On the Class Numbers of Indefinite Binary Quadratic Forms with Discriminant dp 2

A number of results on the average values of the class numbers of indefinite binary quadratic forms with discriminants divisible by a large square are proved. The main result is as follows. Let d = 4n ² + 1. Then

A bound for the representability of large numbers by ternary quadratic forms and nonhomogeneous waring equations

The solvability of the equation n = x ² + y ² + 6pz ² (p is a fixed large prime) is proved under some natural congruential conditions and the assumption nm ¹² > p ²¹. As an implication, the solvability of the equation n = x ² + y ² + u ³ + v ³ + z ⁴ + w ¹⁶ + t ^4k+1 for all sufficiently large n is established. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 5–21.     

Studying the role of temperature and salinity anomalies in the formation of world ocean meridional circulation modes

A numerical model of the ocean circulation is considered for using in a coupled oceanice-atmosphere climatic model. Preliminary experiments conducted on a numerical grid with a coarse spatial resolution over a period of one thousand years reproduce the World ocean dynamics under climatic surface forcing. By introducing anomalies into the surface sources, the sensitivity of the global ocean circulation to thermohaline variability in the northern regions of the Atlantic Ocean is investigated.     

Behavior of the L-functions of cusp forms ats=1

Let f(z) be a Hecke eigenform in the space S_2k(Γ) of holomorphic Γ-cusp forms of even weight 2k, Γ=SL(2,ℤ); let L_f(s) be the L-function of f(z). The goal of this paper is to obtain some results on L_f(1) as k increases. In particular, we prove an analogue of the classical Landau theorem in the theory of Dirichlet L-functions and (under a very plausible hypothesis) an analogue of the famous Siegel theorem. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 37–54. Translated by E. P. Golubeva.     

Asymptotic distribution of integral points on the three-dimensional sphere

Let Q(X)=x ₁ ² +x ₂ ² +x ₃ ² , X=(x₁, x₂, x₃); r(n) let r(n) be the number of integral solutions of the equation

Advance of the interface of liquids with pressure filtration

The article is a continuation of the investigations of a number of authors [1–3] on the advance of the interface between different liquids: it gives a solution of the problem in the case of one-dimensional filtration of liquids with different viscosities in distorted layers of variable permeability; it indicates a method for determining the interface between multicolored liquids, and the method is generalized for the case of a series of two-dimensional flows, connected by a conformai transformation. The article discusses problems that reduce to formation of the functions determining the flow, and to calculation of the integrals.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 185–191, September–October, 1971.     

Combined calculation method of weak exchange interactions in biradicals

A combined calculation method of weak exchange interactions in short biradicals was developed. The combined method includes two stages. The quantum chemical calculations of the high level of the biradical structure and molecular orbitals and binding energies of unpaired electrons are performed at the first stage. Information obtained at the first stage is used further to calculate the exchange interaction between unpaired electrons within the direct exchange model using the asymptotic method. This allows one to estimate the exchange interaction and to determine the dependence of this interaction on the distance between the paramagnetic centers and on their relative orientation. The method developed was used to calculate the exchange interaction in the short nitroxyl biradical containing no conjugate rings between the paramagnetic NO groups. The geometry and electronic structure of the biradical were calculated within the unrestricted DFT method (B3LYP/cc-pvdz) using the ORCA program package.