The generating degree $\text{g}\deg \left( A \right)$ of a topological commutative ring $A$ with char $A\,=\,0$ is the cardinality of the smallest subset $M$ of $A$ for which the subring $\mathbb{Z}\left[ M \right]$ is dense in $A$. For a prime number $p$, ${{\mathbb{C}}_{p}}$ denotes the topological completion of an algebraic closure of the field ${{\mathbb{Q}}_{p}}$ of $p$-adic numbers. We prove that $\text{g}\deg \left( {{\mathbb{C}}_{p}} \right)\,=\,1$, i.e., there exists $t$ in ${{\mathbb{C}}_{p}}$ such that $\mathbb{Z}\left[ t \right]$ is dense in ${{\mathbb{C}}_{p}}$. We also compute $\text{gdeg}\left( A\left( U \right) \right)$ where $A\left( U \right)$ is the ring of rigid analytic functions defined on a ball $U$ in ${{\mathbb{C}}_{p}}$. If $U$ is a closed ball then $\text{gdeg}\left( A\left( U \right) \right)\,=\,2$ while if $U$ is an open ball then $\text{gdeg}\left( A\left( U \right) \right)$ is infinite. We show more generally that $\text{gdeg}\left( A\left( U \right) \right)$ is finite for any affinoid$U$ in ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$ and $\text{gdeg}\left( A\left( U \right) \right)$ is infinite for any wide open subset $U$ of ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$.

Let R be a commutative ring with unity and let M(R) be the multiplicative group of 4 x 4 triangular matrices (aij) over R, where a11 is a unit element of R and aii = 1 for i = 2, 3, 4. If V(=AN2 ∧ N2A) denotes the variety of groups which are both abelian-by-class-2 and class-2-by-abelian, then it is routine to verify that M(R) ∊ V. Here we prove the following,

Theorem. Let F(V) denote the free group of finite or countable infinite rank of the variety V. Then for a suitable choice of R, F(V) is embedded in M(R).

A well known theorem by S. Banach states that a contractive function f: X → X on a complete metric space X has a fixed point, and that this fixed point is unique. This result has a partial extension to multi-functions: every contractive compact-valued multi-function on a complete metric space has a fixed point (see Definition 1 and Theorem 1 below). But simple examples show that this fixed point is no longer unique. We investigate some questions concerned with the properties of the fixed point set Φ of a contractive multi-function φ. Is, e.g., Φ connected if φ is connected-valued? Is Φ convex if φ is convex-valued? The answer is yes if X is the real line (§2), but examples in §3 and §4 show that in general the answer is no.

We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.

We give an elementary proof of the theorem of Saks that states that most functions in C[0, 1] have infinite unilateral derivatives at a continuum many points.

For a simple Lie algebra L over the complex numbers ℂ all irreducible representations admitting a highest weight have been constructed and characterized for example in [3, 6]. In [1] Bouwer considered the family of all irreducible representations of L admitting at least one one-dimensional weight space (this includes, of course, all those having a highest weight space) and showed, by construction, that this is a strictly larger class of representations.

In this paper, another relationship between the quasi-ideal adequate transversals of an abundant semigroup is given. We introduce the concept of a weakly multiplicative adequate transversal and the classic result that an adequate transversal is multiplicative if and only if it is weakly multiplicative and a quasi-ideal is obtained. Also, we give two equivalent conditions for an adequate transversal to be weakly multiplicative. We then consider the case when $I$ and $\Lambda $ (defined below) are bands. This is analogous to the inverse transversal if the regularity condition is adjoined.

It is shown that in a certain extensive class of algebras one can associate with each Lie ideal a corresponding associative ideal which facilitates the study of Lie ideals, especially for simple algebras. We apply this construction to obtain new, simpler proofs of some known results of Herstein [10] and others on the Lie structure of associative rings.

Let $G$ be a $p$-group and let $\unicode[STIX]{x1D712}$ be an irreducible character of $G$. The codegree of $\unicode[STIX]{x1D712}$ is given by $|G:\,\text{ker}(\unicode[STIX]{x1D712})|/\unicode[STIX]{x1D712}(1)$. If $G$ is a maximal class $p$-group that is normally monomial or has at most three character degrees, then the codegrees of $G$ are consecutive powers of $p$. If $|G|=p^{n}$ and $G$ has consecutive $p$-power codegrees up to $p^{n-1}$, then the nilpotence class of $G$ is at most 2 or $G$ has maximal class.

We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

Let R be a left and right perfect right self-injective ring. It is shown that if the radical of R is countably generated as a left ideal then R is quasi-Frobenius. It is also shown that the same conclusion can be drawn if r(A ∩ B) = r(A) + r(B) for all left ideals A and B of R.

Given a finite group $G$, we attach to the character degrees of $G$ a graph whose vertex set is the set of primes dividing the degrees of irreducible characters of $G$, and with an edge between $p$ and $q$ if $pq$ divides the degree of some irreducible character of $G$. In this paper, we describe which graphs occur when $G$ is a solvable group of Fitting height 2.

A projective geometry of dimension (n - 1) can be defined as modular lattice with a spanning n-diamond of atoms (i.e.: n + 1 atoms in general position whose join is the unit of the lattice). The lemma we show is that one could equivalently define a projective geometry as a modular lattice with a spanning n-diamond that is (a) is generated (qua lattice) by this n-diamond and a coordinating diagonal and (b) every non-zero member of this coordinatizing diagonal is invertible. The lemma is applied to describe certain freely generated modular and Arguesian lattices.

Let D(n) denote a function of an integral variable n ≥ 0 such that

1. (1) D(1) = D(0) = 0

2. (2) D(p) = 1 for every prime p

3. (3) D(n1n2) = n1D(n2) + n2D(n1) for every pair of non-negative integers n1, n2.

We study an associative algebra $A$ over an arbitrary field that is a sum of two subalgebras $B$ and $C$ (i.e., $A\,=\,B+C$ ). We show that if $B$ is a right or left Artinian $PI$ algebra and $C$ is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally, we generalize this result for semiprime algebras $A$ . Consider the class of all semisimple finite dimensional algebras $A\,=\,B+C$ for some subalgebras $B$ and $C$ that satisfy given polynomial identities $f\,=\,0$ and $g\,=\,0$ , respectively. We prove that all algebras in this class satisfy a common polynomial identity.

It is shown that if V is a congruence distributive variety whose members have one element subalgebras, then the class of absolute retracts of V is closed under direct products. If V is residually small, then a characterisation of the amalgamation class of V is given.

C(X) (resp. C*(X)) will denote as usual the ring of all (resp. all bounded) continuous functions into the real line R. Define C#(X) to consist of all f∊C(X) whose image M(f) in the residue class ring C(X)jM is real for every maximal ideal M in C(X). Then C# shares with C* the property of being an intrinsically determined subalgebra of C. Then C* shares with C* the property of being an intrinsically determined subalgebra of C. The compactification corresponding to C# (as uniformity determining subalgebra of C*) is thus also an intrinsically determined one. We show that this compactification is well known and "natural" in the cases of several elementary spaces X.

Let X and Y be two Banach spaces and let B(X, Y) denote the set of bounded linear operators with domain X and range in 7. For T∈B(X, Y), let N(T) denote the null space and R(T) the range of T. J. I. Nieto [5, p. 64] has proved the following two interesting results. An operator T∈B(X, Y) has a left regularizer, i.e., there exists a Q∈B(Y, X) such that QT=I+A, where I is the identity on X and A∈B(X, X) is a compact operator, if and only if dim N(T)<∞ and R(T) has a closed complement.

In measure theory and probability it is often useful to be able to extend a set function g to a measure μ. One situation in which such an extension arises is that of obtaining limit measures for inverse (or projective) systems of measure spaces ([1], [5]).

A composition operator CT on L2(X, Σ,m) is a bounded linear transformation induced by a mapping T : X → X via CTf = f∘ T.

If m has no atoms then the co-rank of CT (i.e., dim is either zero or infinite. As a corollary, when m has no atoms, CT is a Fredholm operator iff it is invertible.