On variance of exponents for isolated surface singularities with modality ⩽ 2

Using the theory of the mixed Hodge structure one can define a notion of exponents of a singularity.In 2000,Hertling proposed a conjecture about the variance of the exponents of a singularity.Here,we prove that the Hertling conjecture is true for isolated surface singularities with modality ≤ 2.     

A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers

Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture(see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-dimensional(n≥3) polyhedron. It is well known that getting the estimate of integral points in the polyhedron is equivalent to getting the estimate of the de Bruijn function ψ(x, y), which is important and has a number of applications to analytic number theory and cryptography. We prove the Yau number theoretic conjecture for n = 6. As an application, we give a sharper estimate of function ψ(x, y) for 5≤y 17, compared with the result obtained by Ennola.     

Uniform Boundedness of Torsion Subgroups of Linear Groups

The torsion conjecture says： for any abelian variety A defined over a number field k, the order of the torsion subgroup of A（k） is bounded by a constant C（k,d） which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G（k）. Then the order of Г is bounded by a constant C＇（k, d） which depends only on k and the dimension d of G.     

Singularities of the Radon Transform of a Class of Piecewise Smooth Functions in R2

The Radon transform is the mathematical foundation of Computerized Tomography[1](CT).Its important applications includes medical CT,noninvasive test and etc.If one is specially interested in the places at which the image function changed largely such as the interfaces of two different tissues,tissue and ill tissue and the interfaces of two difierent matters,and want to reconstruct the outlines of the interfaces,one should reconstruct the singularities of the image function.The exact inversion of the Radon transform is valid only for smooth function[2].The singularity places of the reconstructed function should be studied specially.The research includes the propagation and inversion of singularity of the Radon transform.If one use convolutionbackprojection method to reconstruct the image function,the reconstructed function become blurring at the singularity places of the original function.M.Jiang and etc[3]developed a blind deconvolution method deblurring reconstructed image.By[4]and following research,we see that one can use a neighborhood data of the singularities of the Radon transform to inverse the singularity of the Radon transform,and therefore the reconstruction is available for some incomplete data reconstructions.     

A proof of Sethares'''' conjecture

Let (?)(z) be holomorphic in the unit disk △ and meromorphic on △. Suppose / is a Teichmuller mapping with complex dilatation In 1968, Sethares conjectured that f is extremal if and only if either (i)(?) has a double pole or (ii)(?) has no pole of order exceeding two on (?)△. The "if" part of the conjecture had been solved by himself. We will give the affirmative answer to the "only if" part of the conjecture. In addition, a more general criterion for extremality of quasiconformal mappings is constructed in this paper, which generalizes the "if" part of Sethares' conjecture and improves the result by Reich and Shapiro in 1990.     

The spectrum of path factorization of bipartite multigraphs

LetλK_(m,n)be a bipartite multigraph with two partite sets having m and n vertices, respectively.A P_v-factorization ofλK_(m,n)is a set of edge-disjoint P_v-factors ofλK_(m,n)which partition the set of edges ofλK_(m,n).When v is an even number,Ushio,Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P_v-factorization ofλK_(m,n).When v is an odd number,we have proposed a conjecture.Very recently,we have proved that the conjecture is true when v=4k-1.In this paper we shall show that the conjecture is true when v = 4k 1,and then the conjecture is true.That is,we will prove that the necessary and sufficient conditions for the existence of a P_(4k 1)-factorization ofλK_(m,n)are(1)2km≤(2k 1)n,(2)2kn≤(2k 1)m,(3)m n≡0(mod 4k 1),(4)λ(4k 1)mn/[4k(m n)]is an integer.     

Maximum Genus of Strong Embeddings

The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover.Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph.     

On Edge Singularity and Eigenvectors of Mixed Graphs

Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.     

Proof of the Volume Conjecture for Whitehead Doubles of a Family of Torus Knots

A technique to compute the colored Jones polynomials of satellite knots,illus- trated by the Whitehead doubles of knots,is presented.Then the author proves the volume conjecture for Whitehead doubles of a family of torus knots and shows some interesting observations.     

Optimal constant in an L~2 extension problem and a proof of a conjecture of Ohsawa

In this paper,we solve the optimal constant problem in the setting of Ohsawa’s generalized L2extension theorem.As applications,we prove a conjecture of Ohsawa and the extended Suita conjecture,we also establish some relations between Bergman kernel and logarithmic capacity on compact and open Riemann surfaces.     

AVDTC Numbers of Generalized Halin Graphs with Maximum Degree at Least 6

In a paper by Zhang and Chen et al.（see [11]）, a conjecture was made concerning the minimum number of colors Xat（G） required in a proper total-coloring of G so that any two adjacent vertices have different color sets, where the color set of a vertex v is the set composed of the color of v and the colors incident to v. We find the exact values of Xat（G） and thus verify the conjecture when G is a Generalized Halin graph with maximum degree at least 6, A generalized Halin graph is a 2-connected plane graph G such that removing all the edges of the boundary of the exterior face of G （the degrees of the vertices in the boundary of exterior face of G are all three） gives a tree.     

On the Overconvergence of Hermite Interpolating Polynomials

In this paper we study the overconvergence of Hermite interpolating polynomials. A new extension of a theorem of J.L.Walsh is obtained and a conjecture is posed.     

The topology of moment-angle manifolds——on a conjecture of S. Gitler and S. López de Medrano

In this paper,we study the topology of moment-angle manifolds and prove a conjecture of S.Gitler and S.Lopez de Medrano concerned with the behavior of the moment-angle manifold under the surgery 'cutting off a vertex' on a simple polytope.Let P be a simple polytope of dimension n with m facets and Pv be a poly tope obtained from P by cutting off one vertex v.Let Z=Z(P) and Z_v=Z(P_v) be the corresponding moment-angle manifolds.S.Gitler and S.Lopez de Medrano conjectured that:Z_v is diffeomorphic to ?[(Z(-D~(n+m)) × D~2]##_(j=1)~(m-n)(_j~(m-n))(S~(j+2) × S~(m+n-j-1)),and they proved the conjecture in the case m 3 n.In this paper we prove the conjecture in the general case.     

Strong openness of multiplier ideal sheaves and optimal L~2 extension

In this paper, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Koll′ar and Jonsson-Mustat?a implies the truth of twisted versions of the strong openness conjecture; our optimal L~2 extension implies Berndtsson's positivity of vector bundles associated to holomorphic fibrations over a unit disc.     

Proof of Ding’s conjecture on maximal stable sets and maximal cliques in planar graphs

X. Deng et al. proved Chvātal＇s conjecture on maximal stable sets and maximal cliques in graphs. G. Ding made a conjecture to generalize Chvátal＇s conjecture. The purpose of this paper is to prove this conjecture in planar graphs and the complement of planar graphs.     

A Generalized Form of Whitney's Theorem

Whitney's theorem is a famous theorem in the local singularity theory.In this paper,as an application of Malgrange preparation theorem,a generalized form of Whitney's theorem will be derived.     

On Browkin''''s conjecture about the elements of order five in K_2(Q)

Based on the work of Lenstra, a succinct proof of Browkin's conjecture about the elements of order five in K2(Q) is given.     

A Generalized Form of Whitney＇s Theorem

Whitney's theorem is a famous theorem in the local singularity theory.In this paper,as an application of Malgrange preparation theorem,a generalized form of Whitney's theorem will be derived.     

NON-EXISTENCE OF LIMIT CYCLE AROUND A WEAK FOCUS OF ORDER THREE FOR ANY QUADRATIC SYSTEM

In 1980 Professor Ye Yangian proposed a conjecture that around a weak focus oforder 3 of any real quadratic differential system there can exist no limit cycle.It is thepurpose of this paper to give a proof of the conjecture by using the method of continuousvariation of a coefficient.