An extensional vibrorheometer for viscofluid polymers

An extensional vibrorheometer and a system for its automatic control have been developed. The instrument which has been created makes it possible to investigate liquid polymers under conditions of extensional deformation under a regime where =const. Such investigations may be carried out over a wide range of rates and deformations with the simultaneous superimposition on the extensional flow of periodic deformations covering a range of frequencies exceeding three decimal orders.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 336–343, March–April, 1976.     

Boundary feedback stabilizability of parabolic equations

A parabolic equation defined on a bounded domain is considered, with input acting on theboundary expressed as a specifiedfeedback of the solution. Both Dirichlet and mixed (in particular, Neumann) boundary conditions are treated. Algebraic conditions based on the finitely many unstable eigenvalues are given, ensuring the existence ofboundary vectors, for which all the solutions to theboundary feedback parabolic equation decay exponentially to zero ast+ in (essentially) the strongest possible space norm. A semigroup approach is employed.Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-77-3338.A preliminary version of this paper has appeared in the Proceedings of the International Symposium on the Mathematical Theory of Networks and Systems, held at Delft University, The Netherlands, July 3–6, 1979; pp. 428–433.     

Navier–Stokes equations with nonhomogeneous boundary conditions in a convex bi-dimensional domain

We prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a convex and bounded region, ΩR², with nonhomogeneous Dirichlet boundary conditions on ∂Ω. These results are also extended to flow domain surrounding an obstacle.     

Modeling and analysis of multi‐stage transfer lines with unreliable machines and finite buffers

This paper models and analyzes multistage transfer lines with unreliable machines and finite buffers. The machines have exponential operation, failure, and repair processes. First, a mixed vector–scalar Markov process model is presented based on some notations of mixed vector–scalar operations. Then, several steadystate system properties are deduced from this model. These include the reversibility and duality of transfer lines, conservation of flow, and the flow rate–idle time relationship. Finally, a fourstage transfer line case is used to compare and evaluate the accuracy of some approximation methods presented in the literature with the exact numerical solutions this model can provide. The properties and their proofs in this paper lay the theoretic foundation for some widely held assumptions in decomposition techniques of long transfer lines in the area of manufacturing systems engineering.     

On the Mean 3-Rank of Quadratic Fields

The Cohen–Lenstra–Martinet heuristics give precise predictions about the class groups of a random number field. The 3-rank of quadratic fields is one of the few instances where these have been proven. We prove that, in this case, the rate of convergence is at least sub-exponential. In addition, we show that the defect appearing in Scholz';s mirror theorem is equidistributed with respect to a twisted Cohen–Lenstra density.     

Heat Flow for the Yang–Mills–Higgs Field and the Hermitian Yang–Mills–Higgs Metric

For a parameter > 0, we study a type of vortex equations, which generalize the well-known Hermitian–Einstein equation, for a connection A and a section of a holomorphic vector bundle E over a Kähler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang–Mills–Higgs field on E. Assuming the -stability of (E, ), we prove the existence of the Hermitian Yang–Mills–Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.     

Bounded solutions of the Schrödinger equation on noncompact Riemannian manifolds

Necessary and sufficient geometric conditions are proved for the equation u–Q(x)u=0, Q(x)0, to have a bounded nontrivial solution on a noncompact Riemannian manifold. The results imply as corollaries conditions for parabolicity and stochastic completeness of a manifold, previously established by other methods.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 66–77, 1989.     

The Steenrod Algebra and Theories Associated to Hopf Algebras

We show that the homotopy category of products of Z/p-Eilenberg–Mac Lane spaces is an -algebra which algebraically is determined by the Steenrod algebra considered as a Hopf algebra with unstable structure.     

Influence of heat transfer on a peristaltic transport of Herschel–Bulkley fluid in a non-uniform inclined tube

Peristaltic transport in a two-dimensional non-uniform tube filled with Herschel–Bulkley fluid is studied under the assumptions of long wavelength and low Reynold number. The fluid flow is investigated in the wave frame of reference moving with the velocity of the peristaltic wave. Exact solution for the velocity field, the temperature profile, the stream functions and the pressure gradient are obtained. The physical behavior of τ, n, α and on the pressure rise versus flow rate are discussed through graphs. At the end of the article trapping phenomena for Herschel–Bulkley and also for Newtonian, Bingham and power law (which are the special cases of Herschel–Bulkley fluid) fluid are discussed.     

Existence and multiplicity results for some nonlinear problems with singular -Laplacian

Using Leray–Schauder degree theory we obtain various existence and multiplicity results for nonlinear boundary value problems

where l(u,u^′)=0 denotes the Dirichlet, periodic or Neumann boundary conditions on [0,T], is an increasing homeomorphism, (0)=0. The Dirichlet problem is always solvable. For Neumann or periodic boundary conditions, we obtain in particular existence conditions for nonlinearities which satisfy some sign conditions, upper and lower solutions theorems, Ambrosetti–Prodi type results. We prove Lazer–Solimini type results for singular nonlinearities and periodic boundary conditions.     

Structural susceptibility of strength characteristics of polystyrene under uniaxial stretching

The structural susceptibility of the strength characteristics of polystyrene under uniaxial stretching and under test conditions with = const and v_elong = const was studied at 20–100°C. The low structural susceptibility of the durability and durability-equation parameters is due to a change in the elementary fracture volume at different test conditions. It was proposed to characterize the degree of participation of chemical and intermolecular forces by the value U₀/.Moscow Technological Institute of Meat and Milk Industry. Translated from Mekhanika Polimerov, No. 6, pp. 979–984, November–December, 1974.     

Rheological characteristics of melts of low-molecular-weight branched polyethylene in the region of critical molecular weights

The viscosity and the first difference of the normal stresses have been measured at stationary shear rates from 5.68 to 4500 sec^–1 for melts of low-density polyethylenes obtained by thermal degradation of commercial polyethylene in a nitrogen atmosphere. In addition, the complex modulus has been measured on the angular frequency range from 0.396 to 198.5 sec^–1. Three regions of molecular weights differing with respect to the value of the exponent in the relation ₀ = kM have been established. At > normal stresses are observed in the region of shear rates corresponding to a linear dependence of the shear rate on the shear stresses.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga; Plastpolymer Research and Production Association, Leningrad. Translated from Mekhanika Polimerov, No. 6, pp. 963–967, November–December, 1973.     

Sufficient Conditions for Boundedness of Convolution Operators in Rearrangement-Invariant Spaces

We prove equivalence of sufficient conditions for boundedness of convolution integral operators in rearrangement-invariant spaces which were obtained in the articles of S. G. Kren, E. M. Semenov, and the author. The first of these conditions generalizes the Hardy–Littlewood–Sobolev condition and the second is a modification of the Hörmander condition.     

Study of the process of unstable flow of elastomers in open and closed channels and its connection with mechanical characteristics of the material

The unstable flow process has been studied for examples of a series of unvulcanized rubbers of different structures by a complex of methods — capillary, tribometric, and dynamic methods. A connection has been found between the critical parameters which characterize the onset of this process and the dynamic mechanical characteristics. The possibility of modeling the unstable flow process in open channels is shown.A. V. Topchiev Institute for Petrochemical Synthesis, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 530–536, May–June, 1970.     

On predicting unsteady non-Newtonian blood flow

The influence of the non-Newtonian stress–strain relation of blood on the oscillatory shear index (OSI) and mean wall shear stress (WSS) are described. A mathematical non-dimensional model based on the momentum equation for a modified Casson’s fluid is formulated in terms of the dimensionless yield shear stress . An original direct numerical procedure is presented to predict the flow patterns. Results obtained by using a finite difference approach show a difference in OSI when blood is assumed to be a Newtonian fluid instead of a modified Casson’s fluid. The calculation of the OSI in human normal conditions under the Newtonian approach differs in 5% from the result obtained from using the Casson model.     

Conditional functional equations and orthogonal additivity

Summary Some examples of classes of conditional equations coming from information theory, geometry and from the social and behavioral sciences are presented. Then the classical case of the Cauchy equation on a restricted domain is extensively discussed. Some results concerning the extension of local homomorphisms and the implication -additivity implies global additivity are illustrated. Problems concerning the equations[cf(x + y) – af(x) – bf(y) – d][f(x + y) – f(x – f(y)] = 0[g(x + y) – g(x) – g(y)][f(x + y) – f(x) – f(y)] = 0f(x + y) – f(x) – f(y) V (a suitable subset of the range) are presented.The consideration of the conditional Cauchy equation is subsequently focused on the case when it makes sense to interpret as a binary relation (orthogonality):f: (X, +, ) (Y, +);f(x + z) = f(x) + f(z) (x, z Z; x z). A brief sketch on solutions under regularity conditions is given. It is then shown that all regularity conditions can be removed. Finally, several applications (also to physics and to the actuarial sciences) are discussed. In all these cases the attention is focused on open problems and possible extensions of previous results.     

The strength properties of acicular single crystals of polyoxymethylene

The strength properties of acicular formations of polyoxymethylene, obtained by the directed polymerization of trioxane whiskers, have been studied. It has been found that the acicular single crystals obtained in this process exhibit a high strength.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 266–270, March–April, 1970.     

A study of the life of alkyd coatings

The effect of the conditions of formation and aging on the life of alkyd coatings is studied. It is shown that the life of alkyd coatings aged under various conditions can be expressed by the general equation =Ae^–ax The relationship between the life of alkyd coatings and curing temperature is linear. It is observed that the conditions of aging have no effect on the rupture energy or ₀, but cause a substantial change in the distribution of the lines =f(1/T_f). Cross-linkage of alkyd coatings due to the action of heat, light, and moisture, under various conditions of aging, is accompanied by an increase in rupture energy with increase in the internal stresses.Mekhanika Polimerov, Vol. 1, No. 3, pp. 68–76, 1965     

Discreteness of the spectrum and completeness of systems of characteristic and associated vectors of second-order nonself-adjoint differential operators

Conditions are found under which A^–1 _p (A is a nonself-adjoint operator). For differential operators in L₂(–, ) with real coefficients, conditions are found for the existence of a regular point, necessary and sufficient and also sufficient conditions are found for the spectrum to be discrete, and conditions are found for the system of characteristic and associated vectors to be complete.Translated from Matematicheskie Zametki, Vol. 9, No. 4, pp. 401–408, April, 1971.In conclusion the author wishes to thank sincerely his scientific advisor M. A. Naimark for his help in this work.     

On the stability of quasidouble step spline approximations for solutions ofy (n) = f(x,y, ⋯, y (n−1) )

In this paper, we continue the study [4] of the stability question of the quasidouble step spline function approximations,s(x) C ^m–2 , to the initial value problemy ⁽ⁿ⁾ = f(x, y, , y ^(n–1) ). It will be shown that the method is unstable and hence divergent form n + 4.