A reformulation of Le's conjecture

We give a new proof of Le's conjecture on surface germs in ?³ having as link a topological sphere for the case of surface singularities containing a smooth curve. Our proof leads to a reformulation of the general case of the conjecture into a problem of plane curve singularities and their relative polar curves.     

Measurements of total electron content over Ahmedabad and Bombay from Beacon satellite data

The paper outlines the method used at Ahmedabad for determining the total electron content in the ionosphere from differential Faraday rotations of plane polarised radio waves on 40.010 and 41.010 Mc./s. transmitted by NASA satellite 64-64A. The observations made during the period October 1964 to April 1965 are utilised to determine the diurnal variation of the total electron content up to the satellite height which was about 1,000 km. over Ahmedabad and Bombay latitudes. The curve of total electron content up to the satellite height shows similarity to the curve of maximum electron density at the F2 peak. The total electron contents over Bombay are in general higher than those over Ahmedabad.     

Plumbing constructions of connected divides and the Milnor fibers of plane curve singularities

A divide is the image of a generic, relative immersion of a finite number of copies of the unit interval or the unit circle into the unit disk. N. A'Campo defined for each connected divide a link in S³ and proved that the link is fibered. In the present paper we show that the fiber surface of the fibration of a connected divide can be obtained from a disk by a successive plumbing of a finite number of positive Hopf bands. In particular, this gives us a geometric understanding of plumbing constructions of the Milnor fibers of isolated, complex plane curve singularities in terms of certain replacements of the curves of their real morsifications.     

Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve

Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterize those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstraß points and the singularities. Our main tool is the Hodge-theoretic characterization of Teichmüller curves. We deduce from it a finiteness result for the Mordell-Weil group of the family of Jacobians over a Teichmüller curve. The link to the classification of periodic points is provided by interpreting them as sections of the family of curves over a covering of the Teichmüller curve.     

Recursive polynomial curve schemes and computer-aided geometric design

A class of polynomial curve schemes is introduced that may have widespread application to CAGD (computer-aided geometric design), and which contains many well-known curve schemes, including Bézier curves, Lagrange polynomials, B-spline curve (segments), and Catmull-Rom spline (segments). The curves in this class can be characterized by a simple recursion formula. They are also shown to have many properties desirable for CAGD; in particular they are affine invariant, have the convex hull property, and possess a recursive evaluation algorithm. Further, these curves have shape parameters which may be used as a design tool for introducing such geometric effects as tautness, bias, or interpolation. The link between probability theory and this class of curves is also discussed.Communicated by Klaus Höllig.     

On Rational Cuspidal Projective Plane Curves

In 2002, L. Nicolaescu and the fourth author formulated a verygeneral conjecture which relates the geometric genus of a Gorensteinsurface singularity with rational homology sphere link withthe Seiberg--Witten invariant (or one of its candidates) ofthe link. Recently, the last three authors found some counterexamplesusing superisolated singularities. The theory of superisolatedhypersurface singularities with rational homology sphere linkis equivalent with the theory of rational cuspidal projectiveplane curves. In the case when the corresponding curve has onlyone singular point one knows no counterexample. In fact, inthis case the above Seiberg--Witten conjecture led us to a veryinteresting and deep set of ‘compatibility properties’of these curves (generalising the Seiberg--Witten invariantconjecture, but sitting deeply in algebraic geometry) whichseems to generalise some other famous conjectures and propertiesas well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yauinequalities). Namely, we provide a set of ‘compatibilityconditions’ which conjecturally is satisfied by a localembedded topological type of a germ of plane curve singularityand an integer d if and only if the germ can be realized asthe unique singular point of a rational unicuspidal projectiveplane curve of degree d. The conjectured compatibility propertieshave a weaker version too, valid for any rational cuspidal curvewith more than one singular point. The goal of the present articleis to formulate these conjectured properties, and to verifythem in all the situations when the logarithmic Kodaira dimensionof the complement of the corresponding plane curves is strictlyless than 2. 2000 Mathematics Subject Classification 14B05,14J17, 32S25, 57M27, 57R57 (primary), 14E15, 32S45, 57M25 (secondary).     

Oriented matroids and complete-graph embeddings on surfaces

We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable and when one color class of cells consists of triangles only. Precisely for triangular pseudoline arrangements, one element of the image pair of f is a triangular complete-graph embedding on a surface. We obtain all triangular complete-graph embeddings on surfaces this way, when we extend the definition of triangular complete pseudoline arrangements in a natural way to that of triangular curve arrangements on surfaces in which each pair of curves has a point in common where they cross. Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces. Furthermore, we establish the relationship between 2-colorable curve arrangements and Petrie dual maps. A data structure, called intersection pattern is provided for the study of curve arrangements on surfaces. Finally we show that an orientable surface of genus g admits a complete curve arrangement with at most 2g+1 curves in contrast to the non-orientable surface where the number of curves is not bounded.     

SINGULARITIES OF SPHERICAL FOUR-BAR MOTION

The |ink curve of a sphericel four-bar mechanism can be separated into two branches.The condition that the double points of the curve must satisfy is given. The double points maybe the double points of either branch, or the intersection points of the two branches.     

Kinematics of coupler curves for spherical four-bar linkages based on new spherical adjoint approach

The local and global geometric properties of spherical coupler curves constitute spherical kinematics of spherical four-bar linkages, which can be adopted to reveal distribution characteristics of spherical coupler curves. New unified spherical adjoint approach is established in the paper to study both the local and global geometric properties in order to enrich the atlas of spherical coupler curves with geometric characteristics. Since the constraint curve of spherical four-bar linkage is a simple spherical circle and the spherical centrodes imply intrinsic properties of spherical motion of the coupler link, they are in their turn taken as the original curves in spherical adjoint approach to derive the geodesic curvature and analyze the local geometric characteristics of the spherical coupler curves. The conditions for different spherical double points, such as spherical crunodes, tacnodes and cusps of the spherical coupler curve are derived through the spherical adjoint approach. The spherical surface of the coupler link can be divided into several areas by the spherical moving centrode and the spherical tacnode's tracer curve. The points in each area trace spherical coupler curves with a specific shape. The characteristic points, which trace spherical coupler curves with cusp, geodesic inflection point, spherical Ball point, spherical Burmester point, crunode and tacnode can be readily located in the coupler link by the modelling procedure and the derived condition equations. In the end the distribution of spherical coupler curves with both local and global characteristics is elaborated. The research proposes systematic geometric properties of spherical coupler curves based on the new established approach, and provides a solid theoretical basis for the kinematic analysis and synthesis of the spherical four-bar linkages.     

Turaev torsion invariants of 3-orbifolds

We construct a combinatorial invariant of 3-orbifolds with singular set a link that generalizes the Turaev torsion invariant of 3-manifolds. We give several gluing formulas from which we derive two consequences. The first is an understanding of how the components of the invariant change when we remove a curve from the singular set. The second is a formula relating the invariant of the 3-orbifold to the Turaev torsion invariant of the underlying 3-manifold in the case when the singular set is a nullhomologous knot.     

Positive open book decompositions of Stein fillable 3-manifolds along prescribed links

It is known by Loi and Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L^′ to L such that the link L∪L^′ is the binding of a positive open book decomposition of the Stein fillable 3-manifold. To prove the assertion, we will use the divide, which is a generalization of real morsification theory of complex plane curve singularities, and 2-handle attachings along Legendrian curves.     

Geodesic axes in the pants complex of the five-holed sphere

We study the synthetic geometry of the pants graph of the 5-holed sphere, establishing the existence of geodesics connecting any vertex or ideal point to any ideal point. We prove the existence of geodesic axes for sufficiently high powers of any pseudo-Anosov mapping class, and that large link hierarchies from Harvey?s curve graph all induce geodesic paths.     

销售扩散曲线及其数学分析

本文利用概率分布函数来描述销售扩散过程,面对到目前为止的销售扩散曲线,消费者、技术人员和销售人员会有一种"冲动",正是这种"冲动"推动了销售扩散曲线下一步的发展。这种"冲动"可以用统计中生存分析的方法给以描述。我们把这种"冲动"分别与当前的销售扩散曲线、平均销售扩散速度和销售时间联系起来,在数学上构成三类微分方程。利用这些微分方程的解,可以预测出销售最高峰、销售量过50%的时间等。     

Minimal Lagrangian 2-Tori in CP2 Come in Real Families of Every Dimension

It is shown that for every non-negative integer n, there isa real n-dimensional family of minimal Lagrangian tori in CP²,and hence of special Lagrangian cones in C³ whose link is atorus. The proof utilises the fact that such tori arise fromintegrable systems, and can be described using algebro-geometric(spectral curve) data.     

Size dependent resonances in the classical electromagnetic scattering

Some size dependent resonances in the extinction efficiency by small metallic particles have been considered on the basis of the exact calculations on Mie Theory of Scattering. The results may be helpful in explaining some structures in the observed interstellar extinction curve and certain unidentified interstellar diffuse absorption bands.     

Globally controllable systems of rigid bodies with several stable rest states

Natural Langrangian systems, which model mechanisms with the kinematic structure of a tree and have several stable states of rest, are considered. Sufficient are presented for global controllability, that is, the property that the system can be brought, using an admissable control action, from any initial phase state to any given phase state in a finite time. As an example, global controllability is demonstrated foe an object consisting of several point masses connected in series by rods and sliding without friction on a closed smooth curve in a gravitational field, with a single control force applied in the direction of the velocity to the first link.     

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Atiyah and Bott used equivariant Morse theory applied to theYang–Mills functional to calculate the Betti numbers ofmoduli spaces of vector bundles over a Riemann surface, rederivinginductive formulae obtained from an arithmetic approach whichinvolved the Tamagawa number of SL_n. This article attempts tosurvey and extend our understanding of this link between Yang–Millstheory and Tamagawa numbers, and to explain how methods usedover the last three decades to study the singular cohomologyof moduli spaces of bundles on a smooth projective curve over can be adapted to the setting of ¹-homotopy theory to studythe motivic cohomology of these moduli spaces over an algebraicallyclosed field.     

Power and reversal power links for binary regressions: An application for motor insurance policyholders

In binary regression, symmetric links such as logit and probit are usually considered as standard. However, in the presence of unbalancing of ones and zeros, these links can be inappropriate and inflexible to fit the skewness in the response curve and likely to lead to misspecification. This is the case of covering some type of insurance, where it can be observed that the probability of a given binary response variable approaches zero at different rates than it approaches one. Furthermore, when usual links are considered, there is not a skewness parameter associated with the distribution chosen that, regardless of the linear predictor, is easily interpreted. In order to overcome such problems, a proposal for the construction of a set of new skew links is developed in this paper, where some of their properties are discussed. In this context, power links and their reversal versions are presented. A Bayesian inference approach using MCMC is developed for the presented models. The methodology is illustrated considering a sample of motor insurance policyholders selected randomly by gender. Results suggest that the proposed link functions are more appropriate than other alternative link functions commonly used in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Topological imitation of a colored link with the same Dehn surgery manifold

By the topological imitation theory, we construct, from a given colored link, a new colored link with the same Dehn surgery manifold. In particular, we construct a link with a distinguished coloring whose Dehn surgery manifold is a given closed connected oriented 3-manifold except the 3-sphere. As a result, we can naturally generalize the difference between the Gordon–Luecke theorem and the property P conjecture to a difference between a link version of the Gordon–Luecke theorem and the Poincaré conjecture. Similarly, we construct a link with a π₁-distinguished coloring whose Dehn surgery manifold is a given non-simply-connected closed connected oriented 3-manifold. We also construct a link with just two colorings whose Dehn surgery manifolds are the 3-sphere.