Batalin-Vilkovisky Algebras and Cyclic Cohomology of Hopf Algebras

We show that the Connes–Moscovici negative cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a cyclic operad with multiplication is a cocyclic module whose simplicial cohomology is a Batalin–Vilkovisky algebra and whose negative cyclic cohomology is a graded Lie algebra of degree -2. This generalizes the fact that the Hochschild cohomology algebra of a symmetric algebra is a Batalin–Vilkovisky algebra.     

Kinetic theory of strength as a scientific basis for predicting the lifetime of polymers under load

The possibility of using advances in the kinetic theory of strength for predicting the performance of polymers is analyzed. It is shown that the theory provides a scientific basis for predicting the life of polymers under load not only in uniaxial tension at constant stress, when the basic lifetime equation with constant coefficients ₀, U₀, and holds, but also in the presence of various complications — in the region of small stresses and high temperatures, in the presence of structural changes during service, in complex loading regimes (including cyclic loading) and complex states of stress, and, moreover, when the stressed polymer is exposed to radiations or aggressive media.Presented at the All-Union Symposium on Predicting Service Properties of Polymeric Materials, Tbilisi, 1970.Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov No. 1, pp. 98–112, January–February, 1971.     

Fatigue strength of certain materials used in airframe construction

Conclusions The completed tests show that the fatigue strength of carbon-plastic is 2.5 times greater than that of glass-plastic. Consequently, it is best to use carbon-plastic with ED-20 epoxy resin filled with 4–5m glass microspheres for adhesive purposes in load-bearing components. This makes it possible to significantly reduce the weight of airframes and improve their service properties. It was established that, for the composite materials investigated, stiffness is lost during cyclic loading mainly as a result of damage accumulation; the loss in stiffness may serve as a criterion for evaluating the progress of fatigue fracture.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 451–455, May–June, 1980.     

Some algebraic properties of cyclic homology groups

In [10], see also [8], a cyclic homology theory HC _* ^– was introduced. The purpose of this paper is to study algebraically the properties of this version of cyclic homology. First we study its relation to Connes cyclic cohomology theory HC ^* and to the usual cyclic homology theory HC _* studied by Loday and Quillen in [15]. We explain the precise sense in which HC _* ^– is dual to HC ^*. Next we study products and describe a general method for constructing product operations in cyclic homology and cohomology theories. Finally we examine the relation between the theory HC _* ^– and algebraic K-theory.     

Global well-posedness of Korteweg–de Vries equation in

We prove that the Korteweg–de Vries initial-value problem is globally well-posed in and the modified Korteweg–de Vries initial-value problem is globally well-posed in . The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation in H^−3/4 by constructing some special resolution spaces in order to avoid some ‘logarithmic divergence’ from the high–high interactions. Our local solution has almost the same properties as those for H^s (s>−3/4) solution which enable us to apply the I-method to extend it to a global solution.     

Moisture and low temperature impact on the properties of a polyether glass-reinforced plastic

The cyclic action of water and the combined action of water and low temperatures on the physical-mechanical properties of a polyether glass-reinforced plastic are studied. For experimental investigations, methods of sorption, thermogravimetry, and mechanical testing in static bending were used. It was found that the most aggressive external factor for the glass-reinforced plastic was the cyclic action of water, but not the change in its phase state at cooling. The process of sorption-desorption of the glass-reinforced plastic is irreversible.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Institute of Non-Metallic Materials, Siberian Branch of the Russian Academy of Sciences, Yakutsk, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 4, pp. 519–524, July–August, 1998.     

Lifetime and relaxation processes in solid polymers

The relationship between relaxation processes and the fracture of solid polymers is examined with reference to the example of repeat loading with recovery. An investigation of polymethyl methacrylate, KAST-V glass laminate, and a Finnish polyester glass-reinforced plastic has shown that this lifetime may be considerably (several orders) shorter than that at the same level of constant stress. This is attributed to the time dependence of the structure-sensitive coefficient in the lifetime equation and is illustrated by means of a model that includes a Regel' fracture element.Moscow Lenin State Pedagogical Institute, Problem Laboratory of Polymer Physics. Translated from Mekhanika Polimerov, No. 4, pp. 629–635, July–August, 1969.     

Flat cyclic modules

Necessary and sufficient conditions are established for cyclic -modules to be flat, and rings over which any flat cyclic -module is free are investigated.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 245–250, August, 1970.The author wishes to thank L. A. Skornyakov for his advice and suggestions for improving this article.     

Effect of “rest” on creep of glass-reinforced plastic SVAM

The effect of rest on creep of glass-reinforced plastic SVAM is investigated. It is shown that for the cyclic loading regime used (24-h load, 24-h rest) the effect of rest is closely related with orientation of the specimen and with the magnitude of the load.Institute of Mechanics, Academy of Sciences of the Armenian SSR, Erevan. Translated from Mekhanika Polymerov, No. 3, pp. 547–550, May–June, 1974.     

3-Adic Cantor function on local fields and its p-adic derivative

The problem of “rate of change” for fractal functions is a very important one in the study of local fields. In 1992, Su Weiyi has given a definition of derivative by virtue of pseudo-differential operators [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China [series A] 1992;35(7A):826–36. Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5&6):805–24. [2] and [3]]. In Qiu Hua and Su Weiyi [Weierstrass-like functions on local fields and their p-adic derivatives. Chaos, Solitons & Fractals 2006;28(4):958–65. [8]], we have introduced a kind of Weierstrass-like functions in p-series local fields and discussed their p-adic derivatives. In this paper, the 3-adic Cantor function on 3-series field is constructed, and its 3-adic derivative is evaluated, it has at most order. Moreover, we introduce the definition of the Hausdorff dimension [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. [1]] of the image of a complex function defined on local fields. Then we conclude that the Hausdorff dimensions of the 3-adic Cantor function and its derivatives and integrals on 3-series field are all equal to 1.There are various applications of Cantor sets in mechanics and physics. For instance, E-infinity theory [El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. El Naschie MS. Dimensions and Cantor spectra. Chaos, Solitons & Fractals 1994;4(11):2121–32. El Naschie MS. Einstein’s dream and fractal geometry. Chaos, Solitons & Fractals 2005;24(1):1–5. El Naschie MS. The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22(2):495–511. [9], [10], [11] and [12]] is based on random Cantor set which takes the golden mean dimension as shown by El Naschie.     

Effect of ultraviolet radiation on the lifetime of polymers under load in the low-stress region

In the low-stress region the lifetime of polymers placed under load in the presence of UV radiation is appreciably greater in a vacuum than in air. This indicates that the ambient medium plays an important part in the process of restoration of broken bonds and that the knee on the log =f() curve is at least partly due to recombination processes.Ioffe Physico-Technical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 5, pp. 929–931, September–October, 1969.     

Lifetime analysis with allowance for bond stress distribution

A fracture model consisting of a set of bonds assigned various stress concentration coefficients is proposed. The bond lifetime is determined from Zhurkov's equation; the relative number of bonds with a particular _* being taken in accordance with a Weibull distribution. The model explains the similarity of the shapes of the submicrocrack accumulation and creep curves.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 227–231, March–April, 1973.     

The p-rank of Artin-Schreier curves

The groundfield k is algebraically closed and of characteristic p O. The p-rank of an abelian variety A/k is _A if there are _A copies of Z/pZ in the group of points of order p in A(k). The p-rank _X of a curve X/k is the p-rank of its Jacobian. In general the genus of X is _X. X is ordinary if equality holds.Proposition 3.2 proves that the Artin-Schreier curve X_p with equation (x^p–x)(y^p–y)=1 is ordinary. As its genus is (p–1)(p–1) and it has at least 2p. p. (p–1) automorphisms, it is an ordinary counter example of Hurwitz's theorem if p>37. Theorem 3.5 is the inductive step in extending this to smaller characteristics. Both are corollaries of Theorem 4.1 which is our principal result: if YX is a cyclic covering of degree p ramified at n distinct points, then (_Y–1+n)=(_X–1+n)×p. The particular case n=0, the unramiried case, is due to afarevi [7].The preparation of this paper was supported by the Memorial University of Newfoundland and NRC Grant A-8777.     

Effect of starting structure on the lifetime of a polymer under load in the presence of ultraviolet radiation

An investigation of the behavior of the coefficents A_j and _j in the radiation lifetime equation _j=A_je–_j with variation of the starting structure of polycaprolactam has revealed the structure-sensitivity of _j and the structure-insensitivity of A_j. This and the fact that _j is proportional to the structure-sensitive coefficient in the equation for the lifetime without irradiation, together with the previous experimental data, provide a basis for the development of a theory of the mechanism of polymer fracture in the presence of UV radiation and throw light on the physical significance of the coefficients A_j and _j.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 4, pp. 636–641, July–August, 1969.     

Globally Optimized Spherical Point Arrangements: Model Variants and Illustrative Results

The well-balanced distribution of points over the surface of a sphere is of significant interest in various fields of science. The quality of point configurations is typically expressed by criterion functions that have many local optima. A general global optimization framework is suggested to solve such problems. To illustrate the viability of this approach, the model development and solver system LGO is applied to four different model versions. Numerical results – including the visual representation of criterion functions in these models – are presented. The global optimization approach can be tailored to specific problem settings, and it is also applicable to a large variety of other model forms.     

Reasons for the lowering of the lifetime of polymers under cyclic loading

A model is proposed which explains the lowering of the lifetime of polymers on transition from static to cyclic loading by the interaction of local stresses from an external load with residual stresses which arose after preceding loading cycles. The results predicted on the basis of the model have been checked experimentally on polymethyl methacrylate, and a good agreement has been shown.M. I. Kalinin Leningrad Polytechnic Institute. Translated from Mekhanika Polimerov, No. 2, pp. 279–283, March–April, 1976.     

Lifetime levels and fracture of polyethylene terephthalate exposed to γ radiation and an aqueous medium

Conclusion 1. An analysis of the lifetime distribution curves obtained by varying the test conditions has revealed discrete lifetime levels in PETP due to the presence in the material of structural defects with different degrees of dangerousness. These defects are found at the edges, at the surface and in the interior of the specimens.2. Water affects irradiated and unirradiated PETP by reducing the probability of failure, rendering the structural defects less dangerous and, as a result, increasing the lifetime and impeding the fracture process.3. On the dose range investigated, rays affect PETP specimens by causing defects that reduce the lifetime and facilitate the fracture process, the defects produced becoming more dangerous as the dose increases.V. I. Lenin Tadzhik State University, Dushanbe. Translated from Mekhanika Polimerov, No. 6, pp. 1060–1064, November–December, 1978.     

Effect of cyclic cooling to low temperatures on the adhesive strength of fiber-thermoplastic matrix joints

The adhesive strength of the interface between the polyarylene sulfone PSK-1 and a steel wire 150 mm in diameter was measured using a pull-out technique after cyclic cooling of the joints from room temperature down to liquid nitrogen temperature. All specimens were prepared under isothermal conditions (300 °C, 30 min) without applying pressure. The number of cycles N of plunging in liquid nitrogen reached 900. All measurements were taken at room temperature with constant loading rate. The joint areas S were varied within a wide range. It was found that ten cooling cycles do not cause any appreciable change in the values of (Figs. 1–3). After 100 cooling cycles, the adhesive strength losses reached 30–45% (Figs. 1–3). After 810 cycles, the fiber-matrix interface is almost totally destroyed (Fig. 1). As the number of cycles N increases, the character of the dependence of the adhesive strength on the joint area changes: when N < 100, the values of are reduced as the interfacial surface area increases; when N = 810, the adhesive strength does not depend on S. The main factor that governs the reduction in adhesive strength is assumed to be residual thermal stresses. Analysis of the data obtained here and earlier shows that the behavior of polymer-fiber systems after cyclic cooling down to low temperatures is the same for both network and linear polymer matrices. The origin of this behavior also seems to be the same for all polymer-fiber adhesive joints.Report presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 2, pp. 156–162, March–April, 1995.This work was done with the support of the International Science Foundation (the G. Soros Foundation), grant NED000.     

Analysis of possible combinations of local stresses in local strain theory

In determining the strain tensor e_ij instead of all the components of the local stress deviator, it is possible to use only the shear stress _xz acting on a small local area. This fact makes it easy to establish the incremental loading conditions in solving complex loading problems for a plastic material. It is shown that in the local strain theory, distinct from the deformation theory, at degrees of nonlinearity n>3 the effect of the third, as well as the second, invariant of the stress deviator is taken into account.Mekhanika Polimerov, Vol. 3, No. 4, pp. 636–644, 1967     

Pointwise estimation of comonotone approximation

We prove that, for a continuous functionf(x) defined on the interval [–1,1] and having finitely many intervals where it is either nonincreasing or nondecreasing, one can always find a sequence of polynomialsP _n (x) with the same local properties of monotonicity as the functionf(x) and such that ¦f(x)–P _n (x) ¦C₂(f;n^–2+n ^–11–x ²), whereC is a constant that depends on the length of the smallest interval.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1467–1472, November, 1994.The author is grateful to Prof. I. A. Shevchuk for his permanent attention to the work.