Regularity in Sobolev spaces for the fast diffusion and the porous medium equation

The degenerate parabolic equation

     

Reducing Subspaces of Toeplitz Operators on $N_ϕ$-Type Quotient Modules on the Torus

In this paper, we prove that the Toeplitz operator with finite Blaschke product symbol $S_{ψ(z)}$ on $N_ϕ$ has at least $m$ non-trivial minimal reducing subspaces, where $m$ is the dimension of $H^2(Γ_ω) ⊖ ϕ(ω)H^2 (Γ_ω)$. Moreover, the restriction of $S_{ψ(z)}$ on any of these minimal reducing subspaces is unitary equivalent to the Bergman shift $M_z$.     

On the best constant for Hardy's inequality in

Let be a domain in and . We consider the (generalized) Hardy inequality , where . The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth -dimensional domains, , where is the one-dimensional Hardy constant. Moreover it is shown that for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for , it is proved that if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but for convex domains.

     

Dixmier traces are weak dense in the set of all fully symmetric traces

We extend Dixmier's construction of singular traces (see [2]) to arbitrary fully symmetric operator ideals. In fact, we show that the set of Dixmier traces is weak^? dense in the set of all fully symmetric traces (that is, those traces which respect Hardy–Littlewood submajorization). Our results complement and extend earlier work of Wodzicki [22].     

t-Spread strongly stable monomial ideals*

We introduce the concept of t-spread monomials and t-spread strongly stable ideals. These concepts are a natural generalization of strongly stable and squarefree strongly stable ideals. For the study of this class of ideals we use the t-fold stretching operator. It is shown that t-spread strongly stable ideals are componentwise linear. Their height, their graded Betti numbers and their generic initial ideal are determined. We also consider the toric rings whose generators come from t-spread principal Borel ideals.     

On best simultaneous approximation in quotient spaces

We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M and the quotient space W/M.     

几类非Abel群之增广商群的结构

记ZG为有限群G的整群环,△n(G)为增广理想△(G)的n次幂,Qn(G)=△(G)/△n 1(G)为G的增广商群.本文考虑了二面体群D2tk(k 奇)和m次对称群Sm,证明了Qn(D2tk)为秩不超过2t 1的基本2-群以及Qn(Sm)≌Z2.     

线性流形上矩阵方程AXAT=B的极小Frobenius范数对称正交对称解

利用矩阵对的商奇异值分解,给出了线性流形上矩阵方程AXAT=B存在极小Frobe- nius范数对称正交对称解的充要条件及其解的表达式．