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.

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.

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.

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.

Let R denote the rectangle: |t-t0| ≤ a, | x-x0| ≤ b (a,b > 0) in the (t,x) plane and let f(t, x) be a function of two real variables t and x, defined and continuous on R. If I is the interval |t—t0| ≤ d with d = min(a,b/M), where M = max(|f(t, x)|, (t, x) ϵ R), then every solution x = x (t) of the differential equation x' = f(t, x) defined on I and which satisfies the initial condition x(t0) = x0, satisfies the integral equation

1.1

and conversely. In some cases, in order to prove the existence and uniqueness of the solutions of (1.1) on I, one forms the successive approximations

1.2

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.

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

In this paper, we discuss the solvability of the p-k-Hessian inequality $\sigma _{k}^{\frac 1k} ( \lambda ( D_{i} (|Du|^{p-2}$ $ D_{j}u ) ) ) \geq f(u)$ on the entire space $\mathbb {R}^{n}$ and provide a necessary and sufficient condition, which can be regarded as a generalized Keller–Osserman condition. Furthermore, we obtain the optimal regularity of solution.

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).

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 line-coloring of a graph G is an assignment of colors to the lines of G so that adjacent lines are colored differently; an n-line coloring uses n colors. The line-chromatic number χ'(G) is the smallest n for which G admits an n-line coloring.

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].

The purpose of the present note is to give a partial answer to a question raised by Professor Coxeter, namely, if an orthogonal transformation is expressed as a product of orthogonal involutions, how many involutions do we need? Our answer is partial because we are going to consider only non-degenerate symmetric bilinear forms of index 0 and fields of characteristic ≠2. Under these conditions we prove that any orthogonal transformation is the product of at most two orthogonal involutions, which implies that we can write any orthogonal transformation as the product of two involutions.

In [2], a construction for the lower radical class R∘ (η) with respect to a class η of rings was given as the union of an inductively defined ascending transfinite chain of classes of rings. It was shown there that this construction terminates, for associative rings, at ω∘, the first infinite ordinal, in the sense that if {ηα: α an ordinal} is the chain, then R∘ (η) =ηω∘. Also, examples of classes η for which R∘ (η) = η1, η2, η3 were given.

Let $v\,\ge \,k\,\ge \,1$ and $\lambda \,\ge \,0$ be integers. A block design$\text{BD}\left( v,\,k,\,\lambda \right)$ is a collection $\mathcal{A}$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $\lambda $ elements of $\mathcal{A}$. More generally, for a fixed simple graph $G$, a graph design$\text{GD}\left( v,\,G,\,\lambda \right)$ is a collection $\mathcal{A}$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $\lambda $ elements of $\mathcal{A}$. A famous result of Wilson says that for a fixed $ $ and $\lambda $, there exists a $\text{GD}\left( v,\,G,\,\lambda \right)$ for all sufficiently large $ $ satisfying certain necessary conditions. A block (graph) design as above is resolvable if $\mathcal{A}$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence in $v$ of resolvable $\text{BD}\left( v,\,k,\,\lambda \right)$, yet for over twenty years the analogous problem for resolvable $\text{GD}\left( v,\,G,\,\lambda \right)$ has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

Let $\mathbf{A}$ be a density matrix in ${{\mathbb{M}}_{m}}\,\otimes \,{{\mathbb{M}}_{n}}$. Audenaert [J. Math. Phys. 48(2007) 083507] proved an inequality for Schatten $p$-norms:

$$1\,+\,||\mathbf{A}|{{|}_{p}}\,\ge \,{{\left\| \text{T}{{\text{r}}_{1}}\,\mathbf{A} \right\|}_{p}}\,+\,||\text{T}{{\text{r}}_{2}}\,\mathbf{A}|{{|}_{p}},$$

where $\text{T}{{\text{r}}_{1}}$ and $\text{T}{{\text{r}}_{2}}$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality

$$1\,+\,\det \,\mathbf{A}\,\ge \,\det {{\left( \text{T}{{\text{r}}_{1}}\mathbf{A} \right)}^{m}}\,+\,\det {{\left( \text{T}{{\text{r}}_{2}}\mathbf{A} \right)}^{2}}.$$

A selfmapping f of a metric space (X, d) is nonexpansive (ε-nonexpansive) if d(f(x), f(y)) ≤ d(x, y) for all x, y ∊ X (respectively if d(x, y) < ε). In [1], M. Edelstein proved that a nonexpansive mapping f of En admits a fixed point provided the f-closure of En (i.e. the set of all points which are cluster points of {fn(x)} for some x) is nonempty. R. D. Holmes [2] considered commutative semigroups of selfmappings of a metric space and obtained fixed point theorems for such semigroups under certain contractivity conditions.

Let $R$ be a commutative ring with identity. The co-annihilating-ideal graph of $R$, denoted by ${{\mathcal{A}}_{R}}$, is a graph whose vertex set is the set of all non-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever $\text{Ann}\left( I \right)\,\cap \,\text{Ann}\left( J \right)\,=\,\left\{ 0 \right\}$. In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.

In following a suggestion of S. Chowla to apply a method of C. Hooley [3] to obtain an asymptotic formula for the sum ∑ r(n)r(n+a), where r(n) denotes the number of representations of n≤x

n as the sum of two squares and is positive integer, we have had to obtain non-trivial estimates for the error term in the asymptotic expansion of

1