Whe are $${\varvec{}}$$-syzygy modules $${\varvec{}}$$-torsiofree?

Let R be a commutative Noetherian ring. We consider the question of when n-syzygy modules over R are n-torsionfree in the sense of Auslander and Bridger. Our tools include Serre’s condition and certain conditions on the local Gorenstein property of R. Our main result implies the converse of a celebrated theorem of Evans and Griffith.     

Pareto–Fenchel {\epsilon} -subdifferential sum rule and {\epsilon} -efficiency

The paper is centered around a sum rule for the efficient (Pareto) ${\epsilon}$ -subdifferential of two convex vector mappings, having the property to be exact under a qualification condition. Such a formula has not been explored previously. Our formula which holds under the Attouch?CBrézis as well as Moreau?CRockafellar conditions, reveals strangely a primordial presence of the convex (Fenchel) ${\epsilon}$ -subdifferential. This appearance turns out to be rather favorable. This effectively permits to derive approximate efficiency conditions in terms of Pareto subgradient and vectorial normal cone, which completely characterizes an ${\epsilon}$ -efficient solution in constrained convex vector optimization in (partially) ordered spaces. Our sum rule also allows a fundamental deduction of relation between Pareto and Fenchel ${\epsilon}$ -subdifferentials, which, in reality, brings out a certain gap linking ${\epsilon}$ -efficiency with ${\epsilon}$ -optimality. Scalarization approaches in connection with ${\epsilon}$ -subdifferentials are first established by simple proofs. This principle has contributed for a large part, not only for discovering the sum formula, but also for establishing some punctual necessary and/or sufficient conditions for Pareto ${\epsilon}$ -subdifferentiability.     

Perelman’s $${\bar{\lambda}}$$ -invariant and collapsing via geometric characteristic splittings

Any closed, orientable, smooth, nonpositively curved manifold M is known to admit a geometric characteristic splitting, analogous to the JSJ decomposition in three dimensions. We show that when this splitting consists of pieces which are Seifert fibered or pieces each of whose fundamental group has non-trivial centre, then M collapses with bounded curvature and has zero Perelman [`(l)]{\bar{\lambda}} -invariant.     

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在本文中, 令为一列行为混合随机变量阵列. 本文研究了行为混合随机变量阵列加权和的极限行为, 并且一些新的完全收敛性结果被取得, 这些结果推广和改进了相应的已有定理.     

${\mbox{\boldmath $R$}}^N$上奇异非线性多调和方程的正整体解

本文研究形如△((△nu)(p-1) )=f(|x|,u,|(?)u|)u-β,x∈RN的奇异非线性多调和方程在RN上的正整体解,此处P>1,β≥0是常数,n是自然数,f:R × R ×R →R 是一个连续函数, ξδ*:=sign(ξ)·|ξ|δ,,ξ∈R,δ>0,给出了该类方程具有无穷多个其渐进阶刚好为|x|2n的正整体解的充分条件与必要条件.这些结论可以推广到更一般的方程.     

The Gabriel–Roiter Measure for \widetilde{\mathbb{A}}_{\boldsymbol n} II

Let Q be a tame quiver of type $\widetilde{\mathbb{A}}_n$ and Rep(Q) the category of finite dimensional representations over an algebraically closed field. A representation is simply called a module. It will be shown that a regular string module has, up to isomorphism, at most two Gabriel–Roiter submodules. The quivers Q with sink-source orientations will be characterized as those, whose central parts do not contain preinjective modules. It will also be shown that there are only finitely many (central) Gabriel–Roiter measures admitting no direct predecessors. This fact will be generalized for all tame quivers.     

Morse and semistable stratifications of Kähler spacesby ${\Bbb C}^{\ast}$-actions

We prove that the Morse decomposition in the sense of Kirwan and semistable decomposition in the sense of GIT of a -K?hler manifold coincide if the moment map is proper and if the fixed points set has a finite number of connected components. For general K?hler space with holomorphic action of a complex reductive group G, if every component of the moment map is proper, the two decompositions also coincide if each semistable piece is Zariski open in its topological closure and the moment map square is minimal degenerate Morse function in the sense of Kirwan. Author’s address: Department of Mathematics, Tsinghua University, 100084 Beijing, P.R. China     

关于${\cal I}$-覆盖映射与1-${\cal I}$-覆盖映射

In this paper,we introduce the concepts of I-covering mappings and 1-I-covering mappings,discuss the difference between sequence-covering and I-covering mappings by some examples.With those concepts,we get some interesting properties of I-covering(1-I-covering)mappings and some characterizations of I-covering(1-I-covering)and compact mapping images of metric spaces.     

Some Results on the $${\cal F}{\rm{\Phi }}$$ ℱ Φ -Hypercenter of Finite Groups

Mathematical Notes - Asubgroup H of G is said to be $${{\cal M}_p}$$ -supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H with...     

Z∗$$ {\mathcal{Z}}^{\ast } $$-Semilocal Modules and the Proper Class ℛS$$ \mathrm{\mathcal{R}}\mathcal{S} $$

Ukrainian Mathematical Journal - Over an arbitrary ring, a module M is said to be $$ {\mathcal{Z}}^{\ast } $$-semilocal if every submodule U of M has a $$ {\mathcal{Z}}^{\ast } $$ -supplement V in...     

The $${\varphi}$$-Brunn–Minkowski inequality

For strictly increasing concave functions \({\varphi}\) whose inverse functions are log-concave, the \({\varphi}\)-Brunn–Minkowski inequality for planar convex bodies is established. It is shown that for convex bodies in \({\mathbb{R}^n}\) the \({\varphi}\)-Brunn–Minkowski is equivalent to the \({\varphi}\)-Minkowski mixed volume inequalities.     

${\mathbb{Z}}/3{\mathbb{Z}}$-量子群的Gr\"{o}bner-Shirshov 基

通过计算合成, 我们证明了Yamane 给出的关系是 ${\mathbb{Z}}/3{\mathbb{Z}}$-量子群的一个Gr\"{o}bner-Shirshov 基.     

Extrapolation from A_{\infty}^{\rho,\infty}, vector-valued inequalities and applications in the Schrödinger settings

In this paper, we generalize the A _∞ extrapolation theorem (Cruz-Uribe–Martell–Pérez, Extrapolation from A _∞ weights and applications, J. Funct. Anal. 213 (2004), 412–439) and the A _p extrapolation theorem of Rubio de Francia to Schrödinger settings. In addition, we also establish weighted vector-valued inequalities for Schrödinger-type maximal operators by using weights belonging to $A_{p}^{\rho,\infty }$ which includes A _p . As applications, we establish weighted vector-valued inequalities for some Schrödinger-type operators.     

从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)$的加权复合算子

设$u \in H(D), \ \phi$为$D$上的解析自映射,定义$H(D)$上的加权复合算子为$u C_{\phi}(f)=$$uf\circ\phi$, \ $f\in H(D)$.本文得到了从$A^{p}_{\alpha}$到$A^{\infty}(\varphi)\ (A_{0}^{\infty}(\varphi))$的加权复合算子$u C_{\phi}$的有界性和紧性的充要条件.     

Cichoń’s diagram,regularity properties and {\varvec{\Delta}^1_3} sets of reals

We study regularity properties related to Cohen, random, Laver, Miller and Sacks forcing, for sets of real numbers on the \({\varvec{\Delta}^1_3}\) level of the projective hieararchy. For \({\varvec{\Delta}^1_2}\) and \({\varvec{\Sigma}^1_2}\) sets, the relationships between these properties follows the pattern of the well-known Cichoń diagram for cardinal characteristics of the continuum. It is known that assuming suitable large cardinals, the same relationships lift to higher projective levels, but the questions become more challenging without such assumptions. Consequently, all our results are proved on the basis of ZFC alone or ZFC with an inaccessible cardinal. We also prove partial results concerning \({\varvec{\Sigma}^1_3}\) and \({\varvec{\Delta}^1_4}\) sets.     

Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

Cartan-Dieudonné Theorem for {\mathcal {A}}-Modules

Like the classical Cartan-Dieudonné theorem, the sheaf-theoretic version shows that A{\mathcal {A}}-isometries on a convenient A{\mathcal {A}}-module E{\mathcal {E}} of rank n can be decomposed in at most n orthogonal symmetries (reflections) with respect to non-isotropic hyperplanes. However, the coefficient sheaf of \mathbb C{\mathbb {C}}-algebras A{\mathcal {A}} is assumed to be a PID \mathbb C{\mathbb {C}}-algebra sheaf and, if (E,f){(\mathcal {E},\phi)} is a pairing with f{\phi} a non-degenerate A{\mathcal {A}}-bilinear morphism, we assume that E{\mathcal {E}} has nowhere-zero (local) isotropic sections; but, for Riemannian sheaves of A{\mathcal {A}}-modules, this is not necessarily required.     

On $${\mathcal {I}}_{

Tóth  János T.  Filip  Ferdinánd  Bukor  József  Zsilinszky  László 《Periodica Mathematica Hungarica》2021,82(2):125-135