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The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple C*-algebras arising from free product constructions, Canad. J. Math. 50 (1998), 323–341 and of Theorem 3.1 of K J. Dykema, Purely infinite simple C*-algebras arising from free product constructions, II,Math. Scand. 90 (2002), 73–86 are corrected.

It is shown that an old direct argument of Erdős and Heilbronn may be elaborated to yield a result of the current inverse type.

Let π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.

We correct two clerical errors made in the paper ”A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold”.

One result concerning the series representation for the continuous Jacobi transform in Walter and Zayed [1] is improved, the same thought also can be applied to the related results in [1].

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].)

In this paper we will construct an example showing that the problem posed in [1] has a negative answer. Two more theorems on the subject treated in [1] will be included.Let Io = [0, 1], R the reals, and let, for A ⊂ R, Ao be the interior of A. Let {xn} be a sequence in [0, 1> such that 0 = x1 < x2 < … and lim xn = 1. For each n, let In be closed interval having x as its midpoint (except for n = 1 in which case x1 is the left endpoint of I1) such that In ∩ Im= ϕ, and the metric density relative to Io of at 1 is zero. Let Jn be a closed interval in In concentric with In (except for n = 1, where J1 has x1 as its left endpoint) whose length is half that of In

R. A. Brualdi [1] presents a construction yielding matrices whose Birkhoff representation consists of the maximum number of permutation matrices and having 0(2n2) line sum. In this note a counterexample to such a construction is given. Furthermore, a new construction is presented, yielding matrices with lower line sums.

Let M0 be the set of measures μ on the real line such that open sets are μ*-immeasurable. While attempting to find out whether a set μ*-measurable for all μ in Mo is mapped into a similar set by a continuous function of bounded variation, Maurice Sion develops a theory for what he calls variational measure (4). As an application of the theory, he gets conditions on a function f and a set of measures M in order that f map a set, which is μ*-measurable for all μ ∈ M, into a set of the same kind. In particular he proves for his class M2 (def. 2.5), the following theorem (4, § 8.11).

Suppose X is a simply-connected CW- complex whose homotopy groups are finite. For each n, let be the class of torsion groups whose p - primary components are zero for all primes p which do not figure in the homotopy groups πi(X) for i ≤ n.

As usual we denote the number of integers not exceeding n and relatively prime to n by Euler's ϕ function ϕ(n). Lehmer calls the ϕ(n) integersthe totatives of n.Denote by ϕ(k,l,n) the number of a's satisfyingIf nl ≡ 0 (mod k) or n(l + 1) ≡ 0 (mod k) then, since n > k, (n - l/k, n) > 1 and (n(l + 1)/k, n)>1 respectively. Thus ϕ(k, l, n) is the number of totatives of n satisfying

In a recent paper Robinson has obtained an explicit formula for the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices. The object of the present note is to show that this formula may be deduced from known properties of Schur functions, with the aid of a result which the author has proved elsewhere.

The aim of this note is to point out a mistake in the proof of Theorem 2 of Wong's paper [1], We first give an example to show that the theore as stated is not true.Example 1. Consider the cartesian plane and the graph H with equation .

There are several statements in [3] which require clarification.Theorem 1 [3, p. 246] states that U3= U2U4 U2-1U4 U2 U4. In fact this is (essentially) the relation O = PUPU-1PU given in [1, p. 91, (7.35)]. To see this we note that 0 = U3, P = U4, U = U4U2U4 (as explained in [1, p. 88]); since U42, = E

Mason, [1, Theorem 2.6], proved that for any near-ring R, there are no non-trivial injective R-modules. In his proof he embedded R into a simple R -module G of arbitrarily high cardinality.