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.

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

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.

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.

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).

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.

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.)

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.

In the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel–Darboux formulas are presented for the first time.

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.

A hypergraph is a subtree of a tree (SOFT) hypergraph if there exists a tree T such that X=V(T) and for each there is a subtree Ti of T such that Ei = V(Ti). It is shown that H is a SOFT hypergraph if and only if has the Helly property and , the intersection graph of is chordal. Results of Berge and Gavril have previously shown these to be necessary conditions.

Motivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complexmanifold with an almost complex structure. We describe various properties of this structure.

Let R, r, s represent respectively the circumradius, the inradius and the semiperimeter of a triangle with sides a, b, c. Let f(R, r) and F(R, r) be homogeneous real functions. Let q(R, r) and Q(R, r) be real quadratic forms. The latter functions are thus a special case of the former. Our main result is to derive the strongest possible inequalities of the form

1

with equality only for the equilateral triangle.

We prove that for every $k\,>\,1$, there exist $k$-fold coverings of the plane (i) with strips, (ii) with axis-parallel rectangles, and (iii) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two coverings. We also construct for every $k\,>\,1$ a set of points $P$ and a family of disks $D$ in the plane, each containing at least $k$ elements of $P$, such that, no matter how we color the points of $P$ with two colors, there exists a disk $D\,\in \,D$ all of whose points are of the same color.

This paper is concerned with the following elliptic system of Hamiltonian type

$$\left\{ \begin{align} & -\Delta u+V\left( x \right)u={{W}_{v}}\left( x,u,v \right),\,x\in {{\mathbb{R}}^{N}}, \\ & -\Delta v+V\left( x \right)v={{W}_{u}}\left( x,u,v \right),\,x\in {{\mathbb{R}}^{N}}, \\ & u,v\in {{H}^{1}}\left( {{\mathbb{R}}^{N}} \right), \\ \end{align} \right.$$

where the potential $V$ is periodic and $0$ lies in a gap of the spectrum of $-\Delta +V,W\left( x,u,v \right)$ is periodic in $x$ and superlinear in $u$ and $v$ at infinity. We develop a direct approach to finding ground state solutions of Nehari–Pankov type for the above system. Our method is especially applicable to the case when

$$W\left( x,u,v \right)=\sum\limits_{i=1}^{k}{\,\int _{0}^{\left| \alpha iu+\beta iv \right|}\,{{g}_{i}}\left( x,\,t \right)t\text{d}t+\sum\limits_{j=1}^{l}{\int_{0}^{\sqrt{{{u}^{2}}+2{{b}_{juv+aj{{v}^{2}}}}}}{{{h}_{j}}\left( x,t \right)tdt,}}}$$

where ${{\alpha }_{i}},{{\beta }_{i}},{{a}_{j}},{{b}_{j}}\in \mathbb{R}$ with $\alpha _{i}^{2}+\beta _{i}^{2}\ne 0$, and ${{a}_{j}}>b_{j}^{2},{{g}_{i}}\left( x,t \right)$ and ${{h}_{j}}\left( x,t \right)$ are nondecreasing in $t\in {{\mathbb{R}}^{+}}$ for every $x\in {{\mathbb{R}}^{N}}$ and ${{g}_{i}}\left( x,0 \right)={{h}_{j}}\left( x,0 \right)=0$.

A generalization of Schmüdgen’s Positivstellensatz is given which holds for any basic closed semialgebraic set in ${{\mathbb{R}}^{n}}$ (compact or not). The proof is an extension of Wörmann’s proof.

In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, ${{\overline{M}}_{0,0}}\left( {{\mathbb{P}}^{r}},\,d \right)$ , stabilize when $r\,\ge \,d$. We give a complete characterization of the effective divisors on ${{\overline{M}}_{0,0}}\left( {{\mathbb{P}}^{d}},\,d \right)$ . They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

This paper deals with the notion of equational compactness and related concepts in the special case of G-sets for an arbitrary group G. It provides characterizations of pure extensions, pure-essential extensions, and equational compactness in terms of the stability groups of a G-set, proves the general existence of equationally compact hulls, and gives an explicit description of these. Further, it establishes, among other results, that all G-sets are equationally compact iff all subgroups of the group G are finitely generated, that every equationally compact G-set is a retract of a topologically compact one, and that for free groups G with infinite basis there are homogeneous G-sets which are not equationally compact.

We consider here the nonlinear differential equation

(1.1)

where x ∊l= [a, ∞). We will make the following assumptions

1. (A) f is continuous on [a, ∞) × Rn,

2. (B) solutions of initial value problems (I.V.P.'s) are unique and extend throughout [a, ∞).

3. (C)

Let f(n) denote the number of dissections of a regular n-gon into n-2 triangles by n-3 non-intersecting diagonals. It is known that

and that

1

for n = 3, 4, . . . , where f(2) = 1 by definition. (For pertinent references on this and related problems see, e. g. , Motzkin [2].) The object of this note is to obtain a simple expression for g(n) , the number of such dissections remaining when those which differ only by a rotation, reflection, or both are not considered as being different. For convenience we shall let g(2) = 1 and f(k) = 0 when k is not an integer.