An infinite sequence of non-realizable weavings

A weaving is a number of lines drawn in the plane so that no three lines intersect at a point, and the intersections are drawn so as to show which of the two lines is above the other. For each integer n4 we construct a weaving of n lines, which is not realizable as a projection of a number of lines in 3-space, all of whose subfigures are realizable as such projections.     

On homotopy varieties

Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties.     

Contractibility and the clique graph operator

To any graph G we can associate a simplicial complex Δ(G) whose simplices are the complete subgraphs of G, and thus we say that G is contractible whenever Δ(G) is so. We study the relationship between contractibility and K-nullity of G, where G is called K-null if some iterated clique graph of G is trivial. We show that there are contractible graphs which are not K-null, and that any graph whose clique graph is a cone is contractible.     

Characterizations of modalities and lex modalities

A reflective subuniverse in homotopy type theory is an internal version of the notion of a localization in topology or in the theory of ∞-categories. Working in homotopy type theory, we give new characterizations of the following conditions on a reflective subuniverse L: (1) the associated subuniverse $L^{\prime }$ of L-separated types is a modality; (2) L is a modality; (3) L is a lex modality; and (4) L is a cotopological modality. In each case, we give several necessary and sufficient conditions. Our characterizations involve various families of maps associated to L, such as the L-étale maps, the L-equivalences, the L-local maps, the L-connected maps, the unit maps ${\eta }_X$, and their left and/or right orthogonal complements. More generally, our main theorem gives an overview of how all of these classes related to each other. We also give examples that show that all of the inclusions we describe between these classes of maps can be strict.     

Construction of ANR topologies on certain groups

A method is shown to construct ANR-topologies on topological groups of suitable homotopy type.

     

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K⁰(G)=G, Kⁿ⁺¹(G)=K(Kⁿ(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that Kⁿ(G) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G?Kⁿ(G) for all n, moreover Kⁿ(G) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.     

Prototype Proofs in Type Theory

The proofs of universally quantified statements, in mathematics, are given as “schemata” or as “prototypes” which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of λ‐calculus and act as “proof‐schemata”, as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i. e. Girard's system F, where type‐quantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context.     

单叶调和映射

由于在极小曲面理论中的作用，对调和映射的研究已有较长时间。１９８４年以来，经典解析单叶映射的理论被推广至调和单叶映射，并获得许多结论。这些工作引起人们对它的浓厚兴趣。该文介绍这一课题某些重要成果的概貌，并指出一些尚未解决的问题。它共分六个部分：映射定理；单叶调和函数的数值估计；特殊映射；变分方法；境界性质和在极小曲面中的应用。     

单叶解析函数的几个子族之间的关系及应用

在复平面单位圆盘内引入了β型螺形甬数族S<,β>的一个子类S<'β>,<,α>函数族,研究了S<'β>,<,α>族与解析函数族S*,S*(α),K,K(α)及S<,β>之间的关系,利用得到的关系式对S<'β>,<,α>族的第二项系数进行了精确估计,同时得到了K(a)族的第二、三项系数的关系式和S<,β>,族的一个积分表...     

Counterexamples to a fiber theorem

We exhibit a counterexample to a fiber theorem stated by F. Fumagalli in [Francesco Fumagalli, On the homotopy type of the Quillen complex of finite soluble groups, J. Algebra 283 (2) (2005) 639–654] and show how it affects the rest of Fumagalli's paper. As a consequence, whether the poset A_p(G) is homotopy equivalent to a wedge of spheres for any finite solvable group G seems to remain an open question.     

The Existence of Large E(R)-Algebras That Are Sharply Transitive Modules

The Bose–Mesner algebra of the association scheme of the ordinary n-gon has the following remarkable properties:

• (i) It has a P-polynomial structure with respect to every faithful basis element; and

• (ii) Any closed subset generated by a basis element has a P-polynomial structure with respect to this basis element.

C-algebras or table algebras that have these two properties are called perfect P-polynomial C-algebras or table algebras. By applying and extending some of the techniques developed in Xu (2006 Xu , B. ( 2006 ). Table algebras with multiple P-polynomial structures . J. Alg. Combin. 23 : 377 – 393 .[Crossref], [Web of Science ®] , [Google Scholar]), we will give a classification of perfect P-polynomial table algebras in terms of intersection matrices. As a direct consequence of this classification, we will prove that a standard real integral table algebra (A, B) with |B| ≥ 6 is a perfect P-polynomial table algebra if and only if it is exactly isomorphic to the Bose–Mesner algebra of the association scheme of the ordinary (2|B|?2)-gon or (2|B|?1)-gon. This result generalizes part of the main theorem in Xu (2006 Xu , B. ( 2006 ). Table algebras with multiple P-polynomial structures . J. Alg. Combin. 23 : 377 – 393 .[Crossref], [Web of Science ®] , [Google Scholar]). We will present examples revealing that this result is not true if |B| ≤ 5.     

The existence of a kind of bundle map

By means of the theory of Postnikov resolution, a sufficient condition for the existence of a kind of bundle maps is obtained. Some applications of the result are given. Particularly, it is proven that the deleted products as well as configuration spaces of two simply connected manifolds with suitable dimension have the same homotopy type when the original manifolds are homotopically equivalent.     

A model for the homotopy theory of homotopy theory

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory', or more precisely that the category of such models has a well-behaved internal hom-object.

     

Compact open topology and CW homotopy type

The class of spaces having the homotopy type of a CW complex is not closed under formation of function spaces. In 1959, Milnor proved the fundamental theorem that, given a space and a compact Hausdorff space X, the space Y^X of continuous functions X→Y, endowed with the compact open topology, belongs to . P.J. Kahn extended this in 1982, showing that if X has finite n-skeleton and π_k(Y)=0, k>n.

Using a different approach, we obtain a further generalization and give interesting examples of function spaces where is not homotopy equivalent to a finite complex, and has infinitely many nontrivial homotopy groups. We also obtain information about some topological properties that are intimately related to CW homotopy type.

As an application we solve a related problem concerning towers of fibrations between spaces of CW homotopy type.     

亚纯单叶函数的Hadamard乘积

设g(ζ)=ζ+sum from n=0 to ∞b_nζ~(-n)为α级亚纯星形函数(0≤α<1,|ζ|>1),函数ψ(z)=z+sum from n=2 to ∞α_nZ~n为单位园内的凸单叶函数。本文得到,若α∈(1/2,1),则g(ζ)※ζ~2ψ(1/ζ)(|ζ|>1)为α级亚纯星形函数,作为这个结果的一个推论,文[4]中的猜测在α∈(1/2,1)内成立。