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1.
Vladimir V. Sergeichuk 《Linear algebra and its applications》2008,428(1):154-192
We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases:
- •
- F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric;
- •
- F is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F.
2.
A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms for
- •
- nonderogatory complex matrices up to unitary similarity, and
- •
- pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues.
3.
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.
4.
Oscar Rojo 《Linear algebra and its applications》2009,430(1):532-882
A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of:
- the generalized Bethe tree B,
- a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B,
- a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices,
- a graph obtained from the cycle Cr by attaching B, by its root, to each vertex of the cycle, and
- a tree obtained from the path Pr by attaching B, by its root, to each vertex of the path,
- is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.
5.
Masami Sakai 《Topology and its Applications》2012,159(1):308-314
Let F[X] be the Pixley-Roy hyperspace of a regular space X. In this paper, we prove the following theorem.
Theorem.
For a space X, the following are equivalent:
- (1)
- F[X]is a k-space;
- (2)
- F[X]is sequential;
- (3)
- F[X]is Fréchet-Urysohn;
- (4)
- Every finite power of X is Fréchet-Urysohn for finite sets;
- (5)
- Every finite power ofF[X]is Fréchet-Urysohn for finite sets.
6.
Douglas Farenick Vyacheslav Futorny Tatiana G. Gerasimova Vladimir V. Sergeichuk Nadya Shvai 《Linear algebra and its applications》2011,435(6):1356-1369
Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A=[aij] and B=[bij] be upper triangular n×n matrices that
- •
- are not similar to direct sums of square matrices of smaller sizes, or
- •
- are in general position and have the same main diagonal.
7.
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.
8.
Let F be a field and let m and n be integers with m,n?3. Let Mn denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:Mn→Mm that satisfy one of the following conditions:
- 1.
- |F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈Mn and α∈F with ψ(In)≠0.
- 2.
- ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈Mn.
9.
Peter M. Gruber 《Advances in Mathematics》2004,186(2):456-497
Minimum sums of moments or, equivalently, distortion of optimum quantizers play an important role in several branches of mathematics. Fejes Tóth's inequality for sums of moments in the plane and Zador's asymptotic formula for minimum distortion in Euclidean d-space are the first precise pertinent results in dimension d?2. In this article these results are generalized in the form of asymptotic formulae for minimum sums of moments, resp. distortion of optimum quantizers on Riemannian d-manifolds and normed d-spaces. In addition, we provide geometric and analytic information on the structure of optimum configurations. Our results are then used to obtain information on
- (i)
- the minimum distortion of high-resolution vector quantization and optimum quantizers,
- (ii)
- the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures,
- (iii)
- the minimum error of numerical integration formulae for classes of Hölder continuous functions and optimum sets of nodes,
- (iv)
- best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes, and
- (v)
- the minimum isoperimetric quotient of convex polytopes in Minkowski spaces and the form of the minimizing polytopes.
10.
Luoshan Xu 《Topology and its Applications》2006,153(11):1886-1894
In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:
- (1)
- A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;
- (2)
- A poset is continuous iff its Scott topology is completely distributive;
- (3)
- A topological T0 space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;
- (4)
- A topological T1 space is a discrete space iff its topology is completely distributive.
11.
Theorem A ℵ1?.
There is a Boolean algebra B with the following properties:
- (1)
- B is thin-tall, and
- (2)
- B is downward-categorical.
12.
In this paper, we show that for a convex expectation E[⋅] defined on L1(Ω,F,P), the following statements are equivalent:
- (i)
- E is a minimal member of the set of all convex expectations defined on L1(Ω,F,P);
- (ii)
- E is linear;
- (iii)
- two-dimensional Jensen inequality for E holds.
13.
Grega Cigler 《Linear algebra and its applications》2011,435(6):1285-1295
Let G and H be groups of complex n×n matrices. We say that G is an H-like group if every matrix in G is similar to a matrix from H. For several groups H we consider two questions:
- (A)
- Is every H-like group (simultaneously) similar to a subgroup of H?
- (B)
- Is H the only H-like group containing H? Among other results we prove that the symmetric group Sn is the only Sn-like group containing Sn.
14.
Adam J. Prze?dziecki 《Advances in Mathematics》2010,225(4):1893-1913
We construct a functor F:Graphs→Groups which is faithful and “almost” full, in the sense that every nontrivial group homomorphism FX→FY is a composition of an inner automorphism of FY and a homomorphism of the form Ff, for a unique map of graphs f:X→Y. When F is composed with the Eilenberg-Mac Lane space construction K(FX,1) we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null-homotopic maps.We provide several applications of this construction to localizations (i.e. idempotent functors); we show that the questions:
- (1)
- Is every orthogonality class reflective?
- (2)
- Is every orthogonality class a small-orthogonality class?
15.
Marion Scheepers 《Topology and its Applications》2011,158(13):1575-1583
We show that:
- (1)
- Rothberger bounded subgroups of σ-compact groups are characterized by Ramseyan partition relations (Corollary 4).
- (2)
- For each uncountable cardinal κ there is a T0 topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is not a closed subspace of any σ-compact space (Theorem 8).
- (3)
- For each uncountable cardinal κ there is a T0 topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is σ-compact (Corollary 17).
16.
Nobuyuki Kemoto 《Topology and its Applications》2007,154(2):358-363
For an ordinal α, α2 denotes the collection of all nonempty closed sets of α with the Vietoris topology and K(α) denotes the collection of all nonempty compact sets of α with the subspace topology of α2. It is well known that α2 is normal iff cfα=1. In this paper, we will prove that for every nonzero-ordinal α:
- (1)
- α2 is countably paracompact iff cfα≠ω.
- (2)
- K(α) is countably paracompact.
- (3)
- K(α) is normal iff, if cfα is uncountable, then cfα=α.
17.
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)
- .
18.
19.
The following results are obtained.
- -
- An open neighbornet U of X has a closed discrete kernel if X has an almost thick cover by countably U-close sets.
- -
- Every hereditarily thickly covered space is aD and linearly D.
- -
- Every t-metrizable space is a D-space.
- -
- X is a D-space if X has a cover {Xα:α<λ} by D-subspaces such that, for each β<λ, the set ?{Xα:α<β} is closed.
20.
Ali Godjali 《Linear algebra and its applications》2010,432(12):3231-2095
A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let V denote a nonzero finite-dimensional vector space over a field K. We consider an ordered pair of linear transformations A:V→V and A∗:V→V which satisfy both (i) and (ii) below.
- (i)
- There exists a basis for V with respect to which the matrix representing A is Hessenberg and the matrix representing A∗ is diagonal.
- (ii)
- There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A∗ is Hessenberg.