Trigonometric polynomials with many real zeros

This is a contribution to the theory of “incomplete trigonometric polynomials”T _n , but mainly for the case when their zeros are not concentrated at just one point, but are distributed in some intervalI whose length is not too large. We begin with the simple theorem that if ∥T _n ∥ ≤ 1 and ifT _n has ≥θn, 0<θ< 2, zeros at 0, thenT _n (t) must be small on the interval |t|<2 arcsin (θ/2). There are similar but more complicated and more difficult to prove results whenT _n has ≥θn zeros onI. These results have the following application: IfT _n →f a.e., and if ∥T _n >∥_∞<-1, thenf vanishes on a set of the circleT whose measure is controlled by lim sup (N _n /n), whereN _n is the number of zeros ofT _n onT. In turn, this has further applications to series of polynomials, to norms of Lagrange operators, and to Hardy classes.     

The cyclodimerisation of 3-methyl-2-butenenitrile

3-Methyl-2-butenenitrile (1) cyclodimerised on treatment with lithium diisopropylamide in dimethoxyethane at temperatures between ?78°C and 0°C to 3-amino-4-cyano-1,5,5-trimethyl-1,3-cyclohexadiene (2) the structure of which was established by acid hydrolysis to the known 4-cyano-1,5,5-trimethyl-1-cyclohexene-3-one (3).     

An O(n log n) algorithm for finding all repetitions in a string

Any nonempty string of the form xx is called a repetition. An O(n log n) algorithm is presented to find all repetitions in a string of lenght n. The algorithm is based on a linear algorithm to find all the new repetitions formed when two strings are concatenated. This linear algorithm is possible because new repetitions of equal length must occur in blocks with consecutive starting positions. The linear algorithm uses a variation of the Knuth-Morris-Pratt algorithm to find all partial occurrences of a pattern within a text string. It is also shown that no algorithm based on comparisons of symbols can improve O(n log n). Finally, some open problems and applications are suggested.     

Excitation functions of the analyzing power in elastic proton-proton scattering from 0.45 to 2.5 GeV

Excitation functions A_N(p_lab,_c.m.) of the analyzing power in elastic proton-proton scattering have been measured in an internal target experiment at the Cooler Synchrotron COSY with an unpolarized proton beam and a polarized atomic hydrogen target. Data were taken continuously during the acceleration and deceleration for proton kinetic energies T_lab (momenta p_lab) between 0.45 and 2.5 GeV (1.0 and 3.3 GeV/c) and scattering angles 30 ^° _c.m. 90^°. The results provide excitation functions and angular distributions of high precision and internal consistency. The data can be used as calibration standard between 0.45 and 2.5 GeV. They have significant impact on phase shift solutions, in particular on the spin triplet phase shifts between 1.0 and 1.8 GeV.     

Measurement of the analyzing power in p-->d-->(p p)n with a fast forward 1S0 proton pair

A measurement of the analyzing power A(y) of the p-->d--> (p p) + n reaction was carried out at the ANKE spectrometer at COSY at beam energies of 0.5 and 0.8 GeV by detection of a fast forward proton pair of small excitation energy E(pp) < 3 MeV. The S-wave dominance in the fast diproton is experimentally demonstrated in this reaction. While at T(p) = 0.8 GeV the measured analyzing power almost vanishes, it rises to nearly unity at T(p) = 0.5 GeV for neutrons emitted at theta(c.m.)(n) = 167 degrees. The results are compared with a model taking into account one-nucleon exchange, single scattering, and Delta(1232) excitation in the intermediate state. The model describes fairly well the unpolarized cross section obtained earlier and the analyzing power at 0.8 GeV; it fails to reproduce A(y) at 0.5 GeV.     

Über Limitierungsverfahren,Die Von Einem Stieltjes-Integral Abhängen.

Ohne Zusammenfassung     

Conical parametric scattering of a single extraordinary beam in BaTiO3 crystal

In this paper, the parametric scattering of a single extraordinary polarized beam of laser in BaTiO₃ photorefractive crystal has been investigated experimentally and theoretically. The resulting pattern consists of beam fanning, isotropic ring, and anisotropic one. Among all parts of scattering pattern, isotropic ring has not been studied as much as beam fanning and anisotropic ring, and there still are some differences in reports about it. Therefore, the study has mainly focused on this part. In this experimental configuration, isotropic ring is just visible in positive angles although the other parts of parametric scattering pattern can be visible from behind and in front of the crystal. In addition to steady state pattern in forward and backward directions, its transient behavior with the rotation of crystal has been studied. The results of experiments have been analyzed carefully, and their theoretical explanations have been presented based on the standard theory of parametric scattering in photorefractive crystals. It has been shown that this configuration corresponds to the so called parametric B-process scattering.     

Bivariate Gonarov polynomials and integer sequences

Univariate Gonarov polynomials arose from the Gonarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Gonarov polynomials,which form a basis of solutions for multivariate Gonarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Gonarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Gonarov polynomials.     

Modeling of Thin-Layer Chromatographic Separation of Androstane Isomers

JPC – Journal of Planar Chromatography – Modern TLC - Modeling of thin-layer chromatographic separation of androstane isomers to find the optimum mobile phase is described in this...