Distant measurement

Distant correlations are investigated within the framework of quantum mechanics. They are inherent to any physical situation in which two separated quantal systems are described by one composite state vector. Owing to correlations of this kind one can perform a measurement on one of the systems, thereby measuring a certain observable on the other (distant) system without interacting with it. Necessary and sufficient conditions are given for such a distant measurement to take place. It is found which are the observables that can be measured distantly, and which are the states of the distant system obtainable in this way. Solution of these problems is achieved by replacing the composite state vector by two entities equivalent to it: the reduced statistical operator of the system which is directly measured and a correlation operator. The latter gives a connection between states, observables, and probabilities of the two systems. Experimental evidence for distant measurement is discussed.     

Invariant Banach limits and applications

Let ?_∞ be the space of all bounded sequences x=(x₁,x₂,…) with the norm

     

Integrability of higher pentagram maps

We define higher pentagram maps on polygons in $\mathbb{P }^d$ for any dimension $d$ , which extend R. Schwartz’s definition of the 2D pentagram map. We prove their integrability by presenting Lax representations with a spectral parameter for scale invariant maps. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$ -equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. We also study in detail the 3D case, where we prove integrability for both closed and twisted polygons and describe the spectral curve, first integrals, the corresponding tori and the motion along them, as well as an invariant symplectic structure.     

Ordinary reduction of K3 surfaces

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.      

Intramolecular [4+2] cycloaddition of furfurylsubstituted homoallylamines to allylhalides,acryloyl chloride and maleic anhydride

Acylation of 4‐(furyl‐2)‐4‐R‐aminobut‐1‐enes and 4‐R‐4‐furfurylaminobut‐1‐enes with maleic anhydride, acryloyl chloride or allylhalides provided 3‐aza‐10‐oxatricyclo[5.2.1.0^1,5]decenes. The tricycles are formed via an initial amide formation followed by a stereoselective exo‐IMDAF (Intramolecular Diels‐Alder of Furan). In case of competing cycloaddition (for compounds possessing two furan or two dienophilic moieties) the most substituted fivemembered cycle is preferably annulated. Refluxing of 4‐R‐4‐furfurylaminobut‐1‐enes in acetic anhydride led to exo‐3‐aza‐11‐oxatricyclo[6.2.1.0^1,6]undecenes with the pseudoequatorial substituent R‐4. Treatment of 3‐aza‐10‐oxatricyclo[5.2.1.0^1,5]decenes with PPA at 90?110°C promoted cyclic ether opening, aromatization and intramolecular cyclization reactions sequence to give the corresponding tetracyclic compounds — tetrahydroisoindolo[2,1‐a]quinolines and tetrahydroisoindolo[2,1‐b][2]benzazepines in good yields. Unusual products of ipso‐substitution in aromatic ring were obtained on cyclization of N‐p‐R‐substituted 2‐allyl‐4‐oxo‐3‐aza‐10‐oxatricyclo[5.2.1.0^1,5]dec‐8‐enes.     

Refractive-index dispersion of glassy semiconductors in the pseudo-binary As2Se3–SbSI system

The bulk chalcogenide glasses in the pseudo-binary system (As₂Se₃)_100?x(SbSI)_x (with x = 20, 30, 50, 70 and 80 at.%) were prepared from high-purity elemental components by melt-quenching technique. It is a system with the variable ratio of classical amorphous compound As₂Se₃ and the molecule of antimony-sulphoiodide, SbSI, which in the monocrystal form is characterized as a ferroelectric. The refractive-index behaviour (magnitude and spectral dispersion) of investigated glasses was determined by the prism method and analyzed. To calculate and discuss the parameters of dispersion in the band gap region three different approaches were used (Cauchy, Sellmeier and Wemple–DiDomenico single-oscillator model). The comparison of the results shows good agreement between all applied models. It was found that the value of the refractive-index shows normal dispersion behaviour and increases with the increase of Sb (or SbSI) content in the investigated system, whereas the optical gap of these glasses E_g slightly decreases.