Finite Topological Type and Volume Growth

We show that a complete noncompact n-dimensional Riemannian manifold Mwith Ricci curvature Ric _M –(n – 1) and conjugateradius conj _M c > 0 has finite topological type, provided that the volume growth of geodesic balls in M is not very far from that of the balls in an n-dimensional hyperbolic space H ⁿ (–1)of sectional curvature –1. We also show that a complete open Riemannian manifold M with nonnegative intermediate Ricci curvature and quadratic curvature decay has finite topological typeif the volume of geodesic balls of M around the base point grows slowly.     

The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.     

Light absorption and laser-crack development. 2

Conclusions 1. The gray color of a laser crack in PMMA (black in polystyrene) is caused by carbon-black particles that adhere to its walls.2. Further experimentation under an electron microscope is necessary to determine the size of the carbon-black particles. Making use of these new data and the results described here, we can define their absorption index more precisely.3. The absorption index of the gaseous medium between the walls of the crack in a layer 30–100 µ thick is 20 cm^–1; this is 1000 times smaller than that of graphite, but approximately 1000 times greater than that of the ordinary block polymer.4. As a result of photochemical and photothermal reactions, the laser source creates a gray (PMMA) or black (polystyrene) absorbent layer located in the crack and on its walls. Thus, the energy of the laser beam is absorbed due to "self-inductive" opacity. This leads to an increase in the mass of the "gas wedge," to a pressure increase, and ultimately, to crack propagation.For communication 1, see [1].Institute of the Problems of Mechanics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 409–412, May–June, 1977.     

On function-degenerate immersions of a Lobachevsky plane into E4

A surface F² in E⁴ is called function-degenerate if parameters , , a, and b of the ellipse of normal curvature ( and are the coordinates of its center in the normal plane and a and b are the values of the semiaxes) are some functions of a function t(P) of point P of the surface. This class of surfaces is a generalization of the class of surfaces that admit motion with respect to themselves along some family of lines. The following theorem is proved: There does not exist a regular, class C⁶, isometric immersion of the Lobachevsky plane L² into E⁴, with zero Gaussian torsion, in the form of a function-degenerate surface. This theorem includes the Hilbert theorem on surfaces of negative curvature in E³.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 8–18, 1990.     

Curvatures and Currents for Unions of Sets with Positive Reach, II

A class of subsets of ^d which can berepresented as locally finite unions of sets with positive reach isconsidered. It plays a role in PDE's on manifolds with singularities.For such a set, the unit normal cycle (determining the d – 1curvature measures) is introduced as a (d – 1)-currentsupported by the unit normal bundle and its properties are established.It is shown that, under mild additional assumptions, the unit normalcycle (and, hence, also the curvature measures) of such a set can beapproximated by that of a close parallel body or, alternatively, by themirror image of that of the closure of the complement of the parallelbody (which has positive reach). Finally, the mixed curvature measuresof two sets of this class are introduced and a translative integralgeometric formula for curvature measures is proved.     

The dynamic properties of polyethylene and an acrylonitrile — Butadiene — Styrene material in a broad range of temperature and strain-rate variation

Conclusions The mechanical properties of an acrylonitrile-butadiene-styrene material and polyethylenes of densities =0.92 and 0.96 g/cm³ are studied within the temperature range of from 100 to –196°C and at strain rates of from 10^–3 to 1.3 · 10³ sec^–1. It is observed that under low-temperature and high-strain-rate conditions, the laws governing the mechanical properties of the acrylonitrile-butadiene-styrene material and polyethylenes are similar. Regions of strain rates and temperatures are found for which the behavior of the test materials is in good agreement with the Ree-Eyring equation, and, consequently, conforms to a temperature-time analogy. No embrittlement of the polyethylene is observed during tests at strain rates to 1.3 · 10³ sec^–1 and temperatures down to –196°C. In approximating these values of and T, the mechanical properties of polyethylenes will cease to be dependent on initial density.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, 1027–1033, November–December, 1978.     

The curvatures of asymptotic lines on a domain of three-dimensional Lobachevsky space that is immersed intoE 5

It is assumed that a domain of a three-dimensional Lobachevsky space with curvature equal to – 1 is immersed into E ⁵.It is known that four asymptotic lines pass through every point on the immersed domain. It is proved that the second curvature of the asymptotic line is k ₂ =1/cos, where is the angle formed by vector ₃ from the natural frame of the asymptotic line and the normal plane of the submanifold. The third curvature k ₃ satisfies k ₃ |(d/ds) (1/k ₂ ) |and if k ₃ 0, then the fourth curvature is expressed in terms of k ₂ and k ₃.Thus, any curve in E ⁵ may not be an asymptotic line on a domain of a Lobachevsky space immersed in E ⁵.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 5–10, 1992.     

Study of the role of intermolecular interactions (hydrogen bonds) in the initial stage of orientation of polyvinyl alcohol and vinyl alcohol copolymers

The effect of hydrogen bond rupture on the process of destruction of the starting crystallite structure of polyvinyl alcohol and copolymers of vinyl alcohol and N-vinylpyrrolidone and -vinylcaprolactam has been investigated. The differential spectrum method is used to establish the frequency (3200 cm^–1) of the band corresponding to the vibrations of the bound hydroxyl groups; the temperature dependence of the optical density of the 3200 cm^–1 band has been recorded. The strength of the starting structure is calculated from the stress-strain diagram and its temperature dependence is plotted. It is concluded that the strength of the starting structure is chiefly determined by the presence of hydrogen bonds between the polymer macromolecules.Leningrad Kirov Institute of Textile and Light Industry. Translated from Mekhanika Polimerov, No. 2, pp. 327–329, March–April, 1971.     

Extremal Almost-Kähler Metrics and Seiberg–Witten Theory

We show that there exist compact non-Kähler almost-Kähler4-manifolds whose metrics minimize L ²-norm of(2/3) s + 2w among all metrics compatible with a fixeddecomposition H ²(M, = H ⁺ H ^–, where s is the scalar curvature and w is the lowest eigenvalue of self-dual Weyl curvature at each point. In particular, the moduli space of such metrics modulo diffeomorphisms is infinite dimensional. This example also shows that LeBrun's estimate of L ²-norm of (1 – )s + · 6won a compact oriented Riemannian4-manifold with a nontrivial Seiberg–Witten invariant cannot beextended over = 1/3.     

Sound propagation in the nonhomogeneous ocean with currents

One considers the problem of sound propagation in the nonhomogeneous ocean with currents, where the characteristics of the medium vary slowly in the horizontal direction. The solution is constructed in the form of a series of powers of a small parameter , where =v_oc_o–1, is the velocity of the current, c_o is the adiabatic velocity of the sound. In order to determine the coefficients of the series, one applies the space-time ray method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 38–47, 1983.The author is grateful to V. S. Buldyrev for useful discussions of the problem.     

Gauge-Invariant Regularization of Quantum Field Theory on the Light Front

We briefly describe problems of the Hamiltonian approach for quantizing gauge fields on the light front for space–time bounded by the inequality |x ^–|L with periodic boundary conditions in the variable x ^– imposed on all fields (the DLCQ method). With these restrictions, we consider the gauge-invariant ultraviolet regularization by passing to a lattice in transverse coordinates. We remove the remaining ultraviolet divergences in the longitudinal momentum p _– by imposing a gauge-invariant finite-mode regularization. It turns out that the canonical formalism on the light front with such a regularization imposed does not contain the usual most complicated second-class constraints between zero and nonzero modes of fields. The described scheme can be used both to regularize the standard gauge theory and to provide a gauge-invariant formulation of effective low-energy models on the light front. Because the manifest Lorentz invariance is broken in our formalism, the vacuum state is poorly defined. We discuss this problem, in particular, in relation to the problem of passing to the continuous space limit.     

The differential form spectrum of hyperbolic space

Let Hⁿ⁺¹ denote the simply connected complete space of constant curvature –1. The Laplacian , acting on square integrable p-forms of H, is identified up to unitary equivalence.     

On the geometry of principalT 1-bundles over Hodge manifolds

We study Sasakian structures induced in principalT ¹-bundles over Kähler manifolds. A natural model of a Sasakian manifold of constant -holomorphic sectional curvature –3 is constructed.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 824–829, December, 1998.The author is greatly indebted to Professor V. F. Kirichenko for setting the problem, as well as for interest and help during the research.     

On the existence of certain cyclic difference families and difference matrices

A theorem is proved to the effect that if there exists a BIB-schema with parameters (p^m–1,k, k–1), where k¦(p^m–1), p is prime, and m is a natural number, then there exists a BIB-schema (p^mn–1),k, k–1). A consequence is the existnece of a cyclic BIB-schema (p^mn–1, p^m–1, p^m–2) (p^m–1 is prime) that specifies each ordered pair of difference elements at any distance = 1, 2, ..., p^m–2 (cyclically) precisely once. Recursive theorems on the existence of difference matrices and (v, k, k)-difference families in the group Z_v of residue classes mod v are proved, along with a theorem on difference families in an additive abelian group.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 114–119, July, 1992.     

Total gluon propagator in light-cone gauge and the transversality problem of the polarization operator in the infrared region. II

A three-gluon vertex that depends on a gauge vectorn and is free of kinematic and spurion (np)^–1 singularities in the physical region is constructed. It is shown that in the case of integer power-law asymptotic behavior of the gluon propagator there is separation of the singular contributions of the single- and two-loop diagrams in the Dyson—Schwinger equation. Necessary conditions for realization ofp ^–4 asymptotic behavior are obtained, and it is shown that an important role in their fulfillment is played by the parts of the three-gluon vertex that depend on the gauge vector. A solution of these conditions is found; when they are fulfilled, they lead to infrared asymptotic behaviors of the complete Dyson—Schwinger equation and its single-loop approximation that are mutually compatible.Tbilisi State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 14–23, July, 1994.     

Failure of water-filled cylindrical glass-reinforced epoxy shells under internal impulsive loading

Conclusions 1. The explosive failure of water-filled, geometrically similar glass-reinforced epoxy shells, whose dimensions increase by a factor of 1.5–4.4, is not associated with an energetic scale effect. This is attributable to the imperfect similarity of the shells, since although their dimensions changed, the diameter of the glass reinforcing fibers remained the same.2. The relative mass of explosive required to fracture the shells is approximately 0.4%. With respect to this index, glass-reinforced epoxy is comparable to structural steels 20 and 17Mn1Si at R₀160 mm.3. The circumferential deformation of the shells at failure is about 4% and does not depend on the thickness of the shell wall, the dimensions of geometrically similar shells, or the initial strain rate on the interval 0.21 · 10³–1.2 · 10³ sec^–1. Thin-walled shells begin to fail from the outside surface, thick-walled shells from the inside surface.4. The modulus of elasticity of the GRE is 2.4 · 10⁵ kgf/cm² and does not depend on the strain rate on the interval 10^–3–1.5 · 10³ sec^–1. The material deforms elastically up to failure.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 283–289, March–April, 1978.     

The structure of an infinite,complete, convex hypersurface inE 4 with bounded mean curvature

The following theorem is proved. THEOREM. If on an infinite, complete, convex hypersurface F in E⁴ the mean curvature is 1 – H 1, where 0 10^–11, then F is a cylindrical hypersurface.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 8–9, 1991.     

Invariant Metrics on Cohomogeneity One Manifolds

Let G be one of the connected subgroups of the orthogonal group of ⁿ which acts transitively on the unit sphere S ^n–1. We get the necessary and sufficient condition for G-invariant metrics g on ⁿ \{0} to be extendend to the origin. For n=2 this is a classical result of Berard–Bergery. The curvature tensor and the sectional curvature of any such Riemannian G-manifold ( ⁿ , g) are described in terms of the length of the Killing vector fields, as well as the second fundamental form of the regular orbits G(P)=S ^n–1. As an application we describe all G-invariant metrics which are Kähler, hyperKähler or have constant principal curvatures. Some of these results are generalized to the case of any cohomogeneity one G-manifold which, in a neighbourhood of a singular orbit, can be identified with a twisted product.     

Estimation of the field in the shadow zone for diffraction of a spherical wave by a convex,infinitely differentiable surface

Let DR³ be an infinite domain with bounded, smooth boundary having positive Gaussian curvature. Green's function of the Helmholtz operator is studied. The main result is the estimate |G| const exp (– K^1/3– const)( is an arbitrary positive number) in the shadow zone.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 93–118, 1975.     

A characterization of the generalized twisted field planes of characteristic ≥5

Let be a non-Desarguesian semifield plane of orderp ⁿ, p a prime number 5 andn3, and let denote the group induced by the autotopism groupG of on the line at infinity. We prove that is a generalized twisted field plane if, and only if, has an element of order (p ^k–1)((p ⁿ–1)/(p ^m–1)), for some integersk andm, wherek | m, m | n, andm.This work was supported in part by NSF grants RII-9014056, component IV of the EPSCoR of Puerto Rico grant and ARO grant for Cornell MSI