tert-Butylphosphonic acid: from the bulk to the gas phase

The structure of tert-butylphosphonic acid in the solid, in solution, and in the gas phase was studied by single-crystal X-ray diffraction, (1)H and (31)P NMR spectroscopic studies in solution, solid-state (31)P NMR spectroscopy, and electrospray ionization mass spectrometry. In addition, density functional theory (DFT) calculations at the B3LYP/6-31G*, B3LYP/6-31+G*, and B3LYP/6-311+G* level of theory for a large number of H-bonded aggregates of the type (tBuPO(3)H(2))(n) (C(n), P(n); n=1-7) support the experimental work. Crystallization of tBuPO(3)H(2) from polar solvents such as CH(3)CN or THF gives the H-bonded one-dimensional polymer 2, whereas crystallization from the less polar solvent CDCl(3) favors the formation of the H-bonded cluster (tBuPO(3)H(2))(6).CDCl(3) (1). In CDCl(3) the hexamer (tBuPO(3)H(2))(6) (C(6)) is replaced by smaller aggregates down to the monomer with decreasing concentration. DFT calculations and natural bond orbital (NBO) analyses for the clusters C(1)-C(7) and the linear arrays P(1)-P(7) reveal the hexamer C(6) to be the energetically favored structure resulting from cooperative strengthening of the hydrogen bonds in the H-bonded framework. However, the average hydrogen bond strengths calculated for C(6) and P(2) do not differ significantly (42-43 kJ mol(-1)). The average distances r(O.O), r(Obond;H), r(Pdbond;O), and r(Pbond;OH) in C(1)-C(7) and P(1)-P(7) are closely related to the hydrogen bond strength. Electrospray ionization mass spectrometry shows the presence of different anionic species of the type [(tBuPO(3)H(2))(n)-H](-) (A(1)-A(7), n=1-7) depending on the instrumental conditions. DFT calculations at the B3LYP/6-31G* level of theory were carried out for A(1)-A(6). We suggest the dimer [(tBuPO(3)H(2))(2)-H](-) (A(2)) and the trimer [(tBuPO(3)H(2))(3)-H](-) (A(3)) are the energetically favored anionic structures. A hydrogen bond energy of approximately 83 kJ mol(-1) was calculated for A(2). Electrospray ionization mass spectrometry is not suitable to study the assembling process of neutral H-bonded tert-butylphosphonic acid since the removal of a proton from the neutral aggregates has a large influence on the hydrogen bond strength and the cluster structure.     

<Emphasis Type="Italic">P</Emphasis>≠<Emphasis Type="Italic">NP</Emphasis> for Infinite Time Turing Machines

We state different versions of the P=?NP problem for infinite time Turing machines. It is observed that PNP collapses to the fact that there are analytic sets which are not Borel.The author thanks J. D. Hamkins, R. Schipperus, and P. Welch for comments on earlier versions of this paper. He thanks Sanders Seliverstov for pointing out an inaccuracy in the first version of the proof of Theorem 2.10.Received April 19, 2002; in revised form June 10, 2002 Published online May 16, 2003     

Product-trace-rings and a question of G. S. Garfinkel

It is an open question as to whether every left coherent ring satisfying the intersection property for finitely generated left ideals of is a right-product-trace-ring or not. is a right-product-trace-ring iff every product of trace-right--modules (= universally torsionless-right--modules) is a trace-right--module. This question is shown to have a negative answer. Furthermore, looking at all valuation domains, the complete product-trace-rings, the product-trace-rings and the product-content-rings are characterized.

     

Contents of the Supplement S1 to Vol. 50 (2000)

Volume Contents

Contents of the Supplement S1 to Vol. 50 (2000)     

Theoretical solution versus industry standard: Optimal leverage function for CPDOs

In this paper, we derive an optimal leverage function for Constant Proportion Debt Obligations (CPDOs) by using stochastic control techniques. The investor’s goal is to maximise redemption of capital at maturity. The control variable of the problem is the leverage process, i.e. the time dependent notional exposure to the underlying risky index/portfolio. The control problem is solved explicitly with the help of the Legendre transform applied to the HJB equation of stochastic control. A closed form solution is given for the optimal leverage. Contrary to the industry practise, the optimal leverage derived in this paper is a non-linear, bell-shaped function of the CPDO assets value.     

Cogenerators for convex spaces

We give cogenerators for the categories of convex (= finitely superconvex), finitely positively convex, and absolute convex (= finitely totally convex) spaces introduced by Pumplün and Röhrl.Dedicated to our academic teacher Dieter Pumplün on the occasion of his sixtieth birthday.     

Subpicosecond hard x-ray streak camera using single-photon counting

We have developed and characterized a hard x-ray accumulating streak camera that achieves subpicosecond time resolution by using single-photon counting. A high repetition rate of 2 kHz was achieved by use of a readout camera with built-in image processing capabilities. The effects of sweep jitter were removed by using a UV timing reference. The use of single-photon counting allows the camera to reach a high quantum efficiency by not limiting the divergence of the photoelectrons.