The HELP inequality on trees

We establish analogues of Hardy and Littlewood's integro-differential equation for Schrödinger-type operators on metric and discrete trees, based on a generalised strong limit-point property of the graph Laplacian

"Is It Not Our Land?" an Ethnohistory of the Susquehanna-ohio Indian Alliance, 1701-1754

Histor

Conditions for the spectrum associated with a leaky wire to contain the interval [− α2/4, ∞)

The method of singular sequences is used to provide a simple and, in some respects, a more general proof of a known spectral result for leaky wires. The method introduces a new concept of asymptotic straightness

Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map

Guaranteed resonance enclosures and exclosures for atoms and molecules

In this paper, we confirm, with absolute certainty, a conjecture on a certain oscillatory behaviour of higher auto-ionizing resonances of atoms and molecules beyond a threshold. These results not only definitely settle a more than 30 year old controversy in Rittby et al. (1981 Phys. Rev. A24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) and Korsch et al. (1982 Phys. Rev. A26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)), but also provide new and reliable information on the threshold. Our interval-arithmetic-based method allows one, for the first time, to enclose and to exclude resonances with guaranteed certainty. The efficiency of our approach is demonstrated by the fact that we are able to show that the approximations in Rittby et al. (1981 Phys. Rev. A24, 1636–1639 (doi:10.1103/PhysRevA.24.1636)) do lie near true resonances, whereas the approximations of higher resonances in Korsch et al. (1982 Phys. Rev. A26, 1802–1803 (doi:10.1103/PhysRevA.26.1802)) do not, and further that there exist two new pairs of resonances as suggested in Abramov et al. (2001 J. Phys. A34, 57–72 (doi:10.1088/0305-4470/34/1/304))

Proximate and fatty acid composition of 40 southeastern U.S. finfish species

This report describes the proximate compositions (protein, moisture, fat, and ash) and major fatty acid profiles for raw and cooked samples of 40 southeastern finfish species. All samples (fillets) were cooked by a standard procedure in laminated plastic bags to an internal temperature of 70'C (lS8'F). Both summarized compositional data, with means and ranges for each species, and individual sample data including harvest dates and average lengths and weights are presented. When compared with raw samples, cooked samples exhibited an increase in protein content with an accompanying decrease in moisture content. Fat content either remained approximately the same or increased due to moisture loss during cooking. Our results are discussed in reference to compositional data previously published by others on some of the same species. Although additional data are needed to adequately describe the seasonal and geographic variations in the chemical compositions of many of these fish species, the results presented here should be useful to nutritionists, seafood marketers, and consumers.(PDF file contains 28 pages.

Weyl Solutions and jj-selfadjointness for Dirac Operators

We consider a non-selfadjoint Dirac-type differential expression $$(0.1)D(Q)y := J_n {dy \over dx} +Q(x)y$$ with a non-selfadjoint potential matrix $Q$ $\epsilon$ $L$ ^{1}_{loc}(\scr J, \Bbb C^{n \times n}) and a signature matrix $J_n=-J^{-J}_n=-J^*_n$ $\epsilon$ $\Bbb C ^{n \times n}$. Here \scr J denotes either the line $\Bbb R$ or the half-line $\Bbb R_+$. With this differential expression one associates in L^2(\scr J, \Bbb C^n) the (closed) maximal and minimal operators $D_{max}(Q)$ and $D_{min}(Q)$, respectively. One of our main results for the whole line case states that $D_{max}(Q)=D_{min}(Q)$ in $L^2$ $\Bbb R, \Bbb C^n$. Moreover, we show that if the minimal operator $D_{min}(Q)$ in $L^2(\Bbb R, \Bbb C^n)$ is $j$-symmetric with respect to an appropriate involution $j$, then it is $j$-selfadjoint. Similar results are valid in the case of the semiaxis $\Bbb R_+$. In particular, we show that if $n=2p$ and the minimal operator $D^+_{min}(Q)$ in $L^2(\Bbb R_+,\Bbb C^{2p})$ is (\j\)-symmetric, then there exists a $2p \times p-$Weyl-type matrix solution. $\Psi(z,\cdot)$ $\epsilon$ $L^2(\Bbb R_+, \Bbb C^{2p \times p})$ of the equation $D^+_{max}(Q)\Psi(z,\cdot)=z \Psi(z,\cdot)$. A similar result is valid for the expression (0.1) whenever there exists a proper extension $\tilde A$ with dim (dom $\tilde A$/dom $D^+_{min}(Q))=p$ and nonempty resolvent set. In particular, it holds if a potential matrix (\Q\) has a bounded imaginary part. This leads to the existence of a unique Weyl function for the express (0.1). The main results are proven by means of a reduction to the self-adjoint case by using the technique of dual pairs of operators. The differential expression (0.1) is of significance as it appears in the Lax formulation of the vector valued nonlinear Schrödinger equation

The inverse problem for a spectral asymmetry function of the Schrodinger operator on a finite interval

For the Schroedinger equation $−d^2u/dx^2 + q(x)u = λu$ on a finite $x$-interval, there is defined an “asymmetry function” $a(λ; q)$, which is entire of order 1/2 and type 1 in $λ$. Our main result identifies the classes of square-integrable potentials $q(x)$ that possess a common asymmetry function $a(λ)$. For any given $a(λ)$, there is one potential for each Dirichlet spectral sequence