Picosecond third-harmonic light generation inβ-BaB2O4

The type-II phase-matched third-harmonic light generation in a -BaB₂O₄ crystal is studied experimentally. A passively mode-locked Nd: phosphate glass laser is used as a pump source. At a pump pulse peak intensity ofI ₁₀=5×10¹⁰W/cm² a third-harmonic conversion efficiency of a percent is obtained. A theoretical discussion of phase-matched third-harmonic generation in crystals of the symmetry group of -BaB₂O₄ (trigonal class 3) is given. The effective nonlinear susceptibility _eff for type-II phase-matching is determined.On leave from the Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, Shanghai, P.R. China     

An extremal problem related to probability

Summary We consider a walk from a stateA ₁ to a stateA _n+1 in which the probability of remaining atA _i isp _i , and the probability of progressing fromA _i toA _i+1 is 1 –p _i . The probabilityW _nk of reachingA _n+1 fromA ₁ in exactlyn + k steps can then be expressed as a polynomial of degreen + k in then variablesp ₁,,p _n . We determine the maximum value ofW _nk and the (unique) choice (p ₁,,p _n ) for which this extremum occurs.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Why is space three-dimensional and how may groups be seen?

First we propose a model of visual perception essentially based on the Keldysh-Chernavsky-Sossinsky three-channel theorem, from which three-dimensionality of space follows. Second, we associate with a system of subgroups H ₁, ..., H_s of a given group G a geometric object, called a group crystal, in order to visualize G. How this notion works is illustrated via the Burnside problem.     

Synthesis of first nonannulated 1,2,3,4-tetrazine 1,3-dioxides

Thermolysis of o-diazidobenzotetrazine 1,3-dioxides was accompanied by cleavage of the C-C bond of the benzene ring to give nonannulated 1,2,3,4-tetrazine 1,3-dioxides. The structures of these first representatives of nonfused 1,2,3,4-tetrazines were confirmed by ¹³C and ¹⁴N NMR spectroscopy. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1589–1594, September, 2006.     

Cyclizations via Palladium-Catalyzed Allylic Alkylations [New Synthetic Methods (79)]

The history of ring systems in organic chemistry parallels their synthetic accessibility. Transition-metal-catalyzed cyclizations offer a new opportunity to create carbo- and heterocyclic compounds with great facility. Among these methods, allylic alkylations catalyzed by palladium have proven unusually productive because of the extraordinary chemo-, regio-, and diastereoselectivity and the continuing possibility for the development of enantioselectivity. The rules for ring closure differ from those for non-transition-metal-catalyzed reactions. A major benefit is the ability to generate medium (eight-, nine-, ten-, and eleven-membered) and large rings in preference to normal (five-, six- and seven-membered) rings. With the appropriate substrate, efficient macrocyclizations are possible under conditions of normal concentrations. A second major benefit derives from the complementary stereochemistry of the metalcatalyzed substitution (net retention of configuration) compared to non-metal-catalyzed reactions (inversion of configuration). Further, the requirement for the substrate to conform to the transition-metal template may impose a stereochemical preference in the intermediate that ultimately translates into the thermodynamically less stable organic product regardless of the stereochemistry of the starting material. While more work has focused on carbocyclic synthesis, the possibilities for heterocyclic synthesis are just beginning to be tapped. In addition to forming heterocycles by C? C bond formation, use of a heteroatom as a nucleophile has already proven effective for oxygen and nitrogen, with other nucleophiles awaiting investigation. New dimensions for cyclization via allylic alkylation arise by generating the requisite π-allylpalladium intermediates by methods other than palladium(0)-initiated allylic ionizations. In addition, metals other than palladium will clearly expand the possibilities, but as yet remain untapped.     

A new general-purpose isothermal microcalorimeter for use at temperatures up to 200 °C

The design and some properties of a new general-purpose isothermal microcalorimeter are reported. The instrument is a twin thermopile heat conduction calorimeter, which is designed for use up to 200 °C. The calorimetric units and surrounding heat sink are suspended inside a hollow aluminium construction, which is thermostated. Above that unit a second thermostated block is positioned and the whole assembly is suspended inside a Dewar vessel. When the instrument is used at room temperature and below, the thermostated units are cooled by use of an insertion Peltier effect cooler. The instrument can be used with a wide range of different reaction vessels (diameter 14 mm). Baseline experiments have been conducted in the temperature range 15-200 °C. Typical values obtained during 10 h periods at 200 °C are ±3 and ±10 nW for the baseline drift and baseline fluctuations, respectively. The heat detection limit, determined by release of electrical energy, is about 2 μJ. Preliminary stability measurements have been conducted at 100 °C on samples of stabilised and non-stabilised polyamide film.     

Local structure of graphs with λ=μ=2,a 2=4

Several properties of graphs with ==2,a ₂=4 are studied. It is proved that such graphs are locally unions of triangles, hexagons or heptagons. As a consequence, a distance regular graph with intersection array (13,10,7;1,2,7) does not exist.     

Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes

Summary We introduce a new Skorohod topology for functions of several variables. Since ann-variable function may be viewed as a one-variable function with values in the set of (n–1)-variable functions, this topology is defined by induction from the classical Skorohod topology for one-variable functions. This allows us to define the notion of completen-parameter symmetric Markov processes: Such processes are, for any 1pn, rawp-parameter Markov processes (in the sense of our previous paper [17]) with values in the space of (n–p)-variable functions. We prove, for these processes and their Bochner subordinates, a maximal inequality which implies the continuity of additive functionals associated with finite energy measures. We finally present several important examples.     

A Banach space where minimal weak cuscos are generically compact-valued

The space of continuous functions on the double arrow space has long been of interest in differentiability theory since many convex functions on this space are densely but not generically Gâteaux differentiable. We show that this space has the property that minimal weak* cuscos into its dual take compact values at the points of a denseG set.     

Cosmological principle and primordial spinor fields

An exact solution of Dirac's equation in an open Robertson-Walker universe is constructed such that the corresponding energy-momentum densityT _µv obeys the cosmological principle. However, in anopen universe, this principlemust always be violated by the other physical densities of the cosmological solution (e.g. pseudoscalar, (axial-) current and polarization). If the current densityj _µ = isrequired to obey the cosmological principle, then the universe must beflat.