the remaining cases. These conjectures are known, immediate or proved in this paper, except for 17 of them, which remain open.     

The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights

We consider orthogonal polynomials on the real line with respect to a weight and in particular the asymptotic behaviour of the coefficients a_n,N and b_n,N in the three-term recurrence xπ_n,N(x)=π_n+1,N(x)+b_n,Nπ_n,N(x)+a_n,Nπ_n−1,N(x). For one-cut regular V we show, using the Deift-Zhou method of steepest descent for Riemann-Hilbert problems, that the diagonal recurrence coefficients a_n,n and b_n,n have asymptotic expansions as n→∞ in powers of 1/n² and powers of 1/n, respectively.     

Graphic sequences with a realization containing a complete multipartite subgraph

A nonincreasing sequence of nonnegative integers π=(d₁,d₂,…,d_n) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realizeπ. For a given graph H, a graphic sequence π is potentiallyH-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n∈Z⁺, σ(H,n) is defined as the smallest even integer m so that every n-term graphic sequence π with σ(π)≥m is potentially H-graphic. Let denote the complete t partite graph such that each partite set has exactly s vertices. We show that and obtain the exact value of σ(K_j+K_s,s,n) for n sufficiently large. Consequently, we obtain the exact value of for n sufficiently large.     

The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

We study the critical set C of the nonlinear differential operator F(u)=−u^″+f(u) defined on a Sobolev space of periodic functions Hp(S¹), p?1. Let be the plane z=0 and, for n>0, let n be the cone x²+y²=tan²z, |z−2πn|<π/2; also set . For a generic smooth nonlinearity f:R→R with surjective derivative, we show that there is a diffeomorphism between the pairs (Hp(S¹),C) and (R³,Σ)×H where H is a real separable infinite-dimensional Hilbert space.     

Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree

For Banach space operators T satisfying the Tadmor-Ritt condition ||(zI−T)⁻¹||?C|z−1|⁻¹, |z|>1, we prove that the best-possible constant C_T(n) bounding the polynomial calculus for T, ||p(T)||?C_T(n)||p||_∞, deg(p)?n, behaves (in the worst case) as as n→∞. This result is based on a new free (Carleson type) interpolation theorem for polynomials of a given degree.     

Dissipative operators on Banach spaces

A bounded linear operator T on a Banach space is said to be dissipative if ‖etT‖?1 for all t?0. We show that if T is a dissipative operator on a Banach space, then:

(a)

.

(b)

If σ(T)∩iR is contained in [−iπ/2,iπ/2], then

     

Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.