<Emphasis Type="Italic">C</Emphasis>*-simplicity of <Emphasis Type="Italic">n</Emphasis>-periodic products

The C*-simplicity of n-periodic products is proved for a large class of groups. In particular, the n-periodic products of any finite or cyclic groups (including the free Burnside groups) are C*-simple. Continuum-many nonisomorphic 3-generated nonsimple C*-simple groups are constructed in each of which the identity xⁿ = 1 holds, where n ≥ 1003 is any odd number. The problem of the existence of C*-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.     

Pure <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-subgroups of completely decomposable groups

We show a simple way to determine purity for a B(n)-group contained in a completely decomposable group.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

<Emphasis Type="Italic">n</Emphasis>-Flat and <Emphasis Type="Italic">n</Emphasis>-FP-injective modules

In this paper, we study the existence of the n-flat preenvelope and the n-FP-injective cover. We also characterize n-coherent rings in terms of the n-FP-injective and n-flat modules.     

On (<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis>)-semigroups

We generalize Green’s lemma and Green’s theorem for usual binary semigroups to (n,m)-semigroups, define and describe the regularity for an element of an (n,m)-semigroup, give some criteria for an element of an (n,m)-semigroup to be invertible, and further apply the invertibility for (n,m)-semigroups to (n,m)-groups and give some equivalent characterizations for (n,m)-groups. We establish Hosszú-Gluskin theorems for (n,m)-semigroups in two cases, as generalizations of the corresponding theorems for n-groups.     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

Remarks on the <Emphasis Type="Italic">k</Emphasis>-error linear complexity of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-periodic sequences

Recently the first author presented exact formulas for the number of 2 ⁿ -periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k ≥ 2, of a random 2 ⁿ -periodic binary sequence. A crucial role for the analysis played the Chan–Games algorithm. We use a more sophisticated generalization of the Chan–Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for p ⁿ -periodic sequences over prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of p ⁿ -periodic sequences over .      

On <Emphasis Type="Italic">n</Emphasis>-Central Semigroups

A semigroup S is said to be n-central if xⁿ belongs to the center of S for every x S. We prove that every n-central semigroup is a semilattice of archimedean n-central semigroups. We obtain characterizations of simple (0-simple) n-central semigroups and describe subdirectly irreducible n-central semigroups. We also deal with the connection of n-central semigroups and E-k semigroups.     

Groups with less than <Emphasis Type="Italic">n</Emphasis> subgroups of index <Emphasis Type="Italic">n</Emphasis>

We characterise (residually-finite) groups which possess less than n subgroups of index n for almost all n ∈ ℕ.     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

Finite groups with <Emphasis Type="Italic">S</Emphasis>-supplemented <Emphasis Type="Italic">p</Emphasis>-subgroups

Consider a finite group G. A subgroup is called S-quasinormal whenever it permutes with all Sylow subgroups of G. Denote by B _sG the largest S-quasinormal subgroup of G lying in B. A subgroup B is called S-supplemented in G whenever there is a subgroup T with G = BT and B∩T ≤ B _sG . A subgroup L of G is called a quaternionic subgroup whenever G has a section A/B isomorphic to the order 8 quaternion group such that L ≤ A and L ∩ B = 1. This article is devoted to proving the following theorem.     

Mappings of preserving <Emphasis Type="Italic">n</Emphasis>-distance one in <Emphasis Type="Italic">n</Emphasis>-normed spaces

We give a positive answer to the Aleksandrov problem in n-normed spaces under the surjectivity assumption. Namely, we show that every surjective mapping preserving n-distance one is affine, and thus is an n-isometry. This is the first time the Aleksandrov problem is solved in n-normed spaces with only the surjectivity assumption even in the usual case \(n=2\). Finally, when the target space is n-strictly convex, we prove that every mapping preserving two n-distances with an integer ratio is an affine n-isometry.     

Exponentially Many Monochromatic <Emphasis Type="Italic">n</Emphasis>-Matchings in <Emphasis Type="Italic">K</Emphasis><Subscript>3<Emphasis Type="Italic">n</Emphasis>-1</Subscript>

Erdös et al and Gerencsér et al had shown that in any 2-edge-coloring of K _3n-1, there is a n-matching containing edges with the same color(we call such matching monochromatic matching). In this paper we show that for any 2-edge-coloring of K _3n-1 there exists a monochromatic subgraph H of K _3n-1 which contains exponentially many monochromatic n-matchings.     

On <Emphasis Type="Italic">n</Emphasis>-cubic Pyramid Algebras

In this paper we study a class of algebras having n-dimensional pyramid shaped quiver with n-cubic cells, which we called n-cubic pyramid algebras. This class of algebras includes the quadratic dual of the basic n-Auslander absolutely n-complete algebras introduced by Iyama. We show that the projective resolutions of the simples of n-cubic pyramid algebras can be characterized by n-cuboids, and prove that they are periodic. So these algebras are almost Koszul and (n?1)-translation algebras. We also recover Iyama’s cone construction for n-Auslander absolutely n-complete algebras using n-cubic pyramid algebras and the theory of n-translation algebras.     

A note on <Emphasis Type="Italic">n</Emphasis>! modulo <Emphasis Type="Italic">p</Emphasis>

Let p be a prime, \(\varepsilon >0\) and \(0<L+1<L+N < p\). We prove that if \(p^{1/2+\varepsilon }< N <p^{1-\varepsilon }\), then

$$\begin{aligned} \#\{n!\,\,({\mathrm{mod}} \,p);\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon )>0. \end{aligned}$$

We use this bound to show that any \(\lambda \not \equiv 0\ ({\mathrm{mod}}\, p)\) can be represented in the form \(\lambda \equiv n_1!\cdots n_7!\ ({\mathrm{mod}}\, p)\), where \(n_i=o(p^{11/12})\). This refines the previously known range for \(n_i\).

     

Cocyclic <Emphasis Type="Italic">n</Emphasis>-groups

We describe all cocyclic n-groups and the structure of (n, 2)-rings of endomorphisms of cocyclic n-groups. We prove that a cocyclic n-group is defined uniquely by its (n, 2)-ring of endomorphisms.     

The unique trace property of <Emphasis Type="Italic">n</Emphasis>-periodic product of groups

In this paper we prove the unique trace property of C*-algebras of n-periodic products of arbitrary family of groups without involutions. We show that the free Burnside groups B(m, n) and their automorphism groups also possess the unique trace property. Also, we show that every countable group is embedded into some 3-generated group with the unique trace property, while every countable periodic group of bounded period and without involutions is embedded into some 3- generated periodic group G of bounded period with the unique trace property. Moreover, as a group G can be chosen both simple and not simple group.     

<Emphasis Type="Italic">L</Emphasis>-fuzzy roughness of <Emphasis Type="Italic">n</Emphasis>-ary polygroups

In this paper, we consider the relations among L-fuzzy sets, rough sets and n-ary polygroup theory. Some properties of (normal) TL-fuzzy n-ary subpolygroups of an n-ary polygroup are first obtained. Using the concept of L-fuzzy sets, the notion of ϑ-lower and T-upper L-fuzzy rough approximation operators with respect to an L-fuzzy set is introduced and some related properties are presented. Then a new algebraic structure called (normal) TL-fuzzy rough n-ary polygroup is defined and investigated. Also, the (strong) homomorphism of ϑ-lower and T-upper L-fuzzy rough approximation operators is studied.