Strength of glass laminates at low temperatures and high strain rates

The tensile strengths of glass laminates of various compositions under standard, high-speed and impact loading have been experimentally investigated in relation to the effect of low-temperature and stress raisers. As the loading rate increases (from 0.7 to 253 · 10³ kgf/cm². · sec) and the temperature falls to –196° C, the effective stress concentration coefficient decreases, which indicates that the glass laminates will perform adequately under these conditions.All-Union Scientific-Research Institute of Aviation Materials, Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 836–841, September–October, 1970.     

Effect of cycle asymmetry on the cyclic strength of polymers

Cyclic strength diagrams have been obtained for polymethyl methacrylate and kaprolon at different degrees of loading asymmetry (from a pure compression to a pure tension cycle) on the interval from –100 to +80° C at 2·10³ and 3·10⁵ cycles.M. I. Kalinin Leningrad Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 948–951, September–October, 1968.     

Dynamic properties of some polymer materials in loading and unloading

A series of quasi-static tests have been carried out on kapron fiber, vinyl plastic, and fluoroplastic specimens with small cross sections. Stress-strain diagrams have been obtained at strain rates from 1 · 10^–3 to 1 · 10² sec^–1. In the experiments the strain rate changed sign, which corresponded to loading and unloading. In each regime the strain rate remained constant. The experimental data are analyzed using the model of a standard linear viscoelastic solid.Mekhanika Polimerov, Vol. 3, No. 1, pp. 118–122, 1967     

Mass-spectrometric investigation of solid lubricating coatings subjected to friction in a deep vacuum

The life of solid lubricating coatings of the VNII NP type, based on molybdenum disulfides and various binders, has been experimentally investigated under deep vacuum conditions (10^–8–5 · 10^–9 torr) together with the composition of the gas released in the friction process. It is shown that both under atmospheric conditions and in a deep vacuum the life of the coatings depends on the chemical nature of the film-former. The depth of the vacuum also has an important influence on the life of the coatings, both the mechanism and the end result of this effect depending to a large extent on the physicochemical properties of the bind. On the interval 10^–1–10^–2 torr there is a sudden change in the life of the coating.Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR, Khar'kov. Translated from Mekhanika Polimerov, No. 6, pp. 1070–1075, November–December, 1970.     

Deformation of thermoplastic materials at low-cycle loading

The possibility is discussed of deducing analytical equations for evaluating the deformation of polymer materials at a low-cycle periodic loading on the basis of the linear prehistory theory. Finite expressions are given for the case of loading by rectangular and triangular pulses. Comparison of the results of the experimental study of the deformation of low- and high-pressure polyethylenes (LPP and HPP, respectively) with the conclusions of the theory shows satisfactory agreement for tests of up to 5·10³ loading cycles.Leningrad Institute of Aviation Instrument Engineering. Translated from Mekhanika Polimerov, No. 2, pp. 230–234, March–April, 1976.     

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.     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

“Fundamental” and “practical” adhesion in polymer-fiber systems

The relationship between adhesion and bond strength in thin fiber-polymer matrix systems was studied. Adhesive interaction in composite materials was analyzed within the scope of thermodynamic and molecular-kinetic theories of adhesion. Methods based on wetting are shown to give poor estimation of the work of adhesion in fiber-polymer systems, which is due to their low sensibility to donor-acceptor interactions taking place at the interface. Important information about the acidity and basicity of contacting surfaces can be obtained by using inverse gas chromatography to investigate the thermodynamics of adsorption. The calculation of the work of adhesion including acid-base interactions shows the best agreement with the bond strength in the same systems. The local (ultimate) interfacial shear strength is proposed to characterize the quality of fiber-matrix bonding. Analysis of the relationship between the work of adhesion and adhesive pressure for various systems allowed us to differentiate the dispersive and acid-base components of the local bond strength as well as to estimate distances characteristic of these twoTypes of interaction. For dispersive forces, our estimation gives 7–8Å, i.e., of an order of magnitude of the center-to-center distance for van der Waals interactions. At the same time, the acid-baseInteractions have a characteristic range of 4–5Å and can be attributed to hydrogen bonding. The agreement between the calculated distances and literature data is evidence for the applicability of the proposed method to the analysis of the adhesive interaction in fibrous polymer composites.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 4, pp. 431–446, July–August, 1998.     

Strength of boron fibers

The tensile and flexural strength and elastic properties of continuous boron fibers 95–105 µ in diameter obtained by depositing boron on a tungsten filament 10–12 µ in diameter have been investigated. The strength properties are primarily determined by the defects present at the boron sheath-core interface and for the fibers investigated have a limit of approximately 400 kgf/mm². Coarse surface or internal defects are capable of reducing the strength of the fibers to 200–250 kfg/mm².All-Union Scientific-Research Institute of Aviation Materials, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1126–1127, November–December, 1970.     

A study of the special features of stress relaxation in irradiated polytetrafluoroethylene

As a result of experimental studies of stress relaxation in polytetrafluoroethylene irradiated with various doses (from 2×10⁴ to 1×10⁶ rads), data have been obtained which make it possible to judge the rate of occurrence of relaxation processes in the polymer. A number of special features have been revealed which distinguish the character of occurrence of relaxation processes in the irradiated and nonirradiated material.Plastics Scientific-Research Institute, Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 148–150, January–February, 1972.     

A splitting theorem for certain cohomology theories associated to BP* (−)

Let P(n)^*(–) be Brown-Peterson cohomology modulo I_n and put B(n)^*(–)=P(n)^*(–)[1/v_n]. In this note we construct a canonical multiplicative and idempotent operation _n in a suitable completion (n)^*(–) of B(n)^*(–) which has the property that its image is canonically isomorphic to the n-th Morava K-theory K(n)^*(–). In particular, the ring theory K(n)^*(–) is contained as a direct summand in the theory (n)^*(–). A similar result is not true before completing. pleting. Because the completion map B (n)^*(–) (n)^*(–) is injective, the above splitting theorem contains also information about B(n)^*(–). The proof of the theorem depends on a result about the behaviour of formal groups of finite height over complete graded F_p.     

Action of a powerful light beam on polypropylene film

The changes in the supermolecular structure of polypropylene films exposed to powerful light pulses have been investigated by optical microscopy. A pulse lasting 10^–3 sec produces melting of the polymer with subsequent recrystallization and the formation of a finely spherulitic structure. In connection with the change in supermolecular structure there is an increase in the breaking stress of the melted portion of the film. A pulse lasting 2·10^–8 sec causes melting and vaporization of individual parts of the spherulites in the irradiated region. The remainder of the polymer is such that orientation of the structural elements with respect to the radius—the maltese cross pattern of the original spherulites—is preserved.Mekhanika Polimerov, Vol. 2, No. 6, pp. 817–822, 1966     

A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three

We construct a new family of cyclic difference sets with parameters ((3 ^d – 1)/2, (3 ^{d – 1} – 1)/2, (3 ^{d – 2} – 1)/2) for each odd d. The difference sets are constructed with certain maps that form Jacobi sums. These new difference sets are similar to Maschietti's hyperoval difference sets, of the Segre type, in characteristic two. We conclude by calculating the 3-ranks of the new difference sets.     

On optimal simultaneous rational approximation to with being some kind of cubic algebraic function

It is shown that each rational approximant to (ω,ω²)^τ given by the Jacobi–Perron algorithm (JPA) or modified Jacobi–Perron algorithm (MJPA) is optimal, where ω is an algebraic function (a formal Laurent series over a finite field) satisfying ω³+kω-1=0 or ω³+kdω-d=0. A result similar to the main result of Ito et al. [On simultaneous approximation to (α,α²) with α³+kα-1=0, J. Number Theory 99 (2003) 255–283] is obtained.     

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:

We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the L^p–L^q estimates of fundamental solutions for linear damped wave equations [K. Nishihara, L^p–L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The L^p–L^q estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and L^p–L^q estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, L^p–L^q estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].     

Some Asymptotic Formulae for q-Shifted Factorials

The q-shifted factorial defined by (a : q^k) _n = (1 – a) (1 – aq^k)(1 – aq^2k)... (1 – aq^{(n – 1)k}) appears in the terms of basic hypergeometric series. Complete asymptotic expansions as q 1 of some q-shifted factorials are given in terms of polylogarithms and Bernoulli polynomials.     

Some inequalities for algebraic polynomials with the Laguerre weight

There is a series of publications which have considered inequalities of Markov–Bernstein–Nikolskii type for algebraic polynomials with the Jacobi weight (see [N.K. Bari, A generalization of the Bernstein and Markov inequalities, Izv. Akad. Nauk SSSR Math. Ser. 18 (2) (1954) 159–176; B.D. Bojanov, An extension of the Markov inequality, J. Approx. Theory 35 (1982) 181–190; P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995; I.K. Daugavet, S.Z. Rafalson, Some inequalities of Markov–Nikolskii type for algebraic polynomials, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1 (1972) 15–25; A. Guessab, G.V. Milovanovic, Weighted L²-analogues of Bernstein's inequality and classical orthogonal polynomials, J. Math. Anal. Appl. 182 (1994) 244–249; I.I. Ibragimov, Some inequalities for algebraic polynomials, in: V.I. Smirnov (Ed.), Fizmatgiz, 1961, Research on Modern Problems of Constructive Functions Theory; G.K. Lebed, Inequalities for polynomials and their derivatives, Dokl. Akad. Nauk SSSR 117 (4) (1957) 570–572; G.I. Natanson, To one theorem of Lozinski, Dokl. Akad. Nauk SSSR 117 (1) (1957) 32–35; M.K. Potapov, Some inequalities for polynomials and their derivatives, Vestnik Moskov. Univ. Ser. Mat. Mekh. 2 (1960); E. Schmidt, Über die nebst ihren Ableitungen orthogonalen Polynomsysteme und das zugehörige Extremum, Math. Ann. 119 (1944) 165–209; P. Turán, Remark on a theorem of Erhard Schmidt, Mathematica 2 (25) (1960) 373–378]). In this paper we find an inequality of the same type for algebraic polynomials on (0,∞) with the Laguerre weight function e^-xx^α (α>-1).     

Dynamic modulus of polymethylmethacrylate saturated with carbon dioxide

The authors have devised a method for measuring the complex modulus of elasticity of non-plasticized polymethylmethacrylate of grade ST-1-110 in an atmosphere of carbon dioxide. The pressure of the saturated gas in the working chamber was varied between 4·10⁵ and 60·10⁵ N/m², and the experiments were performed at 18–90°C.S. M. Kirov Ural Polytechnic Institute, Sverdlovsk. Translated from Mekhanika Polimerov, No. 2, pp. 370–372, March–April, 1974.     

Distribution of Cardinalities of Row Spaces of Boolean Matrices of Order n

Let R(A) denote the row space of a Boolean matrix A of order n. We show that if n 7, then the cardinality |R(A)| (2^n–1 - 2^n–5, 2^n–1 - 2^n–6) U (2^n–1 - 2^n–6, 2^n–1). This result confirms a conjecture in [1].AMS Subject Classification (1991): 05B20 06E05 15A36Support partially by the Postdoctoral Science Foundation of China.Dedicated to Professor Chao Ko on the occasion of his 90th birthday     

Little-Preemptive Scheduling on Unrelated Processors

We consider the preemptive scheduling of n independent jobs on m unrelated machines to minimize the makespan. Preemptive schedules with at most 2m–3 preemptions are built, which are optimal when the maximal job processing time is no more than the optimal schedule makespan. We further restrict the maximal job processing time and obtain optimal schedules with at most m–1 preemptions. This is better than the earlier known best bound of 4m ²–5m+2 on the total number of preemptions. Without the restriction on the maximal job processing time, our (2m–3)-preemptive schedules have a makespan which is no more than either of the following two magnitudes: (a) the maximum between the longest job processing time and the optimal preemptive makespan, and (b) the optimal nonpreemptive makespan. Our (m–1)-preemptive schedules might be at most twice worse than an optimal one.