Phosphaalkynes as ligands in organometallic chemistry. Syntheses and 31p NMR spectra of platinum(II), palladium(II) and rhodium(I) complexes of dη5-cyclopentadienyltetracarbonyl-μ-(3,3-dimethyl-1-phosphabutyne)dimolybdenum, [Mo2(CO)4(η-c5H5)2tBuCP)]

Syntheses of the complexes trans-[PtCl₂(PR₃)Mo₂(CO)₄(η⁵-C₅H₅)₂(^tBuCP)], (PR₃=PEt₃, PPr₃, PBu₃, PPh₂Me, PPhMe₂) trans-[PdCl₂(PBu₃)Mo₂(CO)₄(η⁵-C₅H₅)₂(^tBuCP)], and trans[RhCl{(PF₂NMe)₂CO}Mo₂(CO)₄(η⁵-C₅H₅)₂(^tBuCP)] are described and their ³¹P NMR spectra presented and discussed.     

Activation of sepharose withN,N′-disuccinimidyl carbonate

A method is described for the activation of Sepharose withN,N′-disuccimidyl carbonate. The activated carbonate reacts smoothly with amine-containing ligands yielding stable carbamate derivatives.     

The Complete Differential Game of Active Target Defense

In the Target–Attacker–Defender differential game, an Attacker missile strives to capture a Target aircraft. The Target tries to escape the Attacker and is aided by a Defender missile which aims at intercepting the Attacker, before the latter manages to close in on the Target. The conflict between these intelligent adversaries is naturally modeled as a zero-sum differential game. The Game of Degree when the Attacker is able to win the Target–Attacker–Defender differential game has not been fully solved, and it is addressed in this paper. Previous attempts at designing the players’ strategies have not been proven to be optimal in the differential game sense. In this paper, the optimal strategies of the Game of Degree in the Attacker’s winning region of the state space are synthesized. Also, the value function is obtained, and it is shown that it is continuously differentiable, and it is the solution of the Hamilton–Jacobi–Isaacs equation. The obtained state feedback strategies are compared to recent results addressing this differential game. It is shown that the correct solution of the Target–Attacker–Defender differential game that provides a semipermeable Barrier surface is synthesized and verified in this paper.

     

The Maximal Number of Geometric Permutations for n Disjoint Translates of a Convex Set in ℝ Is Ω(n)

A geometric permutation induced by a transversal line of a finite family of disjoint convex sets in ℝ^d is the order in which the transversal meets the members of the family. It is known that the maximal number of geometric permutations in families of n disjoint translates of a convex set in ℝ³ is 3. We prove that for d ≥ 3 the maximal number of geometric permutations for such families in ℝ^d is Ω(n).     

A quasistatic contact problem with slip‐dependent coefficient of friction

We consider a mathematical model which describes the bilateral quasistatic contact of a viscoelastic body with a rigid obstacle. The contact is modelled with a modified version of Coulomb's law of dry friction and, moreover, the coefficient of friction is assumed to depend either on the total slip or on the current slip. In the first case, the problem depends upon contact history. We present the classical formulations of the problems, the variational formulations and establish the existence and uniqueness of a weak solution to each of them, when the coefficient of friction is sufficiently small. The proofs are based on classical results for elliptic variational inequalities and fixed point arguments. We also study the dependence of the solutions on the perturbations of the friction coefficient and obtain a uniform convergence result. Copyright © 1999 John Wiley & Sons, Ltd.     

On Polynomially Normal Lattice Configurations

In this paper we extend Champernowne’s construction of a normal sequence in base b to the case and obtain an explicit construction of the generic point of the shift transformation of the set . We prove that the intersection of the constructed configuration with an arbitrary polynomial curve in the plane is a normal sequence in base b.     

A construction of a normal number for the continued fraction transformation

It is shown that if the continued fractions of the rationals $\text{1}\text{2}\text{,}\text{1}\text{3}\text{,}\text{2}\text{3}\text{,}\text{1}\text{4}\text{,}\text{2}\text{4}\text{,}\text{3}\text{4}\text{,}\text{1}\text{5}\text{,}\text{2}\text{5}\text{,}\text{3}\text{5}\text{,}\text{4}\text{5}\text{,…}$ are concatenated, a normal continued fraction is obtained.     

Three-dimensional optical transfer of rotating beams

Systems in which the point spread function (PSF) is a rotating beam have increasing use in three-dimensional (3D) microscopy and depth estimation. We analyze in several ways the 3D optical transfer function (OTF) of Gauss Laguerre modes and rotating beams. This is based on analysis of 3D OTFs of general aperture functions. Consequently, we suggest a criterion for depth resolution based on an effective cutoff of the axial frequency response. This criterion can be used to optimize PSFs explicitly and directly, to maximize axial resolution.     

Indistinguishable photon pairs from independent true chaotic sources

Indistinguishability of events in quantum mechanics is manifested by interference between their probability amplitudes. We report a unique kind of interference occurring between indistinguishable events of photon-pair emission, where each photon of the pair is emitted from a distinct true chaotic light source and has a different energy. The indistinguishability results in an interference which is observed as an ultrafast modulation of the second-order coherence function, measured on a femtosecond time scale by two-photon absorption in a semiconductor photomultiplier tube.     

Unifying model for several classes of two-dimensional phase transition

A relatively simple and physically transparent model based on quantum percolation and dephasing is employed to construct a global phase diagram which encodes and unifies the critical physics of the quantum Hall, "two-dimensional metal-insulator," classical percolation and, to some extent, superconductor-insulator transitions. Using real-space renormalization group techniques, crossover functions between critical points are calculated. The critical behavior around each fixed point is analyzed and some experimentally relevant puzzles are addressed.