By a tournament we mean the outcome of a round-robin tournament in which there are no draws. Such a tournament may be represented by a graph in which the n players are represented by vertices labelled 1, 2, …, n, and the outcomes of the games are represented by directed edges so that every pair of vertices is joined by one directed edge. We call such a graph a complete directed graph. One can also represent such a tournament by an nXn matrix T=(tij) in which tij is 1 if i beats j, and 0 otherwise, so that T is a (0, 1) matrix with tij + tji = 1 for i≠j and (by definition) tii = 0.

A q-analogue of the orthogonality property of the Bessel functions on the zeros is obtained in terms of a q-integral.

The ${{Q}_{p}}$ spaces coincide with the Bloch space for $p\,>\,1$ and are subspaces of $\text{BMOA}$ for $0\,<\,p\,\le \,1$. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into ${{Q}_{p}}$, in particular from the Bloch space into $\text{BMOA}$.

Two types of pseudo-bases, σ-disjoint and σ-discrète, are utilized in this note. In the next section, we show that a first countable Hausdorff space has a σ-disjoint pseudo-base if and only if it has a dense metrizable subspace. This result implies that many first countable spaces have dense metrizable subspaces. In section 3, we show that if X is a Hausdorff space that either is quasi-developable or has a base of countable order, then X has a dense metrizable subspace if and only if it has a dense metrizable Gδ subspace. We give an example to show that the conclusion of this theorem is false for semi-metrizable spaces. Finally, in the last section, we investigate when a quasi-developable (resp. semi-metrizable) space can be embedded as a dense subspace of a quasi-developable (resp. semi-metrizable) Baire space.

This paper deals with the projective objects in the category of all Z-frames, where the latter is a common generalization of different types of frames. The main result obtained here is that a Z-frame is E-projective if and only if it is stably Z-continuous, for a naturally arising collection E of morphisms.

We present a new approach to noncommutative real algebraic geometry based on the representation theory of ${{C}^{*}}$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative ${{C}^{*}}$-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

How many spheres of given diameter can be packed in a cube of given size? Or: What is the maximum diameter of k identical spheres if they can be packed in a cube of given size? These questions are obviously equivalent to the following problem:

Let d(Pi, Pj) denote the distance between the points Pi and Pj, and Γk the set of all configurations of k points Pi (1 ≤ i < j ≤ k) in a closed unit cube C.

A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer ${{C}_{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $S$ of the Schrödinger (Lie) algebra. The simple ${{C}_{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $S$ (over the centre). It is proved that some (prime) factor algebras of $S$ and ${{C}_{S}}(H)$ are tensor homological$/$Krull minimal.

This paper discusses the connection between the local cohomology modules and the Serre classes of $R$-modules. This connection has provided a common language for expressing some results regarding the local cohomology $R$-modules that have appeared in different papers.

We prove that a Minkowski plane is Euclidean if and only if Busemann's or Glogovskij's definitions of angular bisectors coincide with a bisector defined by an angular measure in the sense of Brass. In addition, bisectors defined by the area measure coincide with bisectors defined by the circumference (arc length) measure if and only if the unit circle is an equiframed curve.

A divisor d of a positive integer n is a unitary divisor if d and n/d are relatively prime. An integer is said to be unitary perfect if it equals the sum of its proper unitary divisors. Subbarao and Warren [2] gave the first four unitary perfect numbers: 6, 60, 90 and 87360. In 1969,1 reported [3] that

is also unitary perfect. The purpose of this paper is to show that this last number, which for brevity we denote by W, is indeed the next unitary perfect number after 87360.

We consider ordinary graphs, that is, finite, undirected graphs with no loops or multiple lines. The product (also called cartesian product [4]) G1 × G2 of two graphs G1 and G2 with point sets V1 and V2, respectively, has the cartesian product V1 × V2 as its set of points. Two points (u1, u2) and (v1, v2) are adjacent if u1 = v1 and u2 is adjacent with v2 or u2 = v2 and u1 is adjacent with v1.

The study of general multiplier theorems (Kuhn-Tucker Conditions) for constrained optimization problems has led to extensions of the notion of a differentiable arc. Abadie [1], Varaiya [10], Guignard [5], Zlobec [11] and Massam [12] investigated the so called cone of tangent vectors to a point in a set for optimization purposes.

It is a classical conjecture of E. Artin that any integer a > 1 which is not a perfect square generates the co-prime residue classes (mod ρ) for infinitely many primes ρ. Let E be the set of a > 1, a not a perfect square, for which Artin's conjecture is false. Set E(x) = card(e ∊ E: e ≤ x). We prove that E(x) = 0(log6 x) and that the number of prime numbers in E is at most 6.

A space has the shrinking property if, for every open cover {Va | a ∈ A}, there is an open cover {Wa | a ∈ A} with for each a ∈ A.lt is strangely difficult to find an example of a normal space without the shrinking property. It is proved here that any ∑-product of metric spaces has the shrinking property.

Let D be an integral domain, and let X be an analytic indeterminate. As usual, if I is an ideal of D, set It = ∪{JV = (J-1)-1 | J is a nonzero finitely generated subideal of I}; this defines the t-operation, a particularly useful star-operation on D. We discuss the t-operation on R[[X]], paying particular attention to the relation between t- dim(R) and t- dim(R[[X]]). We show that if P is a t-prime of R, then P[[X]] contains a t-prime which contracts to P in R, and we note that this does not quite suffice to show that t- dim(R[[X]]) ≥ t- dim(R) in general. If R is Noetherian, it is easy to see that t- dim(R[[X]]) ≥ t- dim(R), and we show that we have equality in the case of t-dimension 1. We also observe that if V is a valuation domain, then t-dim(V[[X]]) ≥ t- dim(V), and we give examples to show that the inequality can be strict. Finally, we prove that if V is a finite-dimensional valuation domain with maximal ideal M, then MV[[X]] is a maximal t-ideal of V[[X]].

We prove the following theorem. Suppose that $F\,=\,\left( {{f}_{1}},\,{{f}_{2}} \right)$ is a 2-dimensional, vector-valued modular form on $\text{S}{{\text{L}}_{2}}\left( \mathbb{Z} \right)$ whose component functions ${{f}_{1}}$, ${{f}_{2}}$ have rational Fourier coefficients with bounded denominators. Then ${{f}_{1}}$ and ${{f}_{2}}$ are classical modular forms on a congruence subgroup of the modular group.

G(n) denotes a graph of n vertices and Ḡ(n) denotes its complementary graph. In a complete graph every two distinct vertices are joined by an edge. Let Ck(G(n)) denote the number of complete subgraphs of k vertices contained in G(n). Recently it was proved [1] that for every k

1

where the minimum is over all graphs G(n).

Every continuous mapping T = (T1,. . ., Tn): holomorphic in Bn has a fixed point.

A well-known result, due to Ostrowski, states that if ${{\left\| P-Q \right\|}_{2}}\,<\,\varepsilon $, then the roots $({{x}_{j}})$ of $P$ and $({{y}_{j}})$ of $Q$ satisfy $\left| {{x}_{j}}\,-\,{{y}_{j}} \right|\,\le \,Cn{{\varepsilon }^{1/n}}$, where $n$ is the degree of $P$ and $Q$. Though there are cases where this estimate is sharp, it can still be made more precise in general, in two ways: first by using Bombieri’s norm instead of the classical ${{l}_{1}}$ or ${{l}_{2}}$ norms, and second by taking into account the multiplicity of each root. For instance, if $x$ is a simple root of $P$, we show that $\left| x\,-\,y \right|\,<\,C\varepsilon $ instead of ${{\varepsilon }^{1/n}}$. The proof uses the properties of Bombieri’s scalar product andWalsh Contraction Principle.