McCarthy [4] generalized a necessary and sufficient condition for an arithmetic function to be totally multiplicative to the incidence algebra on a partially ordered set. Several equivalent conditions for an arithmetic function to be totally multiplicative are known [1], [2]. In this paper we generalize several of these (and some apparently new ones) to the regular convolution rings of Narkiewicz [5]. We also investigate the prime factorization of arithmetic functions in a certain subring of some of these regular convolution rings.

In this section we continue to study the indecomposable matrix pairs adopting the same notation as in part I of this paper.

If the matrix A is regular and if it is symmetric or anti-symmetric such that

3.1

and if the matrix pair (X, A) is indecomposable then the corresponding representation space either is indecomposable or it is the direct sum of two indecomposable invariant subspaces. These are operator isomorphic if and only if the minimal polynomial mX of X is equal to (x-δ)μ where

3.2

at any rate there is even a decomposition of the representation space into the direct sum of two isotropic indecomposable invariant subspaces provided the characteristic of F is not 2.

Let X be a closed convex subset of a Banach space and let T: X → X be a nonexpansive mapping, i.e.

In a recent paper, Murray Eden [5] generalized the simple identity for the Eulerian product,

1.1

and obtained the following infinite family of identities:

For A= 1,2, 3,…, let

1.2

where we assume throughout that |x| < 1, empty products equal unity and empty sums equal zero; then

1.3

As Eden noted, Fh(b;x) is the generating function of ph(m, n) which denotes the number of partitions of n into m parts, in which the largest part appears exactly h times and all other parts are distinct:

It is well known that the determination of a (non-isotropic) curve in the euclidean 3-space with given curvature κ(S) and torsion τ(s), where s represents the arc-length, depends upon the integration of a Riccati equation; and that this can be performed only if a particular integral of the equation is known.

The Jacobi coefficients $c_{j}^{\ell }\left( \alpha ,\,\beta \right)\,\left( 1\,\le \,j\,\le \,\ell ,\,\alpha ,\,\beta \,>\,-1 \right)$ are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the Jacobi polynomials $P_{k}^{\left( \alpha ,\,\beta \right)}\,\left( k\,\ge \,0,\,\alpha ,\,\beta \,>\,-1 \right)$ into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed, and a direct trace interpretation of the Maclaurin coefficients is presented.

This paper is devoted to the study of Wirtinger-type inequalities for the Lebesgue $\Delta$-integral on an arbitrary time scale $\mathbb{T}$. We prove a general inequality for a class of absolutely continuous functions on closed subintervals of an adequate subset of $\mathbb{T}$. By using this expression and by assuming that $\mathbb{T}$ is bounded, we deduce that a general inequality is valid for every absolutely continuous function on $\mathbb{T}$ such that its $\Delta$-derivative belongs to $L_{\Delta }^{2}\,([a,\,b)\,\cap \,\mathbb{T})$ and at most it vanishes on the boundary of $\mathbb{T}$.

Suppose throughout that {pn} is a sequence of non-negative numbers with p0>0, that

and that {sn} is a sequence of real numbers.

It is shown that for totally real number fields the Main Conjecture in Iwasawa-Theory for p = 2 proposed by Fédérer implies the 2-primary part of the Birch-Tate-Conjecture in analogy with the case p odd.

Let R be a ring which possesses a unit element, a Lie ideal U ⊄ Z, and a derivation d such that d(U) ≠ 0 and d(u) is 0 or invertible, for all u ∈ U. We prove that R must be either a division ring D or D2, the 2 X 2 matrices over a division ring unless d is not inner, R is not semiprime, and either 2R or 3R is 0. We also examine for which division rings D, D2 can possess such a derivation and study when this derivation must be inner.

We classify the left-invariant metrics with nonnegative sectional curvature on $\text{SO}\left( 3 \right)$ and $U\left( 2 \right)$.

We investigate the bi-orderability of two-bridge knot groups and the groups of knots with 12 or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with 12 or fewer crossings (of which there are 2977), previous theorems were only able to determine bi-orderability of 499 of the corresponding knot groups. With our methods we are able to deal with 191 more.

Brouwer's celebrated fixed point theorem states that every map of a closed n-cell into itself has a fixed point.

A similar theorem is here proved for coincidences between a pair of maps (f, g): In→In, where I denotes a closed n-cell (i.e. a homeomorph of the n-ball) and a coincidence is a point x∊In for which f(x)=g(x). That two arbitrary maps (f, g): In→In need not have a coincidence is shown by the pair f:In→y0, g:In→y1, where y0, y1∊In and y0≠y1. More generally, one can immediately construct a map g so that (f, g) is coincidence free if f is not surjective.

In this paper we propose a new technical tool for analyzing representations of Hilbert ${{C}^{*}}$- product systems. Using this tool, we give a new proof that every doubly commuting representation over ${{\mathbb{N}}^{k}}$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb{R}_{+}^{k}$.

Let M be a free module of rank n over a commutative ring R with unit and let Σn denote the symmetric group acting on a fixed basis of M in the usual way. Let Mm denote the direct sum of m copies of M and let S be the symmetric ring of Mm over R. Then each element of Σn acts diagonally on Mm and consequently on S; denote by Xn the subgroup of Gl(Mm) so defined. The ring of invariants, SΣn, is called the ring of vector invariants by H. Weyl [ 3, Chapter II, p. 27] when R = Q. In this paper a set of generators valid over any ring R is given. This set of generators is somewhat larger than Weyl's. It is interesting to note that, over the integers, his algebra and SΣn have the same Hilbert-Poincaré series, are equal after tensoring with the rationals, and have the same fraction fields, although they are not equal.

We prove Wise's $W$-cycles conjecture. Consider a compact graph $\Gamma '$ immersing into another graph $\Gamma $. For any immersed cycle $\Lambda :{{S}^{1}}\to \Gamma $, we consider the map $\Lambda '$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma '$. Unless $\Lambda '$ is reducible, the degree of the covering map $\mathbb{S}\to {{S}^{1}}$ is bounded above by minus the Euler characteristic of $\Gamma '$. As a corollary, any finitely generated subgroup of a one-relator group has a finitely generated Schur multiplier.

We use the homological perturbation lemma to give an algebraic proof of the cyclic Eilenberg-Zilber theorem for cylindrical modules.

Any term or symbol undefined in this paper is defined in [5]. For graphs F and G, G > F means G contains a subgraph isomorphic to F and E(G) denotes the edge set of G. If E ⊆ E(G), 〈E〉 is the subgraph of G whose edge set is E and whose vertex set is that subset of vertices of G which are incident with edges in E.

It is well known that a symmetric and transitive relation on a set is reflexive wherever it is defined. In this note we show that a converse is true for homomorphic relations in certain classes of algebras.

Consider a class of similar algebras which contains the sub-algebras and quotient algebras of each of its members. Assume also that the direct product A x B of each pair A, B in is also an algebra belonging to . The algebras of , being similar, have the same set of operations. We observe that other operations, called compound operations, may be obtained by composition from the assigned operations.