On total reality of meromorphic functions

We show that if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points then it is conjugate to a real meromorphic function after a suitable projective automorphism of the image.Comment: 13 page

Second harmonic electromagnetic emission of a turbulent magnetized plasma driven by a powerful electron beam

The power of second harmonic electromagnetic emission is calculated for the case when strong plasma turbulence is excited by a powerful electron beam in a magnetized plasma. It is shown that the simple analytical model of strong plasma turbulence with the assumption of a constant pump power is able to explain experimentally observed bursts of electromagnetic radiation as a consequence of separate collapse events. It is also found that the electromagnetic emission power calculated for three-wave interaction processes occurring in the long-wavelength part of turbulent spectrum is in order-of-magnitude agreement with experimental results

Homology class of a Lagrangian Klein bottle

It is shown that an embedded Lagrangian Klein bottle represents a non-trivial mod 2 homology class in a compact symplectic four-manifold $(X,\omega)$ with $c_1(X)\cdot[\omega]>0$. (In versions 1 and 2, the last assumption was missing. A counterexample to this general claim and the first proof of the corrected result have been found by Vsevolod Shevchishin.) As a corollary one obtains that the Klein bottle does not admit a Lagrangian embedding into the standard symplectic four-space.Comment: Version 3 - completely rewritten to correct a mistake; Version 4 - minor edits, added references; AMSLaTeX, 6 page

Finiteness and quasi-simplicity for symmetric K 3-surfaces

We compare the smooth and deformation equivalence of actions of finite groups on K3-surfaces by holomorphic and antiholomorphic transformations. We prove that the number of deformation classes is finite and, in a number of cases, establish the expected coincidence of the two equivalence relations. More precisely, in these cases we show that an action is determined by the induced action in the homology. On the other hand, we construct two examples to show first that, in general, the homological type of an action does not even determine its topological type, and second that K3-surfaces X and X̄ with the same Klein action do not need to be equivariantly deformation equivalent even if the induced action on H2,0(X) is real, that is, reduces to multiplication by ±1

On the number of components of a complete intersection of real quadrics

Our main results concern complete intersections of three real quadrics. We prove that the maximal number B0 2 (N) of connected components that a regular complete intersection of three real quadrics in ℙN may have differs at most by one from the maximal number of ovals of the submaximal depth [(N −1)/2] of a real plane projective curve of degree d = N +1. As a consequence, we obtain a lower bound 1/4 N2 +O(N) and an upper bound 3/8 N2+O(N) for B0 2 (N). © Springer Science+Business Media, LLC 2012

Lines on quartic surfaces

We show that the maximal number of (real) lines in a (real) nonsingular spatial quartic surface is 64 (respectively, 56). We also give a complete projective classification of all quartics containing more than 52 lines: all such quartics are projectively rigid. Any value not exceeding 52 can appear as the number of lines of an appropriate quartic. © 2016, Springer-Verlag Berlin Heidelberg

A search for disordered (glassy) phase in solid 3He deformed in situ

A disordered (glassy) state has been searched in solid 3He deformed in the course of experiment employing precise measurements of pressure. The analysis of the temperature dependence of the crystal pressure measured at a constant volume shows that the main contribution to the pressure is made by the phonon subsystem, the influence of the disordered phase being very weak. Annealing of the deformed crystal does not affect this state. The results obtained differ greatly from the corresponding data for solid 4He measured in the region of supersolid effects where a pressure excessive in comparison to the phonon one was registered. The excess pressure had a quadratic dependence on temperature, which is typical of a disordered system. Absence of the excess pressure in solid 3He is unclear yet, some speculative interpretations are suggested.Comment: 9 pages, 4 figure