scholarly journals Rational cuspidal curves in a moving family of ℙ2

2021 ◽  
Vol 8 (1) ◽  
pp. 125-137
Author(s):  
Ritwik Mukherjee ◽  
Rahul Kumar Singh
Keyword(s):  
Euler Class ◽  
Characteristic Number ◽  
Planar Curves ◽  
Characteristic Numbers ◽  
Rational Degree

Abstract In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3 d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ3 with one cusp (for δ ≤ 2).

scholarly journals Unexpected Solutions of the Nehari Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
A. Gritsans ◽  
F. Sadyrbaev
Keyword(s):  
Differential Equation ◽  
Multiple Solutions ◽  
Integral Functional ◽  
Characteristic Number ◽  
Nonlinear Ordinary Differential Equation ◽  
Oscillatory Solutions ◽  
Nehari Problem ◽  
Characteristic Numbers ◽  
Regular Shape ◽  
Asymmetric Shape

The Nehari characteristic numbers λn(a,b) are the minimal values of an integral functional associated with a boundary value problem (BVP) for nonlinear ordinary differential equation. In case of multiple solutions of the BVP, the problem of identifying of minimizers arises. It was observed earlier that for nonoscillatory (positive) solutions of BVP those with asymmetric shape can provide the minimal value to a functional. At the same time, an even solution with regular shape is not a minimizer. We show by constructing the example that the same phenomenon can be observed in the Nehari problem for the fifth characteristic number λn(a,b) which is associated with oscillatory solutions of BVP (namely, with those having exactly four zeros in (a,b)).

On the KOℤ/2-Euler class, II

1991 ◽  
Vol 117 (1-2) ◽  
pp. 139-154 ◽  
Author(s):  
M. C. Crabb
Keyword(s):  
Vector Bundle ◽  
Finite Subset ◽  
Homotopy Class ◽  
Homotopy Group ◽  
Euler Class ◽  
Class Ii ◽  
Stable Homotopy ◽  
Stiefel Manifold ◽  
Real Vector ◽  
Characteristic Numbers

SynopsisLet ξ be an oriented n-dimensional real vector bundle over an oriented closed m-manifold X. An r-field on ξ defined outside a finite subset of X has an index in the homotopy group πm−l(Vn,r) of the Stiefel manifold of r-frames in ℝn. The principal theorems of this paper relate the d and e-invariants of an associated ℝ/2-equivariant stable homotopy class, in certain cases, to computable cohomology characteristic numbers. Results of this type were first obtained by Atiyah and Dupont [5].

XVII.—On the Asymptotic Expansion of the Characteristic Numbers of the Mathieu Equation

1930 ◽  
Vol 49 ◽  
pp. 210-223 ◽  
Author(s):  
Sydney Goldstein
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Mathieu Equation ◽  
Characteristic Number ◽  
Numerical Approximations ◽  
Characteristic Numbers ◽  
The Difference ◽  
Image Position ◽  
Integral Order

An asymptotic formula has recently been given for the characteristic numbers of the Mathieu equation From tabular values, it will be seen that the formula provides good numerical approximations to the characteristic numbers of integral order; but as pointed out by Ince, it provides better approximations to the characteristic numbers of order (m + ½), where m is a positive integer or zero. In this paper we shall first attempt to find out why this should be so, and then go on to show that the formula is probably an asymptotic expansion, in the Poincaré sense, for any characteristic number. A new asymptotic formula is then found for the difference between two characteristic numbers.

scholarly journals Characteristic numbers of rational curves with cusp or prescribed triple contact

2003 ◽  
Vol 92 (2) ◽  
pp. 223 ◽  
Author(s):  
Joachim Kock
Keyword(s):  
Generating Functions ◽  
Characteristic Number ◽  
Rational Curves ◽  
Characteristic Numbers ◽  
Number Problem

This note pursues the techniques of [Graber-Kock-Pandharipande] to give concise solutions to the characteristic number problem of rational curves in $\boldsymbol P^2$ or $\boldsymbol P^1\times\boldsymbol P^1$ with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials.

scholarly journals CHARACTERISTIC NUMBERS OF NON‐AUTONOMOUS EMDEN‐FOWLER TYPE EQUATIONS

2006 ◽  
Vol 11 (3) ◽  
pp. 243-252 ◽  
Author(s):  
A. Gritsans ◽  
F. Sadyrbaev
Keyword(s):  
Continuous Function ◽  
Boundary Value Problem ◽  
Extremal Problem ◽  
Boundary Value ◽  
Characteristic Number ◽  
Characteristic Numbers ◽  
All Solutions ◽  
Fowler Equation ◽  
Respective Solution

We consider the Emden‐Fowler equation x” = ‐q(t)|x|2εx, ε > 0, in the interval [a,b]. The coefficient q(t) is a positive valued continuous function. The Nehari characteristic number An associated with the Emden‐Fowler equation coincides with a minimal value of the functional [] over all solutions of the boundary value problem x” = ‐q(t)|x|2εx, x(a) = x(b) = 0, x(t) has exactly (n ‐ 1) zeros in (a, b). The respective solution is called the Nehari solution. We construct an example which shows that the Nehari extremal problem may have more than one solution.

scholarly journals Essentially commutative C*-algebras with essential spectrum homeomorphic to S2n−1

2001 ◽  
Vol 70 (2) ◽  
pp. 199-210 ◽  
Author(s):  
Kunyu Guo
Keyword(s):  
Essential Spectrum ◽  
Complete System ◽  
Characteristic Number ◽  
Complete Classification ◽  
C Algebras ◽  
Characteristic Numbers

AbstractThis paper gives a complete classification of essentially commutative C*-algebras whose essential spectrum is homeomorphic to S2n−1 by their characteristic numbers. Let 1, 2 be such two C*-algebras; then they are C*-isomorphic if and only if they have the same n-th characteristic number. Furthermore, let γn() = m then is C*-isomorphic to C*(Mzl, …, Mzn) if m = 0, is C*-isomorphic C*(Tz1, …, Tzn−1, Tznm) if m ≠ 0. Some examples are given to show applications of the classfication theorem. We finally remark that the proof of the theorem depends on a construction of a complete system of representatives of Ext(S2n−1).

scholarly journals Affine Subspaces of Curvature Functions from Closed Planar Curves

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Leonardo Alese
Keyword(s):  
Arc Length ◽  
Planar Curves ◽  
Affine Subspaces ◽  
Planar Curve ◽  
Real Functions ◽  
Discrete Counterpart ◽  
Equivalent Conditions

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

scholarly journals Clothoid fitting and geometric Hermite subdivision

2021 ◽  
Vol 47 (4) ◽  
Author(s):  
Ulrich Reif ◽  
Andreas Weinmann
Keyword(s):  
Building Block ◽  
Numerical Results ◽  
Normal Vector ◽  
Subdivision Schemes ◽  
Planar Curves ◽  
Nonlinear Subdivision ◽  
New Strategy ◽  
Vector Information ◽  
Hermite Subdivision

AbstractWe consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.

scholarly journals EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

2021 ◽  
pp. 1-66
Author(s):  
Tom Bachmann ◽  
Kirsten Wickelgren
Keyword(s):  
Commutative Algebra ◽  
Vector Bundles ◽  
Euler Class ◽  
Complete Intersections ◽  
Bilinear Forms ◽  
Euler Numbers ◽  
Coherent Sheaves ◽  
Linear Subspaces ◽  
Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

Euler classes of vector bundles over manifolds

2021 ◽  
Vol 71 (1) ◽  
pp. 199-210
Author(s):  
Aniruddha C. Naolekar
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Euler Class ◽  
Homology Sphere ◽  
Rational Homology ◽  
Rational Homology Sphere ◽  
Orientable Manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

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