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 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)).

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).

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 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 Tangency quantum cohomology and characteristic numbers

2001 ◽  
Vol 73 (3) ◽  
pp. 319-326
Author(s):  
JOACHIM KOCK
Keyword(s):  
2D Gravity ◽  
Quantum Cohomology ◽  
Rational Curves ◽  
The Other ◽  
Enumerative Geometry ◽  
Poincaré Metric ◽  
Characteristic Numbers ◽  
Systems Of Differential Equations ◽  
Topological Recursion ◽  
Homogeneous Variety

This work establishes a connection between gravitational quantum cohomology and enumerative geometry of rational curves (in a projective homogeneous variety) subject to conditions of infinitesimal nature like, for example, tangency. The key concept is that of modified psi classes, which are well suited for enumerative purposes and substitute the tautological psi classes of 2D gravity. The main results are two systems of differential equations for the generating function of certain top products of such classes. One is topological recursion while the other is Witten-Dijkgraaf-Verlinde-Verlinde. In both cases, however, the background metric is not the usual Poincaré metric but a certain deformation of it, which surprisingly encodes all the combinatorics of the peculiar way modified psi classes restrict to the boundary. This machinery is applied to various enumerative problems, among which characteristic numbers in any projective homogeneous variety, characteristic numbers for curves with cusp, prescribed triple contact, or double points.

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).

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

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