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.

We give a characterization of polynomials with rational coefficients which take integral values on the prime numbers: to test a polynomial of degree n, it is enough to consider its values on the integers from 1 to 2n —1.

We prove that the neighborhood complex $\text{N}\left( G \right)$, the box complex $\text{B}\left( G \right)$, the homomorphism complex $\text{Hom}\left( {{K}_{2}},\,G \right)$ and the Lovász complex $\text{L}\left( G \right)$ have the same simple ${{\mathbb{Z}}_{2}}$ -homotopy type in the sense of Whitehead. We show that these graph complexes are simple ${{\mathbb{Z}}_{2}}$ -universal.

This article discusses our recent proof that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion called integrated super-Brownian excursion (ISE), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof, whose details will appear elsewhere, uses the lace expansion. Here, a related but simpler analysis is applied to show that the scaling limit of a mean-field theory is ISE, in all dimensions. A connection is drawn between ISE and certain generating functions and critical exponents, which may be useful for the study of high-dimensional percolation models at the critical point.

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial $p$-group over a field of characteristic $p$, the ring of vector invariants of $m$ copies of that representation is not Cohen-Macaulay for $m\,\ge \,3$. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group $G$ and a faithful representation of dimension $n$ with $n\,>\,1$, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to $n(\left| G \right|\,-\,1)$. If the ring of invariants is a hypersurface, the upper bound can be improved to $\left| G \right|$.

The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations ${{({{X}_{1}}(u))}^{2}}\,+\,{{({{X}_{2}}(u))}^{2}}\,=\,1$ as special cases, where ${{X}_{1}}\,=\,{{p}_{1}}\partial /\partial {{z}_{1}}\,+\,{{p}_{2}}\partial /\partial {{z}_{2}},\,{{X}_{2}}\,=\,{{p}_{3}}\partial /\partial {{z}_{1}}\,+\,{{p}_{4}}\partial /\partial {{z}_{2}}$ are linearly independent operators with ${{p}_{j}}$ being arbitrary polynomials in ${{\mathbf{C}}^{2}}$.

Put

1

For the Appell function F4 defined by [3, p. 224]

2

Saxena [5, p. 216] proved a reduction formula in the form

3

where

It may be of interest to point out that formula (3) is not correct. Indeed we first establish the following general result.

An oscillation criterion for an even order equation: x(n) + q(t)ƒ(x(t)), x[g(t)]) = 0 is provided. This criterion is an extension of a result established by Yeh for the second order equation ẍ + q(t)ƒ(x(t)), x[g(t)]) = 0.

Bruns' Theorem states that the classical integrals of the gravitational three-body problem generate all algebraic integrals. We show that the first step in his proof, together with Ziglin's non-integrability criterion for complex systems, can be used to prove the non-existence of energy independent algebraic integrals in certain real analytic systems. We also show that this aspect of Bruns' argument is purely algebraic: We offer a proof based on elementary differential algebraic methods.

In [ 2 ] P. Fletcher proved that a finite topological space has a unique compatible quasi-uniformity; C. Barnhill and P. Fletcher showed in [1] that a topological space (X, ), with finite, has a unique compatible quasiuniformity. In this note we give some necessary conditions for unique quasiuniformizability.

We prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure.

In 1857, Kronecker [10] showed that if θ1,…, θn are the roots of the polynomial P(z)= zn|cn-1+ … + cn, where c1, …, cn are integers with cn≠0, and if |θ1| ≤ 1, …, |θ1| ≤1, then θ1, …, θn are roots of unity. The proof is short and ingenious: Consider the polynomials Pm(z) whose roots are The condition on the size of the roots and the fact that the ci are integers implies that there can only be a finite number of different Pm. Thus two distinct powers of each root must coincide and this means that each root is a root of unity.

Let H be a complex Hilbert space and let K = H ⊕ H. Then K can be identified with the set of all column matrices

equipped with componentwise addition and scalar multiplication and the scalar product

In this note we use the method of generating functions to show that there is a unique way of splitting the non-negative integers into two classes in such a way that the sums of pairs of distinct integers will be the same (with same multiplicities) for both classes. We prove a similar theorem for products of positive integers and consider some related problems.

On considère un opérateur U(x) qui transforme l'espace euclidien En en soimême, en général discontinu, et on étudie la convergence d'un processus itératif de la forme xp+1 = xp-μU(x) (μ est une constante numérique positive). Processus de ce type, avec U(x) discontinu, se rencontrent par exemple à l'algorithme de relaxation pour la résolution des systémes d'inéquations [1], [2], de même qu'au calcul des polynômes de la meilleure approximation sur un ensemble fini de points [3].

By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations — the d'Alembert equation, the Wilson equation, and the d'Alembert long equation — on compact groups.

A fundamental construction of a category ζ((2), Appendice) is a triple (S, p, k), where S is a functor from ζ to itself and 2 where p:S2→S and k:1ζ→S are natural transformations such that

We describe two methods to combine sets of lines of different 2-dimensional projective planes into line sets of new 2-dimensional projective planes. Using these methods we describe several ways in which sets of circles of different 2-dimensional circle planes can be combined into circle sets of new 2-dimensional circle planes.

It is an old problem to find how a collection of congruent plane figures should be arranged without overlapping to cover the largest possible fraction of the plane or some region of the plane. If similar figures of arbitrary different sizes are permitted, Vitali's theorem ([7] p. 109) guarantees that packings which cover almost all points are possible. It is natural to study the diameters of figures used in such a packing and we will investigate this for the case of a closed disk packed with smaller open disks.

In this paper we define lower, upper, and symmetric completeness and discuss closure of the sets in products and direct sums. In particular, we introduce suitable bases for these topologies, which leads us to investigate completeness of the direct sum and its components. Some results obtained about $X$-topologies and polars of the neighborhoods.