Measurement of the neutron electric dipole moment by crystal diffraction

An experiment using a prototype setup to search for the neutron electric dipole moment by measuring spin-rotation in a non-centrosymmetric crystal (quartz) was carried out to investigate statistical sensitivity and systematic effects of the method. It has been demonstrated that the concept of the method works. The preliminary result of the experiment is $d_{\rm n}=(2.5\pm 6.5)\cdot 10^{-24}$ e$\cdot$cm. The experiment showed that an accuracy of $\sim 2.5\cdot 10^{-26}$ e$\cdot$cm can be obtained in 100 days data taking, using available quartz crystals and neutron beams.Comment: 13 pages, 4 figure

Algebraic Closed Geodesics on a Triaxial Ellipsoid

We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid $Q$ that are cut out by algebraic surfaces in ${\mathbb R}^3$. Such geodesics are either connected components of spatial elliptic curves or rational curves. Our approach is based on elements of the Weierstrass--Poncar\'e reduction theory for hyperelliptic tangential covers of elliptic curves and the addition law for elliptic functions. For the case of 3-fold and 4-fold coverings, explicit formulas for the cutting algebraic surfaces are provided and some properties of the corresponding geodesics are discussed.Comment: 15 figure

Biphoton ququarts as either pure or mixed states, features and reconstruction from coincidence measurements

Features of biphoton polarization-frequency ququarts are considered. Their wave functions are defined as functions of both polarization and frequency variables of photons with the symmetry obligatory for two-boson states taken into account. In experiments, biphoton ququarts can display different features in dependence on whether experiments involve purely polarization or (alternatively) polarization-frequency measurements. If in experiments one uses only polarization measurements, the originally pure states of ququarts can be seen as mixed biphoton polarization states. Features of such states are described and discussed in details. Schemes of coincidence measurements for reconstruction of the ququart's parameters are suggested and described

On the Grothendieck-Serre Conjecture about principal bundles and its generalizations

Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W\times_kU$ is trivial if it is trivial over the generic fiber of the projection $W\times_kU\to U$. We also simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is trivial if it is trivial over the generic point of $U$. We generalize some other related results from the simple simply-connected case to the case of arbitrary reductive group schemes.Comment: Final version to be published in the Journal of Algebra and Number Theory. We slightly change the isotropy condition as well as the terminology: see Definition 1.1 of strongly locally isotropic reductive groups. We remove the assumption that U is geometrically regular in Theorem 1 (regular is enough). Other minor improvement

Resonances in three-body systems with short and long-range interactions

The complex scaling method permits calculations of few-body resonances with the correct asymptotic behaviour using a simple box boundary condition at a sufficiently large distance. This is also valid for systems involving more than one charged particle. We first apply the method on two-body systems. Three-body systems are then investigated by use of the (complex scaled) hyperspheric adiabatic expansion method. The case of the 2$^+$ resonance in $^6$Be and $^6$Li is considered. Radial wave functions are obtained showing the correct asymptotic behaviour at intermediate values of the hyperradii, where wave functions can be computed fully numerically.Comment: invited talk at the 18th International Conference on Few-Body Problems in Physics, Santos-S.Paulo, August 21-26, 200

On the Grothendieck-Serre conjecture on principal bundles in mixed characteristic

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split.Comment: Minor corrections and improvement

Affine Grassmannians of group schemes and exotic principal bundles over A¹

Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.Comment: Introduction re-written. Other minor improvements. Final versio