Affine Curve Research Articles

Torsion points on the jacobian of a Fermat quotient

Let p ≥ 5 be a prime, ζ a primitive p th root of unity and λ = 1 − ζ . For 1 ≤ s ≤ p − 2 , the smooth projective model C p , s of the affine curve v p = u s ( 1 − u ) is a curve of genus ( p − 1 ) / 2 whose jacobian J p , s has complex multiplication by the ring of integers of the cyclotomic field Q ( ζ ) . In 1981, Greenberg determined the field of rationality of the p -torsion subgroup of J p , s and moreover he proved that the λ 3 -torsion points of J p , s are all rational over Q ( ζ ) . In this paper we determine quite explicitly the λ 3 -torsion points of J p , 1 for p = 5 and p = 7 , as well as some further p -torsion points which have interesting arithmetical applications, notably to the complementary laws of Kummer’s reciprocity for p th powers.

Explicit Chevalley-Weil Theorem for Affine Plane Curves

Let φ : C → C be an unramified morphism of plane affine curves defined over a number field K. In this paper we obtain a quantitative version of the classical Chevalley-Weil theorem for curves giving an effective upper bound for the norm of the relative discriminant of the field K(Q) over K for any integral point P ∈ C(K) and Q ∈ φ−1(P ).

On the Equivariant Cohomology of Subvarieties of a $\mathfrak{B}$ -Regular Variety

By a \(\mathfrak{B}\)-regular variety, we mean a smooth projective variety over \(\mathbb{C}\) admitting an algebraic action of the upper triangular Borel subgroup \(\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}\) such that the unipotent radical in \(\mathfrak{B}\) has a unique fixed point. A result of Brion and the first author [4] describes the equivariant cohomology algebra (over \(\mathbb{C}\)) of a \(\mathfrak{B}\)-regular variety X as the coordinate ring of a remarkable affine curve in \(X \times \mathbb{P}^{1}\). The main result of this paper uses this fact to classify the \(\mathfrak{B}\)-invariant subvarieties Y of a \(\mathfrak{B}\)-regular variety X for which the restriction map iY : H*(X) → H*(Y) is surjective.

Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic

Let π1(C) be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic p > 0 of countable cardinality. Let N be a normal (respectively, characteristic) subgroup of π 1(C). Under the hypothesis that the quotient π 1(C)/N admits an infinitely generated Sylow p-subgroup, we prove that N is indeed isomorphic to a normal (respectively, characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of N is a free profinite group of countable cardinality.

Small amplitude limit cycles for the polynomial Liénard system

We estimate for the maximal number of limit cycles bifurcating from a focus for the Liénard equation x ¨ + f ( x ) x ˙ + g ( x ) = 0 , where f and g are polynomials of degree m and n respectively. These estimates are quadratic in m and n and improve the existing bounds. In the proof we use methods of complex algebraic geometry to bound the number of double points of a rational affine curve.

On the L -functions of the curves y 2 = x ℓ + A

Let > 3 be an odd prime and let A be an integer not divisible by . Let CA be the non-singular projective model over of the affine curve CA: y2 = x + A. In this paper, we use the theory of Hilbert modular Eisenstein series to obtain a formula for the central value of the Hasse�Weil L-function L(CA, s) of the curve CA when the cyclotomic field () has ideal class number 1

Fundamental group in positive characteristic

It is shown that the commutator subgroup of the fundamental group of a smooth irreducible affine curve over a countable algebraically closed field of nonzero characteristic is profinite free of countable rank. A description of the abelianizations of the fundamental groups of affine curves over an algebraically closed field of nonzero characteristic is also given.

Combining Points and Tangents into Parabolic Polygons

Image and geometry processing applications estimate the local geometry of objects using information localized at points. They usually consider information about the tangents as a side product of the points coordinates. This work proposes parabolic polygons as a model for discrete curves, which intrinsically combines points and tangents. This model is naturally affine invariant, which makes it particularly adapted to computer vision applications. As a direct application of this affine invariance, this paper introduces an affine curvature estimator that has a great potential to improve computer vision tasks such as matching and registering. As a proof-of-concept, this work also proposes an affine invariant curve reconstruction from point and tangent data.

An algorithm for checking whether the toric ideal of an affine monomial curve is a complete intersection

Let K be an arbitrary field and { d 1 , … , d n } a set of all-different positive integers. The aim of this work is to propose and evaluate an algorithm for checking whether or not the toric ideal of the affine monomial curve { ( t d 1 , … , t d n ) ∣ t ∈ K } ⊂ A K n is a complete intersection. The algorithm is based on new results regarding the toric ideal of the curve, and it can be seen as a generalization of the classical result of Herzog for n = 3 . Computational experiments show that the algorithm is able to solve large-size instances.

Structures and properties of the tin-doped carbon clusters [formula omitted

A systemic density functional theory study of the tin-doped carbon clusters SnC n / SnC n + / SnC n - ( n = 1 – 10 ) has been carried out using B3LYP method with TZP+ basis set. For each species, the electronic states, relative energies and geometries of various isomers are reported. Except for smaller SnC 2 and the largest SnC 10 / SnC 10 + , the Sn-terminated linear or quasi-linear isomer is the most stable structure for SnC n / SnC n + / SnC n - clusters. The electronic ground state is alternate between 3Σ (for n-odd member) and 1Σ (for the n-even member) for linear SnC n and invariably 2Π for linear SnC n + and SnC n - , except for SnC/SnC +/SnC −, SnC 2 / SnC 2 + , SnC 4 + , SnC 6 + and SnC 10 / SnC 10 + . The incremental binding energy diagrams show that strong even–odd alternations in the cluster stability exist for both neutral SnC n and anionic SnC n - , with their n-even members being much more stable than the corresponding odd n − 1 and n + 1 ones, while for cationic SnC n + , the alternation effect is less pronounced. These parity effects also reflect in the ionization potential and electron affinity curves. By comparing with the fragmentation energies accompanying various channels, the most favorable dissociation channel for each kind of the SnC n / SnC n + / SnC n - clusters are given. All these results are very similar to those obtained previously for the PbC n / PbC n + / PbC n - clusters.

The l-component of the unipotent Albanese map

We prove a finiteness theorem for the local l ≠ p-component of the $${\mathbb{Q}}_p$$ -unipotent Albanese map for curves. As an application, we refine the non-abelian Selmer varieties arising in the study of global points and deduce thereby a new proof of Siegel’s theorem for affine curves over $${\mathbb{Q}}$$ of genus one with Mordell–Weil rank at most one.

Polynomial parametrization of nonsingular complete intersection curves

In this paper we give a new method to compute a polynomial parametrization, in case it exists, of an affine nonsingular complete intersection curve. Our method is based on the study of vector fields on nonsingular algebraic curves. In contrast to the existing methods, our algorithm does not use any projection, and no sample point on the curve is needed. It is also important to stress the fact that the algorithm produces a parametrization with coefficients in the ground field.

Computation of the singularities of parametric plane curves

Given an algebraic plane curve C defined by a rational parametrization P ( t ) , we present formulae for the computation of the degree of C , and the multiplicity of a point. Using the results presented in [Sendra, J.R., Winkler, F., 2001. Tracing index of rational curve parametrizations. Computer Aided Geometric Design 18 (8), 771–795], the formulae simply involve the computation of the degree of a rational function directly determined from P ( t ) . Furthermore, we provide a method for computing the singularities of C and analyzing the non-ordinary ones without knowing its defining polynomial. This approach generalizes the results in [Abhyankar, S., 1990. Algebraic geometry for scientists and engineers. In: Mathematical Surveys and Monographs, vol. 35. American Mathematical Society; van den Essen, A., Yu, J.-T., 1997. The D -resultants, singularities and the degree of unfaithfulness. Proceedings of the American Mathematical Society 25, 689–695; Gutierrez, J., Rubio, R., Yu, J.-T., 2002. D -Resultant for rational functions. Proceedings of the American Mathematical Society 130 (8), 2237–2246] and [Park, H., 2002. Effective computation of singularities of parametric affine curves. Journal of Pure and Applied Algebra 173, 49–58].

Determining the Molecular Basis for the pH-dependent Interaction between the Link Module of Human TSG-6 and Hyaluronan

TSG-6 is an inflammation-associated hyaluronan (HA)-binding protein that has anti-inflammatory and protective functions in arthritis and asthma as well as a critical role in mammalian ovulation. The interaction between TSG-6 and HA is pH-dependent, with a marked reduction in affinity on increasing the pH from 6.0 to 8.0. Here we have investigated the mechanism underlying this pH dependence using a combined approach of site-directed mutagenesis, NMR, isothermal titration calorimetry and microtiter plate assays. Analysis of single-site mutants of the TSG-6 Link module indicated that the loss in affinity above pH 6.0 is mediated by the change in ionization state of a histidine residue (His(4)) that is not within the HA-binding site. To understand this in molecular terms, the pH-dependent folding profile and the pK(a) values of charged residues within the Link module were determined using NMR. These data indicated that His(4) makes a salt bridge to one side-chain oxygen atom of a buried aspartate residue (Asp(89)), whereas the other oxygen is simultaneously hydrogen-bonded to a key HA-binding residue (Tyr(12)). This molecular network transmits the change in ionization state of His(4) to the HA-binding site, which explains the loss of affinity at high pH. In contrast, simulations of the pH affinity curves indicate that another histidine residue, His(45), is largely responsible for the gain in affinity for HA between pH 3.5 and 6.0. The pH-dependent interaction of TSG-6 with HA (and other ligands) provides a means of differentially regulating the functional activity of this protein in different tissue microenvironments.

Miura conjecture on Affine curves

Shinji Miura gave certain multivariable polynomials that express an Affine curve for a given algebraic function field F and its degree one place O, if F contains such an O. Suppose the equations contain t ( 2) variables, and that the pole orders at O are a1, , at 1, where GCD a1, , at = 1. If ai di a1 di 1 N + + ai 1 di 1 N, di = GCD a 1, , ai

Absolute anabelian cuspidalizations of proper hyperbolic curves

In this paper, we develop the theory of “cuspidalizations” of the etale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field. The ultimate goal of this theory is the group-theoretic reconstruction of the etale fundamental group of an arbitrary open subscheme of the curve from the etale fundamental group of the full proper curve. We then apply this theory to show that a certain absolute $p$-adic version of the Grothendieck Conjecture holds for hyperbolic curves “of Belyi type”. This includes, in particular, affine hyperbolic curves over a nonarchimedean mixed-characteristic local field which are defined over a number field and isogenous to a hyperbolic curve of genus zero. Also, we apply this theory to prove the analogue for proper hyperbolic curves over finite fields of the version of the Grothendieck Conjecture that was shown in [Tama].

Effects of temperature and carbon dioxide on green sturgeon blood–oxygen equilibria

There are three Northeast Pacific Rivers still supporting spawning populations of green sturgeon, Acipenser medirostris, but all have been modified hydrologically and thermally by dam construction. Age 1- to 3-year-old green sturgeon, progeny of artificially spawned, wild-caught Klamath River adults, were used to assess the effects of temperature and carbon dioxide on critical hematological parameters related to evolutionary adaptations of this species to its physical environment. In vitro measurement of the effect of temperature and carbon dioxide on blood–oxygen affinity and equilibrium curve shape yielded the following data for the respective temperature treatments (11, 15, 19, and 24°C): half-saturation values (P50’s, kPa, a measure of affinity) 1.26, 1.44, 1.63, 1.69 for low-PCO2 treatments and 2.08, 2.41, 2.74, 2.94 for high-PCO2 treatments; Bohr factors −0.322, −0.327, −0.366, −0.536; and non-bicarbonate buffer values (slykes) −6, −3, −5, −8. Temperature sensitivities (ΔH, kJ mol O 2 −1 ) between these respective temperatures were −34.20, −15.24, −6.74 for low-PCO2 treatments and −20.05, −27.00, and −11.55 for the high-PCO2 treatments. These data suggest that juvenile green sturgeon may tolerate moderate environmental hypoxia, moderate aerobic activity, low to moderate hypercapnia, and moderate temperature changes in their environments.

Molecular pharmacological approach to drug actions on the afferent and efferent fibres of the vagal nerve involved in gastric mucosal protection in rats

Gastric mucosal protection is associated with the actions of anti-ulcer drugs or agents affecting on the afferent and/or efferent nerve fibres of the vagal nerve. 1. To identify the dose-response curves of drugs (compounds) on the afferent vanilloid-receptor (capsaicin or resiniferatoxin-sensitive) and on efferent secretion (atropine, pirenzepine, cimetidine, ranitidine, famotidine, omeprazole, esomeprazole) basal gastric acid and stimulated gastric secretion in relation to the chemically-induced gastric mucosal damage in rats; 2. To determine the ED50 (pD2) and pA2 on the calculation of affinity and intrinsic affinity curves for these agonists/antagonists, as an indication of relative potency of effects. The observations were carried out in rats (30 different models). The ED50 values for affinities of capsaicin, resiniferatoxin were obtained in nmol/kg b.w. range, whereas the values were in the nmol/kg to micromol/kg b.w. ranges for effects on the gastric basal, stimulated (bethanechol, pentagastrin, histamine) gastric secretion, and the gastric mucosal damage-produced by different ulcerogenic agents (ethanol, HCl, aspirin, indomethacin). From the observations, that agents acting on vanilloid (capsaicin) receptors were the most potent inhibitors of acid secretion and gastric lesions from necrotizing agents, suggests that the capsaicin sensitive afferent nerves have a primary place in the efferent regulated events leading to initiation of gastric mucosal damage.

Effective invariants of braid monodromy

In this paper we construct effective invariants for braid monodromy of affine curves. We also prove that, for some curves, braid monodromy determines their topology. We apply this result to find a pair of curves with conjugate equations in a number field but which do not admit any orientation-preserving homeomorphism.

Congruences for Γ1(4)-modular forms of half-integral weight

The main result of this paper gives a complete description of the algebra of level 4, mod p modular forms of half-integral weight. We show that this is the algebra of regular functions of an affine curve over \(\overline{\mathbb F}_p\). The result parallels results of Swinnerton-Dyer (level 1), and Katz (general level) for integral weight forms.