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     

Vanishing of higher-order moments on Lipschitz curves

We study some algebraic and topological objects that appear naturally in the study of the center problem for the ordinary differential equation . In particular, we give a topological characterization of Lipschitz curves defined by the first integrals of the coefficients of this equation such that all moments of order?n, n∈N, vanish on them.     

A new flow on starlike curves in

In this note we find a new evolution equation for starlike curves in . We study the evolution of the subaffine curvature and subaffine torsion under the flow and show that it is completely integrable. The solutions to the evolution which move without changing affine shape are subaffine elastic curves. We integrate the subaffine elastica by quadratures.

     

Positive Solutions of Nonlinear Elliptic Equations in Unbounded Domains in R 2

We study the existence of positive solutions of the nonlinear equation u+f(,u)=0, in D with u=0 on D, where D is an unbounded domain in R ² with a compact nonempty boundary D consisting of finitely many Jordan curves. The aim is to prove an existence result for the above equation in a general setting by using potential theory.     

The elliptic-in-t solutions of the nonlinear Schrödinger equation

Four various anzatzes of the Krichever curves for the elliptic-in-t solutions of the nonlinear Schrödinger equation are considered. An example is given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 188–200, May, 1996.Translated by V. I. Serdobol'skii.     

Division Algebras that Ramify only on a Plane Quartic Curve with Simply Connected Components

A central division algebra over the field of rational functions in two variables with coefficients over an algebraically closed field ramifies along a divisor on P ². If the ramification divisor of is a quartic curve which is the union of simply connected curves, we show that is a symbol algebra and satisfies the index equals exponent equation.     

Primes generated by elliptic curves

For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the -coordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a rational point under an isogeny, we use Siegel's Theorem to prove that only finitely many primes will arise. The same question is considered for elliptic curves in homogeneous form, prompting a visit to Ramanujan's famous taxi-cab equation. Finiteness is provable for these curves with no extra assumptions. Finally, consideration is given to the possibilities for prime generation in higher rank.

     

A Ray Method for the Asymptotic Solution of the Diffusion Equation

A general method is presented for finding asymptotic solutionsof problems posed for linear partial differential equationscontaining lower order (dispersive) terms. Powers of a largeparameter appear in these equations multiplying the lower orderterms. The expansion procedure is a "ray method", i.e. all ofthe functions that appear in the expansion satisfy ordinarydifferential equations along certain curves called rays. Special attention is paid to the equation of heat conduction.In this case, 1/ is related to the thermal conductivity of themedium. For this equation several problems are considered inwhich the parameter enters the data or the inhomogeneous (source)term in various ways.     

A lower bound for the ground state energy of a Schrödinger operator on a loop

Consider a one-dimensional quantum mechanical particle described by the Schrödinger equation on a closed curve of length . Assume that the potential is given by the square of the curve's curvature. We show that in this case the energy of the particle cannot be lower than . We also prove that it is not lower than (the conjectured optimal lower bound) for a certain class of closed curves that have an additional geometrical property.

     

Moduli of twisted spin curves

In this note we give a new, natural construction of a compactification of the stack of smooth -spin curves, which we call the stack of stable twisted -spin curves. This stack is identified with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible -spaces and -line bundles are given as well. The infinitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves.

We construct representable morphisms from the stacks of stable twisted -spin curves to the stacks of stable -spin curves and show that they are isomorphisms. Many delicate features of -spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the -operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.

     

Structure of node polynomials for curves on surfaces

We provide a structural generalization of a theorem by Kleiman–Piene, concerning the enumerative geometry of nodal curves in a complete linear system on a smooth projective surface S. Provided that r, the number of nodes, is sufficiently small compared to the ampleness of the linear system, we show that, under certain assumptions, the number of r‐nodal curves passing through points in general position on S is given by a Bell polynomial in universally defined integers which we identify, using classical intersection theory, as linear, integral polynomials evaluated in four basic Chern numbers. Furthermore, we provide a decomposition of the as a sum of three terms with distinct geometric interpretations, and discuss the relationship between these polynomials and Kazarian's Thom polynomials for multisingularities of maps.     

Humbert surfaces and the Kummer plane

A Humbert surface is a hypersurface of the moduli space of principally polarized abelian surfaces defined by an equation of the form with integers . We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.

     

CM points on products of Drinfeld modular curves

Let be a product of Drinfeld modular curves over a general base ring of odd characteristic. We classify those subvarieties of which contain a Zariski-dense subset of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.

     

On the Points on the Unit Circle with Finite b–Adic Expansions

Let be an arbitrary integer base and let be the number of different prime factors of with , . Further let be the set of points on the unit circle with finite –adic expansions of their coordinates and let be the set of angles of the points . Then is an additive group which is the direct sum of infinite cyclic groups and of the finite cyclic group . If in case of the points of are arranged according to the number of digits of their coordinates, then the arising sequence is uniformly distributed on the unit circle. On the other hand, in case of the only points in are the exceptional points (1, 0), (0, 1), (–1, 0), (0, –1). The proofs are based on a canonical form for all integer solutions of .     

On the Equation xp + y q = z r : A Survey

Let p, q and r be three integers 2. During the last ten years many new ideas have emerged for the study of the Diophantine equation x + y = z. This study is divided into three parts according whether (p, q, r) = is negative, zero or positive. For instance, if we have (p, q, r) < 0, this study is closely connected to the theory of modular representations of dimension 2 in finite characteristic. The approach of associating to this equation Galois representations given by the division points of elliptic curves is very efficient. The aim of this paper is to present this circle of ideas and the known results concerning this equation.     

Linear algebra algorithms for divisors on an algebraic curve

We use an embedding of the symmetric th power of any algebraic curve of genus into a Grassmannian space to give algorithms for working with divisors on , using only linear algebra in vector spaces of dimension , and matrices of size . When the base field is finite, or if has a rational point over , these give algorithms for working on the Jacobian of that require field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to deal with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Florian Hess' 1999 Ph.D. thesis, which works with function fields as extensions of . However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor 1987), superelliptic curves (Galbraith, Paulus, and Smart 2002), and curves (Harasawa and Suzuki 2000); in all those cases, one can attain a complexity of .

     

Explicit Rational Functions on Fermat Curves and a Theorem of Greenberg

This paper is concerned with the arithmetic of curves of the form v^p=u^s(1-u), where p is a prime with p 5 and s is an integer such that 1 s p-2. The Jacobians of these curves admit complex multiplication by a primitive p-th root of unity . We find explicit rational functions on these curves whose divisors are p-multiples of divisors representing (1-)² - and (1-)³-division points on the corresponding Jacobians. This also gives an effective version of a theorem of Greenberg.     

Relaxation of the curve shortening flow via the parabolic Ginzburg -Landau equation

In this paper we study how to find solutions to the parabolic Ginzburg–Landau equation that as have as interface a given curve that evolves under curve shortening flow. Moreover, for compact embedded curves we find a uniform profile for the solution up the extinction time of the curve. We show that after the extinction time the solution converges uniformly to a constant.     

Generalized Hopf bifurcation in Hilbert space

We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations.     

Restrictions on arrangements of ovals of projective algebraic curves of odd degree

This paper investigates the first part of Hilbert's 16th problem which asks about topology of the real projective algebraic curves. Using the Rokhlin-Viro-Fiedler method of complex orientation, we obtain new restrictions on the arrangements of ovals of projective algebraic curves of odd degree , , with nests of depth .