In this paper weak Perron integrals are characterized as n-dimensional interval functions F which are additive, differentiable almost everywhere in the weak sense and which satisfy a new continuity condition concerning the singular set. Before, only one-dimensional Perron integrals were characterized by the theorem of Hake- Alexendrov-Looman, and analogous results for strong Perron integrals (which are best analyzed, but more restrictive) are not available in higher dimensions yet. In order to formulate our continuity condition we introduce an outer measure μ by means of a new weak variation of F which is required to vanish on all null sets. The same condition is also necessary and sufficient for the integral of the weak derivative to yield the original interval function. This “fundamental theorem” is split into two fundamental inequalities of very general nature which contain additional singular terms involving our variation. These inequalities are very useful also for Lebesgue integrals.

The “k-dense” subgraphs of a connected graph G are connected and contain neighbours of all but at most k-1 points. We consider necessary and sufficient conditions that a point be in Γk, the union of the minimal k-dense subgraphs. It is shown that Γk contains all the [m, k]-isthmuses” and [m, k]-articulators“— minimal subgraphs which disconnect the graph into at least k + 1 disjoint graphs—and that an [m, k]-isthmus or [m, k]-articulator of Γk disconnects G. We define “central points,” “degree” of a point, and “chromatic number” and examine the relationship of these concepts to connectivity. Many theorems contain theorems previously proved (1) as special cases.

Let $D\subset \mathbb{C}$ be a domain with $0\in D$. For $R>0$, let $\widehat{\unicode[STIX]{x1D714}}_{D}(R)$ denote the harmonic measure of $D\cap \{|z|=R\}$ at $0$ with respect to the domain $D\cap \{|z|<R\}$ and let $\unicode[STIX]{x1D714}_{D}(R)$ denote the harmonic measure of $\unicode[STIX]{x2202}D\cap \{|z|\geqslant R\}$ at $0$ with respect to $D$. The behavior of the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ near $\infty$ determines (in some sense) how large $D$ is. However, it is not known whether the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ always have the same behavior when $R$ tends to $\infty$. Obviously, $\unicode[STIX]{x1D714}_{D}(R)\leqslant \widehat{\unicode[STIX]{x1D714}}_{D}(R)$ for every $R>0$. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0\in D$ and all $R>0$,

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{D}(R)\geqslant C\widehat{\unicode[STIX]{x1D714}}_{D}(R)?\end{eqnarray}$$

$D$

In this paper, we shall show that if is a nilpotent [5] group and if M, a positive integer, is a uniform bound on the number of conjugates that any element of may have, then there exist “large” integers n for which x → xn is a central endomorphism of . If is not necessarily nilpotent, if the above condition on the conjugates is retained, and if we can find a member of the lower central series [1], every element of which lies in some member of the ascending central series, then we shall show that every non-unity element of the “high” derivatives has finite order.

Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.

In this paper I establish the following bounding formula for the integral of a function of n variables:

(1.1)

In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information about the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.

In the study of direct differential geometry, families of oriented arcs and curves have been employed extensively to define the differentiability of an arc at a point in various kinds of planes; cf. [2]. In [6], P. Scherk used lines in the projective plane; in [3] and [4], N. D. Lane and P. Scherk used circles in the conformai plane; conic-sections in the projective plane were employed in [5] and [7] by N. D. Lane and K. D. Singh; in [1], M. Gupta and N. D. Lane used the graphs of polynomials of degree at most n in the affine plane.

This paper is devoted to the study of convergence and variation of operator spectra with respect to the distance G of Gokhburg and Markus [5] for subspaces and linear operators in a Banach space. We use the convention of Kato [7] and refer to convergence with respect to G as generalizedconvergence.

In the Euclidean plane a curve C has a one-parameter family of parallel involutes and a unique evolute C* which coincides with the locus of the centres of the osculating circles of C. If is parallel to C, C* is also the evolute of . We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.

We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.

In (5),I considered two-term conditions in π-exact couples, of which the exact couple of Federer (7) is an example. Let M(X, Y)be the space of all maps from X to Y with the compact-open topology. Our aim in this paper is to construct a π-exact couple , where Xis a finite-dimensional (in the sense of Lebesgue) metric space and , a certain (rather large) class of spaces. Specifically, is the class of all topological spaces Xwhich possess the following property (P).

(P) Let Y be a (possibly infinite) simplicial complex. There exists x0 ∈ X and y0 ∊ Y such that [X, x0]≃ [Y, y0].

In § 5 it will be seen that contains all CW complexes and all metric absolute neighbourhood retracts (ANR)s.

Soit M un facteur de type IIIλ 0 ≦ λ< 1, agissant sur un espace hilbertien à base dénombrable. Le propos de ce travail est l'étude des états normaux d'un tel facteur à équivalence unitaire près. L'espace quotient n'est pas séparé; l'espace séparé associé est obtenu comme quotient de l'espace des états normaux par la relation d'équivalence R dont les classes sont les fermetures des orbites précédentes.

Pour étudier cet espace, on définit un calcul fonctionnel sur M+* (espace des formes linéaires positives normales sur M) à valeurs dans M. Ce calcul fonctionnel est déterminé par la donnée d'une C* algèbre d'applications de M+ * dans M “continues” en un certain sens, notée C*(M).

Nous allons considérer ici un opérateur de Cauchy à deux variables Fuchsiennes de la forme: Puis nous allons chercher, sous certaines conditions d'hyperbolicité sur les racines caractéristiques Fuchsiennes, des solutions de l'équation 𝓟u = f, dans le cas o ù ƒ est régulière par rapport aux variables Fuchsiennes et analytique par rapport aux autres.

Baouendi et Goulaouic ont traité dans [1] ce même type d'opérateurs dans le cas d'une seule variable Fuchsienne. Dans [3] a été étudié le cas de plusieurs variables Fuchsiennes pour certains opérateurs de Goursat.

Throughout the paper, G will denote an additively written, but not always abelian, group of finite order n; and X = (xij) will denote a square matrix of order n with entries from G and whose rows and columns are numbered 0, 1, …, n − 1. We call X a cartesian array (afforded by G) if

(1.1) The sequence {−xmi + xki, i = 0,…, n – 1} contains all elements of G whenever k ≠ m.

By a theorem of Jungnickel (see Theorem 2.2 in [5]), the transpose of a cartesian array is also a cartesian array. We call G a cartesian group if there is a cartesian array X afforded by G. In this case, we also call (G, X) a cartesian pair.

When a set of axioms is laid down as the basis of any mathematical doctrine, it must be proved that this set never leads to a contradiction. In this note we turn the question around. A set of axioms is given and we wish to adjoin an axiom of a specified type. How far does the demand of non-contradiction limit the choice of the new axiom?

The subsemigroups of the projective group on the line that are described in this paper are those that can be generated by a pair of infinitesimal transformations. One denotes by G the connected component of the identity of this group; Theorem 1 gives a necessary and sufficient condition for a pair of infinitesimal transformations to generate a subsemigroup which is equal to G (and hence is actually a group). This condition is reformulated in a geometric manner in Theorem 1*.

Let be the Boolean algebra of all finite unions of subcells of the plane. Denote by εpthe algebra of all linear bounded transformations of Lp(— ∞, ∞) into itself. Suppose for a moment that p = 2, and let Rp be an involutive abelian subalgebra of εp if Rp is also a Banach space and if Tp ∈ Rp, then:

(i) The family of all homomorphic mappings of into the algebra Rp contains a member EPT such that

(1)

The matrix spaces under consideration are the four main types of irreducible bounded symmetric domains given by Cartan (5). Let z = (zjk) be a matrix of complex numbers, z' its transpose, z* its conjugate transpose and I = I(n) the identity matrix of order n. Then the first three types are defined by

(1)

where z is an n by m matrix (n ≤ m), a symmetric or a skew-symmetric matrix of order n (16). The fourth type is the set of complex spheres satisfying

(2)

where z is an n by 1 matrix. It is known that each of these domains possesses a distinguished boundary B which in the first three cases is given by

(3)

(In the case of skew symmetric matrices the distinguished boundary is given by (2) only if n is even.)

Let X be a normal projective variety of dimension n and G an abelian group of automorphisms such that all elements of $G\setminus \{\operatorname {id}\}$ are of positive entropy. Dinh and Sibony showed that G is actually free abelian of rank $\le n - 1$ . The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\operatorname {rank} G = n - 2$ .

Let A be a commutative real Banach algebra with unit, and MA its maximal ideal space. The existence of the Silov boundary SA for A was established in [5] by resorting to the complexification of A. We give here an intrinsic proof of this result which exhibits the close connection between the absolute values and the real parts of ‘functions’ in A (Theorem 1.3).