The Best Lower Bound for the Permanent of a Correlation Matrix of Rank Two

If 1≤k≤n, then Cor(n,k) denotes the set of all n×n real correlation matrices of rank not exceeding k. Grone and Pierce have shown that if A∈Cor (n, n-1), then per(A)≥n/(n-1). We show that if A∈Cor(n,2), then , and that this inequality is the best possible.     

An inequality involving permanents of certain direct products

Let A denote a decomposable symmetric complex valued n-linear function on C^m. We prove

$\text{6A·A6}{}^2\text{?2}{}^{\mathrm{n}}\text{2n}\text{n}{}^{\mathrm{?1}}\text{6A?A6}{}^2$

, where · denotes the symmetric product and ? the tensor product. As a consequence we have per

$\text{M}\text{M}\text{M}\text{M}\text{?2}{}^{\mathrm{n}}\text{[per(M)]}{}^2$

, where M is a positive semidefinite Hermitian matrix and per denotes the permanent function. A sufficient condition for equality in the matrix inequality is that M is a nonnegative diagonal matrix.     

Some improvements in Neuberger's iteration procedure for solving partial differential equations

If m and n are positive integers then let S(m, n) denote the linear space over R whose elements are the real-valued symmetric n-linear functions on E_m with operations defined in the usual way. If $\text{U}$ is a function from some sphere S in E_m to R then let $\text{U}$⁽ⁱ⁾(x) denote the ith Frechet derivative of $\text{U}$ at x. In general $\text{U}$⁽ⁱ⁾(x)∈S(m,i). In the paper “An Iterative Method for Solving Nonlinear Partial Differential Equations” [Advances in Math. 19 (1976), 245–265] Neuberger presents an iterative procedure for solving a partial differential equation of the form

$\text{AU}{}^{\mathrm{n}}\text{(x)=F(x, U(x), U′(x),…,U}{}^{\mathrm{k}}\text{(x))}$

, where k > n, $\text{U}$ is the unknown from some sphere in E_m to R, A is a linear functional on S(m, n), and F is analytic. The defect in the theory presented there was that in order to prove that the iterates converged to a solution $\text{U}$ the condition $\text{k ?}\text{n}\text{2}$ was needed. In this paper an iteration procedure that is a slight variation on Neuberger's procedure is considered. Although the condition $\text{k ?}\text{n}\text{2}$ cannot as yet be eliminated, it is shown that in a broad class of cases depending upon the nature of the functional A the restriction $\text{k ?}\text{n}\text{2}$ may be replaced by the restriction $\text{k ?}\text{3n}\text{4}$.     

Probing the CH⋅⋅⋅π Weak Hydrogen Bond in Anesthetic Binding: The Sevoflurane–Benzene Cluster

Cooperativity between weak hydrogen bonds can be revealed in molecular clusters isolated in the gas phase. Here we examine the structure, internal dynamics, and origin of the weak intermolecular forces between sevoflurane and a benzene molecule, using multi‐isotopic broadband rotational spectra. This heterodimer is held together by a primary C? H???π hydrogen bond, assisted by multiple weak C? H???F interactions. The multiple nonbonding forces hinder the internal rotation of benzene around the isopropyl C? H bond in sevoflurane, producing detectable quantum tunneling effects in the rotational spectrum.     

Hydrogen Bond Cooperativity and the Three‐Dimensional Structures of Water Nonamers and Decamers

Broadband rotational spectroscopy of water clusters produced in a pulsed molecular jet expansion has been used to determine the oxygen atom geometry in three isomers of the nonamer and two isomers of the decamer. The isomers for each cluster size have the same nominal geometry but differ in the arrangement of their hydrogen bond networks. The nearest neighbor O? O distances show a characteristic pattern for each hydrogen bond network isomer that is caused by three‐body effects that produce cooperative hydrogen bonding. The observed structures are the lowest energy cluster geometries identified by quantum chemistry and the experimental and theoretical O? O distances are in good agreement. The cooperativity effects revealed by the hydrogen bond O? O distance variations are shown to be consistent with a simple model for hydrogen bonding in water that takes into account the cooperative and anticooperative bonding effects of nearby water molecules.