Refinement of the crystal structure of (RS)-carnitine hydrochloride

A single crystal of (RS)-carnitine hydrochloride has been obtained, and its crystal structure has been refined by an x-ray structural experiment.Tashkent Pharmaceutical Institute. Translated from Khimiya Prirodnykh Soedinenii, No. 6, pp. 842–844, November–December, 1991.     

Distribution of eigenvalues for the modular group

The two-point correlation functions of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy-Littlewood method. It is shown that in the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but they have prominent number-theoretical oscillations at larger scale. The results agree well with numerical simulations.     

The Dyson instability and asymptotics of the perturbation series in QED

The probability of the ground state decay at e² < 0 is calculated by the steepest descent method in fermionic electrodynamics. The saddle points are the solution of some equations which have been obtained by calculating the asymptotic of the Dirac operator determinant in a very strong external field. The SO(3) × O(2) solutions are found explicitly. The main contribution from them to the Nth coefficient of perturbation theory is proportional to $\text{(?α)}{}^{\mathrm{<mtext>N</mtext><mtext>2</mtext>}}\text{Γ(}\text{N}\text{2}\text{)}$, where $\text{S = 2}{}^2\text{3}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}\text{π}{}^3$.     

Structure of wave functions of pseudointegrable billiards

Wave functions of pseudointegrable plane polygonal billiards are investigated. It is demonstrated that they have clear structures (superscars) related with families of classical periodic orbits which do not disappear at large energy.     

Periodic Orbits Contribution to the 2-Point Correlation Form Factor for Pseudo-Integrable Systems

Using heuristic arguments based on the trace formulas we relate the 2-point correlation form factor, K ₂(τ), at small values of τ with sums over classical periodic orbits for typical examples of pseudo-integrable systems. The later sums have been explicitly calculated for the following models: (i) plane billiards in the form of right triangles with one angle π/n and (ii) rectangular billiards with the Aharonov-Bohm flux line. In the first model, using the properties of the Veech structure, it is shown that K ₂(0)=(n+ε(n))/(3(n−2)), where ε(n)= 0 for odd n, ε(n)= 2 for even n not divisible by 3, and ε(n)=6 for even n divisible by 3. For completeness we also recall informally the main features of the Veech construction. In the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle. When these ratios are non-commensurable irrational numbers, K ₂(0)=1−3 , where is the fractional part of the flux through the rectangle when and it is symmetric with respect to the line when . The comparison of these results with numerical calculations of the form factor is discussed in detail. The above values of K ₂(0) differ from all known examples of spectral statistics, thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo-integrable systems. Received: 10 January 2000 / Accepted: 18 May 2001     

Refinement of the crystal structure of (RS)-carnitine hydrochloride

A single crystal of (RS)-carnitine hydrochloride has been obtained, and its crystal structure has been refined by an x-ray structural experiment.