In his Geometric Algebra (New York, 1957) E. Artin poses the problem of co-ordinatizing an affine plane in the following terms.

How little do we have to assume, from a geometric point of view, about an affine plane, in order to be able to describe its points by pairs of elements of a field, and its lines by linear equations?

We generalize to a field of characteristic zero certain properties of the invariant functions of the coadjoint representation of solvable Lie algebras with abelian nilradicals, previously obtained over the base field $\mathbb{C}$ of complex numbers. In particular we determine their number and the restricted type of variables on which they depend. We also determine an upper bound on the maximal number of functionally independent invariants for certain families of solvable Lie algebras with arbitrary nilradicals.

We give an elementary potential-theoretic proof of a theorem of G. Johnsson: all solutions of Cauchy’s problems for the Laplace equations with an entire data on a sphere extend harmonically to the whole space RN except, perhaps, for the center of the sphere.

Let $R\,=\,\oplus _{i=1}^{\infty }\,{{R}_{i}}$ be a graded nil ring. It is shown that primitive ideals in $R$ are homogeneous. Let $A\,=\,\oplus _{i=1}^{\infty }\,{{A}_{i}}$ be a graded non-PI just-infinite dimensional algebra and let $I$ be a prime ideal in $A$. It is shown that either $I\,=\,\{0\}$ or $I\,=\,A$. Moreover, $A$ is either primitive or Jacobson radical.

We give characterizations of Toeplitz operators on generalised H2 spaces and derive some properties of the corresponding Toeplitz algebras. The proofs depend essentially on having a "sufficient" supply of inner functions.

In this note we present inequalities relating the successive minima of an $o$-symmetric convex body and the successive inner and outer radii of the body. These inequalities join known inequalities involving only either the successive minima or the successive radii.

Let A be an m × n matrix of complex numbers. Let Aτ and A* denote the transpose and conjugate transpose, respectively, of A. We say A is diagonal if A contains only zeros in all positions (i, j) with i ≠ j. In a recently published paper [4], M.H. Pearl established the following fact: There exist real orthogonal matrices O1 and O2 (O1 m-square, O2 n-square) such that O1AO2 is diagonal, if and only if both AA* and A*A are real. It is the purpose of this paper to show that a theorem substantially stronger than this result of Pearl's is included in the real case of a theorem of N.A. Wiegmann [2]. (For other papers related to Wiegmann's, see [l; 3].)

It is a question of Arhangel'skiĭ [1] (Problem 2) whether the identity ψ(G) = X(G) holds for every minimal Hausdorff topological group G = 〈G,u〉). (Here, as usual, ψ(G), the pseudocharacter of G, is the least cardinal number K for which there is such that and and x(G), the character of G,is the least cardinality of a local base at e for (〈G,u〉.) That 〈G, u〉 is minimal means that, if v is a Hausdorff topological group topology for G and v ⊂ u, then v = u.

In this paper, we give some conditions on G sufficient to ensure a positive response to Arhangel'skiï's question, and we offer an example which responds negatively to a question on minimal groups posed some years ago (cf. [6] (p. 107) and [4] (p. 259)).

We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (i.e., the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimprič regarding $*$-orderability of quantum groups.

‖ A ‖ désignera le cardinal de A. Soient T une théorie complète du premier ordre dans un langage dénombrable, M un modèle de T, X un sous-ensemble de M; soit T(X) l'ensemble des formules closes à paramètres dans X satisfaites par M; on appelle n-type de T sur X un ensemble consistant avec T(X) et maximal de formules à paramètres dans X à n variables libres.

In this note we give a proof of Lomonosov’s extension of Burnside’s theoremto infinite dimensional Banach spaces.

We prove an algebraic “no-go theorem” to the effect that a nontrivial Poisson algebra cannot be realized as an associative algebra with the commutator bracket. Using it, we show that there is an obstruction to quantizing the Poisson algebra of polynomials generated by a nilpotent basic algebra on a symplectic manifold. This result generalizes Groenewold’s famous theorem on the impossibility of quantizing the Poisson algebra of polynomials on ${{\mathbf{R}}^{2n}}$. Finally, we explicitly construct a polynomial quantization of a symplectic manifold with a solvable basic algebra, thereby showing that the obstruction in the nilpotent case does not extend to the solvable case.

Functions employed for interpolation and curve fitting are for the most part polynomials with numerical coefficients. Indeed these are functions whose values for numerically given arguments can be computed directly without resorting to non-algebraic designs. It is known, however, that there are cases where polynomial interpolation does not yield an adequate approximation to a given function (cf. [4], p. 34).

We state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor ${{K}_{2}}$-groups (for function fields) employed in the proof.

Hyadic spaces are the continuous images of a hyperspace of a compact space. We prove that every non-isolated point in a hyadic space is the endpoint of some infinite cardinal subspace. We isolate a more general order-theoretic property of hyerspaces of compact spaces which is also enjoyed by compact semilattices from which the theorem follows.

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 ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$.

We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field $\mathbb{I}$, in particular of the values of $q$-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel's method applied to a system of functional Poincaré-type equations and the connection between the solutions of these functional equations and the generalized Heine series.

$\text{A}$ ring $R$ is said to be $n$-clean if every element can be written as a sum of an idempotent and $n$ units. The class of these rings contains clean rings and $n$-good rings in which each element is a sum of $n$ units. In this paper, we show that for any ring $R$, the endomorphism ring of a free $R$-module of rank at least 2 is 2-clean and that the ring $B\left( R \right)$ of all $\omega \,\times \,\omega$ row and column-finite matrices over any ring $R$ is 2-clean. Finally, the group ring $R{{C}_{n}}$ is considered where $R$ is a local ring.

Mikusinski [1] has extended the operational calculus by methods which are essentially algebraic. He considers the family C of continuous complex valued functions on the half-line [0,∞). Under addition and convolution C becomes a commutative ring. Titchmarsh's theorem [2] shows that the ring has no divisors of zero and, hence, that it may be imbedded in its quotient field Q whose elements are then called operators. Included in the field are the integral, differential and translational operators of analysis as well as certain generalized functions, such as the Dirac delta function. An alternate approach [3] yields a rather interesting result which we shall now describe briefly.

We carry out the computation of the Iwasawa invariants $\rho _{S}^{T},\,\mu _{S}^{T},\,\lambda _{S}^{T}$ associated to abelian $T$-ramified $S$-decomposed $\ell$-extensions over the finite steps ${{K}_{n}}$ of the cyclotomic ${{\mathbb{Z}}_{\ell }}$ -extension ${{K}_{\infty }}/K$ of a number field of CM-type.