Effect of liquid media on the weight,hardness, volume,and deformation of polyurethane

The resistance of polyurethane to aviation fluids and water has been investigated. The deformation of polyurethane directly exposed to these fluids has been determined. It is shown that polyurethane is insoluble in aviation fluids and sensitive to the action of acids and RDV liquid; it has good resistance to water and atmospheric conditions. The deformation that develops in the first hours of exposure subsequently shows little change.Kiev Institute of Civil Aviation Engineers. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 342–344, March–April, 1969.     

Uniform,on the Real Line,Equiconvergence of Spectral Expansions for the Higher-Order Differential Operators

Doklady Mathematics - Result on the uniform, over the entire real line, equiconvergence of spectral expansions related to the self-adjoint extension of a general differential operation of any even...     

Notes on the Feynman integral,II

In the setting of Cameron and Storvick's recent theory, our main result establishes the existence of the analytic Feynman integral for functions on v-dimensional Wiener space of the form F(X) = exp {? ∝_a^b (A(s) X(s), X(s)) ds}. Here X is a $\text{R}$^v-valued continuous function on [a, b] such that X(a) = 0 and {A(s): a ? s ? b} is a commutative family of real, symmetric, positive definite matrices such that the square roots of the eigenvalues are functions of bounded variation on [a, b]. We obtain the existence theorem just referred to without having to construct special spaces, quadratic forms, etc., to fit the particular problem of interest.     

The Generalized Radon Transform on the Plane, the Inverse Transform, and the Cavalieri Conditions

In the two-dimensional case, the generalized Radon transform takes each function supported in a disk to the values of the integrals of that function over a family of curves. We assume that the curves differ only slightly from straight lines and the network formed by these curves has the same topological structure as the network of straight lines. Thus, the generalized Radon transform specifies a function on the set of straight lines. Under these conditions, we obtain a solution of the inversion problem for the generalized Radon transform and indicate a Cavalieri condition describing the range of this transform in the space of functions on the set of straight lines.     

Effect of a filler,aging, and remelting on the distribution of the spherulitic structure in polyformaldehyde

The distribution of the spherulitic structure of a crystalline polymer has been experimentally investigated with reference to the example of polyformaldehyde (PF). Experimental histograms and distribution curves have been constructed for the distances between the centers of the spherulites in unfilled PF and PF filled with mullite particles of varying shape and size before and after aging and multiple melting. Similar data have also been obtained for the angles between the lines of centers of the spherulites. The results are closely correlated with the changes in the structure and the physicochemical and mechanical properties of PF accompanying the above-mentioned processes.Shevchenko Kiev State University, Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 1, pp. 11–16, January–February, 1971.     

Polynomials orthogonal on the semicircle,II

Generalizing previous work [2], we study complex polynomials {π _k },π _k (z)=z ^k +?, orthogonal with respect to a complex-valued inner product (f,g)=∫ ₀ ^π f(e ^iθ)g(e ^iθ)w(e ^iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ ₀ ^π w(e ^iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π _k } in the case of the Jacobi weight functionw(z)=(1?z)^α(1+z)^β, α>?1, and its special case \(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ _n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ _n (z). It has regular singular points atz=1, ?1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.     

The impact of technology,mathematical modeling,and meaning on the content,learning, and teaching of secondary-school algebra

During each day that the secondary subgroup met, discussions were organized around a reaction to the keynote address of that day. Each of these reactions was specially prepared for the secondary group and was intended to address the concerns of secondary-level algebra.     

Effect of molecular mobility,state of stress,orientation, and plasticization on the temperature dependence of the mechanical properties of linear polymers

The temperature dependence of the breaking stress and ultimate creep strain has been investigated in relation to preorientation, plasticization, and state of stress for polymethyl methacrylate, high-pressure polyethylene, polystyrene, and polyvinyl alcohol. The results of the mechanical tests are compared with the NMR data on the molecular mobility. The causes of the observed effects are discussed.Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 30–35, January–February, 1971.     

单位圆周上（0，1，3）插值的一致收敛性

在本文中，我们考虑了将Ｌｅｇｅｎｄｒｅ多项式零点垂直映射到单位圆周时所得到的一组节点集上（０，１，３）插值的一致收敛性．     

Notes on the Length,the Structure and the Cardinality of a Chain

For a linearly ordered set (Z,≤) the length l(Z) of Z is the supremum of all cardinals that can be order-embedded or reverse order-embedded into Z. In this paper we give new proofs of two theorems relating the length and the cardinality of Z. The first one sets the following general inequality: |Z|≤2^l(Z). The second one says that in the case that Z is a scattered chain (i.e. it does not contain rationals) we have |Z|=2^l(Z). Mathematics Subject Classifications (2000) 06A05, 03E04.