Determination of the rheological characteristics of polymer materials from the static σ-ε curves and creep and relaxation curves

A method is proposed for determining the rheological characteristics of polymer materials from the experimental - curves and the creep and relaxation curves, when the behavior of the material is described exactly and approximately by the equation of a standard solid, together with a method of determining the conditional rheological characteristics when only a certain section of these curves is approximately described by the equation of a standard solid. The proposed method makes it possible to eliminate the ambiguity in the determination of the rheological characteristics due to difficulties in the exact determination of the coordinates of the beginning and end of the static curves. The use of conditional rheological characteristics makes it possible to describe the behavior of polymer materials over a broad time interval under static loading conditions by means of the equation of a standard solid without resorting to the use of complex spectral functions. A relation is established between the spectral viscoelastic functions and the conditional rheological characteristics.Mekhanika Polimerov, Vol. 3, No. 6, pp. 977–988, 1967     

Zeta functions of trinomial curves and maximal curves

We determine the zeta functions of trinomial curves in terms of Jacobi sums, and obtain an explicit formula of the genus of a trinomial curve over a finite field, and we study the conditions for this curve to be a maximal curve over a finite field.     

Relationship between polymer creep curves and stress-strain diagrams

For polymers whose () diagram has the form of a monotonically increasing curve the creep rate is monotonically damped with time. If, however, the stress-strain diagram has a maximum, the creep curve becomes stepped: after first slowing, the creep sharply accelerates, and then slows down again. The strains corresponding to characteristic points on the stress-strain curves coincide with those corresponding to singular points on the creep curves.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 787–792, September–October, 1968.     

Hypergeometric functions and elliptic curves

We provide uniform formulas for the real period and the trace of Frobenius associated to an elliptic curve in Legendre normal form. These are expressed in terms of classical and Gaussian hypergeometric functions, respectively. 2000 Mathematics Subject Classification Primary—11G05, 33C05 This research was supported by K. Ono’s NSF grant     

Simple functions on space curves

We classify simple singularities of functions on space curves. We show that their bifurcation sets have the same properties as those of functions on smooth manifolds and complete intersections [3, 4]: thek(π, 1)-theorem for the bifurcation diagram of functions is true, and both this diagram and the discriminant are saito's free divisors. Department of Mathematical Sciences Division of Pure Mathematics, The University of Liverpool. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol, 34, No. 2, pp. 63–67, April–June, 2000. Translated by V. V. Goryunov     

Traces of pluriharmonic functions on curves

We prove that, if γ is a simple smooth curve in the unit sphere inC ⁿ, the space o pluriharmonic functions in the unit ball, continuous up to the boundary, has a trace of finite cof dimension in the space of all continuous functions on the curve. First author partially supported by the Swedish Natural Science Research Council. Second author partially supported by CICYT grant PB85-0374.     

Hilbert functions of Gorenstein monomial curves

It is a conjecture due to M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. In this article, we show that the Hilbert function is non-decreasing for local Gorenstein rings with embedding dimension four associated to monomial curves, under some arithmetic assumptions on the generators of their defining ideals in the non-complete intersection case. In order to obtain this result, we determine the generators of their tangent cones explicitly by using standard basis computations under these arithmetic assumptions and show that the tangent cones are Cohen-Macaulay. In the complete intersection case, by characterizing certain families of complete intersection numerical semigroups, we give an inductive method to obtain large families of complete intersection local rings with arbitrary embedding dimension having non-decreasing Hilbert functions.

     

Traces of pluriharmonic functions on curves

We prove that, if γ is a simple smooth curve in the unit sphere inC ⁿ, the space o pluriharmonic functions in the unit ball, continuous up to the boundary, has a trace of finite cof dimension in the space of all continuous functions on the curve.     

Ultradifferentiable functions on smooth plane curves

The regularity, in the sense of ultradifferentiability, of real functions of two variables is determined in terms of the regularity of their restrictions to a given family of smooth plane curves. The special case of line segments reduces to the main result in [Proc. Amer. Math. Soc. 127 (1999) 2099-2104]. As a consequence, the Bochnak-Siciak theorem on real analyticity is obtained. A formal analog of one of the results provides a generalization of the two-variable case of the Abhyankar and Moh [J. Reine Angew. Math. 241 (1970) 27-33] theorem.     

Invariant analytic curves for entire functions

Kharkov. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 3, pp. 3–8, May–June, 1989.     

Elliptic functions and equations of modular curves

Let be a prime. We show that the space of weight one Eisenstein series defines an embedding into ${mathbb P}^{(p-3)/2}X_1(p)$ for the congruence group that is scheme-theoretically cut out by explicit quadratic equations. Received: 8 November 2000 / Published online: 17 August 2001     

Dimensions of the coordinate functions of space-filling curves

The graphs of coordinate functions of space-filling curves such as those described by Peano, Hilbert, Pólya and others, are typical examples of self-affine sets, and their Hausdorff dimensions have been the subject of several articles in the mathematical literature. In the first half of this paper, we describe how the study of dimensions of self-affine sets was motivated, at least in part, by these coordinate functions and their natural generalizations, and review the relevant literature. In the second part, we present new results on the coordinate functions of Pólya's one-parameter family of space-filling curves. We give a lower bound for the Hausdorff dimension of their graphs which is fairly close to the box-counting dimension. Our techniques are largely probabilistic. The fact that the exact dimension remains elusive seems to indicate the need for further work in the area of self-affine sets.     

The Igusa local zeta functions of elliptic curves

We determine the explicit form of the Igusa local zeta function associated to an elliptic curve. The denominator is known to be trivial. Here we determine the possible numerators and classify them according to the Kodaira-Néron classification of the special fibers of elliptic curves as determined by Tate's algorithm.