We prove that a separable, nuclear, purely infinite, simple ${{C}^{*}}$-algebra satisfying the universal coefficient theorem is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.

Let R be a commutative ring with identity. A subgroup S of GLn(R), where n ≥ 2, is said to be standard if and only if S contains all the q-elementary matrices and all conjugates of those matrices by products of elementary matrices, where q is the ideal in R generated by Xij,xii — Xjj(i ≠ j), for all (xij) ∊ S. It is known that, when n ≧ 3, the standard subgroups of GLn(R) are precisely those normalized by the elementary matrices. To demonstrate how completely this result can break down for n = 2 we prove that GL2(Z), where Z is the ring of rational integers, has uncountably many non-normal, standard subgroups and uncountably many non-standard, normal subgroups.

All finite simple self 2-distance graphs with no square, diamond, or triangles with a common vertex as subgraph are determined. Utilizing these results, it is shown that there is no cubic self 2-distance graph.

We extend the comprehensive treatment of k-spaces and sequential spaces provided by Franklin's refined notion of a natural cover to kR-spaces and sR-spaces. For this purpose, an apparently unstudied class of maps of topological spaces, the class of R-quotient maps, is introduced.

Let $H$ be an exceptional, adjoint group of type ${{E}_{6}}$ and split rank 2, over a $p$-adic field $k$. In this article we discuss the restriction of the minimal representation of $H$ to a dual pair $P{{D}^{\times }}\,\times \,{{G}_{2}}\left( k \right)$, where $D$ is a division algebra of dimension 9 over $k$. In particular, we discover an interesting class of supercuspidal representations of ${{G}_{2}}\left( k \right)$.

Let $\chi $ be a locally compact metric space and let $\mathcal{A}$ be any of the Lipschitz algebras $\text{Li}{{\text{p}}_{\alpha }}\text{ }\!\!\chi\!\!\text{ }$, $\text{Li}{{\text{p}}_{\alpha }}\text{ }\!\!\chi\!\!\text{ }$, or $\text{lip}_{\alpha }^{0}\,\chi $. In this paper, we show, as a consequence of rather more general results on Banach algebras, that $\mathcal{A}$ is $C$-character amenable if and only if $\chi $ is uniformly discrete.

Let $R$ be a dense subring of $\text{End}\left( _{D}V \right)$, where $V$ is a left vector space over a division ring $D$. If $\dim{{\,}_{D}}V\,=\,\infty $, then the range of any nonzero polynomial $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ on $R$ is dense in $\text{End}\left( _{D}V \right)$. As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\,\ne \,a\,\in \,R$. If $af{{\left( {{x}_{1}},\ldots ,{{x}_{m}} \right)}^{n\left( {{x}_{i}} \right)}}\,=\,0$ for all ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$, where $n\left( {{x}_{i}} \right)$ is a positive integer depending on ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$, then $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.

We present examples showing that certain results on the invariance of link polynomials under generalized mutation are the best possible. They show, moreover, that this generalized mutation cannot be effected by a sequence of ordinary mutations. One of the examples also shows that the reduced Jones polynomial can be a more sensitive invariant than the Jones polynomial itself.

A binary structure $S$ has the pigeonhole property $\left( P \right)$ if every finite partition of $S$ induces a block isomorphic to $S$. We classify all countable tournaments with $\left( P \right)$; the class of orders with $\left( P \right)$ is completely classified.

Dans ce qui suit f(X) désigne un polynöme irréductible de degré n de et un polynöme de tel que a≠0 et r désigne un entier non nul.

On montre dans cet article que les polynÔmes ne sont pas irréductibles sur pour m≥3.

Quantum tori with graded involution appear as coordinate algebras of extended affine Lie algebras of type ${{\text{A}}_{1}}$, $\text{C}$ and $\text{BC}$. We classify them in the category of algebras with involution. From this, we obtain precise information on the root systems of extended affine Lie algebras of type $\text{C}$.

Let R be an integral domain. It is proved that if a nonzero ideal I of R can be generated by n < ∞ elements, then I is invertible (i.e., flat) if and only if I(∩ Rai) = ∩ Iai for all { a1, . . ., a n﹜ ⊂ I. The article's main focus is on torsion-free R-modules E which are LCM-stable in the sense that E(Ra ∩ Rb) = Ea ∩ Eb for all a, b ∈ R. By means of linear relations, LCM-stableness is shown to be equivalent to a weak aspect of flatness. Consequently, if each finitely generated ideal of R may be 2-generated, then each LCM-stable R-module is flat. Finally, LCM-stableness of maximal ideals serves to characterize Prüfer domains, Dedekind domains, principal ideal domains, and Bézout domains amongst suitably larger classes of integral domains.

S. M. Nikolski [4, Theorem 1; cf. 3, esp. pp. 144 and 148] considered the remainder term in the approximation by the n-th Fejér mean, σn(x), to a function, f(x), of period 2π satisfying a Lipschitz condition of order α, 0<α≤1. In this connection, he introduced the quantity

1

where the maximum is taken over all x and the supremum is taken over all functions of period 2π, bounded by 1 (a notational convenience only) and satisfying a Laps chitz condition of order α.

Let A(n) be the number of partitions of { 1 , … , n} into two sets A, B of cardinality n/2 such that . Then there is the asymptotic result

Let $\mathcal {B}(\mathcal {H})$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal {H}$ . In this paper, we first establish several sharp improved and refined versions of the Bohr’s inequality for the functions in the class $H^{\infty }(\mathbb {D},\mathcal {B}(\mathcal {H}))$ of bounded analytic functions from the unit disk $\mathbb {D}:=\{z \in \mathbb {C}:|z|<1\}$ into $\mathcal {B}(\mathcal {H})$ . For the complete circular domain $Q \subset \mathbb {C}^{n}$ , we prove the multidimensional analogues of the operator valued Bohr-type inequality which can be viewed as a special case of the result by G. Popescu [Adv. Math. 347 (2019), 1002–1053] for free holomorphic functions on polyballs. Finally, we establish the multidimensional analogues of several improved Bohr’s inequalities for operator valued functions in Q.

Let ψ be a non-decreasing continuous subadditive function defined on [0, ∞) and satisfy ψ(x) = 0 if and only if x = 0. The space H(ψ) is defined as the set of analytic functions in the unit disk which satisfy

and the space H+ (ψ) is the space of a f ∊ H(ψ) for which

where almost everywhere.

In this paper we study the H(ψ) spaces and characterize the continuous linear functionals on H+ (ψ).

Let $U(n)$ be the group of $n\,\times \,n$ unitary matrices. We show that if $\phi $ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ is a multiple of $n$, and $\phi $ has the form

$$A\,\mapsto \,V[(A\,\otimes \,{{I}_{s}})\,\otimes \,({{A}^{t}}\,\otimes \,{{I}_{r}})]W$$

for some $V,\,W\,\in \,U(m)$. From this result, one easily deduces the characterization of linear operators that map $U(n)$ into itself obtained by Marcus. Further generalization of the main theorem is also discussed.

A ring R with unity is called right (left) self injective if the right (left) R-module R is injective [7]. The purpose of this note is to prove the following: Let R be a prime ring with a maximal annihilator right (left) ideal. If R is right (left) self injective then R is a primitive ring with a minimal one-sided ideal. If R satisfies the maximum condition on annihilator right (left) ideals and R is right (left) self injective then R is a simple ring with the minimum condition on one-sided ideals.

Let there be given two sets of points, P = {P1,…, Pm}1 m and Q={Q1,…, Qn}, such that joining each pair of points (Pi, Qk), for i=1,…, m and k=1…, n, is a line oriented towards one, and only one, of the pair. Such a configuration will be called an m×n bipartite tournament. If the line joining Pi to Qk is oriented towards Qk we may indicate this by pi→Qk, and similarly if the line is oriented in the other sense. The points Pi, Pj, Qk, and Ql. will be said to form a 4-cycle if either Pi→Qk→Pj→Ql→Pi or Pi→Ql→Pj→Qk→Pi. C(m, n), the number of 4-cycles in a given m×n bipartite tournament, provides, in some sense, a measure of the degree of transitivity of the relationship indicated by the orientation of the lines, and the complete configuration may be thought of as representing the outcome of comparing each member of one population with each member of a second population, and making a decision, upon some basis, as to which component of each pair is the preferred one.

The line - graph of an ordinary graph G is that graph whose points can be put in one-to-one correspondence with the lines of G in such a way that two points of are adjacent if and only if the corresponding lines of G are adjacent. This concept originated with Whitney [ 5 ], has the property that its (point) chromatic number equals the line chromatic number of G, where the point (line) chromatic number of graph is the minimum number of colors required to color the points (lines) of the graph such that adjacent points (lines) are colored differently.