Convergence Radii for Eigenvalues of Tri--diagonal Matrices

Consider a family of infinite tri--diagonal matrices of the form $L+ zB,$ where the matrix $L$ is diagonal with entries $L_{kk}= k^2,$ and the matrix $B$ is off--diagonal, with nonzero entries $B_{k,{k+1}}=B_{{k+1},k}= k^\alpha, 0 \leq \alpha < 2.$ The spectrum of $L+ zB$ is discrete. For small $|z|$ the $n$-th eigenvalue $E_n (z), E_n (0) = n^2,$ is a well--defined analytic function. Let $R_n$ be the convergence radius of its Taylor's series about $z= 0.$ It is proved that $$ R_n \leq C(\alpha) n^{2-\alpha} \quad \text{if} 0 \leq \alpha <11/6.$

Gold-Catalyzed Diastereoselective Cycloisomerization of Alkylidene-Cyclopropane-Bearing 1,6-Diynes

An unprecedented gold-catalyzed diastereoselective cycloisomerization of 1,6-diynes bearing an alkylidene cyclopropane moiety has been developed. This methodology enables rapid access to a variety of 1,2-trimethylenenorbornanes, important building blocks in the preparations of abiotic and sesquiterpene core structures

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schroedinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive the similar self-adjoint operator and also find the associated "charge conjugation" operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.Comment: 27 page

Versatile pulse programmer for pulsed nuclear magnetic resonance spectroscopy

A description of the sequence of events and the decisions leading to the design of a versatile pulse programmer for pulsed NMR are presented. Background and application information is discussed in order that the reader might better understand the role of the pulse programmer in a NMR spectrometer. Various other design approaches are presented as a basis for comparison. Specifications for this design are proposed, the hardware implementation of the specifications is discussed, and the software operating system is presented