Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ of $ {\mathbb {R}}^{\alpha },$ and we study the Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative $\partial _s\mathcal {T}(s;\varepsilon,\,a)$ tends to $-\infty$ as $(s,\,\varepsilon )\to (0^{+},\,0)$ uniformly on compact subsets of $A.$ This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.

L2 Boundedness of Discrete Double Hilbert Transform Along polynomials

We will show $L^{2}$ boundedness of Discrete Double Hilbert Transform along polynomials satisfying some conditions. Double Hilbert exponential sum along polynomials:$\mu(\xi)$ is Fourier multiplier of discrete double Hilbert transform along polynomials. In chapter 1, we define the reverse Newton diagram. In chapter 2, We make approximation formula for the multiplier of one valuable discrete Hilbert transform by study circle method. In chapter 3, We obtain result that $\mu(\xi)$ is bounded by constants if $|D|\geq2$ or all $(m,n)$ are not on one line passing through the origin. We study property of $1/(qt^{n})$ and use circle method (Propsotion 2.1) to calculate sums. We also envision combinatoric thinking about $\mathbb{N}^{2}$ lattice points in j-k plane for some estimates. Finally, we use geometric property of some inequalities about $(m,n)\in\Lambda$ to prove Theorem 3.3. In chapter 4, We obtain the fact that $\mu(\xi)$ is bounded by sums which are related to $\log_{2}({\xi_{1}-a_{1}\slash {q}})$ and $\log_{2}({\xi_{2}-a_{2}\slash {q}})$ and the boundedness of double Hilbert exponential sum for even polynomials with torsion without conditions in Theorem 3.3. We also use $\mathbb{N}^{2}$ lattice points in j-k plane and Proposition 2.1 which are shown in chapter 2 and some estimates to show that Fourier multiplier of discrete double Hilbert transform is bounded by terms about $\log$ and integral this with torsion is bounded by constants.

Radial kinetic nonholonomic trajectories are Riemannian geodesics!

Nonholonomic mechanics describes the motion of systems constrained by nonintegrable constraints. One of its most remarkable properties is that the derivation of the nonholonomic equations is not variational in nature. However, in this paper, we prove (Theorem 1.1) that for kinetic nonholonomic systems, the solutions starting from a fixed point q are true geodesics for a family of Riemannian metrics on the image submanifold $${{\mathcal {M}}}^{nh}_q$$ M q nh of the nonholonomic exponential map. This implies a surprising result: the kinetic nonholonomic trajectories with starting point q, for sufficiently small times, minimize length in $${{\mathcal {M}}}^{nh}_q$$ M q nh !

ON BOREL DIRECTION CONCERNING SMALL FUNCTIONS

In this paper, we shall prove Theorem 1 Let $f$ be nonconstant meromorphic in $\mathbb{C}$ with finite positive order $\lambda$, $\lambda(r)$ be a proximate order of $f$ and $U(r, f)=r^{\lambda(r)}$, then for each number $\alpha$,$0<\alpha<\pi/2$, there exists a number $\phi_0$ with $0\le \phi_0 < 2\pi$ such that the inequality \[ \limsup_{r\to\infty}\sum_{i=1}^3 n(r, \phi_0, \alpha, f=a_i(z))/U(r, f)>0,\] holds for any three distinct meromorphic function $a_i(z)(i=1, 2, 3)$ with $T(r,a_i)=o(U(r, f))$ as $r\to\infty$.

Regge-Wheeler Type Equations

This chapter explores estimates for Regge-Wheeler type wave equations used in Theorem M1. It first proves basic Morawetz estimates for ψ‎. The chapter then proves rp-weighted estimates in the spirit of Dafermos and Rodnianski for ψ‎. In particular, it obtains as an immediate corollary the proof of Theorem 5.17 in the case s = 0 (i.e., without commutating the equation of ψ‎ with derivatives). It also uses a variation of the method of [5] to derive slightly stronger weighted estimates and prove Theorem 5.18 in the case s = 0. Finally, commuting the equation of ψ‎ with derivatives, the chapter completes the proof of Theorem 5.17 by controlling higher order derivatives of ψ‎.

Initialization and Extension (Theorems M6, M7, M8)

This chapter focuses on the proof for Theorem M6 concerning initialization, Theorem M7 concerning extension, and Theorem M8 concerning the improvement of higher order weighted energies. It first improves the bootstrap assumptions on decay estimates. The chapter then improves the bootstrap assumptions on energies and weighted energies for R and Γ‎ relying on an iterative procedure which recovers derivatives one by one. It also outlines the norms for measuring weighted energies for curvature components and Ricci coefficients. To prove Theorem M8, the chapter relies on Propositions 8.11, 8.12, and 8.13. Among these propositions, only the last two involve the dangerous boundary term.

Decay Estimates for α‎ and α‎ (Theorems M2, M3)

This chapter investigates the proof for Theorems M2 and M3. It relies on the decay of q to prove the decay estimates for α‎ and α‎. More precisely, the chapter relies on the results of Theorem M1 to prove Theorem M2 and M3. In Theorem M1, decay estimates are derived for q defined with respect to the global frame constructed in Proposition 3.26. To recover α‎ from q, the chapter derives a transport equation for α‎ where q is on the RHS. It then derives as Teukolsky–Starobinsky identity a parabolic equation for α‎. The chapter also improves bootstrap assumptions for α‎.

Existence of Many Large Orbits

A plaid polygon is called N-fat if it is not contained in any strip of width N. As a related notion, a plaid polygon is called N-long if it has diameter at least N. This chapter will prove Theorem 0.8. Section 22.2 studies equidistribution properties of the plaid PET map Φ‎A, as a function of A. Section 22.3 uses these equidistribution properties to show that the N-fat polygons essentially appear everywhere in the planar plaid model. The result is called the Ubiquity Lemma. Section 22.4 examines how the plaid model interacts with the grid of all lines of capacity at most K. Section 22.5 uses the Rectangle Lemma on many scales to show the existence of many distinct N-fat polygons. Section 2.6 discusses some properties of continued fractions and circle rotations. Finally, Section 22.7 proves the Grid Supply Lemma.

The Plaid Master Picture Theorem

This chapter aims to prove Theorem 1.4 and Theorem 0.3, the Plaid Master Picture Theorem. Both of these results are deduced from Theorem 8.2, which says that the union PLA of plaid polygons is generated by an explicitly defined tiling classifying space (ලA, XP). Moreover, there is a nice space X which has the individual spaces XP as rational slices. The space X has a partition into convex polytopes, and one obtains the partition of XP by intersecting the relevant slice with this partition. Section 8.2 describes the space X. Section 8.3 describes the partition of X into convex integer polytopes. The partition is called the checkerboard partition. Section 8.4 explains the classifying map Φ‎A : Π‎ → XP. Section 8.5 states Theorem 8.2 and deduces Theorem 1.4 and Theorem 0.3 from it.

Graph Master Picture Theorem

This chapter aims to prove Theorem 0.4, the Graph Master Picture Theorem. Theorem 0.4 is proven in two different ways, the first proof is discussed here; it deduces Theorem 0.4 from Theorem 13.2, which is a restatement of [S1, Master Picture Theorem] with minor cosmetic changes. The chapter is organized as follows. Section 13.2 discusses the special outer billiards orbits on kites. Section 13.3 defines the arithmetic graph, which is an arithmetical way of encoding the behavior of a certain first return map of the special orbits. Section 13.4 states Theorem 13.2, a slightly modified and simplified version of [S1, Master Picture Theorem]. Section 13.5 deduces Theorem 0.4 from Theorem 13.2 and one extra piece of information. Finally, Section 13.6 lists the polytopes comprising the partition associated to Theorems 13.2 and 0.4.