The convergence rate for the strong law of large numbers: General lattice distributions

LetX ₁,X ₂, ... be a sequence of independent random variables with common lattice distribution functionF having zero mean, and let (S _n ) be the random walk of partial sums. The strong law of large numbers (SLLN) implies that for any and >0

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Block renormalization group for Euclidean Fermions

Block renormalization group transformations (RGT) for lattice and continuum Euclidean Fermions in d dimensions are developed using Fermionic integrals with exponential and -function weight functions. For the free field the sequence of actionsD _k generated by the RGT from D, the Dirac operator, are shown to have exponential decay; uniform ink, after rescaling to the unit lattice. It is shown that the two-point functionD ^–1 admits a simple telescopic sum decomposition into fluctuation two-point functions which for the exponential weight RGT have exponential decay. Contrary to RG intuition the sequence of rescaled actions corresponding to the -function RGT do not have uniform exponential decay and we give examples of initial actions in one dimension where this phenomena occurs for the exponenential weight RGT also.     

On the solution of singular linear systems of algebraic equations by semiiterative methods

Summary For a square matrixT ^n,n , where (I–T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀ ⁿ ,x _k T c _k–1+c(k1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M–N (M nonsingular) which leads tox=M ^–1 N x+M ^–1 bT x+c. Even ifx=T x+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues 1 with ||1, or if =1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=T x+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector which can be described in terms of the Drazin inverse of (I–T). We further give conditions under which is a solution or a least squares solution of (I–T)x=c.Research supported in part by the Alexander von Humboldt-Stiftung     

Quantum relativistic action at a distance

A well-known relativistic action at a distance interaction of two unequal masses is altered so as to yield purely Newtonian radial forces with fixed particle rest masses in the system center-of-momentum inertial frame. Although particle masses experience no kinematic mass increase in this frame, speeds are naturally restricted to less than the speed of light. We derive a relation between the center-of-momentum frame total Newtonian energy and the composite rest mass. In a new proper time quantum formalism, we obtain an L²(R⁴ R⁴, C) Hilbert space by varying individual particle rest masses. We propose the use of density operators, recognizing that the auxiliary proper time parameter is not an observable. The quantum formalism is applied to our altered version of the relativistic harmonic oscillator. Our generalized coherent states yield four-dimensional wave packets which follow the correct classical world lines. Appendices contain reviews of classical Hamiltonian reparametrization (incorporating our notion of manifest covariance), and a comparison of this work with the literature.     

Diastereoselective [4+2] annulation of vinyl carbodiimides with N-alkyl imines. Asymmetric synthetic access to the batzelladine alkaloids

A diastereoselective [4+2] annulation of vinyl carbodiimides with chiral N-alkyl imines has been developed to access the stereochemically rich tricyclic core of the batzelladine alkaloids. Its application to the asymmetric synthesis of batzelladine D permitted the use of long-range, directed hydrogenation and stereoselective intramolecular iodoamination as additional key steps to establish the remaining stereocenters within the natural product with excellent stereocontrol.     

Photodynamic Therapy of 9L Gliosarcoma with Liposome-Delivered Photofrin

Abstract— The effect of Photofrin encapsulated in a liposome delivery vehicle for photodynamic therapy (PDT) of the 9L gliosarcoma and normal rat brain was tested. We hypothesized that the liposome vehicle enhances therapeutic efficacy, possibly by increasing tumor tissue concentration of Photofrin. Male Fisher rats bearing a 9L gliosarcoma were treated 16 days after intracerebral tumor implantation with either Photofrin in dextrose (n = 5) or Photofrin in liposome (n = 6). Nontumor-bearing animals were treated with Photofrin delivered either in dextrose (n = 4) or liposome (n = 4) vehicle. Tissue concentrations of Photofrin delivered either in dextrose (n = 4) or liposome (n = 4) vehicle were measured in tumor, brain adjacent to tumor and in normal brain tissue. Photofrin was administered (intraperitoneally) at a dose of 12.5 mg/kg and PDT (17 J/cm₂ of 632 nm light at 100 mW/cm₂) was performed 24 h after Photofrin administration. Brains were removed 24 h after PDT and stained with hematoxylin and eosin for analysis of cellular damage. The PDT using Photofrin in the liposome vehicle caused significantly more damage to the tumor ( P < 0.001) than did PDT with Photofrin in dextrose. The PDT of tumor with Photofrin delivered in liposomes caused a 22% volume of cellular necrosis, while PDT of tumor with Photofrin delivered in dextrose caused only scattered cellular damage. Photofrin concentration in tumors was significantly higher ( P = 0.021) using liposome (33.8 ± 18.9 μg/g) compared to dextrose delivery (5.5 ± 1.5 μg/g). Normal brain was affected similarly in both groups, with only scattered cellular necrosis. Our data suggest that the liposome vehicle enhances the therapeutic efficacy of PDT treatment of 9L tumors.     

Chemical Reactions in the Manufacture of Waveguides for Long Distance Optical Data Transmission

One of the modern high technologies which has advanced enormously in the last few years is glass fiber technology. This is used in the manufacture of glass fibers for lighting purposes and for the optical transfer of analog and digital data with a high transfer density. The technical demands made on the glass fibers required for data transfer, the optical waveguides, are extremely high and are already fulfilled to a large extent by industry. At present about four million kilometers of fiber, worth ca. 800 million DM are produced worldwide (10% in the Federal Republic of Germany). Numerous chemical processes take place during the manufacture of optical waveguides. However, in contrast both to the high and constantly growing demands on the quality and to the increasing production volume of such fibers, little is in fact known about the reactions involved. The present article will attempt to develop a picture of the multifarious reactions occurring in the course of this technical process on the basis of literature data and our own studies.