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1.
Emma D'Aniello 《Topology and its Applications》2010,157(5):954-960
Let M be the Cantor space or an n-dimensional manifold with C(M,M) the set of continuous self-maps of M, and . We prove the following:
- (1)
- If α≠∞, then Sα(M) is a nowhere dense subset of M×C(M,M) that contains no isolated points.
- (2)
- If α?β, then .
2.
Mohamed Aziz Taoudi 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(1):478-3452
In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A:M→X and B:X→X are two weakly sequentially continuous mappings satisfying:
- (i)
- AM is relatively weakly compact;
- (ii)
- B is a strict contraction;
- (iii)
- .
3.
4.
Christopher Mouron 《Topology and its Applications》2009,156(3):558-576
Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:
- (1)
- X is non-Suslinean.
- (2)
- If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.
- (3)
- If X is G-like, then X is indecomposable.
- (4)
- If all lie in the same ray and X is finitely cyclic, then X is indecomposable.
5.
Axel Hultman 《Journal of Combinatorial Theory, Series A》2011,118(7):1897-1906
Let W be a finite Coxeter group. For a given w∈W, the following assertion may or may not be satisfied:
- (?)
- The principal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w.
- (1)
- The criterion only involves the order ideal of w as an abstract poset. In this sense, (?) is a poset-theoretic property.
- (2)
- For W of type A, another characterisation of (?), in terms of pattern avoidance, was previously given in collaboration with Linusson, Shareshian and Sjöstrand. We obtain a short and simple proof of that result.
- (3)
- If W is a Weyl group and the Schubert variety indexed by w∈W is rationally smooth, then w satisfies (?).
6.
Friedrich Knop 《Advances in Mathematics》2007,214(2):571-617
We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,δ) depending on a degree function δ. Assume that all objects have only finitely many subobjects. Then our results are as follows:
- 1.
- Let N be the maximal proper tensor ideal of T(A,δ). We show that T(A,δ)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories.
- 2.
- Using lattice theory, we give a simple numerical criterion for the vanishing of N.
- 3.
- We determine all degree functions for which T(A,δ)/N is Tannakian. As a result, we are able to interpolate the representation categories of many series of profinite groups such as the symmetric groups Sn, the hyperoctahedral groups , or the general linear groups GL(n,Fq) over a fixed finite field.
7.
Linus Kramer 《Advances in Mathematics》2005,193(1):142-173
Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that and let Γ be a uniform lattice in G.
- (a)
- If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.
- (b)
- If CH fails, then Γ has 22ω asymptotic cones up to homeomorphism.
8.
We present a new construction of non-classical unitals from a classical unital U in . The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model Π of with the following three properties:
- (i)
- points of Π are those of ;
- (ii)
- lines of Π are certain lines and conics of ;
- (iii)
- the points in U form a non-classical B-M unital in Π.
9.
Yankui Song 《Topology and its Applications》2012,159(5):1462-1466
10.
Julio Becerra Guerrero 《Journal of Functional Analysis》2008,254(8):2294-2302
We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties:
- (1)
- Every member of R has the Daugavet property.
- (2)
- It Y is a member of R, then, for every Banach space X, both the space L(X,Y) (of all bounded linear operators from X to Y) and the complete injective tensor product lie in R.
- (3)
- If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, and for most vector space topologies τ on Y, the space C(K,(Y,τ)) (of all Y-valued τ-continuous functions on K) is a member of R.
- (4)
- If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, most C(K,Y)-superspaces (in the sense of [V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003) 195-206]) are members of R.
- (5)
- All dual Banach spaces without minimal M-summands are members of R.
11.
Let G be a 2-dimensional connected, compact Abelian group and s be a positive integer. We prove that a classification of s-sheeted covering maps over G is reduced to a classification of s-index torsionfree supergroups of the Pontrjagin dual . Using group theoretic results from earlier paper we demonstrate its consequences. We also prove that for a connected compact group Y:
- (1)
- Every finite-sheeted covering map from a connected space over Y is equivalent to a covering homomorphism from a compact, connected group.
- (2)
- If two finite-sheeted covering homomorphisms over Y are equivalent, then they are equivalent as topological homomorphisms.
12.
In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:
- (i)
- G contains no sequence such that {0}∪{±xn∣n∈N} is infinite and quasi-convex in G, and xn?0;
- (ii)
- one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;
- (iii)
- G contains an open compact subgroup of the form or for some cardinal κ.
13.
Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be a Cowen-Douglas operator if there exist Ω, a connected open subset of complex plane C, and n, a positive integer, such that
- (a)
- (b)
- for z in Ω;
- (c)
- ; and
- (d)
- for z in Ω.
14.
Andrei C?ld?raru 《Advances in Mathematics》2005,194(1):34-66
We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild-Kostant-Rosenberg theorem. The main contributions of the present paper are:
- •
- we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds;
- •
- we give a proof of the fact that the natural Chern character map K0(X)→HH0(X) becomes, after the HKR isomorphism, the usual one ; and
- •
- we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture.
15.
Su Gao 《Advances in Mathematics》2008,217(2):814-832
A group G?Sym(N) is cofinitary if g has finitely many fixed points for every g∈G except the identity element. In this paper, we discuss the definability of maximal cofinitary groups and some related structures. More precisely, we show the following two results:
- (1)
- Assuming V=L, there is a set of permutations on N which generates a maximal cofinitary group.
- (2)
- Assuming V=L, there is a mad permutation family in Sym(N).
16.
The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of the Laplacian, signless Laplacian and adjacency matrices of a weighted rooted graph G obtained from a weighted generalized Bethe tree of k levels and weighted cliques in which
- (1)
- the edges connecting vertices at consecutive levels have the same weight,
- (2)
- each set of children, in one or more levels, defines a weighted clique, and
- (3)
- cliques at the same level are isomorphic.
17.
For a space X, X2 denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K(X) denotes the collection of all non-empty compact sets of X with the subspace topology of X2. The following are known:
- •
- ω2 is not normal, where ω denotes the discrete space of countably infinite cardinality.
- •
- For every non-zero ordinal γ with the usual order topology, K(γ) is normal iff whenever cf γ is uncountable.
- (1)
- ω2 is strongly zero-dimensional.
- (2)
- K(γ) is strongly zero-dimensional, for every non-zero ordinal γ.
18.
Heybetkulu Mustafayev 《Journal of Functional Analysis》2007,248(2):428-447
A bounded linear operator T on a Banach space is said to be dissipative if ‖etT‖?1 for all t?0. We show that if T is a dissipative operator on a Banach space, then:
- (a)
- .
- (b)
- If σ(T)∩iR is contained in [−iπ/2,iπ/2], then
19.
Puiman Ng 《Journal of Pure and Applied Algebra》2011,215(10):2378-2384
This paper studies the existence of Auslander-Reiten sequences in subcategories of mod(Λ), where Λ is a finite dimensional algebra over a field. The two main theorems give necessary and sufficient conditions for the existence of Auslander-Reiten sequences in subcategories.
Theorem.
LetMbe a subcategory ofmod(Λ)closed under extensions and direct summands, and letMbe an indecomposable module inMsuch thatfor someinM. Then the following are equivalent.
- (i)
- has an-precover in the stable category,
- (ii)
- There is an Auslander-Reiten sequence0→X→Y→M→0inM.
20.
M. Jahangiri 《Journal of Pure and Applied Algebra》2009,213(4):573-581
Let R=?n≥0Rn be a homogeneous Noetherian ring, let M be a finitely generated graded R-module and let R+=?n>0Rn. Let b?b0+R+, where b0 is an ideal of R0. In this paper, we first study the finiteness and vanishing of the n-th graded component of the i-th local cohomology module of M with respect to b. Then, among other things, we show that the set becomes ultimately constant, as n→−∞, in the following cases:
- (i)
- and (R0,m0) is a local ring;
- (ii)
- dim(R0)≤1 and R0 is either a finite integral extension of a domain or essentially of finite type over a field;
- (iii)
- i≤gb(M), where gb(M) denotes the cohomological finite length dimension of M with respect to b.