If X is a matrix with non-negative entries then X′X is positive semi-definite with non-negative entries. Conversely, if A is positive semi-definite then there exist matrices Y, not necessarily with non-negative entries, such that Y′Y = A. In the present paper we investigate whether, given a positive semidefinite matrix A with non-negative entries, the equation X′X = A has a solution X with non-negative entries. An equivalent statement of the problem is: Can a positive semi-definite matrix with non-negative entries be expressed as a sum of rank 1 positive semi-definite matrices with non-negative entries? We answer the question in the affirmative for n≦4 and quote the following example due to M.

In an earlier paper (5) a description was given in set-theoretic terms of the semigroup generated by the idempotents of a full transformation semigroup , one of the results being that if X is finite then every element of that is not bijective is expressible as a product of idempotents. In view of this it was natural to ask whether by analogy every singular square matrix is expressibleas a product of idempotent matrices. This is indeed the case, as was shown by J. A. Erdos (2). Magill (6) has considered products of idempotents in thesemigroup of all continuous self-maps of a topological space X, but a comparable characterization of products of idempotents in this case appears to be extremelydifficult, and no solution is available yet.

In [6] Brown and Spencer noted that internal categories within the category of groups are equivalent to crossed modules. As they remarked, this result was known to various others before them, but it had not until then appeared in print. That paper led me to investigate the question of which algebraic categories, C, were such that a similar result held i.e. internal categories in C are equivalent to crossed modules of the appropriate type. The resulting work was written up in 1980 but was not submitted for publication.

We establish a general sharp inequality for warped products in real space form. As applications, we show that if the warping function $f$ of a warped product $N_1\times_fN_2$ is a harmonic function, then

(1) every isometric minimal immersion of $N_1\times_fN_2$ into a Euclidean space is locally a warped-product immersion, and

(2) there are no isometric minimal immersions from $N_1\times_f N_2$ into hyperbolic spaces.

Moreover, we prove that if either $N_1$ is compact or the warping function $f$ is an eigenfunction of the Laplacian with positive eigenvalue, then $N_1\times_f N_2$ admits no isometric minimal immersion into a Euclidean space or a hyperbolic space for any codimension. We also provide examples to show that our results are sharp.

AMS 2000 Mathematics subject classification: Primary 53C40; 53C42; 53B25

A C*-dynamical system is called topologically free if the action satisfies a certain natural condition weaker than freeness. It is shown that if a discrete system is topologically free then the ideal structure of the crossed product algebra is related to that of the original algebra. One consequence is that a minimal topologically free discrete system has a simple reduced crossed product. Sharper results are obtained when the algebra is abelian.

In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their $r$-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their $r$-mean curvatures are linearly related or one of them is constant.

AMS 2000 Mathematics subject classification: Primary 53C42. Secondary 53B30; 53C50; 53Z05; 83C99

The aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the (p, q)-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bounds and regularity up to the boundary. The positivity of the solution is shown by applying a new comparison principle, which is established here.

Motivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the form

$$ u(t)=\gamma(t)\alpha[u]+\int_{G}k(t,s)f(s,u(s))\,\mathrm{d}s, $$

where $G$ is a compact set in $\mathbb{R}^{n}$. Here $\alpha[u]$ is a positive functional and $f$ is positive, while $k$ and $\gamma$ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. Wedescribe a variant of the Cuntz–Krieger relations which applies to graphs with sources, and describe a local convexitycondition which characterizes the higher-rank graphs that admit a non-trivial Cuntz–Krieger family. We then proveversions of the uniqueness theorems and classifications of ideals for the $C^*$-algebras generated by Cuntz–Krieger families.

AMS 2000 Mathematics subject classification: Primary 46L05

In this paper, we consider interpolants on h·ℤn from the closure of the space spanned by translates of the function (‖·‖2 + 1)β/2 (β>−n and not an even nonnegative integer) along h·ℤn. We show that these interpolants approximate a function, whose Fourier transform satisfies certain asymptotic conditions, up to an error of order hp, on any compact domain in ℝn, where p is only restricted by the smoothness of the function.

Some new inequalities for quotients of modified Bessel functions of the first and second kinds are deduced. Moreover, some developments on bounds for modified Bessel functions of the first and second kinds, higher-order monotonicity properties of these functions and applications to a special function that arises in finite elasticity, are summarized. The key tool in our proofs is a frequently used criterion for the monotonicity of the quotient of two Maclaurin series.

The object of this paper is to introduce certain functions analogous to the circular functions. The functions will be denoted by

Formulæ analogous to

will be obtained, and the use of the functions in symbolical solutions of certain differential equations exemplified. The connection of the functions with generalised Bessel-functions of order half an odd integer will be shown.

We show that the representations of the Cuntz $C^*$-algebras $\mathcal{O}_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work.

AMS 2000 Mathematics subject classification: Primary 46L45; 47A20; 46L60; 42C40; 81P15

A geometric hypothesis is presented under which the cohomology of a group G given by generators and defining relators can be computed in terms of a group H defined by a subpresentation. In the presence of this hypothesis, which is framed in terms of spherical pictures, one has that H is naturally embedded in G, and that the finite subgroups of G are determined by those of H. Practical criteria for the hypothesis to hold are given. The theory is applied to give simple proofs of results of Collins-Perraud and of Kanevskiĭ. In addition, we consider in detail the situation where G is obtained from H by adjoining a single new generator x and a single defining relator of the form xaxbxεc, where a, b, c ∈ H and |ε| = 1.

1. If we write

where a is arbitrary, but s is a positive integer, then Vandermonde's Theorem is

Divide by , and put a for q, s + γ for p.

1. Many investigations have been concerned with a squaro matrix P with non-negative coefficients (elements). It is remarkable that many interesting properties of P are determined by the set Σ of index pairs of positive (i.e. non-zero) coefficients of P, the actual values of these coefficients being irrelevant. Thus, for example, the number of characteristic roots equal in absolute value to the largest non-negative characteristic root p depends on Σ alone, if P is irreducible. If P is reducible, then Σ determines the standard forms of P (cf. § 3). The multiplicity of p depends on Σ, and on the set S of indices of those submatrices in the diagonal in a standard form of P which have p as a characteristic root. It has apparently not been considered before whether Σ and S also determine the elementary divisors associated with p. We shall show that, in general, the elementary divisors do not depend on these sets alone, but that necessary and sufficient conditions may be found in terms of Σ and S (a) for the elementary divisors associated with p to be simple, and (b) that there is only one elementary divisor associated with p.

We consider the Sturm–Liouville equation

$$ -y''+qy=\lambda y\quad\text{on }[0,1], $$

subject to the boundary conditions

$$ y(0)\cos\alpha=y'(0)\sin\alpha,\quad\alpha\in[0,\pi), $$

and

$$\frac{y'}{y}(1)=a\lambda+b-\sum_{k=1}^N\frac{b_k}{\lambda-c_k}. $$

Topics treated include existence and asymptotics of eigenvalues, oscillation of eigenfunctions, and transformations between such problems.

AMS 2000 Mathematics subject classification: Primary 34B24; 34L20

The object of this paper is to give a generalisation to vector valued functions of the classical mean value theorem of differential calculus. In that theorem we have

for some c in the open interval a, b when f is a real valued function which is continuous on the closed interval a, b and differentiable on the open interval. The counterpart to (1) when f has values in an n-dimensional vector space turns out to be

where cka, b, 0k, and .

An integral equation of the first kind, with kernel involving a hypergeometric function, is discussed. Conditions sufficient for uniqueness of solutions are given, then conditions necessary for existence of solutions. Conditions sufficient for existence of solutions, only a little stricter than the necessary conditions, are given; and with them two distinct forms of explicit solution. These two forms are associated at first with different ranges of the parameters, but their validity in the complementary ranges is also discussed. Before giving the existence theory a digression is made on a subsidiary integral equation.

Corresponding theorems for another integral equation resembling the main one are deduced from some of the previous theorems. Two more equations of similar form, less closely related, will be considered in another paper. Special cases of some of these four integral equations have been considered recently by Erdélyi, Higgins, Wimp and others.

In this paper we will study the properties of a natural partial order which may bedefined on an arbitrary abundant semigroup: in the case of regular semigroups werecapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1, 2] and further studied by Chacron [7]. Burmistroviˇ [6] investigated Sussman's order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5].