Volume 11, Issue 2

We construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases and the same presentation P/I.

Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.

Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.

The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.

We consider isometry groups of a fairly general class of non standard products of metric spaces. We present sufficient conditions under which the isometry group of a non standard product of metric spaces splits as a permutation group into direct or wreath product of isometry groups of some metric spaces.

After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K 3/ℚ, i.e. $$ S_{l,K_3 } (x) = \sum\nolimits_{m \leqslant x} {M^l (m)} $$, where M(m) denotes the number of integral ideals of the field K 3 of norm m and l ∈ ℕ. We improve the previous results for $$ S_{2,K_3 } (x) $$ and $$ S_{3,K_3 } (x) $$.

We investigate the visibility parameter, i.e., the number of visible pairs, first for words over a finite alphabet, then for permutations of the finite set {1, 2, …, n}, and finally for words over an infinite alphabet whose letters occur with geometric probabilities. The results obtained for permutations correct the formula for the expectation obtained in a recent paper by Gutin et al. [Gutin G., Mansour T., Severini S., A characterization of horizontal visibility graphs and combinatorics on words, Phys. A, 2011, 390 (12), 2421–2428], and for words over a finite alphabet the formula obtained in the present paper for the expectation is more precise than that obtained in the cited paper. More importantly, we also compute the variance for each case.

We deal with rectangular m×n boards of square cells, using the cut technics of the height function. We investigate combinatorial properties of this function, and in particular we give lower and upper bounds for the number of essentially different cuts. This number turns out to be the cardinality of the height function’s range, in case the height function has maximally many rectangular islands.

If a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity of G is ℈Δ/2⌊ if Δ ≥ 17, where G is an IC-planar graph and Δ is the maximum degree of G.

We consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Carathéodory topology to another pointed hyperbolic domain. Using these we describe an equivalent condition to Carathéodory convergence which is formulated in terms of Riemann mappings to standard slit domains.

It is shown that every metrizable consonant space is a Cantor set-selector. Some applications are derived from this fact, also the relationship is discussed in the framework of hyperspaces and Prohorov spaces.

The paper concerns topologies introduced in a topological space (X, τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated.

We prove that if the composition operator F generated by a function f: [a, b] × ℝ → ℝ maps the space of bounded (p, k)-variation in the sense of Riesz-Popoviciu, p ≥ 1, k an integer, denoted by RV(p,k)[a, b], into itself and is uniformly bounded then RV(p,k)[a, b] satisfies the Matkowski condition.

We show that Tanahashi’s argument on best possibility of the grand Furuta inequality has an additional consequence.

In this short note a correct proof of Theorem 3.3 from [Tărnăuceanu M., Solitary quotients of finite groups, Cent. Eur. J. Math., 2012, 10(2), 740–747] is given.