Volume 63, Issue 2

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set I of all countable graphs (since every graph in I is isomorphic to an induced subgraph of R). In this paper we describe a general recursive construction which proves the existence of a countable universal graph for any induced-hereditary property of countable general graphs. A general construction of a universal graph for the set of finite graphs in a product of properties of graphs is also presented. The paper is concluded by a comparison between the two types of construction of universal graphs.

We introduce the category SSN σ Frm of super strong nearness σ-frames and show the existence of a completion for a super strong nearness σ-frame unique up to isomorphism by the similar construction presented in [WALTERS-WAYLAND, J. L.: Completeness and Nearly Fine Uniform Frames. PhD Thesis, Univ. Catholique de Louvain, 1996] and [WALTERS-WAYLAND, J. L.: A Shirota Theorem for frames, Appl. Categ. Structures 7 (1999), 271–277]. Completion is also shown to be a coreflection in SSN σ Frm. We also engage with the notion of total boundedness for nearness σ-frames and provide a characterization of the Samuel compactification of a nearness σ-frame alternative to the description in [NAIDOO, I.: Samuel compactification and uniform coreflection of nearness σ-frames, Czechoslovak Math. J. 56(131) (2006), 1229–1241].

We test the methods for computing the Picard group of a K3 surface in a situation of high rank. The examples chosen are resolutions of quartics in P 3 having 14 singularities of type A 1. Our computations show that the method of R. van Luijk works well when sufficiently large primes are used.

Assume that f is a function defined on some interval I ⊂ ℝm. Literature offers several equivalences of the type: f has a property P (like absolute continuity, bounded variation, etc.) on I if and only if f has P on each (closed) null subset of I. Such results feature a pretty important role in the integration theory. We make a brief review of these results and then provide an example showing that they can break down if considered in the uniform version, that is, for sequences of functions instead of a particular function.

In this paper, sufficient conditions are obtained for oscillation of a class of nonlinear fourth order mixed neutral differential equations of the form (E)$$\left( {\frac{1} {{a\left( t \right)}}\left( {\left( {y\left( t \right) + p\left( t \right)y\left( {t - \tau } \right)} \right)^{\prime \prime } } \right)^\alpha } \right)^{\prime \prime } = q\left( t \right)f\left( {y\left( {t - \sigma _1 } \right)} \right) + r\left( t \right)g\left( {y\left( {t + \sigma _2 } \right)} \right)$$ under the assumption $$\int\limits_0^\infty {\left( {a\left( t \right)} \right)^{\tfrac{1} {\alpha }} dt} = \infty .$$ where α is a ratio of odd positive integers. (E) is studied for various ranges of p(t).

In this paper, we establish some new sufficient conditions for oscillation of the second-order neutral functional dynamic equation $$\left[ {r\left( t \right)\left[ {m\left( t \right)y\left( t \right) + p\left( t \right)y\left( {\tau \left( t \right)} \right)} \right]^\Delta } \right]^\Delta + q\left( t \right)f\left( {y\left( {\delta \left( t \right)} \right)} \right) = 0$$ on a time scale $$\mathbb{T}$$ which is unbounded above, where m, p, q, r, T and δ are real valued rd-continuous positive functions defined on $$\mathbb{T}$$. The main investigation of the results depends on the Riccati substitutions and the analysis of the associated Riccati dynamic inequality. The results complement the oscillation results for neutral delay dynamic equations and improve some oscillation results for neutral delay differential and difference equations. Some examples illustrating our main results are given.

We have recently shown that, for 2 < p < ∞, a locally compact group G is compact if and only if the convolution multiplication f * g exists for all f, g ∈ L p(G). Here, we study the existence of f * g for all f, g ∈ L p(G) in the case where 0 < p ≤ 2. Also, for 0 < p < ∞, we offer some necessary and sufficient conditions for L p(G) * L p(G) to be contained in certain function spaces on G.

In this paper, we give the characterization of S-essential spectra, we define the S-Riesz projection and we investigate the S-Browder resolvent. Finally, we study the S-essential spectra of sum of two bounded linear operators acting on a Banach space.

For a normal space X, α (i.e. the nonempty player) having a winning strategy (resp. winning tactic) in the strong Choquet game Ch(X) played on X is equivalent to α having a winning strategy (resp. winning tactic) in the strong Choquet game played on the hyperspace CL(X) of nonempty closed subsets endowed with the Vietoris topology τ V. It is shown that for a non-normal X where α has a winning strategy (resp. winning tactic) in Ch(X), α may or may not have a winning strategy (resp. winning tactic) in the strong Choquet game played on the Vietoris hyperspace. If X is quasi-regular, then having a winning strategy (resp. winning tactic) for α in the Banach-Mazur game BM(X) played on X is sufficient for α having a winning strategy (resp. winning tactic) in BM(CL(X), τ V), but not necessary, not even for a separable metric X. In the absence of quasi-regularity of a space X where α has a winning strategy in BM(X), α may or may not have a winning strategy in the Banach-Mazur game played on the Vietoris hyperspace.

Sharp bounds are given that connect split points — conic singularities of a special type — of a Morse form with the global structure of its foliation.

We deal with a construction of some difference posets via a method of a pasting of MV-algebras. We generalize Greechie diagrams used in MV-algebra pastings. We give necessary and sufficient conditions under which the resulting pasting of an admissible system MV-algebras is a lattice-ordered D-poset.

Using Mawhin’s continuation theorem we obtain some existence results of periodic solutions for a type of p-Laplacian neutral Liénard equation $$\left( {\phi _p \left( {\left( {x\left( t \right) - c\left( t \right)x\left( {t - \tau } \right)} \right)^\prime } \right)} \right)^\prime + f\left( {x\left( t \right)} \right)x'\left( t \right) + g\left( {x\left( {t - \gamma \left( t \right)} \right)} \right) = e\left( t \right).$$ It is worth noting that c(t) is no longer a constant which is different from the corresponding ones of past work.