On the maximal resolvability of monotonically normal spaces

We continue the work started in [6] and show that all monotonically normal (in short: MN) spaces are maximally resolvable if and only if all uniform ultrafilters are maximally decomposable. As a consequence we get that the existence of an MN space which is not maximally resolvable is equi-consistent with the existence of a measurable cardinal. We also show that it is consistent (modulo the consistency of a measurable cardinal) that there is an MN space X with |X| = Δ(X) = ? _ω which is not ω ₁-resolvable. It follows from the results of [6] that this is best possible.     

Monolithic spaces and D-spaces revisited

We introduce the classes of monotonically monolithic and strongly monotonically monolithic spaces. They turn out to be reasonably large and with some nice categorical properties. We prove, in particular, that any strongly monotonically monolithic countably compact space is metrizable and any monotonically monolithic space is a hereditary D-space. We show that some classes of monolithic spaces which were earlier proved to be contained in the class of D-spaces are monotonically monolithic. In particular, Cp(X) is monotonically monolithic for any Lindelöf Σ-space X. This gives a broader view of the results of Buzyakova and Gruenhage on hereditary D-property in function spaces.     

A note on monotone covering properties

In this note, we show that a monotonically normal space that is monotonically countably metacompact (monotonically meta-Lindelöf) must be hereditarily paracompact. This answers a question of H.R. Bennett, K.P. Hart and D.J. Lutzer. We also show that any compact monotonically meta-Lindelöf T₂-space is first countable. In the last part of the note, we point out that there is a gap in Proposition 3.8 which appears in [H.R. Bennett, K.P. Hart, D.J. Lutzer, A note on monotonically metacompact spaces, Topology Appl. 157 (2) (2010) 456-465]. We finally give a detailed proof of how to overcome the gap.     

Some weak covering properties and infinite games

We show that (I) there is a Lindelöf space which is not weakly Menger, (II) there is a Menger space for which TWO does not have a winning strategy in the game G_fin(O,D_o). These affirmatively answer questions posed in Babinkostova, Pansera and Scheepers [Babinkostova L., Pansera B.A., Scheepers M., Weak covering properties and infinite games, Topology Appl., 2012, 159(17), 3644–3657]. The result (I) automatically gives an affirmative answer of Wingers’ problem [Wingers L., Box products and Hurewicz spaces, Topology Appl., 1995, 64(1), 9–21], too. The selection principle S₁(D_o,D_o) is also discussed in view of a special base. We show that every subspace of a hereditarily Lindelöf space with an ortho-base satisfies S₁(D_o,D_o).     

Set-theoretic constructions of nonshrinking open covers

A family {M_α|α?A} is a shrinking of a cover {O_α|α?A} of a topological space if {M_α|α?A} also covers and M_α?O_α for all α?A.?⁺⁺ implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking.? implies that there is a P-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [17] to show that any space which has a normal product with every P-space is metrizable.     

Weak convergence of linear forms in D[0, 1]

Convergence in probability of the linear forms Σ_k=1^∞a_nkX_k is obtained in the space D[0, 1], where (X_k) are random elements in D[0, 1] and (a_nk) is an array of real numbers. These results are obtained under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements (X_n) on subintervals of a partition of [0, 1]. Since the hypotheses are in general much less restrictive than tightness (or convex tightness), these results represent significant improvements over existing weak laws of large numbers and convergence results for weighted sums of random elements in D[0, 1]. Finally, comparisons to classical hypotheses for Banach space and real-valued results are included.     

The two-machine open shop problem: To fit or not to fit, that is the question

We consider the open shop scheduling problem with two machines. Each job consists of two operations, and it is prescribed that the first (second) operation has to be executed by the first (second) machine. The order in which the two operations are scheduled is not fixed, but their execution intervals cannot overlap. We are interested in the question whether, for two given values D₁ and D₂, there exists a feasible schedule such that the first and second machine process all jobs during the intervals [0,D₁] and [0,D₂], respectively.We formulate four simple conditions on D₁ and D₂, which can be verified in linear time. These conditions are necessary and sufficient for the existence of a feasible schedule. The proof of sufficiency is algorithmical, and yields a feasible schedule in linear time. Furthermore, we show that there are at most two non-dominated points (D₁,D₂) for which there exists a feasible schedule.     

A monotone version of the Sokolov property and monotone retractability in function spaces

We introduce the monotone Sokolov property and show that it is dual to monotone retractability in the sense that X   is monotonically retractable if and only if _Cp(X)$C_p(X)$ is monotonically Sokolov. Besides, a space X   is monotonically Sokolov if and only if _Cp(X)$C_p(X)$ is monotonically retractable. Monotone retractability and monotone Sokolov property are shown to be preserved by R$\mathbb{R}$-quotient images and _Fσ$F_{\sigma }$-subspaces. Furthermore, every monotonically retractable space is Sokolov so it is collectionwise normal and has countable extent. We also establish that if X   and _Cp(X)$C_p(X)$ are Lindelöf Σ-spaces then they are both monotonically retractable and have the monotone Sokolov property. An example is given of a space X   such that _Cp(X)$C_p(X)$ has the Lindelöf Σ-property but neither X   nor _Cp(X)$C_p(X)$ is monotonically retractable. We also establish that every Lindelöf Σ-space with a unique non-isolated point is monotonically retractable. On the other hand, each Lindelöf space with a unique non-isolated point is monotonically Sokolov.     

Products with linear and countable type factors

The basic theorem presented shows that the product of a linearly ordered space and a countable (regular) space is normal. We prove that the countable space can be replaced by any of a rather large class of countably tight spaces. Examples are given to prove that monotone normality cannot replace linearly ordered in the base theorem. However, it is shown that the product of a monotonically normal space and a monotonically normal countable space is normal.

     

Monotone normality and neighborhood assignments

We prove that every neighborhood assignment for a monotonically normal space has a kernel which is homeomorphic to some subspace of an ordinal. As a corollary, every monotone neighborhood assignment for a monotonically normal space has a discrete kernel, which gives a partial answer to a question posed in Buzyakova et al. (2007) [5]. We also give an example of a regular space which has a neighborhood assignment with no kernels homeomorphic to any subspace of an ordinal.     

Longest path partitions in generalizations of tournaments

We consider the so-called Path Partition Conjecture for digraphs which states that for every digraph, D, and every choice of positive integers, λ₁,λ₂, such that λ₁+λ₂ equals the order of a longest directed path in D, there exists a partition of D into two digraphs, D₁ and D₂, such that the order of a longest path in D_i is at most λ_i, for i=1,2.We prove that certain classes of digraphs, which are generalizations of tournaments, satisfy the Path Partition Conjecture and that some of the classes even satisfy the conjecture with equality.     

Okounkov bodies and restricted volumes along very general curves

Given a big divisor D on a normal complex projective variety X, we show that the restricted volume of D along a very general complete-intersection curve C⊂X can be read off from the Okounkov body of D with respect to an admissible flag containing C. From this we deduce that if two big divisors D₁ and D₂ on X have the same Okounkov body with respect to every admissible flag, then D₁ and D₂ are numerically equivalent.     

Normality of products of monotonically normal spaces with compact spaces

Let S be the class of all spaces, each of which is homeomorphic to a stationary subset of a regular uncountable cardinal (depending on the space). In this paper, we prove the following result: The product X×C of a monotonically normal space X and a compact space C is normal if and only if S×C is normal for each closed subspace S in X belonging to S. As a corollary, we obtain the following result: If the product of a monotonically normal space and a compact space is orthocompact, then it is normal.     

Kernels in quasi-transitive digraphs

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively.A kernel N of D is an independent set of vertices such that for every w∈V(D)-N there exists an arc from w to N. A digraph is called quasi-transitive when (u,v)∈A(D) and (v,w)∈A(D) implies (u,w)∈A(D) or (w,u)∈A(D). This concept was introduced by Ghouilá-Houri [Caractérisation des graphes non orientés dont on peut orienter les arrêtes de maniere à obtenir le graphe d’ un relation d’ordre, C.R. Acad. Sci. Paris 254 (1962) 1370-1371] and has been studied by several authors. In this paper the following result is proved: Let D be a digraph. Suppose D=D₁∪D₂ where D_i is a quasi-transitive digraph which contains no asymmetrical infinite outward path (in D_i) for i∈{1,2}; and that every directed cycle of length 3 contained in D has at least two symmetrical arcs, then D has a kernel. All the conditions for the theorem are tight.     

On a strong metric on the space of copulas and its induced dependence measure

Using the one-to-one correspondence between copulas and Markov operators on L¹([0,1]) and expressing the Markov operators in terms of regular conditional distributions (Markov kernels) allows to define a metric D₁ on the space of copulas C that is a metrization of the strong operator topology of the corresponding Markov operators. It is shown that the resulting metric space (C,D₁) is complete and separable and that the induced dependence measure ζ₁, defined as a scalar times the D₁-distance to the product copula Π, has various good properties. In particular the class of copulas that have maximum D₁-distance to the product copula is exactly the class of completely dependent copulas, i.e. copulas induced by Lebesgue-measure preserving transformations on [0,1]. Hence, in contrast to the uniform distance d_∞, Π cannot be approximated arbitrarily well by completely dependent copulas with respect to D₁. The interrelation between D₁ and the so-called ∂-convergence by Mikusinski and Taylor as well as the interrelation between ζ₁ and the mutual dependence measure ω by Siburg and Stoimenov is analyzed. ζ₁ is calculated for some well-known parametric families of copulas and an application to singular copulas induced by certain Iterated Functions Systems is given.     

On weakly nil-clean rings

We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring \(\mathbb{M}_n (R)\) is weakly nil-clean, and to show that the endomorphism ring End _D (V) over a vector space V _D is weakly nil-clean if and only if it is nil-clean or dim(V) = 1 with D?= ?₃.     

When is every linear transformation a sum of two commuting invertible ones?

A well-known result of Wolfson [7] and Zelinsky [8] says that every linear transformation of a vector space V over a division ring D   is a sum of two invertible linear transformations except when dim(V)=1$\dim (V)=1$ and D=_F2$D={\mathbb{F}}_2$. Indeed, many of these linear transformations satisfy a stronger property that they are sums of two commuting invertible linear transformations. The goal of this note is to prove that every linear transformation of a vector space V over a division ring D   is a sum of two commuting invertible ones if and only if |D|?3$|D|?3$ and dim(V)<∞$\dim (V)<\infty$. As a consequence, a sufficient and necessary condition is obtained for a semisimple module to have the property that every endomorphism is a sum of two commuting automorphisms.     

Collectionwise normality and selections into Hilbert spaces

A T₁-space X is countably paracompact and collectionwise normal if and only if every l.s.c. mapping from X into a Hilbert space with closed and convex point-images has a continuous selection. This settles a conjecture posed by M. Choban, V. Gutev and S. Nedev [M. Choban, S. Nedev, Continuous selections for mappings with generalized ordered domain, Math. Balkanica (N.S.) 11 (1-2) (1997) 87-95].     

An extension theorem for D-normal spaces

A second countable developable T₁-space $\text{D}$₁ is defined which has the following properties: (1) $\text{D}$₁ is an absolute extensor for the class of perfect spaces. (2) $\text{D}$₁^?₀ is a universal space for second countable developable T₁-spaces.