Uniqueness theorem for algebraic curves on compact Riemann surfaces

In this paper, we prove a uniqueness theorem for algebraic curves from a compact Riemann surface into complex projective spaces.     

Existence and extremal solutions of parabolic variational–hemivariational inequalities

We consider quasilinear parabolic variational–hemivariational inequalities in a cylindrical domain $Q=\Omega \times (0,\tau )$ of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q j^o(x,t, u;v-u)\,dxdt\ge 0,\ \ \forall \ v\in K, \end{aligned}$$ where $K\subset X_0=L^p(0,\tau ;W_0^{1,p}(\Omega ))$ is some closed and convex subset, $A$ is a time-dependent quasilinear elliptic operator, and $s\mapsto j(\cdot ,\cdot ,s)$ is assumed to be locally Lipschitz with $(s,r)\mapsto j^o(x,t, s;r)$ denoting its generalized directional derivative at $s$ in the direction $r$ . The main goal of this paper is threefold: first, an existence and comparison principle is proved; second, the existence of extremal solutions within some sector of appropriately defined sub-supersolutions is shown; third, the equivalence of the above parabolic variational–hemivariational inequality with an associated multi-valued parabolic variational inequality of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q \eta \, (v-u)\,dxdt\ge 0,\ \ \forall \ v\in K \end{aligned}$$ with $\eta (x,t)\in \partial j(x,t, u(x,t))$ is established, where $s\mapsto \partial j(x,t, s)$ denotes Clarke’s generalized gradient of the locally Lipschitz function $s\mapsto j(\cdot ,\cdot ,s)$ .     

Finiteness property of pairs of 2×2 sign-matrices via real extremal polytope norms

This paper deals with the joint spectral radius of a finite set of matrices. We say that a set of matrices has the finiteness property if the maximal rate of growth, in the multiplicative semigroup it generates, is given by the powers of a finite product.Here we address the problem of establishing the finiteness property of pairs of $2\times 2$ sign-matrices. Such problem is related to the conjecture that pairs of sign-matrices fulfil the finiteness property for any dimension. This would imply, by a recent result by Blondel and Jungers, that finite sets of rational matrices fulfil the finiteness property, which would be very important in terms of the computation of the joint spectral radius. The technique used in this paper could suggest an extension of the analysis to $n\times n$ sign-matrices, which still remains an open problem.As a main tool of our proof we make use of a procedure to find a so-called real extremal polytope norm for the set. In particular, we present an algorithm which, under some suitable assumptions, is able to check if a certain product in the multiplicative semigroup is spectrum maximizing.For pairs of sign-matrices we develop the computations exactly and hence are able to prove analytically the finiteness property. On the other hand, the algorithm can be used in a floating point arithmetic and provide a general tool for approximating the joint spectral radius of a set of matrices.     

Moduli for pairs of elliptic curves¶with isomorphic N-torsion

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a “determinant” map from this moduli surface to (Z/N Z)^*; its fibers are the components of the surface. We define spaces of modular forms on these components and Hecke correspondences between them, and study how those spaces of modular forms behave as modules for the Hecke algebra. We discover that the component with determinant −1 is somehow the “dominant” one; we characterize the difference between its spaces of modular forms and the spaces of modular forms on the other components using forms with complex multiplication. In addition, we prove Atkin–Lehner-style results about these spaces of modular forms. Finally, we show some simplifications that arise when N is prime, including a complete determination of such CM-forms, and give numerical examples. Received: 20 September 2000 / Revised version: 7 February 2001     

Is there an easy algebraic characterisation of¶universal proper G-spaces?

We shall study properties of groups having finite cohomological dimension relative to the family of all finite subgroups. We also compare these groups with those satisfying various suggested algebraic analogues to group-actions on finite dimensional proper G-spaces. Received: 15 March 2000     

Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities

We use the formalism of the Rényi entropies to establish the symmetry range of extremal functions in a family of subcritical Caffarelli–Kohn–Nirenberg inequalities. By extremal functions we mean functions that realize the equality case in the inequalities, written with optimal constants. The method extends recent results on critical Caffarelli–Kohn–Nirenberg inequalities. Using heuristics given by a nonlinear diffusion equation, we give a variational proof of a symmetry result, by establishing a rigidity theorem: in the symmetry region, all positive critical points have radial symmetry and are therefore equal to the unique positive, radial critical point, up to scalings and multiplications. This result is sharp. The condition on the parameters is indeed complementary of the condition that determines the region in which symmetry breaking holds as a consequence of the linear instability of radial optimal functions. Compared to the critical case, the subcritical range requires new tools. The Fisher information has to be replaced by Rényi entropy powers, and since some invariances are lost, the estimates based on the Emden–Fowler transformation have to be modified.     

Liénard systems for quadratic systems with invariant algebraic curves

We study the character of the friction function f(x) and the restoring force g(x) in the Liénard system to which a quadratic system with an invariant second-order algebraic curve (an ellipse that is a limit cycle, a hyperbola defining two separatrix cycles, or a parabola) or fourth-order algebraic curve with an oval being a limit cycle can be reduced. Invariant curves are constructed for quadratic systems in a five-parameter canonical family, which can readily be reduced to Liénard systems.     

On the Cohen-Macaulay property for monomial curves in ℙ K 3

The arithmetical Cohen-Macaulay property for monomial curves in _K ³ with generic zero was shown in [2] to be true forn ₃>(n ₂–1)(n ₂–n ₁),n ₃>n ₂, (n ₁,n ₂,n ₃)=1. Here we establish necessary and sufficient arithmetic conditions in terms ofn ₁,n ₂,n ₃ in order for the indicated curves not to be arithmetically Cohen-Macaulay.     

Relative equidimensionality and stability of actions¶of a reductive algebraic group

In order to inquire into invariants of non-semisimple groups, we introduce and study relative versions of equidimensionality and stabilty, which are called relative quasi-equidimensionality and relative stability, of actions of affine algebraic groups, especially of reductive groups, on affine varieties. As an application of our results, for complex reductive groups of semisimple rank one, we characterize, respectively, relatively stable representations and relatively equidimensional representations and, consequently, show that every equidimensional representation is cofree. Received: 23 October 1998     

Consistencies and inconsistencies in students' solutions to algebraic ‘single-value’ inequalities

This paper describes students’ solutions to a commonly taught and not commonly taught inequality. The findings showed students’ difficulties. Participants implicitly and explicitly exhibited two intuitive beliefs: inequalities must result in inequalities and solving inequalities and equations are the same process. Following the analysis of students’ written solutions, individual interviews were conducted that gave a better insight into their reasoning and provided some ideas for teaching. The concluding section of the paper offers relevant educational implications.     

Néron-Tate heights on algebraic curves and subgroups of the modular group

Combining Arakelov theory with Belyis theorem we derive that the values of the Néron-Tate height pairing for divisors on algebraic curves defined over number fields are essentially given by linear combinations of scattering constants associated to finite index subgroups of the modular group PSL₂().Mathematics Subject Classification (2000): 14G40, 11G40, 11G50Supported by the TMR-Network Arithmetic Algebraic Geometry     

Smashing subcategories and the telescope conjecture – an algebraic approach

We prove a modified version of Ravenel’s telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pure-injective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Oblatum 23-XI-1998 & 19-V-1999 / Published online: 5 August 1999     

Invariant algebraic curves and rational first integrals of holomorphic foliations in CP(2)

In this paper, using the tools of algebraic geometry we provide sufficient conditions for a holomor-phic foliation in CP(2) to have a rational first integral. Moreover, we obtain an upper bound of the degrees of invariant algebraic curves of a holomorphic foliation in CP(2). Then we use these results to prove that any holomorphic foliation of degree 2 does not have cubic limit cycles.     

Mixed boundary value problems for concentric cylinders—an algebraic operator formulation

Unitary operators are introduced which act on the Fourier transforms of the boundary values on concentric cylinders of a harmonic function in E ₃. The boundary values on the concentric cylinders are determined from given mixed boundary conditions on the same cylinders, and the solution for them is expressed explicitly and simply through the use of these unitary operators plus certain other self adjoint positive operators. The boundary values on the cylinders then determine the harmonic function everywhere in E ₃.     

The Nother and Riemann-Roch type theorems for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nother type theorems for Cμpiecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible Cμpiecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the Cμpiecewise algebraic curve is established.     

Pre- and in-service teachers’ perceived value of an experimental real analysis course for teachers

Prospective secondary mathematics teachers are usually required to complete several university advanced mathematics courses before being certified to teach secondary mathematics. However, teachers usually do not find these courses to be valuable for their teaching. We designed an experimental real analysis course with the goal of making real analysis content useful and relevant to teaching. Our approach was to ground the real analysis content in pedagogical situations that problematized a secondary mathematics topic, where the nuances of teaching secondary mathematics could be informed by the real analysis that was covered. The experimental course was implemented in a graduate teacher education programme with 32 pre- and in-service teachers (PISTs). After the course, we conducted focus group interviews with 20 of these PISTs to get feedback on how the course was valuable to their teaching practice. Many PISTs found the course to be valuable for teaching secondary mathematics, as well as for their understanding of secondary mathematics and real analysis.     

Some inequalities and convergence theorems for Choquet integrals

Fuzzy measure (or non-additive measure), which has been comprehensively investigated, is a generalization of additive probability measure. Several important kinds of non-additive integrals have been built on it. Integral inequalities play important roles in classical probability and measure theory. In this paper, we discuss some of these inequalities for one kind of non-additive integrals—Choquet integral, including Markov type inequality, Jensen type inequality, Hölder type inequality and Minkowski type inequality. As applications of these inequalities, we also present several convergence concepts and convergence theorems as complements to Choquet integral theory.