Topological invariants of plane curve singularities: Polar quotients and Łojasiewicz gradient exponents

In this paper, we study polar quotients and Łojasiewicz exponents of plane curve singularities, which are not necessarily reduced. We first show that, for complex plane curve singularities, the set of polar quotients is a topological invariant. We next prove that the Łojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. For real plane curve singularities, we also give a formula computing the Łojasiewicz gradient exponent via real polar branches. As an application, we give effective estimates of the Łojasiewicz exponents in the gradient and classical inequalities of polynomials in two (real or complex) variables.

Irregular Hodge numbers of confluent hypergeometric differential equations

We give a formula computing the irregular Hodge numbers for a confluent hypergeometric differential equation. Comment: 9 pages. V2: typos corrected

Computing certain Gromov-Witten invariants of the crepant resolution of ℙ(1, 3, 4, 4)

We prove a formula computing the Gromov-Witten invariants of genus zero with three marked points of the resolution of the transversalA3-singularity of the weighted projective space ℙ(1,3,4,4) using the theory of deformations of surfaces withAn-singularities. We use this result to check Ruan’s conjecture for the stack ℙ(1,3,4,4).

Computing certain Gromov-Witten invariants of the crepant resolution of ℙ(1, 3, 4, 4)

We prove a formula computing the Gromov-Witten invariants of genus zero with three marked points of the resolution of the transversal A3-singularity of the weighted projective space ℙ(1,3,4,4) using the theory of deformations of surfaces with An-singularities. We use this result to check Ruan’s conjecture for the stack ℙ(1,3,4,4).

Application and Non-Homogeneous Index Grey Model NIGM(1,1,k)

To improve the modeling accuracy of grey model and broaden its application fields, a non-homogeneous index grey model (termed NIGM(1,1,k)) was built, which is based on the non-homogeneous dispersion index function and the formula computing the parameters of grey model NIGM(1,1,k) was proposed through the least square method. The function of the time response sequence of the proposed grey model was solved by taking differential equations as a deductive reasoning tool. The proposed grey NIGM(1,1,k) model has the characteristic of high precision as well as high adaptability. Examples validate the practicability and reliability of the proposed model.