A left FPF ring is a ring R such that every finitely generated faithful left R -module generates the category of left R-modules. It is shown that such rings split into R = A⊕B, where A is a two sided ideal, and A contains the left singular ideal of R as an essential submodule. If R is FPF on both sides B is two sided too, and R is the product of A and B. An example shows this is the best possible and that right FPF does not imply left FPF.

Let {Xn} be a sequence of random variables, not necessarily independent or identically distributed, put and Mn =max0≤k≤n|Sk|. Effective bounds on in terms of assumed bounds on , are used to identify conditions under which an extended-valued stopping time τ exists. That is these inequalities are used to guarantee the existence of the stopping time τ such that E(ST/aτ) = supt ∈ T∞ E(|Sτ|/at), where T∞ denotes the class of randomized extended-valued stopping times based on S1, S2, … and {an} is a sequence of constants. Specific applications to stochastic processes of the time series type are considered.

Let R be a prime ring and d≠0 a derivation of R. We examine the relationship between the structure of R and that of d(R). We prove that if R is an algebra over a commutative ring A such that d(R) is a finitely generated submodule then R is an order in a simple algebra finite dimensional over its center.

Let H(x) be a homogeneous polynomial in n indeterminates over an algebraically closed field K. A necesssary and sufficient condition is given for H(x) to admit a factorization of the form

a, b∈ Kn, and “∘” is the usual inner product. This condition involves the linear derivatives of H(x).

The question of what sets of integers may be the score sets of bipartite tournaments was posed recently by K. B. Reid. The main theorem of this paper establishes a sufficient condition for pairs of sets to be bipartite score sets. This simple condition yields an immediate affirmative answer for a large class of pairs of sets.

This note determines the quadratic field generated by the square root of the discriminant of the modular equation satisfied by the special value j(α) of the modular function α for a an integer in an imaginary quadratic field.

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.

The main purpose of this paper is to give a characterization of a Veronese manifold, as a generalization of a Veronese surface, in terms of isotropic immersions. This is an improvement of Itoh and Ogiue's results.

In 1929 J. Hjelmslev introduced a geometry over the dual numbers ℝ+tℝ with t2 = Q. The dual numbers form a Hjelmslev ring, that is a local ring whose (unique) maximal ideal is equal to the set of 2 sided zero divisors and whose ideals are totally ordered by inclusion. This paper first shows that if we endow the dual numbers with the product topology of ℝ2, then we obtain the only locally compact connected hausdorfT topological Hjelmslev ring of topological dimension two. From this fact we establish that Hjelmslev's original geometry, suitably topologized, is the only locally compact connected hausdorfr topological desarguesian projective Hjelmslev plane to topological dimension four.

In this paper we consider how much we can say about an irreducible symmetric space M which admits a hypersurface N with at most two distinct principal curvatures. Then we will obtain that (1) if N is locally symmetric, then M must be a sphere, a real projective space and their noncompact duals (2) if N is Einstein, then M must be rank 1.

Ordered partitions are enumerated by Fn = Σk k !S(n, k) where S(n, k) is the Stirling number of the second kind. We give some comments on several papers dealing with ordered partitions and turn then to ordered Fibonacci partitions of {1, ߪ, n}: If d is a fixed integer, the sets A appearing in the partition have to fulfill i, j ∈ A, i ≠ j ⟹ |i-j| ≥ d. The number of ordered Fibonacci partitions is determined.

Questions concerning the convergence of Padé and best rational approximations are considered from a categorical point of view in the complete metric space of entire functions. The set of functions for which a subsequence of the mth row of the Padé table converges uniformly on compact subsets of the complex plane is shown to be residual.

The speed of convergence of best uniform rational approximations and Padé approximations on the unit disc is compared. It is shown that, in a categorical sense, it is expected that subsequences of these approximants will converge at the same rate.

Likewise, it is expected that the poles of certain sequences of best uniform rational approximations wil be dense in the entire plane.

The classical Levi problem in several complex variables characterizes domains of holomorphy in terms of a boundary condition called pseudo convexity. The purpose of this note is to give a characterization of those domains D in ℂ×ℝ, where one can always solve the -problem with C∞ parameters, in terms of a certain kind of convexity condition on their boundaries.

This note contains a strengthened version of the following well-known theorem: there exists a continuous function whose Fourier series diverges at a point.

Let R be an integral domain with quotient field K and let I be a nonzero ideal of R. We show (1) that I is invertible if and only if for every nonempty collection {Bα} of ideals of R and (2) that I is flat if and only if I(B ∩ C) = IB∩IC for each pair of ideals B and C of R.

It is shown that suitable conditions on the components of a measure in the algebra M(G) ensure that this element lies outside of all maximal bi-ideals.

We survey results on partitioning some common sets into m congruent pieces, and prove that a ball in Rn cannot be so partitioned if 2 ≤ m ≤ n.

It is shown that the nilpotency of a derivation on a 2-torsion free semiprime ring is always an odd number. Examples are provided to show the necessity of the assumptions.

Let X be a locally compact, completely regular Hausdorff space. In this paper it is shown that all compact metric spaces are remainders of X if and only if the quotient ring C*(X)/C∞(X) contains a subring having no primitive idempotents.

Recently Goodman and Pollack [3] have proved a theorem which can be applied to encode in a compact form the order properties of an arbitrary configuration of points in ℝd. In this note we generalise this theorem to oriented matroids.