∞—∞与0·∞的几何解释

学生初次接触无穷大的概念,感到它难以捉摸。从一般常识去理解,往往想不通。例如,学生问道:“同数相减剩余为0,为什么∞—∞就不等于0呢?”还会问:“既然0乘任何数都等于0。那为什么0·∞却是不定型呢?” 下面的两个几何图形有助于学生理解上述问题。 在图1中,随着点P的无限右移,线段AP,GP,AE,CE,CP,PE都将无限地变长。因此,差值AP-GP,AE-CE,CE-CP均属于∞—∞的型式。     

A ∞-simplicial objects andA ∞-topological groups

The notion ofA _∞-topological group is introduced. It is proved that, if a space is obtained by deformation retraction of a topological groupG, then it has the structure of anA _∞-topological group, and theA _∞-homotopy equivalence holds. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 913–919, December, 1999.     

Accessible ∞-cosmoi

We introduce the notion of an accessible ∞-cosmos and prove that these include the basic examples of ∞-cosmoi and are stable under the main constructions. A consequence is that the vast majority of known examples of ∞-cosmoi are accessible. By the adjoint functor theorem for homotopically enriched categories which we proved in an earlier paper, joint with Luká? Vok?ínek, it follows, for instance, that all such ∞-cosmoi have flexibly weighted homotopy colimits.     

Some Characterizations of the Properties LB^-∞ and LB∞

The aim of this paper is to give some properties of the linear topological invariant Using these results we show that a nuclear Fréchet space F has the property LB _∞ if and only if every separately holomorphic function on an open subset U × V of E × F* has a local Dirichlet representation, where E is a nuclear Fréchet space with the property having a basis.     

Homomorphisms onC ∞ (E) andC ∞-bounding sets

We prove, for the class of real locally convex spacesE that are continuously and linearly injectable into somec ₀(), that every non-zero homomorphism on the algebraC (E) ofC -functions onE is given by a point evaluation at some point ofE. Furthermore, if every real-valuedC -function on the weak topology of a quasi-complete locally convex spaceE is bounded on a subsetA ofE, thenA is relatively weakly compact.     

D ∞-Differentials and A ∞-Structures in Spectral Sequences

We introduce the notion of D -differential and use this notion for studying the structure of differential A -algebra on multiplicative spectral sequences. We review basic notions, constructions, and results of our previous papers, where the homotopy invariance of D -differentials was proved, D -differentials and differential perturbations of chain complexes were compared, and the connection between D -differentials and differentials of multiplicative spectral sequences was established. As a consequence of this connection, we describe a method of construction of the structure of A -coalgebra on the Milnor coalgebra dual to the Steenrod algebra immediately by the differentials of the Adams spectral sequence. We describe the method of construction of the structure of A -comodule over a Milnor A -coalgebra on homologies of an arbitrary spectrum.     

Some Characterizations of the Properties ■~∞ and LB_∞

The aim of this paper is to give some properties of the linear topological invariant .Using these results we show that a nuclear Fréchet space F has the property LB_∞ if and only if everyseparately holomorphic function on an open subset U×V of E×F~* has a local Dirichlet representation,where E is a nuclear Fréchet space with the property ~∞ having a basis.     

JB_∞- 环

研究了JB∞-环,即满足R/J(R)是QB∞-环,得到了很多J B∞-环的判定条件:R是JB∞-环当且仅当对任意满足条件a R+b R=R的a,b∈R,存在y∈R使得a+by∈R-1J∞当且仅当对任意满足条件a R+b R=d R的a,b,d∈R,存在y∈R,u∈R-1∞使得a+by=du.另外还讨论了替换环是JB∞-环的充分必要条件,这些结论对QB∞-环提供了一些研究基础.     

L ∞ Algebra Representations

It is shown that the direct sum of an L algebra and a module over it has a natural L algebra structure.     

On QB ∞-Rings

In this article, we introduce a new class of rings, the QB _∞-rings. We establish various properties of this concept. These show that, in several respects, QB _∞-rings behave like QB-rings. We prove that the notion of QB _∞-rings is a Morita invariant property of rings and every finite subdirect product of QB _∞-rings is a QB _∞-ring. We also exhibit examples to point out that the class of QB _∞-rings is much larger than the class of QB-rings.     

关于∞／∞型的L‘Hospital法则的一个改进

本对现行的数学分析、微积分及其高等数学的教材中的∞/∞型的L'Hospital法则进行了一个改进,论证了该法则在无须考虑分子是否趋于∞的条件时,仍旧成立,利用了这个减弱了的∞/∞型的L'Hospital法则给出了一大类的极限问题的简单解法。     

Weakly compact operators onH ∞

We prove that a weakly compact operator fromH or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH or any of its even duals either fixes a copy ofl or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH or any of its even duals either contains a copy ofl or is super-reflexive; (3) every subspace ofL ¹/H ₀ ¹ or any of its even duals either contains a complemented copy ofl ¹ or is super-reflexive.     

Rectifications of A∞-Algebras

In this article, in the setting of connected DG-modules, we prove that, for any A _∞-algebra (M, {m _i } _i≥1), there is a chain contraction from a DG-algebra A _M onto the DG-module M such that the A _∞-algebra structure induced by perturbation theory on M is precisely the original one. In fact, the mentioned DG-algebra can be considered a rectification of the A _∞-algebra in the sense of (Boardman and Vogt, 1973 Boardman , J. M. , Vogt , R. M. ( 1973 ). Homotopy Invariant Algebraic Structures on Topological Spaces . Lecture Notes in Mathematics . Springer-Verlag , pp. 347 . [Google Scholar]). Appropiate dual results are given for A _∞-coalgebras.     

Ergodic Control in L∞

The infinite horizon discounted L ^∞ problem is studied as the discount factor goes to zero. It is related to the limit of the finite horizon problem as the horizon becomes infinite. The deterministic problem with and without a running cost is considered.     

迷惑的“∞”

<正> 有限数及其运算在中小学里早已熟悉了。在进入大学后,必然会接触到无穷,同学们对无穷的认识是很模糊的,这就发生对其他概念的不清,而无穷又是高等数学的重要概念,不能对无穷有一个初步的了解,学习数学是困难的。     

Cyclic modules with ∞-simplicial faces and the cyclic homology of A ∞-algebras

Mathematical Notes - A chain bicomplex for A ∞-algebras, which generalizes the Tsygan chain bicomplex in the theory of cyclic homology of associative algebras, is constructed by using the...     

Uniqueness of optimal controls inL ∞

We consider those functionsuL (a, b) which satisfyX=Ax+Bu, whereX satisfies certain prescribed interpolatory constraints. We show that there is au with smallest sup norm and that suchu are uniquely determined on a certain subset of [a, b]; more restrictions allow us to conclude thatX is uniquely determined on this set. An extension is given to certain linear control systems on the real line.Research supported in part by National Science Foundation Grant Nos. GP-43955 and MCS 75-05591.Research supported in part by National Science Foundation Grant Nos. GP-43955 and MPS 74-02292-A01.     

∞-Tilting Subcategories in Extriangulated Categories

In this paper, the authors introduce a new definition of ∞-tilting(resp. cotilting) subcategories with infinite projective dimensions(resp. injective dimensions) in an extriangulated category. They give a Bazzoni characterization of ∞-tilting(resp. cotilting)subcategories. Also, they obtain a partial Auslander-Reiten correspondence between ∞-tilting(resp. cotilting) subcategories and coresolving(resp. resolving) subcategories with an E-projective generator(resp. E-injective cogenerator) in an ex...     

Supremal Representation of L∞ Functionals

We characterize the maps F=F(u,A) defined for u ∈ W^1,∞ and A open, which can be written as supremal functionals of the form F(u,A)=ess sup_{x ∈ A} f(x,u(x),Du(x)).