Cycles of covers

We initially consider an example of Flynn and Redmond, which gives an infinite family of curves to which Chabauty’s Theorem is not applicable, and which even resist solution by one application of a certain bielliptic covering technique. In this article, we shall consider a general context, of which this family is a special case, and in this general situation we shall prove that repeated application of bielliptic covers always results in a sequence of genus 2 curves which cycle after a finite number of repetitions. We shall also give an example which is resistant to repeated applications of the technique. E. V. Flynn thanks the International Center for Transdisciplinary Studies at Jacobs University Bremen for its hospitality during July 2007, and thanks EPSRC for support: grant number EP/F060661/1.     

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form Y ²=f ₃ X ⁶+f ₂ X ⁴+f ₁ X ²+f ₀, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα : ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ. Received: 27 November 1998 / Revised version: 4 June 1999     

Application of covering techniques to families of curves

Much success in finding rational points on curves has been obtained by using Chabauty's Theorem, which applies when the genus of a curve is greater than the rank of the Mordell-Weil group of the Jacobian. When Chabauty's Theorem does not directly apply to a curve , a recent modification has been to cover the rational points on by those on a covering collection of curves , obtained by pullbacks along an isogeny to the Jacobian; one then hopes that Chabauty's Theorem applies to each . So far, this latter technique has been applied to isolated examples. We apply, for the first time, certain covering techniques to infinite families of curves. We find an infinite family of curves to which Chabauty's Theorem is not applicable, but which can be solved using bielliptic covers, and other infinite families of curves which even resist solution by bielliptic covers. A fringe benefit is an infinite family of Abelian surfaces with non-trivial elements of the Tate-Shafarevich group killed by a bielliptic isogeny.     

On certain remarkable curves of genus five

The aim of this note is twofold. First to show the existence of genus five curves having exactly twenty four Weierstrass points, which constitute the set of fixed points of three distinct elliptic involutions on them. Second to characterize these curves, in fact we prove that all such curves are bielliptic double cover of Fermat's quartic.     

Scrollar invariants and resolutions of certaind-gonal curves

In this note we compute the scrollar invariants of certaind-gonal curves (e.g. Castelnuovo curves and bielliptic curves) by using appropriate plane models. Ford=4 andg(C)≥10, we show that those invariants discriminate bielliptic curves among tetragonal ones.

---

Sunto In questa nota si determinano gli invarianti scrollari per alcuni tipi di curved-gonaliC (ad esempio curve di Castelnuovo, curve biellittiche) tramite appropriati modelli piani. Perd=4 eg(C)≥10, si mostra che tali invarianti individuano le curve biellittiche fra le tetragonali.

---

---

An erratum to this article is available at .     

Some remarks on reachability of infinite-dimensional linear systems

In this paper we shall be concerned with the question of reachability when allowing distribution inputs. We show that a certain class of systems accept distribution inputs, but, in general, they cannot be exactly reachable. We shall also consider the problem of the uniqueness of canonical realizations in relation to exact reachability, and show that Matsuo's result on uniqueness (Ph.D. dissertation, in preparation) does not apply to the example given in Baras, Brockett, and Fuhrmann [IEEE Trans. Automatic ControlAC-19 (1974), 693–700].     

Geometric Syzygies of Elliptic Normal Curves and their Secant Varieties

We show that the linear syzygy spaces of elliptic normal curves, their secant varieties and of bielliptic canonical curves are spanned by geometric syzygies.     

Fractional Weyl-Riesz Integrodifferentiation of Periodic Functions of Two Variables via the Periodization of the Riesz Kernel

We consider the periodization of the Riesz fractional integrals (Riesz potentials) of two variables and show that already in this case we come across different effects, depending on whether we use the repeated periodization, first in one variable, and afterwards in another one, or the so called double periodization. We show that the naturally introduced doubly-periodic Weyl-Riesz kernel of order 0< f <2 in general coincides with the periodization of the Riesz kernel, the repeated periodization being possible for all 0< f <2 , while the double one is applicable only for 0< f <1 . This is obtained as a realization of a certain general scheme of periodization, both repeated and double versions. We prove statements on coincidence of the corresponding periodic and nonperiodic convolutions and give an application to the case of the Riesz kernel.     

Multiplication formulas for Kubert functions

The aim of this paper is two folds. First, we shall prove a general reduction theorem to the Spannenintegral of products of （generalized） Kubert functions. Second, we apply the special case of Carlitz＇s theorem to the elaboration of earlier results on the mean values of the product of Dirichlet L-functions at integer arguments. Carlitz＇s theorem is a generalization of a classical result of Nielsen in 1923. Regarding the reduction theorem, we shall unify both the results of Carlitz （for sums） and Mordell （for integrals）, both of which are generalizations of preceding results by Frasnel, Landau, Mikolas, and Romanoff et al. These not only generalize earlier results but also cover some recent results. For example, Beck＇s lamma is the same as Carlitz＇s result, while some results of Maier may be deduced from those of Romanoff. To this end, we shall consider the Stiletjes integral which incorporates both sums and integrals. Now, we have an expansion of the sum of products of Bernoulli polynomials that we may apply it to elaborate on the results of afore-mentioned papers and can supplement them by related results.     

Finite element error estimates for non-linear elliptic equations of monotone type

Summary In this paper we shall consider the application of the finite element method to a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient, and the derivation of error estimates for the finite element approximations. Such problems arise in many practical situations — for example, in shock-free airfoil design, seepage through coarse grained porous media, and in some glaciological problems. By making use of certain properties of the nonlinear coefficients, we shall demonstrate that the variational formulations associated with these boundary value problems are well-posed. We shall also prove that the abstract operators accompanying such problems satisfy certain continuity and monotonicity inequalities. With the aid of these inequalities and some standard results from approximation theory, we show how one may derive error estimates for the finite element approximations in the energy norm.     

Cauchy problems for Hilfer fractional evolution equations on an infinite interval

It is well known that the classical Ascoli-Arzelà theorem is powerful technique to give a necessary and sufficient condition for investigating the relative compactness of a family of abstract continuous functions, while it is limited to finite compact interval. In this paper, we shall generalize the Ascoli-Arzelà theorem on an infinite interval. As its application, we investigate an initial value problem for fractional evolution equations on infinite interval in the sense of Hilfer type, which is a generalization of both Riemann-Liuoville and Caputo fractional derivatives. Our methods are based on the Hausdorff theorem, classical/generalized Ascoli-Arzelà theorem, Schauder fixed point theorem, Wright function, and Kuratowski measure of noncompactness. We obtain the existence of mild solutions on an infinite interval when the semigroup is compact as well as noncompact.     

Curvature flow with driving force on fixed boundary points

In this paper, we consider the curvature flow with driving force on fixed boundary points in the plane. We give a general local existence and uniqueness result of this problem with \(C^2\) initial curve. For a special family of initial curves, we classify the solutions into three categories. Moreover, in each category, the asymptotic behavior is given.     

On some classes of uniqueness for the solution of integral geometry problems

We consider the problem of determining a function from a knowledge of integrals of the function along families of curves with a known weight function. For sufficiently general assumptions on the family of curves and on the weight function the problem is reduced to the solution of an integrodifferential equation. We establish the uniqueness of the solution of this equation in certain classes of functions.     

Graev metric groups and Polishable subgroups

We prove a conjecture of Hjorth: There is an uncountable Polish group all of whose abelian subgroups are discrete. We first construct directly a witness to Hjorth's conjecture. Then we consider an existing example in the literature. The example is the metric completion of a free topological group constructed by Graev. We give a definition slightly more general than Graev's and prove some properties of the Graev metrics which seem to be unknown previously. We also consider the problem of finding Polishable subgroups of the Graev metric groups with arbitrarily high Borel rank. In doing this we prove some general theorems on extensions of Polish groups with this property.     

Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin

Let T denote the unit circle in the plane. For various simple sets Λ in the plane we shall study the question whether (T,Λ) is a Heisenberg uniqueness pair. For example, we shall consider the cases where Λ is a circle or a union of two straight lines. We shall also use a theorem of Beurling and Malliavin.     

On the monodromy of weierstrass points

Summary In this paper we consider the family of curves of genus g=2m with a g ₃ ¹ lying on a particular rational normal scroll S in P^g– 1(C). We define a covering of this family representing the Weierstrass points and we study the monodromy. Applying the techniques of [3] we prove that if g=4 the monodromy is the full symmetric group and for general g=2m it is transitive. We show also that the generic curve of the family has only normal Weierstrass points generalizing a classical result. We work always over the complex numbers.Partially supported by: Ministero della Pubblica Istruzione - Italia; Consiglio Nazionale delle Ricerche — Italia.     

Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable. The prototypical example of such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third‐order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as multipoint conditions , and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e., it admits a unique solution. For the second‐order case, we also give criteria that guarantee well posedness.     

Long-Step Path-Following Algorithm for Convex Quadratic Programming Problems in a Hilbert Space

We develop an interior-point technique for solving quadratic programming problems in a Hilbert space. As an example, we consider an application of these results to the linear-quadratic control problem with linear inequality constraints. It is shown that the Newton step in this situation is basically reduced to solving the standard linear-quadratic control problem.     

Quasi-orthogonal decompositions of structured frames

A decomposition of a Hilbert space into a quasi-orthogonal family of closed subspaces is introduced. We shall investigate conditions in order to derive bounded families of corresponding quasi-projectors or resolutions of the identity operator. Given a local family of atoms, or generalized stable basis, for each subspace, we show that the union of the local atoms can generate a global frame for the Hilbert space. Corresponding duals can be calculated in a flexible way by means of systems of quasi-projectors. An application to Gabor frames is presented as example of the use of this technique, for calculation of duals and explicit estimates of lattice constants.     

Local analytic reduction of families of diffeomorphisms

We study local analytic simplification of families of analytic maps near a hyperbolic fixed point. A particularly important application of the main result concerns families of hyperbolic saddles, where Siegel's theorem is too fragile, at least in the analytic category. By relaxing on the formal normal form we obtain analytic conjugacies. Since we consider families, it is more convenient to state some results for analytic maps on a Banach space; this gives no extra complications. As an example we treat a family passing through a 1:−1 resonant saddle.