A generalized version of Branner-Hubbard conjecture for rational functions

In 1992, Branner and Hubbard raised a conjecture that the Julia set of a polynomial is a Cantor set if and only if each critical component of its filled-in Julia set is not periodic. This conjecture was solved recently. In this paper, we generalize this result to a class of rational functions.     

NOTES ON POLYNOMIAL LEVELABILITY

We consider the polynomial levelability with respect to approximation algorithms (PLAA).A set A is PLAA if given any approximation algorithm a for A and a polynomial p,there are another approximation algorithm β for A and a polynomial q such that for infinitely many inputs x,a accepts x but has ruuning time greater than p(|x|) and β accepts x within time q(|x|).In this paper,an algorithm a is called an approximation algorithm for A if the symmetric difference A△L(a) is sparse,where L(a) is the set of strings recegnized by a.We prove that all natural NP-complete sets are PLAA unless P=NP and all EXP-complete sets are PLAA.     

Quasisymmetric geometry of the Julia sets of McMullen maps Dedicated to the Memory of Professor Lei Tan

We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z~m+ λ/z~?,where λ∈ C \ {0} and ? and m are positive integers satisfying 1/? + 1/m 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet(which is also standard in some sense).If the free critical points are not escaped, we give a sufficient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.     

An algorithm for decomposing a polynomial system into normal ascending sets

We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal ascending sets.If the polynomial system is zero dimensional,the set of the zeros of the polynomials is the union of the sets of the zeros of the normal ascending sets.     

On the set of grouplikes of a coring

We focus our attention to the set Gr(■) of grouplike elements of a coring ■ over a ring A.We do some observations on the actions of the groups U(A) and Aut(■) of units of A and of automorphisms of corings of ■,respectively,on Gr(■),and on the subset Gal(■) of all Galois grouplike elements.Among them,we give conditions on ■ under which Gal(■) is a group,in such a way that there is an exact sequence of groups {1} → U(Ag) → U(A) → Gal(■) → {1},where Ag is the subalgebra of coinvariants for some g ∈ Gal(■).     

Automorphism groups of pseudo-real Riemann surfaces of low genus

A pseudo-real Riemann surface admits anticonformal automorphisms but no anticonformal involution.We obtain the classifcation of actions and groups of automorphisms of pseudo-real Riemann surfaces of genera 2,3 and 4.For instance the automorphism group of a pseudo-real Riemann surface of genus 4 is eitherC4orC8or the Fro¨benius group of order 20,and in the case ofC4there are exactly two possible topological actions.Let MK P R,g be the set of surfaces in the moduli space MK g corresponding to pseudo-real Riemann surfaces.We obtain the equisymmetric stratifcation of MK P R,g for generag=2,3,4,and as a consequence we have that MK P R,gis connected forg=2,3 but MK P R,4has three connected components.     

Toeplitz算子在Dirichlet空间上的亚正规性

In this paper,we prove that the necessary and sufficient condition for a Toeplitz operator Tu on the Dirichlet space to be hyponormal is that the symbol u is constant for the case that the projection of u in the Dirichlet space is a polynomial and for the case that u is a class of special symbols,respectively.We also prove that a Toeplitz operator with harmonic polynomial symbol on the harmonic Dirichlet space is hyponormal if and only if its symbol is constant.     

Julia集的淹没分支

In this note,it is shown that if a rational function fofdegree≥2 has a nonempty set of buried points ,then for a generic choice of the point z in the Julia set ,z is a buried point ,and if the Julia set is disconnected,it has uncountably many buried components.     

Conditions for polynomial Liénard centers

In 1999, Christopher gave a necessary and sufficient condition for polynomial Li′enard centers, which requires coupled functional equations, where the primitive functions of the damping function and the restoring function are involved, to have polynomial solutions. In order to judge whether the coupled functional equations are solvable, in this paper we give an algorithm to compute a Gr¨obner basis for irreducible decomposition of algebraic varieties so as to find algebraic relations among coefficients of the damping function and the restoring function. We demonstrate the algorithm for polynomial Li′enard systems of degree 5, which are divided into 25 cases. We find all conditions of those coefficients for the polynomial Li′enard center in 13 cases and prove that the origin is not a center in the other 12 cases.     

Estimation of the misclassification error for multicategory support vector machine classification

The purpose of this paper is to provide an error analysis for the multicategory support vector machine （MSVM） classificaton problems. We establish the uniform convergency approach for MSVMs and estimate the misclassification error. The main difficulty we overcome here is to bound the offset vector. As a result, we confirm that the MSVM classification algorithm with polynomial kernels is always efficient when the degree of the kernel polynomial is large enough. Finally the rate of convergence and examples are given to demonstrate the main results.     

Restricted parameter range promise set cover problems are easy

The restricted parameter range set cover problem is a weak form of the NP-hard set cover problem with the restricted range of parameters. We give a polynomial time algorithm for this problem by lattices.     

A Biordered Set Representation of Regular Semigroups

In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the w-ideals of E, we construct a fundamental regular semigroup WE called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that WE can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to WE whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of WE. Moreover, when E is a biordered set of a semilattice Eo, WE is isomorphic to the Munn-semigroup TEo; and when E is the biordered set of a band B, WE is isomorphic to the Hall-semigroup WB.     

Tree wavelet approximations with applications

We construct a tree wavelet approximation by using a constructive greedy scheme (CGS). We define a function class which contains the functions whose piecewise polynomial approximations generated by the CGS have a prescribed global convergence rate and establish embedding properties of this class. We provide sufficient conditions on a tree index set and on bi-orthogonal wavelet bases which ensure optimal order of convergence for the wavelet approximations encoded on the tree index set using the bi-orthogonal wavelet bases. We then show that if we use the tree index set associated with the partition generated by the CGS to encode a wavelet approximation, it gives optimal order of convergence.     

A Characterization of a Class of Symmetric Graphs of Twice Prime Valency

Let г be a connected undirected graph aud G a group of automorphisms ofг.G is said to act symmetrically on г and г is said to be G-symmetric, if G istransitive on the set {(α,β) |α,β adjacent vertices} of 1-arcs of г. Thus G is tran-sitive on vertices and the stablizer G_a of a vertex a is transitive on the set г_1(a)     

A NOTE ON SENSITIVITY ANALYSIS OF SHAPE-FROM-MOMENTS

The shape-from-moments problem is to reconstruct a planar polygon from a set of its complex moments. To reconstruct a polygon means to estimate the vertices and the ordering of the vertices. We notice that some coefficients are very important in finding out the ordering of the vertices. We introduce sensitive factors for the coefficients and use it to analyze sensitivity. These factors are also useful for the sensitivity of the vertices.     

OPTIMALLY ACCURATE PETROV-GALERKIN METHOD OF FINITE ELEMENTS

We perform analysis for a finite elements method applied to the singular self-adjoint problem.This method uses continuous piecewise polynomial spaces for the trial and the test spaces.We fit the trial polynomial space by piecewise exponentials and we apply so exponentially fitted Galerkin method to singular self-adjomt problem by approximating driving terms by Lagrange piecewise polynomials,linear,quadratic and cubic.Wt measure the erroe in max norm.We show that method is optimal of the first order in the error estimate,We also give numerical results for the Galerkin approximation.     

Dynamical Properties of Morphisms on Higher Dimensional Projective Space over Local Field北大核心CSCD

Let Cv be an algebraically closed non-archimedean field, complete with respect to a valuation v. Let ϕ : PN → PN be a morphism of degree greater than one defined over Cv, Φ a lift of ϕ. Let GΦ be the Green function of Φ and ρ the chordal metric on PN(Cv). In this paper, we first study the properties of reduction of points in high dimensional projective space and reduction of automorphisms of PN with degree one. With the help of Green function GΦ of Φ, we introduce the arithmetic distance of morphisms and investigate its property. The necessary and sufficient condition which Φ has good reduction is obtained in this paper. We also describe explicitly the Filled Julia set of Φ by its Green function. © 2022 Chinese Academy of Sciences. All rights reserved.     

Mather theory for piecewise smooth Lagrangian systems

We establish the Mather theory for a type of piecewise smooth and positive definite Lagrangian systems.It models a mechanical system subject to external impulsive forcing.We show the existence of the minimal measure and the Lipschitz property of Aubry set.In addition,the weak KAM solution to this kind of piecewise smooth Lagrangian is also established.