Sharp nonlogarithmic Kneser theorems for fourth-order elliptic equations

Let D(?) be the Doob's class containing all functions f(z) analytic in the unit disk Δ such that f(0) = 0 and lim $\text{inf}\text{¦f(z) ¦ ? 1}$ on an arc A of ?Δ with length $\text{¦A ¦? ?}$. It is first proved that if f?D(?) then the spherical norm ∥ f ∥ = sup_z?Δ$\text{(}\text{1 ? ¦z¦}{}^2\text{)¦f′(z)¦}\text{(1 + ¦f(z)¦}{}^2\text{) ? C}{}_1\text{sin}\text{(π ? (}\text{?}\text{2}\text{))/ (π ? (}\text{g9}\text{2}\text{))}$, where C₁ = lim_n→∞$\text{∥ z}{}^{\mathrm{n}}\text{∥}\text{and}\text{1}\text{2}\text{< C}{}_1\text{<}\text{2}\text{e}$. Next, U represents the Seidel's class containing all non-constant functions f(z) bounded analytic in Δ such that $\text{¦tf(e}{}^{\mathrm{i0}}\text{)¦ = 1}$ almost everywhere. It is proved that inf_f?U∥f∥ = 0, and if f has either no singularities or only isolated singularities on ?Δ, then ∥f∥ ? C₁. Finally, it is proved that if f is a function normal in Δ, namely, the norm ∥f∥< ∞, then we have the sharp estimate ∥f^p∥ ? p∥f∥, for any positive integer p.     

The approximation of flows

Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space $\text{X}$ with corresponding infinitesimal generators S, T. We prove the following: ∥U_t, ? V_t ∥ = O(t), t → 0, if and only if U = V; ∥U_t ? V_t∥ = O(t^α), t → 0; with 0 ? α ? 1, if and only if $\text{S = Ω(T + P)Ω}{}^{\mathrm{?1}}$, where Ω, P, are bounded operators on $\text{X}$ such that $\text{∥U}{}_{\mathrm{t}}\text{Ω ? ΩU}{}_{\mathrm{t}}\text{∥ = O(t}{}^{\mathrm{\alpha }}\text{), ∥U}{}_{\mathrm{t}}\text{P ? PU}{}_{\mathrm{t}}\text{∥ = ?O(t}{}^{\mathrm{\alpha }}\text{), t → 0; ∥U}{}_{\mathrm{t}}\text{? V}{}_{\mathrm{t}}\text{∥ = O(t)}$ if and only if $\text{S}{}^1\text{? T}{}^1$ has a bounded extension to $\text{X}$¹. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras.     

Brownian motion,Lp properties of Schrödinger operators and the localization of binding

We use Brownian motion ideas to study Schrödinger operators $\text{H =}\text{built?1}\text{2}\text{Δ + V}\text{on}\text{L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{v}}\text{)}$. In particular: (a) We prove that lim_t→∞t^?1In ∥ e^?tH ∥_p,p is p-independent for a very large class of V's where ∥ A ∥_p,p = norm of A as an operator from L^pto L^p. (b) For v ? 3 and $\text{V ? L}{}^{\mathrm{<mtext>v</mtext><mtext>2\; ?\; ?</mtext>}}\text{∩ L}{}^{\mathrm{<mtext>v</mtext><mtext>2\; +\; ?</mtext>}}$, we show that sup ∥ e^?tH ∥_∞,∞ < ∞ if and only if H has no negative eigenvalues or zero energy resonances. (c) We relate the “localization of binding” recently noted by Sigal to Brownian hitting probabilities.     

Almost commuting matrices

It is shown that if A and B are n × n complex matrices with $\text{A = A}{}^1\text{and}\text{∥AB ? BA∥</}\text{2?}{}^2\text{(n ? 1)}$, then there exist n × n matrices A′ and B′ with $\text{A′ = A′}{}^1\text{such that}\text{A′B′ = B′A′}\text{and}\text{∥A ? A′∥? ?, ∥B ? B′∥? ?}$.     

The resolvent of a singularly perturbed elliptic operator

Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?^{2(m ? n)}A + B + λ²ⁿI)u_? = f and (B + λ²ⁿI)u = f in a bounded domain Ω in R^k with a smooth boundary ?Ω. The estimate is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given.     

Generalized Schur complements

Let A be an n×n complex matrix. For a suitable subspace $\text{M}$ of Cⁿ the Schur compression A _$\text{M}$ and the (generalized) Schur complement A_/$\text{M}$ are defined. If A is written in the form

$\text{A=}\text{B}\text{C}\text{S}\text{T}$

according to the decomposition $\text{C}{}^{\mathrm{n}}\text{=}\text{M}\text{⊕}\text{M}{}^{\mathrm{\perp }}$ and if B is invertible, then

$\text{A}{}_{\mathrm{<mtext>M</mtext>}}\text{=}\text{B}\text{C}\text{S}\text{SB}{}^{\mathrm{?1}}\text{C}$

and

$\text{A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{=}\text{0}\text{0}\text{0}\text{T?SB}{}^{\mathrm{?1}}\text{C}\text{·}$

The commutativity rule for Schur complements is proved:

$\text{(A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{)}{}_{\mathrm{<mtext>/</mtext><mtext>N</mtext>}}\text{=(A)}{}_{\mathrm{<mtext>/</mtext><mtext>N</mtext>}}\text{)}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{·}$

This unifies Crabtree and Haynsworth's quotient formula for (classical) Schur complements and Anderson's commutativity rule for shorted operators. Further, the absorption rule for Schur compressions is proved:

$\text{(A}{}_{\mathrm{<mtext>/</mtext><mtext>M</mtext>}}\text{)}{}_{\mathrm{<mtext>N</mtext>}}\text{=(A}{}_{\mathrm{<mtext>N</mtext>}}\text{)}{}_{\mathrm{<mtext>M</mtext>}}\text{=A}{}_{\mathrm{<mtext>M</mtext>}}\text{whenever}\text{M}\text{?}\text{N}$

.     

Anti-Hadamard matrices

An anti-Hadamard matrix may be loosely defined as a real (0, 1) matrix which is invertible, but only just. Let A be an invertible (0, 1) matrix with eigenvalues λ_i, singular values σ_i, and inverse B = (b_ij). We are interested in the four closely related problems of finding λ(n) = min_{A, i}|λ_i|, σ(n) = min_{A, i}σ_i, χ(n) = max_{A, i, j} |b_ij|, and μ(n) = max_AΣ_ijb²_ij. Then A is an anti-Hadamard matrix if it attains μ(n). We show that λ(n), σ(n) are between $\text{(2n)}{}^{\mathrm{?1}}\text{(}\text{n}\text{4}\text{)}{}^{\mathrm{<mtext>?n</mtext><mtext>2</mtext>}}$ and c√n (2.274)^?n, where c is a constant, $\text{c}\text{(2.274)}{}^{\mathrm{n}}\text{?χ(n)?2(}\text{n}\text{4}\text{)}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, and $\text{c}\text{(5.172)}{}^{\mathrm{n}}\text{?μ(n)?4n}{}^2\text{(}\text{n}\text{4}\text{)}{}^{\mathrm{n}}$. We also consider these problems when A is restricted to be a Toeplitz, triangular, circulant, or (+1, ?1) matrix. Besides the obvious application—to finding the most ill-conditioned (0, 1) matrices—there are connections with weighing designs, number theory, and geometry.     

An inertial decomposition and related range results

With quasicommutative n-square complex matrices A₁,…,A_s and s-square hermitian G=(g_ij), relationships are given between the image $\text{Σ}{}^{\mathrm{s}}{}_{\mathrm{i,j=1}}\text{g}{}_{\mathrm{ij}}\text{A}{}_{\mathrm{i}}\text{HA}{}^1{}_{\mathrm{j}}$ of a linear transformation on $\text{H}$_n being positive definite and the action of H on generalized inertial decompositions of $\text{C}$ⁿ.     

One-sided inequalities for ternary forms: II

For an indefinite quadratic form f(x₁, …, x_n) let P(f) denote the greatest lower bound of the positive values assumed by f for integers x₁, …, x_n. This paper investigates the values of $\text{P}{}^3\text{∥d∥}$ for nonzero ternary forms of signature ?1 and finds two new classes of forms with $\text{P}{}^3\text{∥d∥}\text{>}\text{1}\text{2}$.     

On the convergence of the Neumann series in interval analysis

Let $\text{A}$ be a real or complex n × n interval matrix. Then it is shown that the Neumann series $\text{Σ}{}^{\mathrm{\infty }}{}_{\mathrm{k=0}}\text{A}{}^{\mathrm{k}}$ is convergent iff the sequence {$\text{A}$^k} converges to the null matrix $\text{O}$, i.e., iff the spectral radius of the real comparison matrix $\text{B}$ constructed in [2] is less than one.     

Multiplicativity of lp norms for matrices. II

For 1 ? p ? ∞, let

$\text{|A|}{}_{\mathrm{p}}\text{=}\text{Σ}\text{i=1}\text{m}\text{Σ}\text{j=1}\text{n}\text{, |α}{}_{\mathrm{ij}}\text{|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$

, be the l_p norm of an m × n complex A = (α_ij) ?C_{m × n}. The main purpose of this paper is to find, for any p, q ? 1, the best (smallest) possible constants τ(m, k, n, p, q) and σ(m, k, n, p, q) for which inequalities of the form

$\text{|AB|}{}_{\mathrm{p}}\text{? τ(m, k, n, p, q) |A|}{}_{\mathrm{p}}\text{|B|}{}_{\mathrm{q}}\text{, |AB|}{}_{\mathrm{p}}\text{? σ (m, k, n, p, q)|A|}{}_{\mathrm{q}}\text{|B|}{}_{\mathrm{p}}$

hold for all A?C_{m × k}, B?C_{k × n}. This leads to upper bounds for inner products on C^k and for ordinary l_p operator norms on C_{m × n}.     

Lower bounds for the norm of the symmetric product

Suppose each of m, n, and k is a positive integer, k ? n, A is a (real-valued) symmetric n-linear function on E_m, and B is a k-linear symmetric function on E_m. The tensor and symmetric products of A and B are denoted, respectively, by A ?B and A?B. The identity

$\text{6A · B6}{}^2\text{=}\text{∑}\text{q=0}\text{n}\text{(}\text{n}\text{k}\text{)}\text{(}\text{n+k}\text{k}\text{)}\text{6A?}{}_{\mathrm{q}}\text{B6}{}^2$

is proven by Neuberger in [1]. An immediate consequence of this identity is the inequality

$\text{6A · B 6}{}^2\text{?}\text{n+k}\text{n}{}^{\mathrm{?1}}\text{6A · B 6}{}^2$

In this paper a necessary and sufficient condition for

$\text{6A · B 6}{}^2\text{=}\text{n+k}\text{n}{}^{\mathrm{?}}\text{6A · B 6}{}^2$

is given. It is also shown that under certain conditions the inequality can be considerably improved. This improvement results from an analysis of the terms 6A?_qB6, 1?q?n, appearing in the identity.     

On the matrix monotonicity of generalized inversion

Let A, B be two matrices of the same order. We write A>B(A>?B) iff A? B is a positive (semi-) definite hermitian matrix. In this paper the well-known result if

$\text{A>B>θ, then B}{}^{\mathrm{?1}}\text{>A}{}^{\mathrm{?1}}\text{> θ}$

(cf. Bellman [1, p.59]) is extended to the generalized inverses of certain types of pairs of singular matrices A,B?θ, where θ denotes the zero matrix of appropriate order.     

A regularity result for singular nonlinear elliptic systems in inverse-power weighted Sobolev spaces

The compactness method to weighted spaces is extended to prove the following theorem:Let H_2,s¹(B₁) be the weighted Sobolev space on the unit ball in Rⁿ with norm

$\text{6ν6}{}^1{}_{\mathrm{2,s}}\text{=}\text{∫}{}_{\mathrm{B1}}\text{(}\text{1}\text{r}{}^{\mathrm{s}}\text{)|ν|}{}^2\text{dx + ∫}{}_{\mathrm{B1}}\text{(}\text{1}\text{r}{}^{\mathrm{s}}\text{)|Dν|}{}^2\text{dx.}$

Let n ? 2 ? s < n. Let u? [H_2,s¹(B₁) ∩ L^∞(B₁)]^N be a solution of the nonlinear elliptic system

$\text{∫}{}_{\mathrm{B1}}\text{1}\text{r}{}^{\mathrm{s}}\text{,}\text{∑}\text{i,j=1}\text{n}\text{,}\text{∑}\text{h,K=1}\text{N}\text{A}{}^{\mathrm{hK}}{}_{\mathrm{ij}}\text{(x,u) D}{}^{\mathrm{i}}\text{u}{}_{\mathrm{h}}\text{D}{}^{\mathrm{j\psi }}{}_{\mathrm{K}}\text{dx=0}$

, $\text{∨}\text{ψ}\text{?}\text{¦C}{}_0{}^1\text{(B}{}_1\text{)¦}{}^{\mathrm{N}}\text{,}\text{where}\text{¦A}{}_{\mathrm{ij}}{}^{\mathrm{hk}}\text{¦ ? L, A}{}_{\mathrm{ij}}{}^{\mathrm{hk}}$ are uniformly continuous functions of their arguments and satisfy:

$\text{|η|}{}^2\text{=}\text{∑}\text{i=1}\text{n}\text{,}\text{∑}\text{j=1}\text{N}\text{|η}{}^{\mathrm{i}}{}_{\mathrm{j}}\text{|}{}^2\text{?}\text{∑}\text{i,j=1}\text{n}\text{,}\text{1}\text{r}{}^{\mathrm{s}}\text{,}\text{∑}\text{h,K=1}\text{N}\text{A}{}^{\mathrm{hK}}{}_{\mathrm{ij}}\text{η}{}^{\mathrm{i}}{}_{\mathrm{h}}\text{η}{}^{\mathrm{i}}{}_{\mathrm{k}}\text{,}\text{?η?R}{}^{\mathrm{Nn}}$

. Then there exists an R₁, 0 < R₁ < 1, and an α, 0 < α < 1, along with a set $\text{Ω ? B}{}_1$ such that (1) $\text{H}{}_{\mathrm{n\; ?\; 2}}\text{(Ω) = 0}$, (2) Ω does not contain the origin; Ω does not contain B_R1, (3) $\text{B}{}_1\text{? Ω}$ is open, (4) u is $\text{Lip}{}_{\mathrm{\alpha }}\text{(B}{}_1\text{? Ω)}$; u is Lip_αB_R1.     

On the continuity of the map ϑ → ¦ϑ¦ from the predual of a W1-algebra

Let ?, ψ be elements in the predual of a W¹-algebra. For their absolute value parts ¦?¦, ¦ψ¦, the estimate $\text{∥¦?¦ ? ¦ψ¦∥ ? (2 ∥? + ψ∥ ∥? ? ψ∥)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is obtained.     

Maximum zero strings of Bell numbers modulo primes

For any prime p, the sequence of Bell exponential numbers B_n is shown to have p ? 1 consecutive values congruent to zero (mod p), beginning with B_m, where $\text{m ≡ 1 ?}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}{}^2$ ($\text{mod}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}$). This is an improvement over previous results on the maximal strings of zero residues of the Bell numbers. Similar results are obtained for the sequence of generalized Bell numbers A_n generated by .     

Function algebra invariants and the conformal structure of planar domains

Let A(S) be the sup-normed Banach algebra of analytic functions with continuous boundary values on the compact bordered Riemann surface S.For (?) in $\text{A(S)}{}^{\mathrm{?1}}\text{exp}\text{(A(S))}$, the colength of (?) is defined by $\text{∥(?)∥ =}\text{1}\text{2}\text{log inf}\text{{∥ g ∥ ∥ g}{}^{\mathrm{?1}}\text{∥; g ? (?)}}$. Colength is shown to induce a norm on the cohomology group H¹(S,R) dual to the norm induced on the homology group H₁(S,R) by harmonic length, or, equivalently, dual to the norm on Re A(S)^⊥.The existence and uniqueness of extremal functions for the colength functional is demonstrated. The aforementioned norms are shown to determine the conformal structure of S (up to reflection) and to be related to the mapping properties of S.