A separable metric space X is called rigid if the identity 1X is the only autohomeomorphism, and homogeneous if, for any points x, y of X, there is an (onto) homeomorphism h: X → X such that h(x) = y.

In this note, we show that this onto-ness of the homeomorphism h could not be removed in the definition of homogeneity, by constructing a continuum X which is rigid and has many embeddings, that is, for any two points x, y, there is an embedding (= into homeomorphism) h: X→X such that h(x) = y.

We examine a class of functions which can serve as Mellin multipliers in the setting of the spaces Fp,μ which we have used extensively in other papers. The conditions to be satisfied by such a multiplier h do not involve h′ explicitly. This means that multipliers involving T-functions can be handled by means of the asymptotics of Γ(z) alone, without the need to study ψ = Γ′/Γ, thereby saving effort in the case of complicated multipliers.

In this article we investigate the average order of the arithmetical function

where p1(t), p2(t) are polynomials in Z [t], of equal degree, positive and increasing for t ≥ 1. Using the modern method for the estimation of exponential sums ("Discrete Hardy-Littlewood Method"), we establish an asymptotic result which is as sharp as the best one known for the classical divisor problem.

In this note we show that if 𝓛 is a commutative subspace lattice, then every trace-class operator in Alg 𝓛 lies in the norm-closure of the span of rank-one operators in Alg 𝓛. We also give an elementary proof of a recent result of Davidson and Pitts that if 𝓛 is a CSL generated by completely distributive lattice and finitely many commuting chains, then 𝓛 is compact in the strong operator topology if and only if 𝓛 is completely distributive.

For every infinite cardinal k we construct a space Ck universal for all compact T1 -spaces of weight ≤ k. It follows, in particular, that there are only 2k topologically different compact T1 -spaces. We show that Cw is universal for all second countable developable T1 -spaces. The existence of closely universal compact T1 -spaces is discussed.

A unique factorisation theorem is obtained for tensor products of finite lattices of commuting projections in a factor. This leads to unique tensor product factorisations for reflexive subalgebras of the hyperfinite II1 factor which have irreducible finite commutative invariant projection lattices. It is shown that the finite refinement property fails for simple approximately finite C*-algebras, and this implies that there is no analogous general result for finite lattice subalgebras in this context.

The main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

Let l(N, r) be the minimum number of terms needed to express r as a sum of distinct divisors of N. Let l(N) = max{l(N, r) : 1 ≤ r ≤ N}. Then for Vose's sequence improving the result of M. Vose.

In this paper we study some aspects of the behaviour of p-lattice summing operators. We prove first that an operator T from a Banach space E to a Banach lattice X is p-lattice summing if and only if its bitranspose is. Using this theorem we prove a characterization for 1 -lattice summing operators defined on a C(K) space by means of the representing measure, which shows that in this case 1 -lattice and ∞-lattice summing operators coincide. We present also some results for the case 1 ≤ p < ∞ on C(K,E).

For a prime ring R and σ ∊ Aut(R), we determine the group of Rstabilizing automorphisms of the skew polynomial ring R[x; σ]. In the case where R is simple, we characterize the X-inner automorphisms of R[x; σ]. We also provide necessary and sufficient conditions for a σ -commuting derivation of a prime ring R to extend to a derivation of R[x; σ].

We study the properties of a new class SCB(L, B) of bounded linear maps, called symmetrically completely bounded maps, from a linear subspace L of a C* -algebra to another C*-algebra B. This class contains the class of all completely bounded linear maps from L to B. In particular, we obtain a representation theorem for maps in SCB(L, B) when B is the algebra of all bounded linear operators on a Hilbert space.

It is shown that each commutative Artin local ring having each of its ideals generated by two elements is the homomorphic image of a one-dimensional local complete intersection ring which also has each of its ideals generated by two elements. It is indicated how this can be applied to show that the property that each ideal is projective over its endomorphism ring does not pass to homomorphic images, and in determining the commutative group rings with the two-generator property.

This paper deals with the Stefan-type problem with a zone of coexistence of both phases. We formulate the problem in the enthalpy form and show that the interfaces between the liquid and the mushy, the mushy and the solid phase are smooth. Our approach is to study the structures of the level sets of the solution via Sard's Lemma and the implicit function theorem.