Discriminant Curve Research Articles

Discriminant of a generic projection of a minimal normal surface singularity

Let ( S,0) be a rational complex surface singularity with reduced fundamental cycle, also known as a minimal singularity. Using a fundamental result of M. Spivakovsky, we explain how to describe the equisingularity type of the discriminant curve for a generic projection of ( S,0) onto ( C 2,0) from the resolution of ( S,0). To cite this article: R. Bondil, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Topological invariants of higher order for a pair of plane curve germs

Let ( g, f) be an analytic map germ from ( C 2,0) into ( C 2,0) and denote by ( u, v) the canonical coordinates in (g,f)( C 2) ; it is ( g( x, y), f( x, y))=( u, v). In [J. London Math. Soc. (2) 59 (1999) 207–226], we showed that the set constituted of the first (not necessarily characteristic) Puiseux exponent (in the ( u, v)-coordinates) of each branch δ of the discriminant curve of ( g, f) is a topological invariant of ( g, f). Here we prove that for each branch δ there exists an integer k( δ) such that the set constituted of the first (not necessarily characteristic) k( δ) exponents of the Puiseux series in the ( u, v)-coordinates of each δ is a topological invariant of ( g, f). We give different ways to compute these invariants.

Knots, braids and BPS states in M-theory

In previous work we considered M-theory five branes wrapped on elliptic Calabi-Yau threefold near the smooth part of the discriminant curve. In this paper, we extend that work to compute the light states on the worldvolume of five-branes wrapped on fibers near certain singular loci of the discriminant. We regulate the singular behavior near these loci by deforming the discriminant curve and expressing the singularity in terms of knots and their associated braids. There braids allow us to compute the appropriate string junction lattice for the singularity and,hence to determine the spectrum of light BPS states. We find that these techniques are valid near singular points with N=2 supersymmetry.

SEVERI VARIETIES AND BRANCH CURVES OF ABELIAN SURFACES OF TYPE (1, 3)

A polarized abelian surface (A, L) of type (1, 3) induces a 6:1 covering of A onto the projective plane with branch curve, a plane curve B of degree 18. The main result of the paper is that for a general abelian surface of type (1, 3), the curve B is irreducible and reduced and admits 72 cusps, 36 nodes or tacnodes, each tacnode counting as 2 nodes, 72 flexes and 36 bitangents. The main idea of the proof is that for a general (A, L) the discriminant curve in the linear system |L| coincides with the closure of the Severi variety of curves in |L| admitting a node and is dual to the curve B in the sense of projective geometry.

Five-brane BPS states in heterotic M-theory

We present explicit methods for computing the discriminant curves and the associated Kodaira type fiber degeneracies of elliptically fibered Calabi-Yau threefolds. These methods are applied to a specific three-family, SU(5) grand unified theory of particle physics within the context of Heterotic M-Theory. It is demonstrated that there is always a region of moduli space where a bulk space five-brane is wrapped on a pure fiber in the Calabi-Yau threefold. Restricting the discussion to the smooth parts of the discriminant curve, we explore the properties of the N = 2 BPS supermultiplets that arise on the worldvolume of this five-brane due to the degeneration of the elliptic fiber. The associated degenerating M membranes are shown to project to string junctions in the base space. We use string junction techniques to explicitly compute the light BPS hyper- and vector multiplet spectrum for each Kodaira type fiber near the smooth part of the discriminant curve in the SU(5) GUT theory.

Relationships between the discriminant curve and other bifurcation diagrams

The parameter dependence of the number and type of the stationary points of an ODE is considered. The number of the stationary points is determined by the saddle-node (SN) bifurcation set and their type (e.g., stability) is given by another bifurcation diagram (e.g., Hopf bifurcation set). The relation between these bifurcation curves on the parmeter plane is investigated. It is shown that the ‘cross-shaped diagram’, when the Hopf bifurcation curve makes a loop around a cusp point of the SN curve, is typical in some sense. It is proved that the two bifurcation curves meet tangentially at their common points (Takens–Bogdanov point), and these common points persist as a third parameter is varied. An example is shown that exhibits two different types of 3-codimensional degenerate Takens–Bogdanov bifurcation.

Braid monodromy factorizations and diffeomorphism types

In this paper we prove that if two cuspidal plane curves and have equivalent braid monodromy factorizations, then and are smoothly isotopic in . As a consequence, we obtain that if , are surfaces of general type embedded in a projective space by means of a multiple canonical class and if the discriminant curves (the branch curves) , of some smooth projections of , to have equivalent braid monodromy factorizations, then and are diffeomorphic (as real four-dimensional manifolds).

On Chisini's conjecture

Chisini's conjecture asserts that if is a cuspidal curve, then a generic morphism , , of a smooth projective surface to branched along is unique up to isomorphism. In this paper we prove that Chisini's conjecture is true for if is greater than the value of some function depending on the degree, genus and the number of cusps of . This inequality holds for almost all generic morphisms. In particular, on a surface with ample canonical class, it holds for generic morphisms defined by a linear subsystem of the -canonical class, .Moreover, we present examples of pairs (, ) of plane cuspidal curves such that (i) , and these curves have homeomorphic tubular neighbourhoods in , but the pairs and are not homeomorphic;(ii) is the discriminant curve of a generic morphism , , where are surfaces of general type;(iii) the surfaces and are homeomorphic (as four-dimensional real manifolds);(iv) the morphism is defined by a three-dimensional linear subsystem of the -canonical class of .

Discriminant of a Germ Φ: (C2 , 0)→(C2 , 0) and Seifert Fibred Manifolds

Let f and g be two analytic function germs without common branches. We define the Jacobian quotients of (g, f), which are ‘first order invariants’ of the discriminant curve of (g, f), and we prove that they only depend on the topological type of (g, f). We compute them with the help of the topology of (g, f). If g is a linear form transverse to f, the Jacobian quotients are exactly the polar quotients of f and we affirm the results of D. T. Lê, F. Michel and C. Weber.

Subnormal operators of finite type II. Structure theorems

This paper concerns pure subnormal operators with finite rank self-commutator, which we call subnormal operators of finite type. We analyze Xia's theory of these operators [21–23] and give its alternative exposition. Our exposition is based on the explicit use of a certain algebraic curve in \mathbb C^2 , which we call the discriminant curve of a subnormal operator, and the approach of dual analytic similarity models of [26]. We give a complete structure result for subnormal operators of finite type which corrects and strenghtens the formulation that Xia gave in [23]. Xia claimed that each subnormal operator of finite type is unitarily equivalent to the operator of multiplication by z on a weighted vector H^2 -space over a “quadrature Riemann surface” (with a finite rank perturbation of the norm. We explain how this formulation can be corrected and show that, conversely, every “quadrature Riemann surface” gives rise to a family of subnormal operators. We prove that this family is parametrized by the so-called characters. As a departing point of our study, we formulate a kind of scattering scheme for normal operators, which includes Xia's model as a particular case.

Dynamics of the velocity gradient tensor invariants in isotropic turbulence

The evolution of the invariants (R and Q) of the velocity gradient tensor in homogeneous isotropic turbulence is investigated using data from direct numerical simulation (DNS). The concepts of conditional average time rate of change of the invariants and conditional mean trajectories (CMT) in invariant phase space are introduced to study the dynamics of this flow. The resulting dynamical system in the (R,Q) phase space is a clockwise spiral with a stable focus at the origin, illustrating that in the mean, the cyclic sequence of topological evolution following a fluid particle is unstable-node/saddle/saddle (UN/S/S)→stable-node/saddle/saddle (SN/S/S)→stable-focus/stretching (SF/S)→unstable-focus/contracting (UF/C). The mean rates of change of R and Q, i.e., Ṙ, Q̇, are found to be negligible near the right branch of the null discriminant (D=0) curve, indicating that this curve is an attractor in the (R,Q) space. The effects of both the diffusion term and the anisotropic part of the pressure Hessian term on the dynamics of the invariants have also been analyzed using the conditional averages. Both contributions are found to be important in the dynamics of the velocity gradient invariants. Based on these results the extent of the validity of the model equations governing the evolution of R and Q proposed by Cantwell [Phys. Fluids A 4, 782 (1992)] and Dopazo et al. [“Velocity gradients in turbulent flows. Stochastic models,” Ninth Symposium on “Turbulent Shear Flows,” Kyoto, Japan, 1993, pp. 26-2-1–26-2-5] are discussed.

Determination of saturation parameter in iodine and precise molecular linewidth measurements in He-Ne/I 2 standard at 633 nm

A new method for linewidth studies in a primary wavelength standard is proposed, resulting in the determination of a saturation parameter in iodine, the physical quantity that defines the amplitude and the power broadening of the inverted Lamb dip in the absorption medium. Precise linewidth information is obtained by comparison of the experimental beat-frequency data of the laser locked to the zero crossings of the discriminant curve, with the results of calculations that use the high-power gas laser theory with Gaussian intensity distribution beam profile. The method is based on the variation principle, the saturation parameter value being found by the computer fit, which gives a constant value of the homogeneous linewidth of iodine for several laser modulation width values. The saturation parameter in iodine has been determined for the first time. Its dependence on iodine pressure is found to be parabolic. The relation between the saturation parameter and the laser output power is presented, giving the possibility to calculate the saturation parameter of an auxiliary He-Ne/I 2 laser at 633 nm.

A rationality criterion for conic bundles

It is that a three-dimensional variety that is a conic bundle in the Mori sense has a base with at most double rational singularities of type . A rationality criterion is proved subject to this assumption in the case when the discriminant curve is large enough, for example, for the case when .

The Rationality of the Moduli Space of Bielliptic Curves of Genus Five

Bulletin of the London Mathematical SocietyVolume 28, Issue 4 p. 356-362 Notes and papers The Rationality of the Moduli Space of Bielliptic Curves of Genus Five G. Casnati, G. Casnati Dipartimento di Matematica Pura ed Applicata Università degli Studi di Padova, via Belzoni 7 35131 Padova ItalySearch for more papers by this authorA. Del Centina, A. Del Centina Dipartimento di Matematica Università degli Studi di Ferrara, via Machiavelli 35 44100 Ferrara ItalySearch for more papers by this author G. Casnati, G. Casnati Dipartimento di Matematica Pura ed Applicata Università degli Studi di Padova, via Belzoni 7 35131 Padova ItalySearch for more papers by this authorA. Del Centina, A. Del Centina Dipartimento di Matematica Università degli Studi di Ferrara, via Machiavelli 35 44100 Ferrara ItalySearch for more papers by this author First published: June 1996 https://doi.org/10.1112/blms/28.4.356Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume28, Issue4June 1996Pages 356-362 RelatedInformation

A new, cellular automaton-based, nearest neighbor pattern classifier and its VLSI implementation

A new, parallel, nearest-neighbor (NN) pattern classifier, based on a 2D Cellular Automaton (CA) architecture, is presented in this paper. The proposed classifier is both time and space efficient, when compared with already existing NN classifiers, since it does not require complex distance calculations and ordering of distances, and storage requirements are kept minimal since each cell stores information only about its nearest neighborhood. The proposed classifier produces piece-wise linear discriminant curves between clusters of points of complex shape (nonlinearly separable) using the computational geometry concept known as the Voronoi diagram, which is established through CA evolution. These curves are established during an "off-line" operation and, thus, the subsequent classification of unknown patterns is achieved very fast. The VLSI design and implementation of a nearest neighborhood processor of the proposed 2D CA architecture is also presented in this paper.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Topology of the quasi-ordinary surface singularities

IN this paper we are interested in one type of (local) complex surface singularities (nonisolated in general, (2.4.3) ex. 2) the investigation of which is due essentially to Lipman [20]. Let F be the surface in an open set in C3 defined by the vanishing of a holomorphic function in three variables. Assume that the origin 0 is a singular (nonsmooth) point of F. If there is a projection of F to a complex plane so that the discriminant (branch) curve has a smooth point at the image of 0, then F is equisingular at 0 (along the preimage of the discriminant curve) [27]. The surface singularities that we are interested in are those which are next in order of complexity, namely those whose images in some (local) projections are ordinary double points of the discriminant curves. Such singularities are said to be quasi-ordinary or Jungian (2.1). See [21] for an introduction to the quasi-ordinary singularities and a summary of the fundamental work [20]. Quasi-ordinary singularities are amenable to detailed analysis since they can be “parametrized” by fractional power series in two variables [19, p. 2071, [Z], which are called the parametrizations of the singularities (cf. the irreducible plane curve singularities can always be parametrized by fractional power (Puiseux) series in one variable). From a suitably normalized parametrization of a quasi-ordinary singularity one can single out (in the same way one obtains the characteristic pairs of plane curves) some exponents. called the distinguished pairs, which are shown in [20] to depend only on the analytic type ( = isomorphism type of the analytic local ring) of the singularity. Moreover, like the characteristic pairs for plane curves, the distinguished pairs determine quite a lot of geometry of the singularity [21, p. 1631. The main result of this paper is that the distinguished pairs of (F, 0) depend only on the topological type of (C3, F) at 0. In other words, the local topology of a quasi-ordinary singularity determines its distinguished pairs. Since the distinguished pairs determine the local topology (2.2.3), it follows that topo!ogically the quasi-ordinary singulariries are completely classified by rheir disringuished pairs (2.2.4). As a corollary, the question of Zariski about the topological invariance of multiplicity is answered affirmatively for quasi-ordinary singularities (2.2.5). Another consequence of the main result is that the local topology determines the Zariski tangent cone (up to isomorphism) (2.2.5)-ii, this answers a question of L(LTeissier [ 18, p. 107]in the quasi-ordinary case. The following question may be of some interest to topologists: Question (Lipman [ZO, p. 1683). Is there an explicit topological interpretation of the distinguished pairs, as one has via compound torus knots for the characteristic pairs of plane curves? The main result: “local topology determines the distinguished pairs (2.2.1)” will be proved

Functional recovery of cortical neurons as related to degree and duration of ischemia.

Simultaneous recordings of spontaneous single cell activity and local cerebral blood flow were obtained from 72 cortical neurons and adjacent brain in 54 cats before, during, and after ischemia induced by reversible occlusion of the middle cerebral artery. In most cells spontaneous electrical activity ceased at flow values of about 0.18 ml/gm/min (range, 0.06 to 0.22 ml/gm/min). No signal was obtained from 28 neurons during reperfusion following ischemia of varying degree and duration. Overall, neurons exposed to a residual flow of 0.14 ml/gm/min or less for more than 45 minutes had a poorer prognosis compared to any other combination of degree and duration of ischemia. A discriminant curve was estimated to define the border line between recovering and nonrecovering cells. Regions showing irreversible neuronal failure contained selective neuronal necrosis or areas of infarction by histological examination. Reperfusion restored neuronal function in 44 cells. In this group of neurons, there was a joint interaction of duration of ischemia, ischemic residual flow, and recovery time: cells exposed to moderate ischemia (0.09 to 0.22 ml/gm/min) for up to 20 minutes recovered rapidly; most neurons subjected either to extreme ischemia (less than 0.09 ml/gm/min) of short duration (less than 20 minutes) or to moderate ischemia (0.09 to 0.22 ml/gm/min) for longer periods (20 to 141 minutes) required from 19 to 50 minutes for recovery. A few resistant neurons tolerated less than 0.09 ml/gm/min for more than 20 minutes.(ABSTRACT TRUNCATED AT 250 WORDS)

Behaviour of lead electrodes in sulphate electrolytes. I. Effect of electrode preparation on discriminant curve cyclic voltammograms

A new technique called discriminant curve cyclic voltammetry (DCCV) has been developed to study the cycling behaviour of battery plate materials. The method is used to investigate the effect of various electrode pretreatments on the cycling behaviour of pure lead in sulphate electrolytes. It is found that three distinctive patterns of response are observed depending upon the surface condition of the lead. The results are correlated with photographic evidence obtained by scanning electron microscopy.

Die Zahl der Spitzen und die Jacobi-Algebra einer isolierten Hyperfl�chensingularit�t

Let f∈ℂ{xo,...,xn} define a germ of a complex analytic hypersurface (Xo,0) with isolated singularity. We show that the number of cusps of the unfolded discriminant curve is an invariant of the Jacobian algebra ℂ{x,o},...,xn/(∂f/∂xo,...,∂f/∂xn) of f. Moreover we show that this number + 1 equals the sum of the Milnor numbers of (Xo,0) and of the polar curve of (Xo,0). Our result generalizes formulas of Iversen and Le for plane curves to arbitrary dimensions.

Concerning the stability of some classes of orbits of artificial earth satellites: Ye. P. Aksenov∗, Ye. A. Grebenikov and V. G. Demin: (pp. 163–172)

This work re-examines some classical results in the kinematics of points in space using modern vector–matrix methods. In particular, some very simple Lie theory allows the velocities and accelerations of points to be found in terms of the instantaneous twist of the motion and its derivative. From these results many of the classical results follow rather simply.Although most of the results are well known, some new material is presented. In particular, the discriminant curve that separates cases with one or three real acceleration axes is found and plotted. Another new result concerns the chords to the cubic of inflexion points. It is shown that for points on such a chord the osculating planes of the point's trajectories are parallel. Also a new result is found which distinguishes between cases where the Bresse hyperboloid of points whose velocities and accelerations are perpendicular, has one or two sheets.