卷 294, 编号 1 (2016)

We prove an asymptotic formula for the second moment of primitive L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift, and twist.

An asymptotic formula is obtained for the number of solutions to a symmetric Diophantine equation with reciprocals, and its applications to problems related to the distribution of values of short Kloosterman sums are presented.

The following problem is studied: If a finite sum of ridge functions defined on an open subset of Rⁿ belongs to some smoothness class, can one represent this sum as a sum of ridge functions (with the same set of directions) each of which belongs to the same smoothness class as the whole sum? It is shown that when the sum contains m terms and there are m − 1 linearly independent directions among m linearly dependent ones, such a representation exists.

We describe a method of translating a Lambek grammar with one division into an equivalent context-free grammar whose size is bounded by a polynomial in the size of the original grammar. Earlier constructions by Buszkowski and Pentus lead to exponential growth of the grammar size.

We classify nodal Fano threefolds that are double covers of smooth quadrics branched over intersections with quartics and are acted on by finite simple non-abelian groups. We also study their rationality.

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² − v²)/(uB(v) − vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) − v²A(u))/(uA(v)² − vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) − v²A(u))/(uB(v) − vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ − (3A₁² − 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.

We study the problem of consistency of equations of continuum dynamics (using the Euler equations and the continuity equation as examples) and thermodynamic equations of state (for the specific free energy, entropy, and volume). We propose a variant of the Hamiltonian formulation of a model that combines the fluid dynamics of a potential flow of a compressible fluid or gas and local equilibrium thermodynamics into a unified field theory. Thermodynamic equations of state appear in this model as second-class constraint equations. As a consistency condition, there arises another second-class constraint requiring that the product of density and temperature should be independent of time. The model provides an in-principle possibility of finding the time dependence of the specific entropy of the arising dynamical system.

We consider the dynamics of an open three-level quantum degenerate system. One of the levels in this system is degenerate. The system interacts with three reservoirs (quantum fields) and a classical external field. We show that nondecaying so-called dark states are generated in this system. Since the interactions of the degenerate level with two different reservoirs are different (correspond to different spaces of dark states), we can describe excitation and manipulations for this kind of states (in particular, observation in spectroscopical experiments). Possible applications of this model in quantum optics, quantum computations, and quantum photosynthesis are discussed.

We show that a sequence of improving upper bounds to the ground state energy of the quantized Fröhlich polaron model can be obtained in a regular way by means of combining a variational method originated from the theory of coherent states with a generalized variational approach in quantum mechanics. Due to their variational nature, these bounds hold for arbitrary strength of the electron–phonon interaction.

Order estimates are obtained for the Kolmogorov, linear, and Gel’fand widths of the image of the unit ball of the space l_p under the action of a two-weight summation operator with weights of special kind. Some limit conditions on the parameters defining the weights are considered.