In 1892, Sylvester [7] proved that in the set of integers n, n+l,…, n+k—1, n> k > 1, there is a number containing a prime divisor greater than k. This theorem was rediscovered, in 1929, by Schur [6]. More recent results include an elementary proof by Erdös [1] and a proof of the following theorem by Faulkner [2]: Let pk be the least prime ≥2k; if n≥pk then has a prime divisor ≥pk with the exceptions and In that paper the author uses some deep results of Rosser and Schoenfeld [5] on the distribution of primes.

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.

Given an essentially unitary contraction and an arbitrary unitary dilation of it, there is a naturally associated spectral flow that is shown to be equal to the index of the operator. This result is interpreted in terms of the K-theory of an associated mapping cone. It is then extended to connect Z2 indices of odd symmetric Fredholm operators to a Z2-valued spectral flow.

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.

Suppose is a sequence of functions on ℝn with Δfk = fk+1 (where Δ is the Laplacian) that satisfies the growth condition: |fk(x)| ≤ Mk{1 + |x|)a where a ≥ 0 and the constants have sublinear growth Then Δf0 = —f0- This characterizes eigenfunctions f of Δ with polynomial growth in terms of the size of the powers Δkf, —∞ < k < ∞. It also generalizes results of Roe (where a = 0, Mk = M, and n = 1 ) and Strichartz (where a = 0, Mk = M for n). The analogue holds for formally self-adjoint constant coefficient linear partial differential operators on ℝn.

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.

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.

The proximal subgradient formula is a refinement due to Rockafellar of Clarke's fundamental proximal normal formula. It expresses Clarke's generalized gradient of a lower semicontinuous function in terms of analytically simpler proximal subgradients. We use the infinite-dimensional proximal normal formula recently given by Borwein and Strojwas to derive a new version of the proximal subgradient formula in a reflexive Banach space X with Frechet differentiable and locally uniformly convex norm. Our result improves on the one given by Borwein and Strojwas by referring only to the given norm on X.

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

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.

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 $\mathbf{A}$ 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

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

We use the Carleson measure-embedding theorem for weighted Bergman spaces to characterize the positive Borel measures $\unicode[STIX]{x1D707}$ on the unit disc such that certain analytic function spaces of Dirichlet type are embedded (compactly embedded) in certain tent spaces associated with a measure $\unicode[STIX]{x1D707}$. We apply these results to study Volterra operators and multipliers acting on the mentioned spaces of Dirichlet type.

We give a “wall-crossing” formula for computing the topology of the moduli space of a closed $n$-gon linkage on ${{\mathbb{S}}^{2}}$. We do this by determining the Morse theory of the function ${{\rho }_{n}}$ on the moduli space of $n$-gon linkages which is given by the length of the last side—the length of the last side is allowed to vary, the first $\left( n\,-\,1 \right)$ side-lengths are fixed. We obtain a Morse function on the $\left( n\,-\,2 \right)$-torus with level sets moduli spaces of $n$-gon linkages. The critical points of ${{\rho }_{n}}$ are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of ${{\rho }_{n}}$ at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

Proofs of commutativity theorems for general rings usually employ the Jacobson structure theory; however, alternative approaches to the "xn = x theorem" [ l, 2] suggest that the power of the Jacobson theory is not required. In this note we prove two commutativity theorems of Herstein in an elementary way. Both proofs involve establishing first that the rings under consideration are duo-rings - rings in which every one-sided ideal is two-sided.

It is known that if f is a real valued function on a rational vector space V, δ > 0,

1

and if f is unbounded then f(x + y) = f(x)f(y) for all x, y ∊ V. In response to a problem of E. Lukacs, in this paper we study the bounded solutions of (1). For example, it is shown that if f is a bounded solution of (1) then - δ ≤ f(x) ≤ (1 + (1 + 4δ)1/2)/2 for all x ∊ V and these bounds are optimal.

Let X and Y be Banach spaces, L(X, Y) the space of bounded linear operators from X to Y and C(X, Y) its subspace of the compact operators. A sequence {Ti} in C(X, Y) is said to be an unconditional compact expansion of T ∈ L (X, Y) if ∑ Tix converges unconditionally to Tx for every x ∈ X. We prove: (1) If there exists a non-compact T ∈ L(X, Y) admitting an unconditional compact expansion then C(X, Y) is not complemented in L(X, Y), and (2) Let X and Y be classical Banach spaces (i.e. spaces whose duals are some LP(μ) spaces) then either L(X, Y) = C(X, Y) or C(X, Y) is not complemented in L(X, Y).

In this article, we give an explicit formula to compute the non abelian twisted sign-determined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.

By an r-graph G we mean a finite set V(G) of elements called vertices and a set E(G) of some of the r-subsets of V(G) called edges. This paper defines certain colour classes of r-graphs which connect the material of a variety of recent graph theoretic literature in that many existing results may be reformulated as structural properties of the classes for some special cases of r-graphs. It is shown that the concepts of Ramsey Numbers, chromatic number and index may be defined in terms of these classes. These concepts and some of their properties are generalized. The final subsection compares two existing bounds for the chromatic number of a graph.

A sequence of non-negative numbers, μ0 μ1 … μn, is called completely monotonic [5, p. 108] if (-1)n Δn μk ≧ 0 for n, k = 0, 1, 2, …. Such sequences occur in many connexions, such as the Hausdorff moment problem and Hausdorff summability [1, Chapter XI, 5, Chapters III and IV].

It is natural to inquire as to the circumstances under which the inequality "≧" above can be strengthened to ">". As it happens, this can be done always, except for sequences all of whose terms past the first are identical. The formal statement follows. (In it, as above, Δnμk is the n-th forward difference, i. e. Δ0μk = μk; Δnμk = Δn-1 μk+1 - Δn-1μk.)