On the Fell topology

Let 2 ^X denote the closed subsets of a Hausdorff topological space <X, {gt}>. The Fell topology τ_F on 2 ^X has as a subbase all sets of the form {A ∈ 2 ^X :A ∩V ≠ 0}, whereV is an open subset ofX, plus all sets of the form {A ∈ 2 ^X :A ?W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for τ_F in terms of topological properties for τ. Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.     

A finiteness criterion and asymptotics for codimensions of generalized identities

Let A be an associative algebra over a field of characteristic zero. Then either all codimensions gc _n (A) of its generalized polynomial identities are infinite or A is the sum of ideals I and J such that dim _F I < ∞ and J is nilpotent. In the latter case, there exist numbers n ₀ ∈ ?, C ∈ ?₊, and t ∈ ?₊ for which gc _n (A) < +∞ if n ≥ n ₀ and gc _n (A) ～ Cn ^t d ⁿ as n → ∞, where d = PIexp(A) ∈ ?₊. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.     

A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Let G be a connected simple graph, let X?V (^G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that d _T (x) for all x ∈ X, where d _T (x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that d _T (x) ≥ f(x) for all x ∈ X if and only if for any nonempty subset S ? X, |N _G (S) ? S| ? f(S) + 2|S| ? ω _G (S) ≥, where ω _G (S) is the number of components of the subgraph induced by S.     

Orbits and invariants of visible group actions

We consider actions G?×?X?→?X of the affine, algebraic group G on the irreducible, affine, variety X. If [k[X] ^G ]?=?[k[X]] ^G we call the action visible. Here [A] denotes the quotient field of the integral domain A. If the action is not visible we construct a G-invariant, birational morphism φ: Z?→?X such that G?×?Z?→?Z is a visible action. We use this to obtain visible open subsets U of X. We also discuss visibility in the presence of other desirable properties: What if G?×?X?→?X is stable? What if there is a semi-invariant f ∈ k[X] such that G?×?X _f ?→?X _f is visible? What if X is locally factorial? What if G is reductive?     

Canonical forms of Borel-measurable mappings Δ: [ω]ω → R

We show that for every Borel-measurable mapping Δ: [ω]^ω → $\text{R}$ there exists A ∈ [ω]^ω and there exists a continuous mapping Γ: [A]^ω → [A]^?ω with Γ(X) ? X such that for all X, Y ∈ [A]^ω it follows that Δ(X) = Δ(Y) if Γ(X) = Γ(Y). In a sense, this is generalization of the Erdös-Rado canonization theorem     

Note: A conjecture on partitions of groups

We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A _n A _n ^?1 ) = |G| for each n ∈ ω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.     

On nonmeasurable unions

For a Polish space X and a σ-ideal I of subsets of X which has a Borel base we consider families A of sets in I with the union ?A not in I. We determine several conditions on A which imply the existence of a subfamily A^′ of A whose union ?A^′ is not in the σ-field generated by the Borel sets on X and I. Main examples are X=R and I being the ideal of sets of Lebesgue measure zero or the ideal of sets of the first Baire category.     

On the minimum number of disjoint pairs in a family of finite sets

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2ⁿ. Then there is a family $\text{F}$, |$\text{F}$| = m, such that F ∈ $\text{F}$, G ? X, | G | > | F | implies G ∈ $\text{F}$ and $\text{F}$ minimizes the number of pairs (F₁, F₂), F₁, F₂ ∈ $\text{F}$ F₁ ∩ F₂ = ? over all families consisting of m subsets of X.     

Acyclic edge coloring of planar graphs without adjacent cycles

A proper edge coloring of a graph G is said to be acyclic if there is no bicolored cycle in G.The acyclic edge chromatic number of G,denoted byχ′a(G),is the smallest number of colors in an acyclic edge coloring of G.Let G be a planar graph with maximum degree.In this paper,we show thatχ′a(G)+2,if G has no adjacent i-and j-cycles for any i,j∈{3,4,5},which implies a result of Hou,Liu and Wu(2012);andχ′a(G)+3,if G has no adjacent i-and j-cycles for any i,j∈{3,4,6}.     

LK property for σ-ideals

An ideal J of subsets of a Polish space X has (LK) property whenever for every sequence (An) of analytic sets in X, if lim supn∈HAn∉J for each infinite H then _?n∈G∉J for some infinite G. In this note we present a new class of σ-ideals with (LK) property.     

A note on an open question on ω- and τ- Matrices

In a recent paper by Engel and Schneider, it was asked if, for every n ? 1, A ∈ τ_<n> implies (A+D) ∈ τ_<n> for every D = diag[d₁, d₂,… d_n] with d_i ? 0, 1 ? i ? n. We answer this question in the negative. More precisely, we show that for, any n ? 3, the set

(τ _{< n>}): = {D ∈C^n,n:(A+D)∈τ < n> for all A∈τ_<n>} is exactly given by

(Gt_<n>) = {γI_n:γ ? 0}.     

Best constants for metric space inversion inequalities

For every metric space (X, d) and origin o ∈ X, we show the inequality I _o (x, y) ≤ 2d _o (x, y), where I _o (x, y) = d(x, y)/d(x, o)d(y, o) is the metric space inversion semimetric, d _o is a metric subordinate to I _o , and x, y ∈ X \ {o} The constant 2 is best possible.     

An existence theorem for probability measures invariant under a Markov kernel

LetP be a Markov kernel defined on a measurable space (X,A). A probability measure μ onA is said to beP-invariant if μ(A=∫P(x,A)dμ(x) for allA ∈A∈A. In this note we prove a criterion for the existence ofP-invariant probabilities which is, in particular, a substantial generalization of a classical theorem due to Oxtoby and Ulam ([5]). As another consequence of our main result, it is shown that every pseudocompact topological space admits aP-invariant Baire probability measure for any Feller kernelP.     

Logical laws for existential monadic second-order sentences with infinite first-order parts

We consider existential monadic second-order sentences ?X φ(X) about undirected graphs, where ?X is a finite sequence of monadic quantifiers and φ(X) ∈ +_∞ω^ω is an infinite first-order formula. We prove that there exists a sentence (in the considered logic) with two monadic variables and two first-order variables such that the probability that it is true on G(n, p) does not converge. Moreover, such an example is also obtained for one monadic variable and three first-order variables.     

On I-Lindelöf sets

A space (X, T) is called I-Lindelöf [1] if every cover A of X by regular closed subsets of the space (X, T) contains a countable subfamily A′ such that X = ∪{int (A): A ∈ A′}. In this work we introduce the class of I-Lindelöf sets as a proper subclass of rc-Lindelöf sets [3]. We study various properties of I-Lindelöf sets and investigate the relationship between I-Lindelöf sets and I-Lindelöf subspaces. We give a new characterization of I-Lindelöf spaces in terms of this type of sets. Also, we study spaces (X, T) in which every I-Lindelöf set in (X, T) is closed.     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

MULTIPLICATIVELY CLOSED SETS OF IDEALS AND RESIDUAL DIVISION

Let R be a ring (commutative with identity), let Γ be a multiplicatively closed set of finitely generated nonzero ideals of R, for an ideal I of R let I _Γ = ∪ {I : G; G ∈ Γ}, and for an R-algebra A such that GA ≠ (0) for all G ∈ Γ let A ^Γ = ∪ {A : _T GA; G ∈ Γ}, where T is the total quotient ring of A. Then I _Γ is an ideal in R, I → I _Γ is a semi-prime operation (on the set I of ideals I of R) that satisfies a cancellation law, and it is a prime operation on I if and only if R = R ^Γ. Also, A ^Γ is an R-algebra and A → A ^Γ is a closure operation on any set A = {A; A is an R-algebra, R ? A, and if C is a ring between R and A, then regular elements in C remain regular in A}. Finally, several results are proved concerning relations between the ideals I _Γ and (IA)_ΓA and between the R-algebras R ^Γ and A ^Γ.

     

Differences of Idempotents In <Emphasis Type="Italic">C</Emphasis>*-Algebras and the Quantum Hall Effect

Let ? be a trace on the unital C*-algebra A and M _? be the ideal of the definition of the trace ?. We obtain a C*analogue of the quantum Hall effect: if P,Q ∈ A are idempotents and P ? Q ∈ M _? , then ?((P ? Q)²ⁿ⁺¹) = ?(P ? Q) ∈ R for all n ∈ N. Let the isometries U ∈ A and A = A*∈ A be such that I+A is invertible and U-A ∈ M _? with ?(U-A) ∈ R. Then I-A, I?U ∈ M _? and ?(I?U) ∈ R. Let n ∈ N, dimH = 2n + 1, the symmetry operators U, V ∈ B(H), and W = U ? V. Then the operator W is not a symmetry, and if V = V*, then the operator W is nonunitary.     

A note on the Engelking-Karłowicz theorem

We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Kar?owicz (in [?]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected iff for some a ∈ X ∩ Y the order-type of a ∩ X is different from that of a ∩ Y.In addition to the chromatic number x(G) of these graphs we study χ _κ (G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph.