Narrow Results

Let $A$ be the ring of elements in an algebraic function field $K$ over ${𝔽}_q$ which are integral outside a fixed place $\infty$. In contrast to the classical modular group $S{L}_2(\mathbb{Z})$ and the Bianchi groups, the Drinfeld modular group$G=G{L}_2(A)$ is not finitely generated and its automorphism group $\mathrm{\text{Aut}}(G)$ is uncountable. Except for the simplest case $A={𝔽}_q[t]$ not much is known about the generators of $\mathrm{\text{Aut}}(G)$ or even its structure. We find a set of generators of $\mathrm{\text{Aut}}(G)$ for a new case.

On the way, we show that every automorphism of $G$ acts on both the cusps and the elliptic points of $G$. Generalizing a result of Reiner for $A={𝔽}_q[t]$ we describe for each cusp an uncountable subgroup of $\mathrm{\text{Aut}}(G)$ whose action on $G$ is essentially defined on the stabilizer of that cusp. In the case where $\delta$ (the degree of $\infty$) is $1$, the elliptic points are related to the isolated vertices of the quotient graph $G\setminus {𝒯}$ of the Bruhat–Tits tree. We construct an infinite group of automorphisms of $G$ which fully permutes the isolated vertices with cyclic stabilizer.

Large samples of K^± → 3π and K^± → π⁺π^-e^±ν (K_e4) decays have been collected by the NA48/2 experiment at CERN SPS.

At the π⁺π^- threshold, the π⁰π⁰ invariant mass spectrum of the decay K^± → π^±π⁰π⁰ xhibits a Wigner cusp, from which the S-wave ππ scattering lengths are extracted with high precision.

The same scattering lengths are also independently determined from the accurate measurement of the form factors in the K_e4 decay K^± → π⁺π^-e^±ν.

I present analytical expressions for the massive cusp anomalous dimension in QCD through three loops, first calculated in 2014, in terms of elementary functions and ordinary polylogarithms. I observe interesting relations between the results at different loops and provide a conjecture for the n-loop cusp anomalous dimension in terms of the lower-loop results. I also present numerical results and simple approximate formulas for the cusp anomalous dimension relevant to top-quark production.

In this work, the evolution of a family of open-to-closed curves of saddle-node bifurcations of periodic orbits is numerically studied in Chua's equation. This family is organized by open and closed curves of Shil'nikov homoclinic connections in the presence of a nontransversal T-point. We remark the important role of cusp catastrophes of periodic orbits in the complete mechanism of creation/destruction of such closed curves of saddle-node bifurcations.

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold–Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold–heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems.

In this paper, we study the number of limit cycles of a kind of polynomial Liénard system with a nilpotent cusp and obtain some new results on the lower bound of the maximal number of limit cycles for this kind of systems.

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

This paper is concerned with the expansion of the first-order Melnikov function for general Hamiltonian systems with a cuspidal loop having order m. Some criteria and formulas are derived, which can be used to obtain first-order coefficients in the expansion. In particular, we deduce the first-order coefficients for the case m = 3 and give the corresponding conditions of existing several limit cycles. As an application, we study a Liénard system of type (n, 9) and prove that it can have 14 limit cycles near a cuspidal loop of order 3 for n = 8.

Using the theory of minimal models of quasi-projective surfaces we give a new proof of the theorem of Lin–Zaidenberg which says that every topologically contractible algebraic curve in the complex affine plane has equation Xⁿ = Y^m in some algebraic coordinates on the plane. This gives also a proof of the theorem of Abhyankar–Moh–Suzuki concerning embeddings of the complex line into the plane. Independently, we show how to deduce the latter theorem from basic properties of ℚ-acyclic surfaces.

A resonance-like structure as narrow as 10 MeV is observed in the K⁻p invariant mass distributions in ${\mathrm{\Lambda }}_c^{+}\to p{K}^{-}{\pi }^{+}$. This precise measurement is based on a data sample of about 1.5 million events, and the bin width of K⁻p invariant mass is only 1 MeV. The narrow peak precisely lies on the Λη threshold, because of which it is natural to identify it as a threshold cusp. Being different from the common two-body unitary cusp, we find that the narrowness of this cusp can be induced by a nearby triangle singularity of the Λa₀(980) or ηΣ(1660) rescattering process.

Taking advantage of the high spatial-resolution and global coverage of Defense Meteorological Satellite Program/Special Sensor Ultraviolet Spectrographic Imager (DMSP/SSUSI) observations, we investigated the critical interplanetary factors in controlling the cusp auroral emission by dividing the midday auroras into the gap (weak emission) and non-gap (intense emission) events. Although the cusp auroral intensity is essentially determined by a parameter (dФ_MP/dt = v^4/3B_T^2/3sin^8/3(θ/2)) which well predicts the global solar wind power input and is calculated from the interplanetary magnetic field (IMF) direction, IMF magnitude ((B_y² + B_z²)^1/2), and solar wind speed (V), we found that the cusp aurora is statistically weak under southward IMF but intense when the V and the IMF |B_y| are greater. Further, we confirmed that even with V > 600 km/s, the intense-aurora events still show a minimal occurrence near the IMF |B_y| = 0. These results demonstrate that the IMF |B_y| is critical in controlling the cusp auroral intensity, most likely by producing an electric field through V×B_y.

In this paper we consider smooth affine plane curves of genus two having one place at infinity. We identify them with projective plane curves of genus two having only one cusp as their singular points and meeting with the line at infinity only at the cusp. We characterize such curves by the self-intersection number of the strict transform of them via the minimal embedded resolution of their cusp.