Qualitative analysis of finite-gap solutions of the KgΦ equation by the automorphic approach

By the automorphic approach and the use of computer graphics, we carry out a qualitative analysis of finite-gap periodic solutions of the Kg equation. Graphs are constructed for multiphase solutions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 31–41, 1987.     

Addendum to the paper “on a theorem of Miranda”

An existence theorem is proved for closed convex surfaces whose principal radii of curvature regarded as functions of the unit normal vectorn satisfy the equation

On a method of constructing the solutions of boundary-value problems of the theory of bendings of surfaces of positive curvature

A new method is proposed for constructing the solutions of boundary-value problems of Riemann-Hilbert type for noncanonical linear and quasilinear first-order elliptic systems in a simply connected bounded region of the plane. For a linear boundary condition we obtain complete results; for a nonlinear boundary condition we study the solvability in a neighborhood of zero. Applications are given to the problem of isometric transformations of a surface diffeomorphic to the disk and having positive curvature all the way to the boundary under prescribed boundary conditions.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 56–82.     

Estimates for the fractal dimension and the number of determining modes for invariant sets of dynamical systems

Majorants of the fractal dimension and of the number of determining modes for unbounded sets, invariant with respect to operators of semigroups of classes 1 and 2, are obtained. They are computed for the Navier-Stokes equations (two- and three-dimensional) under the first boundary condition and under periodicity conditions in the spaces and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 105–129, 1987.     

On immersions ofn-dimensional Lobachevskií space in 2n-dimensional Euclidean space withn principal direction fields

In this article we prove that the principal direction fields are holonomic and use them to introduce curvilinear coordinates on an immersed region in terms of which the linear element of Lobachevskií space is written in the form where The fundamental system of equations is established for an immersion ofn-dimensional Lobachevskií space into 2n-dimensional Euclidean space withn principal direction fields for the functions _i and, and a way of constructing an arbitrary local analytic immersion is shown.Translated from Ukrainskií Geometricheskií Sbornik, No. 28, pp. 3–8, 1985.     

Cubic surfaces in AG(3, q) and projective planes of order q 3

Spread sets of projective planes of order q ³ are represented as sets of q ³ points in A AG(3, q ³). A line through the origin in A can be interpreted as a space A ₀ AG(3, q), and the spread set induces a cubic surface L in A ₀. If the projective plane is a semifield plane of dimension 3 over its kernel, then L has the property that it misses a plane of A ₀. Determining all such surfaces L leads to a complete classification of the semifield planes of order q ³, whose spread sets are division algebras of dimension 3.An alternative proof of a result due to Menichetti, that finite division algebras of dimension 3 are associative or are twisted fields, follows with the classification.     

Properties of nonlinear waves in dissipative-dispersive media with instability

Wave processes in dissipative-dispersive media with instability described by a fourth-order nonlinear evolution equation are considered. Analytic solutions in the form of solitary and cnoidal waves are obtained. The existence of a critical value of the dispersion coefficient beyond which an initial disturbance (in particular, white noise) is transformed into a structure is demonstrated by numerical modeling.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 130–136, July–August, 1990.     

Synthesis and crystal and molecular structure of newN-substituted γ-hydroxybutyramides

TheN-(2-phenylethyl)amide of-(1, 1-ethylenedioxy)-ethyl--hydroxybutyric acid (II) and theN-(p-methoxybenzyl)amide of-(1, 1-ethylenedioxy)-ethyl--hydroxybutyric acid (III) have been synthesized and the crystal structure ofIII has been solved. CompoundIII, C₁₆H₂₃O₅N, crystallizes in the monoclinic space groupP2₁/c witha=12.572(2),b=11.149(2),c=12.317(2) Å and=106.53(1)°. The structure was solved by direct methods, and refined by full-matrix least squares to giveR=0.040 (R _w =0.047) for 2004 intensities. The dioxolane ring has a deformed half-chair conformation.     

Crystal structure and vibrational spectra of hydrazinium (2+) tri-μ-fluoro bis[pentafluorozirconate (IV)] fluoride and vibrational spectra of its hafnium analogue

Monoclinic (N₂H₆)₃Zr₂F₁₃·F crystallizes in space group P2₁-C ₂ ² (No. 4) with unit cell dimensionsa=5.670(1),b=10.984(2),c=10.601(2) Å,=93.88(1)°,V=658.7(4) ų andZ=2. Two different types of N₂H₆ ²⁺ ions are present. One is involved in strong H-bonds to F^– ions in infinite chains running along the a axis (the shortest N-F distance is 2.437(5) Å), and the other links the structure through weaker bi- and trifurcated H-bonds to fluorine ligands of the Zr₂F₁₃ ^5– ions. The N-N bond lengths range from 1.430(5) to 1.446(5) Å with apparently no meaningful correlation to the type of N₂H₂ ²⁺ ions. The Zr₂F₁₃ ^5– ions have very nearly C₂ point symmetry and are formed by joining two distorted bicapped trigonal prisms of ZrF₈-units through a common face. Distances of Zr-F terminal bonds range from 2.015(2) to 2.112(2) Å and of bridging bonds from 2.133(2) to 2.212(2) Å. (N₂H₆)₃Hf₂F₁₃·F is isomorphous. The vibrational spectra of the two compounds are nearly identical, with the exception of a strong infrared band, which is assigned to a stretching mode with the moving central atom within the anion. The anion part of the spectrum is simple, showing broad unresolved bands. The cation part shows two types of N₂H₆ ^– ions. H-Bonding is strongly present in the spectra, but no simple correlations with the H-bond strength is evident.     

Behavior on the real line of entire functions represented by Dirichlet series with complex exponents

It is shown, in particular, that if _{n k} when n k, Re _n > 0, and , then an entire function F that is bounded on the real line and represented by a Dirichlet series d_n exp (_nz) that is uniformly and absolutely convergent on each compactum in is identically zero.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 882–888, July, 1990.