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A space X is called *functionally countable* if ƒ (X) is countable for any continuous function ƒ : X → Ø. Given an infinite cardinal k, we prove that a compact scattered space K with d(K) > k must have a convergent k^{+}-sequence. This result implies that a Corson compact space K is countable if the space (K × K) \ Δ_{K} is functionally countable; here Δ_{K} = {(x, x): x ϵ K} is the diagonal of K. We also establish that, under Jensen’s Axiom ♦, there exists a compact hereditarily separable non-metrizable compact space X such that (X × X) \ Δ_{X} is functionally countable and show in ZFC that there exists a non-separable σ-compact space X such that (X × X) \ Δ_{X} is functionally countable.

We provide necessary and sufficient conditions for the coincidence, up to equivalence of the norms, between strong and weak Orlicz spaces. Roughly speaking, this coincidence holds true only for the so-called *exponential* spaces.

We also find the exact value of the embedding constant which appears in the corresponding norm inequality.

Suppose that K and K' are knots inside the homology spheres Y and Y', respectively. Let X = Y (K, K') be the 3-manifold obtained by splicing the complements of K and K' and Z be the three-manifold obtained by 0 surgery on K. When Y' is an L-space, we use the splicing formula of [1] to show that the rank of ^{2}) = 0 and is bounded below by rank(

Let *k* ≥ 1. A *Sperner k-family* is a maximum-sized subset of a finite poset that contains no chain with *k* + 1 elements. In 1976 Greene and Kleitman defined a lattice-ordering on the set *S _{k}
*(

*P*) of Sperner

*k*-families of a fifinite poset

*P*and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of

*S*(

_{k}*P*),” adding, “This has apparently not been done even for the case

*k*= 1.”

In this article, the case *k* = 1 is done.

The aim of this paper is to prove some uncertainty inequalities for the continuous Hankel wavelet transform, and study the localization operator associated to this transformation.

In this note, we show that the result [1, Proposition 5.2] is inaccurate. We further give and prove the correct modification of such a result. Some applications are also given.

Let K = ℚ(α) be a number field generated by a complex root a of a monic irreducible polynomial ƒ (x) = x^{36} − m, with m ≠ ±1 a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≠ ±1 (mod 9) then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡±1 (mod 9), then the number field K is not monogenic.

In this paper, we prove that the ∗-Ricci tensor of a real hypersurface in complex projective plane ℂ*P*
^{2} or complex hyperbolic plane ℂ*H*
^{2} is cyclic parallel if and only if the hypersurface is of type (A). We find some three-dimensional real hypersurfaces having non-vanishing and non-parallel ∗-Ricci tensors which are cyclic parallel.

We prove that

for all integers n ≥ 1 and ɵ ≤ 8 ≤ π. This result refines inequalities due to Jackson (1911) and Turán (1938).

Let *D* be a weighted oriented graph, whose underlying graph is *G*, and let *I (D)* be its edge ideal. If *G* has no 3-, 5-, or 7-cycles, or *G* is Kőnig, we characterize when *I (D)* is unmixed. If *G* has no 3- or 5-cycles, or *G* is Kőnig, we characterize when *I (D)* is Cohen–Macaulay. We prove that *I (D)* is unmixed if and only if *I (D)* is Cohen–Macaulay when *G* has girth greater than 7 or *G* is Kőnig and has no 4-cycles.