We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

Averaging interpolation generalizes polynomials of Lagrange interpolation and also the next-to-interpolatory polynomials. This notion has recently been introduced by Motzkin, Sharma and Straus [3]. Later in [5], Saxena and Sharma have considered the convergence problem of the averaging interpolators on Tchebycheff abscissas. It appears that averaging interpolators have convergence properties similar to those of Lagrange interpolation. It is therefore reasonable to look for an extension of these operators along the lines of Hermite-Fejér interpolation. This can be done in three ways: (i) by taking assigned averages of function-values and by taking the derivatives to be zero, (ii) by taking assigned function-values and by taking the averages of derivatives to be zero, or (iii) by taking averages of function-values and by taking the averages of derivatives to be zero. The object of this note is to take the first approach. The second approach has been the subject of study by M. Botto and A. Sharma [2].

We shall present examples of Schauder bases in the preduals to the hyperfinite factors of types $\text{I}{{\text{I}}_{1}},\,\text{I}{{\text{I}}_{\infty }},\,\text{II}{{\text{I}}_{\lambda }},\,0\,<\,\lambda \,\le \,1$. In the semifinite (respectively, purely infinite) setting, these systems form Schauder bases in any associated separable symmetric space of measurable operators (respectively, in any non-commutative ${{L}^{p}}$-space).

It is shown that if 1<n ≡ 0 or 1 (mod 2 m), then the edges of Kn may be partitioned into isomorphic copies of a graph D3(m) and also of a graph D4(m), graphs consisting respectively of a triangle with an attached path of m - 3 edges or a quadrilateral with an attached path of m - 4 edges. If m is a power of 2 then the above condition is shown to be necessary and sufficient for the existence of such a partition.

In 1968, Galvin conjectured that an uncountable poset $P$ is the union of countably many chains if and only if this is true for every subposet $Q\,\subseteq \,P$ with size ${{\aleph }_{1}}$. In 1981, Rado formulated a similar conjecture that an uncountable interval graph $G$ is countably chromatic if and only if this is true for every induced subgraph $H\,\subseteq \,G$ with size ${{\aleph }_{1}}$. Todorčević has shown that Rado's conjecture is consistent relative to the existence of a supercompact cardinal, while the consistency of Galvin's conjecture remains open. In this paper, we survey and collect a variety of results related to these two conjectures. We also show that the extension of Rado's conjecture to the class of all chordal graphs is relatively consistent with the existence of a supercompact cardinal.

A wavefront set is defined to be an image of Legendrian mapping. In this note we prove that generic wavefront sets have stable local topological structures by using Mather's stratification theory.

Necessary and sufficient conditions are given for a ring to be a union of finitely many proper two-sided ideals.

This is a continuation of the author's article [3], and it contains further problems connected with the theory of convex sets in En. To the list of general references in [3] may be added the recent book [2] on convex polyhedra.

1) Let A and B be two convex bodies in E2 and let a packing P = {B1, B2, …} be an infinite sequence of homothetic images of B sucn that:

1. a) each Bn is a subset of A,

2. b) no two of them share interior points,

3. c) Area (A) =

The existence of such packings is guaranteed by Vitali's Theorem. Let D(X) be the diameter of the set X and put .

We obtain an asymptotic formula for the number of square-free integers in $N$ consecutive values of polynomials on average over integral polynomials of degree at most $k$ and of height at most $H$, where $H\,\ge \,{{N}^{k-1+\varepsilon }}$ for some fixed $\varepsilon \,>\,0$. Individual results of this kind for polynomials of degree $k\,>\,3$, due to A. Granville (1998), are only known under the $ABC$-conjecture.

The invariant operator range lattices of a wide class of uniformly closed algebras (including C*-algebras) are stable under weak closures. There is an algebra whose invariant operator range lattice contains properly the corresponding lattice of its norm closure. An operator range transitive algebra is operator range n -transitive for all n. A normal operator is algebraic if and only if each of its invariant operator ranges is the range of some operator commuting with it.

In this note we study dependence in matroids as an exercise in combinational algebra. Because the work seems to have little connection with graph theory we will not use Tutte′s approach (1) which uses dual concepts. To define a matroid we use Edmunds (2).

Let B be a Boolean algebra and let ℳ and n be two systems of subsets of B, both containing all finite subsets of B. Let us assume further that the join ∨M of every set M∊ℳ and the meet ∧N of every set N∊n exist. Several authors have treated the question under which conditions there exists an isomorphism φ between B and a field δ of sets, satisfying the conditions:

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

We introduce a new type of differentiability, called cofinite Fréchet differentiability. We show that the convex point-of-continuity property of Banach spaces is dual to the cofinite Fréchet differentiability of all equivalent norms. A corresponding result for dual spaces with the weak* convex point-of-continuity property is also established.