Solutions of multiple vector refinement equations with infinite mask

This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（α（α））α∈Z^s is an infinitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M^-n =0, with m = detM. Some properties about the solutions of refinement equations axe obtained.     

The Least Regular Order with Respect to a Regular Congruence on Ordered Γ-Semigroups

The motivation mainly comes from the conditions of congruences to be regular that are of importance and interest in ordered semigroups. In 1981, Sen has introduced the concept of the Γ-semigroups. We can see that any semigroup can be considered as a Γ-semigroup. In this paper, we introduce and characterize the concept of the regular congruences on ordered Γ-semigroups and prove the following statements on an ordered Γ-semigroup M : (1) Every ordered semilattice congruences is a regular congruence. (2) There exists the least regular order on the Γ-semigroup M/ρ with respect to a regular congru- ence ρ on M . (3) The regular congruences are not ordered semilattice congruences in general.     

Lp-Solutions of Vector Refinement Equations with General Dilation Matrix

The purpose of this paper is to investigate the solutions of refinement equations of the form ψ（x）∑α∈Z α（α）ψ（Mx-α）,x∈R, where the vector of functions ψ = （ψ1,..., ψr）^T is in （Lp（R^n））^r, 0 〈 p≤∞, α（α）, α ∈ Z^n, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn→∞M^-n=0, In this article, we characterize the existence of an Lp=solution of the refinement equation for 0〈 p ≤∞, Our characterizations are based on the p-norm joint spectral radius.     

关于Г-半环

Just as Nobusawa has extended the concept of ring to the concept of Γ-ring. we introduce in this paper a new concept of Γ-hemiring based on the concept of abelian additive hemigroup. And then, we extend the foundamental results associated with hemiring which are developed by Bourne and Zassenhaus to some analogous results associated with Γ-hemiring, such as the notions of ideal, zeroid, homomorphism and Jacobsonian radical and semi-radical in the case of Γ-hemiring. Finally, we have given some conditions for a Γ-hemiring being a Γ-ring.     

关于亚直接不可约完全左对称Г-环

A left ideal L of a Г-ring M is called left symmetric if aabβb∈L implies aacβb∈L, where a,b, c∈ M, a ,β∈Γ. A Γ-ring M is called to be a Γ-division ring if it has the left unity, and if left oprator ring R is a division ring.In this paper, the following results are pvoved:Suppose that every left ideal of a Γ-ring M is left symmetiic.If A is subdirectly irreducible with more than one element and with no nonzero nilpotents then it is a Γ-division ring.If M is subdirectly irreducible and if D≠M then M/D is a Γ-division ring where D=(a∈M|aΓb=0,0≠b∈M).     

POSITIVE SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS WITH MIXED BOUNDARY CONDITIONS

We consider the nonlinear elliptic equation with mixed boundary conditions where Ω is a smooth bounded domain in R~N (N≥3), whose boundary Ω is made of two open sets Γ_0 and Γ_1, p= N+2/N-2. We prove the existence of a positive solution of (P)when b (x) and λ satisfy suitable conditions.     

Difference of a Hauptmodul for Γ0(N ) and certain Gross-Zagier type CM value formulas

In this work, we show that the difference of a Hauptmodul for a genus zero group Γ₀(N) as a modular function on Y₀(N) × Y₀(N) is a Borcherds lift of type(2, 2). As applications, we derive the monster denominator formula like product expansions for these modular functions and certain Gross-Zagier type CM value formulas.     

Estimates for eigenvalues of stochastic matrices

It is well-known that the eigenvalues of stochastic matrices lie in the unit circle and at least one of them has the value one. Let {1, r 2 , ··· , r N } be the eigenvalues of stochastic matrix X of size N × N . We will present in this paper a simple necessary and sufficient condition for X such that |r j | < 1, j = 2, ··· , N . Moreover, such condition can be very quickly examined by using some search algorithms from graph theory.     

Deviation matrix and asymptotic variance for GI/M/1-type Markov chains

We investigate deviation matrix for discrete-time GI/M/1-type Markov chains in terms of the matrix-analytic method, and revisit the link between deviation matrix and the asymptotic variance. Parallel results are obtained for continuous-time GI/M/1-type Markov chains based on the technique of uniformization. We conclude with A. B. Clarke＇s tandem queue as an illustrative example, and compute the asymptotic variance for the queue length for this model.     

ON NODE ENUMERATION AND NODE PATH LENGTH OF ORDERED TREES

Considering the class (n, m) of ordered trees with m leaves and n-m internal nodes, a set of generating functions are established for the following problems: (1) the total number nodes with degree r over Γ(n, m), (2) the total path length of nodes over Γ(n, m), and (3) the total number of nodes over Γ(n, m) on level k. Some particular counting fomulas are derived from them.     

On Hamiltonian Property of Cayley Digraphs

Let G be a finite group generated by S and C(G,S) the Cayley digraphs of G with connection set S.In this paper,we give some sufficient conditions for the existence of hamiltonian circuit in C(G,S),where G=Z_m ×H is a semiproduct of Z_m by a subgroup H of G.In particular,if m is a prime,then the Cayley digraph of G has a hamiltonian circuit unless G=Z_m×H.In addition,we introduce a new digraph operation,called φ-semiproduct of Γ₁ by Γ₂ and denot...     

Vector subdivision schemes in (Lp(Rs))r(1 ≤ p ≤∞) spaces

The purpose of this paper is to investigate the refinement equations of the form ψ(x) = ∑α∈Zs a(α)ψ(Mx - α), x ∈ Rs,where the vector of functions ψ=(ψ1,…,ψr)T is in (Lp(Rs))r, 1≤p≤∞,a(α),α∈Zs,is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix suchthat lim n→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vectorof compactly supported functions ψ0 ∈ (Lp(Rs))r and use the iteration schemes fn := Qnaψ0,n = 1,2,…,where Qa is the linear operator defined on (Lp(Rs))r given by Qaψ:= ∑α∈Zs a(α)ψ(M·- α),ψ∈ (Lp(Rs))r. This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of somelinear operators determined by the sequence a and the set B restricted to a certain invariant subspace, wherethe set B is a complete set of representatives of the distinct cosets of the quotient group Zs/MZs containing 0.     

Browder spectra and essential spectra of operator matrices

Let Mc = （ A0CB ） be a 2 × 2 upper triangular operator matrix acting on the Banach space X × Y. We prove that
σr（A） ∪ σr（ B）= σr （Mc） ∪ W ,
where W is the union of certain of the holes in σr（Mc） which happen to be subsets of σr（A） ∩ σr（B）, and σr（A）, σr（B）, σr（Mc） can be equal to the Browder or essential spectra of A, B, Mc, respectively. We also show that the above result isn＇t true for the Kato spectrum, left （right） essential spectrum and left （right） spectrum.     

The morphism induced by Frobenius push-forward

Let X be a smooth projective curve of genus g 2 over an algebraically closed field k of characteristic p0,and F:X→X(1)the relative Frobenius morphism.Let M s X(r,d)(resp.M ss X(r,d))be the moduli space of(resp.semi-)stable vector bundles of rank r and degree d on X.We show that the set-theoretic map S ss Frob:M ss X(r,d)→M ss X(1)(rp,d+r(p-1)(g-1))induced by[E]→[F(E)]is a proper morphism.Moreover,the induced morphism S s Frob:M s X(r,d)→M s X(1)(rp,d+r(p-1)(g-1))is a closed immersion.As an application,we obtain that the locus of moduli space M s X(1)(p,d)consisting of stable vector bundles whose Frobenius pull backs have maximal Harder-Narasimhan polygons is isomorphic to the Jacobian variety Jac X of X.     

关于A+,A+MN的表达式及其应用

For A ∈ Cm×nr, let M and N be Hermitian positive definite matrices oforder m and n respectively. We derived the representation of the Moore-PenroseA+MN in terms of maximal nonsin-inverse A+ and weighted Moore-Penrose inverse +gular submatrices of A. In our notation,A+ = | detA[plq]|2A+pq1/vol2(A) (p,q)∈N(A)AM+N = 1/vol2(A)(p,q)∈N(A) |detA[p|q]|2N-1/2A+pqM1/2where A=M1/2 AN -1/2. From this, we propose a new method to calculate A+A+MN. The results generalize that of Moore-Penrose inverse in [2][3].     

On Green’s Relations, 2~0-regularity and Quasi-ideals in Γ-semigroups

In this paper, we introduce the definition of ( m, n) 0-regularity in Γ-semigroups. We investigate and characterize the 20-regular class of Γ-semigroups using Green's relations. Extending and generalizing the Croisot's Theory of Decomposition for Γ-semigroups, we introduce and study the absorbent and regular absorbent Γ-semigroups. We approach this problem by examining quasi-ideals using Green's relations.     

Some Notes on Submanifolds of an Euclidean Space with Conformal Gauss Map

Let (M,g) be an m-dimensional Riemannian manifold and i:M→E~n an isometricimmersion of (M,g) into an n-dimensional Euclidean space E~n. Let VM be anopen set in which the immersion i:M→E~n is given by x~h=x~h(y~a), (h=1,…,n; α=1,…,m). Here and in the sequel x~h(h=1,…,n) are rectangular coordinates of E~nand y~α(α=1,…,m) are local coordinates of a generic point in V. The tangent planeiM_(ip)=i(M_p), P∈V, of iM can be considered after a suitable parallel displacementas a point Γ_i(P) of the Grassmann manifold G(m,n-m).The mapping Γ:iM→G(m,n-m), ipΓ(iP) =Γ_i(P) is called the Gauss map. The mapping Γ_i:M→G(m,n-m),p_i(P) is called the Gauss map associated with the immersion i, and _i(M)=F(iM) the Gauss image of M.     

Multivariate refinement equation with nonnegative masks

This paper is concerned with multivariate refinement equations of the type where (?) is the unknown function defined on the s-dimensional Euclidean space Rs, a is a finitely supported nonnegative sequence on Zs, and M is an s×s dilation matrix with m := |detM|. We characterize the existence of L2-solution of refinement equation in terms of spectral radius of a certain finite matrix or transition operator associated with refinement mask a and dilation matrix M. For s = 1 and M = 2, the sufficient and necessary conditions are obtained to characterize the existence of continuous solution of this refinement equation.     

Infinitely Many Solutions for Schr?dinger–Choquard–Kirchhoff Equations Involving the Fractional p-Laplacian

In this article,we show the existence of infinitely many solutions for the fractional pLaplacian equations of Schr?dinger-Kirchhoff type equation ■ ,where(-△)_p~s is the fractional p-Laplacian operator,[u]_(s,p) is the Gagliardo p-seminorm,0 s 1 q p N/s,α∈(0,N),M and V are continuous and positive functions,and k(x) is a non-negative function in an appropriate Lebesgue space.Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma,we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.     

Biorthogonal multiple wavelets generated by vector refinement equation

Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis.In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form (?)(x)=(?)a(α)(?)(Mx-α),x∈R~s, where the vector of functions(?)=((?)_1,...,(?)_r)~T is in(L_2(R~s))~r,a=:(a(α))_(α∈Z~s)is a finitely supported sequence of r×r matrices called the refinement mask,and M is an s×s integer matrix such that lim_(n→∞)M~(-n)=0.Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.