We shall give a simple (basically) purely in characteristic p proof of the irreducibility of the Igusa tower over Shimura varieties of PEL type. Our result covers Shimura variety of type A and type C classical groups, in particular, the Siegel modular varieties, the Hilbert-Siegel modular varieties, Picard surfaces and Shimura varieties of inner forms of unitary and symplectic groups over totally real fields.

We shall give a simple (basically) purely in characteristic p proof of the irreducibility of the Igusa tower over Shimura varieties of PEL type. Our result covers Shimura variety of type A and type C classical groups, in particular, the Siegel modular varieties, the Hilbert–Siegel modular varieties, Picard surfaces and Shimura varieties of inner forms of unitary and symplectic groups over totally real fields.

 Using known C ^α -multiplier result, we give necessary and sufficient conditions for the second order delay equations:

to have maximal regularity in Hölder continuous function spaces C ^α (?, X), where X is a Banach space, A is a closed operator in X, F, G ? ?(C([−r, 0],X), X) are delay operators for some fixed r > 0.

Using known C^α-multiplier result,we give necessary and sufficient conditions for the secondorder delay equations:?(t)=Au(t)+Fu^t+Gút+f(t),t∈Rto have maximal regularity in Holder continuous function spaces C^α(R,X),where X is a Banach space,A is a closed operator in X,F,G∈￡(C([-r,0],X),X)are delay operators for some fixed r>0.

The aim of this paper is to study the stability problem of the generalized sine functional equations as follows:

.

The aim of this paper is to study the stability problem of the generalized sine functional equations as follows:Namely,we have generalized the Hyers–Ulam stability of the(pexiderized)sine functional equation.

 The study of symmetric property in the L ²-sense for the non-positive definite operator is motivated by the theory of probability and analysis. This paper presents some sufficient conditions for the existence of symmetric measure for Lévy type operator. Some new examples are illustrated. The present study is an important step for considering various ergodic properties and functional inequalities of Lévy type operator.

The study of symmetric property in the L²-sense for the non-positive definite operator ismotivated by the theory of probability and analysis.This paper presents some sufficient conditions forthe existence of symmetric measure for Léevy type operator.Some new examples are illustrated.The present study is an important step for considering various ergodic properties and functional inequalities of Léevy type operator.

 In this paper, a Lebesgue type theorem on the structure of graphs embedded in the surface of characteristic σ ≤ 0 is given, that generalizes a result of Borodin on plane graphs. As a consequence, it is proved that every such graph without i-circuits for 4 ≤ i ≤ 11 − 3σ is 3-choosable, that offers a new upper bound to a question of Y. Zhao.

In this paper,a Lebesgue type theorem on the structure of graphs embedded in the surface of characteristicσ≤0 is given,that generalizes a result of Borodin on plane graphs.As a consequence, it is proved that every such graph without i-circuits for 4≤i≤11-3σis 3-choosable,that offiers a new upper bound to a question of Y.Zhao.

 In this paper, we prove that the 2D Navier-Stokes equations possess a global attractor in H ^k (Ω, R ²) for any k ≥ 1, which attracts any bounded set of H ^k (Ω, R ²) in the H ^k -norm. The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator, together with a classical existence theorem of global attractor. This extends Ma, Wang and Zhong’s conclusion.

In this paper,we prove that the 2D Navier–Stokes equations possess a global attractor inH^k(Ω,R²)for any k≥1,which attracts any bounded set of H^k(Ω,R²)in the H^k-norm.The result is established by means of an iteration technique and regularity estimates for the linear semigroup of operator,together with a classical existence theorem of global attractor.This extends Ma,Wang and Zhong's conclusion.

In this paper, we get a lower bound of inner radius of univalency of Schwarzian derivative by means of the norm of pre-Schwarzian derivative. Furthermore, we apply the theory of Universal Teichmuller Space to explain its geometric meaning which shows the relationship between the inner radius in Universal Teichmuller Space embedded by Schwarzian derivative and the norm defined in Universal Teichmuller Space embedded by pre-Schwarzian derivative.

In this paper,we get a lower bound of inner radius of univalency of Schwarzian derivativeby means of the norm of pre-Schwarzian derivative.Furthermore,we apply the theory of Universal Teichmuller Space to explain its geometric meaning which shows the relationship between the inner radius in Universal Teichmuller Space embedded by Schwarzian derivative and the norm defined in Universal Teichmuller Space embedded by pre-Schwarzian derivative.

 Borwein and Choi conjectured that a polynomial P(x) with coefficients ±1 of degree N − 1 is cyclotomic iff

, where N = p ₁ p ₂ … p _r and the p _i are primes, not necessarily distinct. Here Φ _p (x):= (x ^p − 1)/(x − 1) is the p-th cyclotomic polynomial. They also proved the conjecture for N odd or a power of 2. In this paper we introduce a so-called E-transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the conjecture.

Borwein and Choi conjectured that a polynomial P(x)with coecients±1 of degree N-1 is cyclotomiciff P(x)=±Φ_p1(±x)Φ_p2(±x^p1)···Φ_pr(±x^{p1,p2···pr-1}),where N=p1p2···pr and the pi are primes,not necessarily distinct.HereΦp(x):=(x^p-1)/(x-1)is the p-th cyclotomic polynomial.They also proved the conjecture for N odd or a power of 2.In this paper we introduce a so-called E-transformation,by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the conjecture.

Let x: M ⁿ → R ⁿ⁺¹ be an n(≥ 2)-dimensional hypersurface immersed in Euclidean space R ⁿ⁺¹. Let σ _i (0 ≤ i ≤ n) be the ith mean curvature and Q _n = Σ _i=0 ⁿ (−1) ⁱ⁺¹( _i ⁿ )σ ₁ ⁿ⁻ⁱ σ _i . Recently, the author showed that W _n (x) = ∫ _M Q _n dM is a conformal invariant under conformal group of R ⁿ⁺¹ and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional W _n is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in R ⁴ which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.

Let x:Mⁿ→Rⁿ⁺¹ be an n(≥2)-dimensional hypersurface immersed in Euclidean spaceRⁿ⁺¹.Letσi(0≤i≤n)be the ith mean curvature and Q_n=Σⁿ_i=₀(-1)^{i+1 n}_iσ^n-i₁σ_i.Recently,theauthor showed that W_n(x)=∫_M Q_ndM is a conformal invariant under conformal group of Rⁿ⁺¹ and called it the nth Willmore functional of x.An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface.The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in R⁴ which have good geometric behaviors.The ordinary diffierential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.

In this paper, we prove Beurling’s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.

In this paper,we prove Beurling's theorem for the Jacobi transform,from which we derive some other versions of uncertainty principles.

  Let G be an abelian group, B the G-graded λ-Hopf algebra with λ being a bicharacter on G. By introducing some new twisted algebras (coalgebras), we investigate the basic properties of the graded antipode and the structure for B. We also prove that a G-graded λ-Hopf algebra can be embedded in a usual Hopf algebra. As an application, it is given that if G is a finite abelian group then the graded antipode of a finite dimensional G-graded λ-Hopf algebra is invertible.

Let G be an abelian group,B the G-gradedλ-Hopf algebra withλbeing a bicharacteron G.By introducing some new twisted algebras(coalgebras),we investigate the basic properties ofthe graded antipode and the structure for B.We also prove that a G-gradedλ-Hopf algebra can beembedded in a usual Hopf algebra.As an application,it is given that if G is a finite abelian group thenthe graded antipode of a finite dimensional G-gradedλ-Hopf algebra is invertible.

 In this paper, we consider the Prandtl system for the non-stationary boundary layer in the vicinity of a point where the outer flow has zero velocity. It is assumed that U(t, x, y) = x ^m U ₁(t, x), where 0 ≤ x ≤ L and m ≥ 1. We establish the global existence of the weak solution to this problem. Moreover the uniqueness of the weak solution is proved.

In this paper,we consider the Prandtl system for the non-stationary boundary layer in thevicinity of a point where the outer flow has zero velocity.It is assumed that U(t,x,y)=x^mU₁(t,x),where 0≤x≤L and m≥1.We establish the global existence of the weak solution to this problem.Moreover the uniqueness of the weak solution is proved.

We consider the questions of boundary regularity for weak solutions of second-order nonlinear elliptic systems under the natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. The proof yields directly the optimal regularity for the solution in this neighborhood. This result is new for the situation under the natural growth conditions.

We consider the questions of boundary regularity for weak solutions of second-order non-linear elliptic systems under the natural growth condition.We obtain a general criterion for a weaksolution to be regular in the neighborhood of a given boundary point.The proof yields directly theoptimal regularity for the solution in this neighborhood.This result is new for the situation under thenatural growth conditions.

In this paper, we start at a random evolution system on biological particles, which is described by a Markov jump system. Under a suitable scaling, we perform a proper approximation procedure. Then the so-called weak convergence of Markov processes and Martingales allow us to establish a (deterministic) two species competitive Lotka-Volterra equation.

In this paper,we start at a random evolution system on biological particles,which isdescribed by a Markov jump system.Under a suitable scaling,we perform a proper approximationprocedure.Then the so-called weak convergence of Markov processes and Martingales allow us toestablish a(deterministic)two species competitive Lotka–Volterra equation.

设$\Lambda$是特征不等于２的数域ｋ上的Grassmann代数，$\wt{\Lambda}$是$\Lambda$的$\mathbb{Z}_2$-Galois覆盖代数．本文清晰地计算了$\wt{\Lambda}$的各阶Hochschild同调群和上同调群的ｋ－维数，并由此证明了$\oplus_{i\geq 1}HH^i(\wt{\Lambda})$与$\oplus_{i\geq 1}HH^i(\Lambda)$同构．而且，在基础数域的特征为零时，我们也计算了$\wt{\Lambda}$的循环同调群的ｋ－维数