Linear orbits of arbitrary plane curves

The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of \PGL(3). In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane curve, and give explicit formulas for plane curves with irreducible singularities. The main tool is an intersection@-theoretic study of the projective normal cone of a scheme determined by the curve in the projective space $\P^8$ of $3\times 3$ matrices; this expresses the degree of the orbit closure in terms of the degrees of suitable loci related to the limits of the curve. These limits, and the degrees of the corresponding loci, have been established in previous work.Comment: 33 pages, AmS-TeX 2.

Mach Bands: How Many Models are Possible? Recent Experiemental Findings and Modeling Attempts

Mach bands are illusory bright and dark bands seen where a luminance plateau meets a ramp, as in half-shadows or penumbras. A tremendous amount of work has been devoted to studying the psychophysics and the potential underlying neural circuitry concerning this phenomenon. A number of theoretical models have also been proposed, originating in the seminal studies of Mach himself. The present article reviews the main experimental findings after 1965 and the main recent theories of early vision that have attempted to discount for the effect. It is shown that the different theories share working principles and can be grouped in three clsses: a) feature-based; b) rule-based; and c) filling-in. In order to evaluate individual proposals it is necessary to consider them in the larger picture of visual science and to determine how they contribute to the understanding of vision in general.Air Force Office of Scientific Research (F49620-92-J-0334); Office of Naval Research (N00014-J-4100); COPPE/UFRJ, Brazi

Detecting quantum critical points at finite temperature via quantum teleportation: further models

In [Phys. Rev. A 107, 052420 (2023)] we showed that the quantum teleportation protocol can be used to detect quantum critical points (QCPs) associated with a couple of different classes of quantum phase transitions, even when the system is away from the absolute zero temperature (T=0). Here, working in the thermodynamic limit (infinite chains), we extend the previous analysis for several other spin-1/2 models. We investigate the usefulness of the quantum teleportation protocol to detect the QCPs of those models when the temperature is either zero or greater than zero. The spin chains we investigate here are described by the XXZ model, the XY model, and the Ising model, all of them subjected to an external magnetic field. Specifically, we use a pair of nearest neighbor qubits from an infinite spin chain at thermal equilibrium with a reservoir at temperature T as the resource to execute the quantum teleportation protocol. We show that the ability of this pair of qubits to faithfully teleport an external qubit from the chain is dramatically affected as we cross the QCPs related to the aforementioned models. The results here presented together with the ones of [Phys. Rev. A 107, 052420 (2023)] suggest that the quantum teleportation protocol is a robust and quite universal tool to detect QCPs even when the system of interest is far from the absolute zero temperature.Comment: 15 pages, 23 figures, double column, RevTex4; v2: published versio

Binary differential equations at parabolic and umbilical points for 22-parameter families of surfaces

We determine local topological types of binary differential equations of asymptotic curves at parabolic and flat umbilical points for generic $2$-parameter families of surfaces in $\mathbb P^3$ by comparing our projective classification of Monge forms and classification of general BDE obtained by Tari and Oliver. In particular, generic bifurcations of the parabolic curve are classified. The flecnodal curve is also examined by direct computations, and we present new bifurcation diagrams in typical examples.Comment: 20 page

Determining Critical Points of Handwritten Mathematical Symbols Represented as Parametric Curves

We consider the problem of computing critical points of plane curves represented in a finite orthogonal polynomial basis. This is motivated by an approach to the recognition of hand-written mathematical symbols in which the initial data is in such an orthogonal basis and it is desired to avoid ill-conditioned basis conversions. Our main contribution is to assemble the relevant mathematical tools to perform all the necessary operations in the orthogonal polynomial basis. These include implicitization, differentiation, root finding and resultant computation

Gravitational Nanolensing from Subsolar Mass Dark Matter Halos

We investigate the feasibility of extracting the gravitational nanolensing signal due to the presence of subsolar mass halos within galaxy-sized dark matter halos. We show that subsolar mass halos in a lensing galaxy can cause strong nanolensing events with shorter durations and smaller amplitudes than microlensing events caused by stars. We develop techniques that can be used in future surveys such as Pan-STARRS, LSST and OMEGA to search for the nanolensing signal from subsolar mass halos.Comment: 12 pages, 10 figures. Replaced with version accepted for publication in ApJ. Very minor changes from version

On the Characterization Diagrams of Planar Cubic Hybrid Hyperbolic Polynomial Curve

根据文献[9](WAng g z,yAng Q M.PlAnAr CubIC HybrId HyPErbOlIC POlynOMIAl CurVE And ITS SHAPE ClASSIfICATIOn.PrOgrESS In nATurAl SCIEnCE,2004,14(1):41-46)中提出的H-曲线带奇点或拐点的条件,利用H-曲线奇点、拐点的仿射不变性,给出H-曲线几何特征图的判别法,并找到了不同特征图在三维空间中的关系.该判别法完善了H-曲线的奇异点检测理论,提升了几何特征图维数.In this paper,we propose the geometric characterization diagrams of planar cubic H-curves based on the conditions of singularities for planar cubic H-curves presented by Reference(Wang G Z,Yang Q M.Planar cubic hybrid hyperbolic polynomial curve and its shape classification.Progress In Natural Science,2004,14(1): 41-46) Since inflection points and singularities(loops or cusps) of curves are affinely invariant,we find out the geometrically intuitive relationship of these different geometric characterization diagrams in a common 3-dimension characterization space.This approach completes the theory of detecting singularities of H-curves in the point of elevating geometric characterization diagrams dimension.国家自然科学基金(60773179;60970079;60933008

Phase transitions in rotating neutron stars cores: back bending, stability, corequakes and pulsar timing

The back-bending phenomenon for compact stars is studied by means of analytical equations of state, for both constant-pressure phase transitions and the transitions through the mixed-phase region. We restrict ourselves to the case of normal rotating configurations, with baryon mass below the maximum allowable baryon mass for non-rotating stars. We use high-precision 2-D multi-domain spectral code LORENE to search the parameter space for possible instability regions, and possible changes in the stability character of rotating stars with phase transitions in their cores. Conditions on the density jump in constant-pressure phase transitions, leading to the existence of the unstable segments in the evolutionary sequences of spinning down isolated normal neutron stars, are derived. Conjectures concerning the existence of two disjoint families of non-rotating and rotating stationary configurations of neutron stars are formulated. Particular case of EOSs leading to marginal instability of static and rotating configurations is also studied: marginal instability point in non-rotating configurations continues to exist in all evolutionary spin-down tracks. The fate of rotating stars entering the region of instability is discussed. The change in radius, energy release, and spin-up associated with the corequake in rotating neutron star, triggered by the instability, are calculated. The energy release is found to be very weakly dependent on the angular momentum of collapsing star.Comment: 13 pages, 15 figures, accepted by A&

PTOPO: A Maple package for the topology of parametric curves

International audiencePTOPO is a MAPLE package computing the topology and describing the geometry of a parametric plane curve. The algorithm behind PTOPO constructs an abstract graph that is isotopic to the curve. PTOPO exploits the benefits of the parametric representation and performs all computations in the parameter space using exact computing. PTOPO computes the topology and visualizes the curve in less than a second for most examples in the literature