Poles of Zeta and Eta Functions for Perturbations¶of the Atiyah–Patodi–Singer Problem

The zeta and eta functions of a differential operator of Dirac-type on a compact n-dimensional manifold, provided with a well-posed pseudodifferential boundary condition, have been shown in [G99] to be meromorphic on ℂ with simple or double poles on the real axis. Extending results from [G99] we show how perturbations of the boundary condition of order −J affect the poles; in particular they preserve a possible regularity of zeta at 0 and a possible simple pole of eta at 0 when J≥n. This applies to perturbations of spectral boundary conditions, also when the structure is non-product and the problem is non-selfadjoint. Received: 4 October 1999 / Accepted: 7 July 2000     

A method for the determination of speed-dependent semi-classical vector correlations from sliced image anisotropies

We present analytical expressions relating the bipolar moment β(Q)(K)(k(1)k(2)) parameters of Dixon to the measured anisotropy parameters of different pump/probe geometry sliced ion images. In the semi-classical limit, when there is no significant coherent contribution from multiple excited states to fragment angular momentum polarization, the anisotropy of the images alone is sufficient to extract the β(Q)(K)(k(1)k(2)) parameters with no need to reference relative image intensities. The analysis of sliced images is advantageous since the anisotropy can be directly obtained from the image at any radius without the need for 3D-deconvolution, which is not applicable for most pump/probe geometries. This method is therefore ideally suited for systems which result in a broad distribution of fragment velocities. The bipolar moment parameters are obtained for NO(2) dissociation at 355 nm using these equations, and are compared to the bipolar moment parameters obtained from a proven iterative fitting technique for crushed ion images. Additionally, the utility of these equations in extracting speed-dependent bipolar moments is demonstrated on the recently investigated NO(3) system.     

Regularity in Lp Sobolev spaces of solutions to fractional heat equations

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P:

(*)$Pu+{?}_{t}u=f(x,t),\text{ for }x\in \mathrm{\Omega }?{\mathbb{R}}^n,t\in I?\mathbb{R}.$

1) For strongly elliptic pseudodifferential operators (ψdo's) P on ${\mathbb{R}}^n$ of order $d\in {\mathbb{R}}_{+}$, a symbol calculus on ${\mathbb{R}}^{n+1}$ is introduced that allows showing optimal regularity results, globally over ${\mathbb{R}}^{n+1}$ and locally over $\mathrm{\Omega }\times I$:

$f\in H_{p,\mathrm{loc}}^{(s,s/d)}(\mathrm{\Omega }\times I)?u\in H_{p,\mathrm{loc}}^{(s+d,s/d+1)}(\mathrm{\Omega }\times I),$

for $s\in \mathbb{R}$, $1<p<\infty$. The $H_p^{(s,s/d)}$ are anisotropic Sobolev spaces of Bessel-potential type, and there is a similar result for Besov spaces.2) Let Ω be smooth bounded, and let P equal ${(?\mathrm{\Delta })}^a$ ($0<a<1$), or its generalizations to singular integral operators with regular kernels, generating stable Lévy processes. With the Dirichlet condition $\mathrm{supp}u?\stackrel{￣}{\mathrm{\Omega }}$, the initial condition $u{|}_{t=0}=0$, and $f\in L_p(\mathrm{\Omega }\times I)$, (*) has a unique solution $u\in L_p(I;H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }}))$ with ${?}_{t}u\in L_p(\mathrm{\Omega }\times I)$. Here $H_p^{a(2a)}(\stackrel{￣}{\mathrm{\Omega }})={\dot{H}}_p^{2a}(\stackrel{￣}{\mathrm{\Omega }})$ if $a<1/p$, and is contained in ${\dot{H}}_p^{2a?\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ if $a=1/p$, but contains nontrivial elements from $d^a{\stackrel{￣}{H}}_p^a(\mathrm{\Omega })$ if $a>1/p$ (where $d(x)=\mathrm{dist}(x,?\mathrm{\Omega })$). The interior regularity of u is lifted when f is more smooth.     

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

In this paper, we combine results on extensions of operators with recent results on the relation between the M ‐function and the spectrum, to examine the spectral behaviour of boundary value problems. M ‐functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M ‐function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On strang’s fifth bound for eigenvalues of the jordan product of two self-adjoint operators on a finite dimensional unitary space

An almost coordinate-free proof of the algebraic version of the Mohr circles theorem is given. An alternative and somewhat simplified version of the expression for the famous fifth bound of Strang for the Jordan product of two self-adjoint operators on a finite-dimensional unitary space is found and it is proved that the associated extremum is attained in a two dimensional subspace. Details of the calculations leading to the extreme values are given.     

Reduction of the fundamental initial-boundary-value problems for Stokes equations to initial-boundary-value problems for parabolic systems of pseudodifferential equations

It is shown that an initial-boundary-value problem for Stokes' system, in which on the boundary one prescribes the vector field of velocities , or the stress field, or the normal component of the velocity and the tangential stresses, reduces to an initial-boundary-value problem for a system of the form , where the operator A contains a nonlocal term (the so-called singular Green operator). For the solutions of these problems, coercive estimates in the spaces W₂ ^{l, l/2} and also estimates of the norm of the resolving operator in W₂ ^r are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 37–48, 1987.     

Adsorption and surface dynamics of short DNA and LNA oligonucleotides on single-crystal Au(1 1 1) electrode surfaces

We have studied the surface dynamics of a double-strand decanucleotide (HS-10AT^L) with 10 adenine-thymine base pairs linked to a Au(1 1 1)-electrode surface via a hexamethylene thiol linker. The study is based on a combination of voltammetry, interfacial capacitance data, electrochemical in situ scanning tunnelling microscopy, and X-ray photoelectron spectroscopy. The thymine bases of the oligonucleotide are connected to furanoses locked in a C3′-endo configuration called LNA (locked nucleic acid). Hybridization in solution is effected prior to linking to the Au(1 1 1)-surface. The melting point of the linker-free locked decanucleotide, 10AT^L is >63 °C. However, voltammetric reductive desorption of the adsorbed thiol-modified double-strand decanucleotide, HS-10AT^L, gives almost the same charge as single-strand HS-10A, 29 ± 3 and 27 ± 5 μC cm⁻², respectively. In situ STM after HS-10AT^L-immobilization also gives images showing highly ordered domains, virtually indistinguishable from those of immobilized HS-10A. X-ray photoelectron spectroscopy gives an N/P ration of 5.0 for HS-10AT^L in line with the expected value for single-strand HS-10A (5.0).All three sets of data suggest that HS-10AT^L hybridized in solution is significantly dissociated on binding to the Au(1 1 1)-electrode surface. This points to an adsorption mechanism in which a stable high density of Au-S bonds is achieved but at the expense of significant unzipping of the more voluminous duplex form.     

Deformation in lamellar and crystalline structures: in situ simultaneous small‐angle X‐ray scattering and wide‐angle X‐ray diffraction measurements on polyethylene terephthalate fibers

Changes in the lamellar and crystalline structures were followed as a function of applied stress to understand the influence of the interactions between the crystalline and amorphous domains on the fiber properties. We observed a reversible transformation from a structure giving a four‐point small‐angle pattern to a structure giving a two‐point pattern; these structures corresponded to the lamellae with oblique and normal lamellar surfaces, respectively. The characteristics of these two structures such as the stack diameter, stack height, and tilt angle were different and were determined by the processing conditions and did not change when the fiber was elastically deformed. The structure giving a two‐point pattern was probably the load‐carrying lamellar entity in these fibers. The diameter of the lamellar stacks, tilt angle of the lamellae, and the strain in the lamellar spacing appeared to have the most influence on properties such as tenacity and dimensional stability. The long‐spacing strain, which is about the same as the fiber strain, determined the modulus at low elongation as well as ultimate elongation. These indicate that the lamellar stacks have at least as much influence as the interfibrillar chains on fiber properties. Structural features that determine the performance in semicrystalline polymers were investigated by analyzing four generations of polyethylene terephthalate fibers. Some of the fiber properties correlate with changes in the crystalline domains such as the crystalline orientation, size, and unit cell dimensions. Fibers in which the crystalline strain was large because of their strong linkages to the amorphous chains, and better load transfer, had the highest modulus and lowest ultimate elongation. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1538–1553, 2003     

Lamellar morphology of polyethylene: Electron microscopy of a melt-crystallized sharp fraction

The morphology of linear polyethylene of M_w/M_n 1.19 crystallized isothermally from the melt has been investigated. Electron microscopy (EM) of stained thin sections gave exceptionally well-defined images. There were no qualitative differences between Regime I and Regime II crystallization, but the lamellae grown at lower supercooling were thicker, larger, and had fewer spiral growths per unit area of fold surface. Small-angle x-ray scattering indicated a much sharper distribution of lamellar thicknesses than EM and Raman LAM did. This is evidence of the complementary nature of the techniques and not a contradiction. EM also revealed nonrepresentative features in detail. There were extremely thin lamellae formed on cooling, indicating molecular weight segregation even in this fractionated material, and there were extremely thick lamellae, attributed to isothermal thickening. As the representative lamellae are also supposed to have thickened isothermally, this raises the issue of when and how the lamellae which are the primary products of crystallization can be identified.