量子代数模的张量积与不可分解模

令M是Z[v]的由v-1和奇素数p生成的理想,U是A=Z[v]M上相伴于对称Cartan矩阵的量子代数.k是特征为零的代数闭域,A→k(v(?)ξ)是环同态.U_k=U(?)_Ak,u_k是U_k的无穷小量子代数.令ξ是1的p次本原根.本文证明了:若有限维可积U_k模M,V中至少有一个是内射模,或者M,V中有一个模作为u_k模是平凡的,则有U_k模同构M(?)V≌V(?)M.我们还证明了:若有限维可积U_k模V作为u_k模是不可分解的,有限维可积U_k模M是不可分解的,且M|_(uk)是平凡的,则V(?)M是不可分解U_k模.令V和M是有限维可积U_k模,作为u_k模是同构的且具有单基座,本文证明V和M作为U_k模也是同构的.由此得到:不可分解内射u_k模提升为U_k模是唯一的.     

二面体群上Hopf路余代数的结构分类

从Hopf quiver出发,借助于右kZu(C)-模的直积范畴ПC∈K(G) MkZu(C)与kG-Hopf双模范畴kG kG M kG kG之间的同构,当G是二面体群D3时,给出了Hopf路余代数kQc的同构分类及其子Hopf代数kG[kQ1]结构.     

二面体群上Hopf路余代数的结构分类

从Hopf quiver出发,借助于右kZ_u(c)-模的直积范畴■ Mkz_(u(C))与kG-Hopf双模范畴kG/kG M kG/kG之间的同构,当G是二面体群D_3时,给出了Hopf路余代数kQ~c的同构分类及其子Hopf代数kG[kQ_1]结构.     

一类具有规范变换的拟Hopf代数

设H是拟Hopf代数,A是左H-模代数,F∈H■H是规范变换,本文给出了代数同构A#H≌A_F~(-1)#H_F,M是右A#H-模的一个充分必要条件,并且证明了范畴同构_(A,H)M≌A_F~(-1),H_FM和_HM_A≌H_FM_A_F~(-1).     

Kantorovich型Meyer-K(o)nig-Zeller算子的点态逼近定理

1引 言 1960年Meyer-K(o)nig W.和Zeller K.在[6]中提出了Meyer-K(o)nig-Zeller算子 Mn(f,x)=∞∑k=0f(k/(n+k))mn,k(x),0≤x＜1,Mn(f,1):=f(1),mn,k(x)=(n+kk)xk(1-x)n+1,在[1,2,5,7,9,10,12]中对于此算子的逼近性质及各种修正了的Meyer-K(o)nig-Zeller算子作了研究,其中重要的变形是Kantorovich型的积分算子： M*n(f;x)=∞∑k=0((n+k)(n+k+1))/n∫(k+1)/(n+k+1)k/(n+k)f(u)dumn,k(x),x∈[0,),其中Mn(f,1)：＝f(1),mn,k(x)＝(n+kk)xk(1+x)n+1,mn,-1(x)：=0. V.Totik在[8]中给出了M*n(f;x)的Lp-逼近(1≤p＜∞),王建力在[11]研究了其加权Lp-逼近(1≤p＜∞).本文引进新的K+泛函,利用Ditzian-Totik模ω2ψ(f,t)研究了该算子的点态逼近性质,得到了它的逼近正、逆及等价定理.     

单位积决定的若当矩阵代数(英文)

本文研究了单位积决定的若当矩阵代数M=Mn(R)的条件及分类问题.利用基矩阵及巧妙对对称双线性映射{·,·}进行构造和扩充,用初等矩阵的方法,获得了一系列新的同样重要的定义,结论与证明(与参考文献[1]相比较),推广了参考文献[1]的结论,作为其应用可以进一步证明了Mn(R)上的任意可逆线性映射都是保单位积的.     

量子代数的内射模与Tilting模

A=Z[ν] m ' m是 Z[ν]的由ν- 1和奇素数 p生成的理想 .U是 A上的量子代数 .设 k是特征为零的代数闭域 .A→ K (ν|→ξ)是代数同态 ,并假定ξ不是 1的根或ξ是 p次本原根 .命Uk=U k A.J是 UK- Tilting模范畴 .对 λ∈ X+,M(λ)表首权为 λ的不可分解 UK- Tilting模 .本文证明了 ,对每个λ∈ X+,M(λ)作为 Uk 模是内射的当且仅当λ- (p- 1 )ρ∈ X+.我们还给出了内射 Uk模的若干充要条件 .     

U_(r,t)在量子平面上的模代数结构

当q不是单位根,且所在的域是复数域时,给出了U_(r,t)在量子平面上模代数结构的完全分类,并描述了这些表示.有趣的是,在某些情况下,有C_q[x,y]=∞⊕n=0C_q[x,y]_n,其中C_q[x,y]_n是量子平面的n次齐次部分,同构于U_(r,t)的某种不可约模V_(1,n,γ_1)n/2.     

Global well-posedness and blow-up for the hartree equation

For 2 γ min{4, n}, we consider the focusing Hartree equation iu_t+ △u +(|x|~(-γ)* |u|~2)u = 0, x ∈ R~n.(0.1)Let M [u] and E [u] denote the mass and energy, respectively, of a solution u, and Q be the ground state of-△ Q + Q =(|x|~(-γ)* |Q|~2)Q. Guo and Wang [Z. Angew. Math.Phy.,2014] established a dichotomy for scattering versus blow-up for the Cauchy problem of(0.1) if M [u]~(1-s_c)E [u]~(s_c) M [Q]~(1-s_c)E [Q]~(s_c)(s_c=(γ-2)/2). In this paper, we consider the complementary case M [u]~(1-s_c)E [u]~(s_c)≥ M [Q]~(1-s_c)E [Q]~(s_c) and obtain a criteria on blow-up and global existence for the Hartree equation(0.1).     

一类条件不等式探源

文[1]、[2]、[3]、[4]、[5]、[6]分别介绍了下列不等式:若a,b>0且a+b=1则3/2<1/(a~3+1)+1/(b~3+1)≤(16)/9(1)若a,b,c>0且a+b+c=1则1/(1+a~2)+1/(1+b~2)+1/(1+c~2)≤(27)/(10)(2)     

矩阵代数的Kadison-Singer格的分类

研究了矩阵代数M_n(C)的KS格,证明了每个生成M_3(C)的KS格都相似于(?)_0或I-(?)_0,其中(?)_0为M_3(C)的一个极大对角投影套和一个赋值全非零的秩1投影所生成的KS格,从而M_3(C)的对角平凡的KS代数都是4维的.同时,还给出了几个生成M_4(C)但非同构的KS格的例子.     

矩阵代数上的拟三重Jordan可导映射

设R是一个含单位元的可交换2-无挠环,且M_n(R)是R上的n×n阶矩阵代数.本文证明了M_n(R)(n≥2)上的满足Φ(ABA)=Φ(A)BA+AΦ(B)A+ABΦ(A)的映射Φ具有形式:存在T∈M_n(R)和R上的一个可加导子φ,使得对任意A= (a_(ij))∈M_n(R),有Φ(A)=AT-TA+A_φ,这里A_φ=(φ(a_(ij))).     

The trace graph of the matrix ring over a finite commutative ring

Let R be a commutative ring and let \({n >1}\) be an integer. We introduce a simple graph, denoted by \({\Gamma_t(M_n(R))}\), which we call the trace graph of the matrix ring \({M_n(R)}\), such that its vertex set is \({M_n(R)^{\ast}}\) and such that two distinct vertices A and B are joined by an edge if and only if \({{\rm Tr} (AB)=0}\) where \({ {\rm Tr} (AB)}\) denotes the trace of the matrix AB. We prove that \({\Gamma_t(M_n(R))}\) is connected with \({{\rm diam}(\Gamma_{t}(M_{n}(R)))=2}\) and \({{\rm gr} (\Gamma_t(M_n(R)))=3}\). We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of \({\Gamma_t(M_n(R))}\). Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal.     

Computable Homogeneous Boolean Algebras and a Metatheorem

We consider computable homogeneous Boolean algebras. Previously, countable homogeneous Boolean algebras have been described up to isomorphism and a simple criterion has been found for the existence of a strongly constructive (decidable) isomorphic copy for such. We propose a natural criterion for the existence of a constructive (computable) isomorphic copy. For this, a new hierarchy of -computable functions and sets is introduced, which is more delicate than Feiner's. Also, a metatheorem is proved connecting computable Boolean algebras and their hyperarithmetical quotient algebras.     

Lipschitz-Nikolski\imath constants and asymptotic simultaneous approximation on theM n -operatorsconstants and asymptotic simultaneous approximation on theM n -operators

This note is a study of approximation of classes of functions and asymptotic simultaneous approximation of functions by theM _n -operators of Meyer-König and Zeller which are defined by $$(M_n f)(x) = (1 - x)^{n + 1} \sum\limits_{k = 0}^\infty {f\left( {\frac{k}{{n + k}}} \right)} \left( \begin{array}{l} n + k \\ k \\ \end{array} \right)x^k , n = 1,2,....$$ Among other results it is proved that for 0<α≤1 $$\mathop {\lim }\limits_{n \to \infty } n^{\alpha /2} \mathop {\sup }\limits_{f \in Lip_1 \alpha } \left| {(M_n f)(x) - f(x)} \right| = \frac{{\Gamma \left( {\frac{{\alpha + 1}}{2}} \right)}}{{\pi ^{1/2} }}\left\{ {2x(1 - x)^2 } \right\}^{\alpha /2} $$ and if for a functionf, the derivativeD ^m+2 f exist at a pointx∈(0, 1), then $$\mathop {\lim }\limits_{n \to \infty } 2n[D^m (M_n f) - D^m f] = \Omega f,$$ where Ω is the linear differential operator given by $$\Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$      

On a tensor-analogue of the Schur product

We consider the tensorial Schur product \(R \circ ^\otimes S = [r_{ij} \otimes s_{ij}]\) for \(R \in M_n(\mathcal {A}), S\in M_n(\mathcal {B}),\) with \(\mathcal {A}, \mathcal {B}~\text{ unital }~ C^*\)-algebras, verify that such a ‘tensorial Schur product’ of positive operators is again positive, and then use this fact to prove (an apparently marginally more general version of) the classical result of Choi that a linear map \(\phi :M_n \rightarrow M_d\) is completely positive if and only if \([\phi (E_{ij})] \in M_n(M_d)^+\), where of course \(\{E_{ij}:1 \le i,j \le n\}\) denotes the usual system of matrix units in \(M_n (:= M_n(\mathbb C))\). We also discuss some other corollaries of the main result.     

Map determined by rank-$$\varvec{}$$ matrice for relatively mall $$\varvec{}$$

Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\), then \(g(x)=f(x)+g(0)\), \(x\in M_n(\mathbb {K})\), for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\). Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \).     

Quantum Weyl symmetric polynomials and the center of quantum group U_q(sl_4)

Suppose that q is not a root of unity, it is proved in this paper that the center of the quantum group Uq(sl4) is isomorphic to a quotient algebra of polynomial algebra with four variables and one relation.     

HOPF代数的右伴随作用构造的上循环模(英文)

本文研究了上循环模,对于特征为O的域k上满足S~2=id_H的Hopf代数H,和左H-模代数A,利用日的右伴随作用以及H在A上的模作用,构造了上循环模(C)_H~#(A),并且证明了由H的右伴随作用和左伴随作用分别诱导的上循环模(C)_H~(#)(A)和(C)_H~(#)(A)足同构的.     

A discrete representation of free MV‐algebras

We prove that the m ‐generated free MV‐algebra is isomorphic to a quotient of the disjoint union of all the m ‐generated free MV⁽ⁿ⁾‐algebras. Such a quotient can be seen as the direct limit of a system consisting of all free MV⁽ⁿ⁾‐algebras and special maps between them as morphisms (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)