A simplified proof for the limit of a tower over a cubic finite field

Recently Bezerra, Garcia and Stichtenoth constructed an explicit tower F=(Fn)_n?0 of function fields over a finite field Fq³, whose limit λ(F)=limn→∞N(Fn)/g(Fn) attains the Zink bound λ(F)?2(q²−1)/(q+2). Their proof is rather long and very technical. In this paper we replace the complex calculations in their work by structural arguments, thus giving a much simpler and shorter proof for the limit of the Bezerra, Garcia and Stichtenoth tower.     

The Galois closure of Drinfeld modular towers

In this article we study Drinfeld modular curves X₀(pn) associated to congruence subgroups Γ₀(pn) of GL(2,Fq[T]) where p is a prime of Fq[T]. For n>r>0 we compute the extension degrees and investigate the structure of the Galois closures of the covers X₀(pn)→X₀(pr) and some of their variations. The results have some immediate implications for the Galois closures of two well-known optimal wild towers of function fields over finite fields introduced by Garcia and Stichtenoth, for which the modular interpretation was given by Elkies.     

On rational embeddings of curves in the second Garcia–Stichtenoth tower

In 1995, Garcia and Stichtenoth explicitly constructed a tower of projective curves over a finite field with q² elements which reaches the Drinfeld–Vlăduţ bound. These curves are given recursively by covers of Artin–Schreier type where the curve on the nth level of the tower has a natural model in . In this paper, for q an even prime power, we use point projections in order to embed these curves into projective space of the lowest possible dimension.     

A Class of Artin-Schreier Towers with Finite Genus

We study the asymptotic behaviour of the genus in some Artin-Schreier towers of function fields over a finite field, and we present a new class of Artin-Schreier towers having finite genus.     

Curves with Many Points and Multiplication Complexity in Any Extension of q

From the existence of algebraic function fields having some good properties, we obtain some new upper bounds on the bilinear complexity of multiplication in all extensions of the finite field _q, where q is an arbitrary prime power. So we prove that the bilinear complexity of multiplication in the finite fields _qⁿ is linear uniformly in q with respect to the degree n.     

A sequence of one-point codes from a tower of function fields

We construct a sequence of one-point codes from a tower of function fields whose relative minimum distances have a positive limit. Our tower is characterized by principal divisors. We determine completely the minimum distance of the codes from the first field of our tower. These results extend those of Stichtenoth [IEEE Trans Inform Theory (1988), 34(15):1345–1348], Yang and Kumar [Lecture Notes in Mathematics, 1518, (1991), Springer-Verlag, Berlin Hidelberg New York, pp. 99–107], and Garcia [Comm. Algebra, 20(12): 3683–3689]. As an application, we show that the minimum distance corresponds to the Feng–Rao bound.     

Zeta function of the projective curveaY 21 =bX 21 +cZ 21 over a class of finite fields, for odd primesl

Letp andl be rational primes such thatl is odd and the order ofp modulol is even. For such primesp andl, and fore = l, 2l, we consider the non-singular projective curvesaY ²¹ =bX ²¹ +cZ ²¹ defined over finite fields F_q such thatq = p ^α? l(mode).We see that the Fermat curves correspond precisely to those curves among each class (fore = l, 2l), that are maximal or minimal over F_q. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. Fore = 2l, we explicitly determine the ζ -function(s) for this class of curves, over F_q, as rational functions in the variablet, for distinct cases ofa, b, andc, in F _q ^* . Theζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves. Fore = l, 2l, we determine the class numbers for the function fields associated to each class of curves over F_q. As a consequence, when the field of definition of the curve(s) is fixed, this provides concrete information on the growth of class numbers for constant field extensions of the function field(s) of the curve(s).     

Some remarks on the Picard curves over a finite field

In this paper we study the Newton polygon of the L ‐polynomial L (t) associate to the Picard curves y³ = x⁴ – 1, y³ = x⁴ – x defined over a finite field ??_p . In the former case we get a complete classification. In the latter case we obtained a partial result. As a consequence of our result we obtain a criterion to find a supersingular Picard curves for the above two cases. Our main results are stated in Theorems 3.1 and 4.1. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Rees algebra of certain codimension two perfect ideals

 Suppose ? is a set of arbitrary number of smooth points in ℙ² its defining ideal. In this paper, we study the Rees algebras of the ideals generated by I _t , t ≥α. When the points of ? are general, we give a set of defining equations for the Rees algebra . When the points of ? are arbitrary, we show that for all t≫ 0, the Rees algebra is Cohen-Macaulay and its defining ideal is generated by quadratics. A cohomological characterization for arithmetic Cohen-Macaulayness of subvarieties of a product space is also given. Received 4 April 2001     

On Subfields of the Hermitian Function Field

The Hermitian function field H= K(x,y) is defined by the equationy ^q+ y=x ^q+1(q being a powerof the characteristic of K). OverK= q ² it is a maximalfunction field; i.e. the numberN(H)of q²-rationalplaces attains the Hasse--Weil upper boundN(H)=q ²+1+2g(H)·q.All subfields K EHare also maximal.In this paper we construct a large number of nonrational subfields EH, by considering the fixed fieldsH under certaingroups type="Italic">g0 that occur as the genus of some maximal function field over q ².     

Impulsive control systems with commutative vector fields

We consider variational problems with control laws given by systems of ordinary differential equations whose vector fields depend linearly on the time derivativeu=(u ¹,...,u ^m ) of the controlu=(u ¹,...,u ^m ). The presence of the derivativeu, which is motivated by recent applications in Lagrangian mechanics, causes an impulsive dynamics: at any jump of the control, one expects a jump of the state.The main assumption of this paper is the commutativity of the vector fields that multiply theu . This hypothesis allows us to associate our impulsive systems and the corresponding adjoint systems to suitable nonimpulsive control systems, to which standard techniques can be applied. In particular, we prove a maximum principle, which extends Pontryagin's maximum principle to impulsive commutative systems.     

Regularity and Related Rings

A ring R is a Garcia ring provided that the product of two regular elements is unit-regular. We prove that every regular element in a Garcia ring R is the sum/difference of an idempotent and a unit. Furthermore, we prove that every regular element in a weak Garcia ring is the sum of an idempotent and a one-sided unit. These extend several known theorems on (one-sided) unit-regular rings to wider classes of rings with sum summand property.     

A system of elliptic equations arising in Chern-Simons field theory

We prove the existence of topological vortices in a relativistic self-dual Abelian Chern-Simons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R². We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R² limit from the bounded-domain solutions to obtain a topological solution in R².     

Constructive proofs of theorems relating to:F(x) = y,with applications

Some theorems are given which relate to approximating and establishing the existence of solutions to systemsF(x) = y ofn equations inn unknowns, for variousy, in a region of euclideann-space E ⁿ . They generalize known theorems.Viewing complementarity problems and fixed-point problems as examples, known results or generalizations of known results are obtained.A familiar use is made of homotopies H: E ⁿ × [0, 1]E ⁿ of the formH(x, t) = (1 –t)F ⁰ (x) + t[F(x) – y] where theF ⁰ in this paper is taken to be linear. Simplicial subdivisionsT ^k of E ⁿ × [0, 1] furnish piecewise linear approximatesG ^k toH. The basic computation is via the generation of piecewise linear curvesP ^k which satisfyG ^k (x, t) = 0. Visualizing a sequence {T ^k } of such subdivisions, with mesh size going to zero, arguments are made on connected, compact limiting curvesP on whichH(x, t) = 0.This paper builds upon and continues recent work of C.B. Garcia.The authors respectively: A. Charnes, research partially supported by Proj. No. NR047-021 Contract N00014-75-C-0269; C.B. Garcia, C.E. Lemke, research partially supported by NSF Grant No. MPS75-09443.     

Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators

The object of this paper is threefold. First, we investigate in a Hilbert space setting the utility of approximate source conditions in the method of Tikhonov–Phillips regularization for linear ill‐posed operator equations. We introduce distance functions measuring the violation of canonical source conditions and derive convergence rates for regularized solutions based on those functions. Moreover, such distance functions are verified for simple multiplication operators in L²(0, 1). The second aim of this paper is to emphasize that multiplication operators play some interesting role in inverse problem theory. In this context, we give examples of non‐linear inverse problems in natural sciences and stochastic finance that can be written as non‐linear operator equations in L²(0, 1), for which the forward operator is a composition of a linear integration operator and a non‐linear superposition operator. The Fréchet derivative of such a forward operator is a composition of a compact integration and a non‐compact multiplication operator. If the multiplier function defining the multiplication operator has zeros, then for the linearization an additional ill‐posedness factor arises. By considering the structure of canonical source conditions for the linearized problem it could be expected that different decay rates of multiplier functions near a zero, for example the decay as a power or as an exponential function, would lead to completely different ill‐posedness situations. As third we apply the results on approximate source conditions to such composite linear problems in L²(0, 1) and indicate that only integrals of multiplier functions and not the specific character of the decay of multiplier functions in a neighbourhood of a zero determine the convergence behaviour of regularized solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Doubling and Tripling Constructions for Defining Sets in Steiner Triple Systems

 A minimal defining set of a Steiner triple system on v points (STS(v)) is a partial Steiner triple system contained in only this STS(v), and such that any of its proper subsets is contained in at least two distinct STS(v)s. We consider the standard doubling and tripling constructions for STS(2v+1) and STS(3v) from STS(v) and show how minimal defining sets of an STS(v) gives rise to minimal defining sets in the larger systems. We use this to construct some new families of defining sets. For example, for Steiner triple systems on 3 ⁿ points, we construct minimal defining sets of volumes varying by as much as 7 ⁿ⁻² . Received: September 16, 2000 Final version received: September 13, 2001 RID="*" ID="*" Research supported by the Australian Research Council A49937047, A49802044