The Yang-Feldman formalism vising the Feynman-like Green's functions is set up. The corresponding free fields have non-trivial commutation relations and contain information about the scattering. S-matrix elements are simply the matrix elements of anti-normal products of the field φF′(x). These are evaluated, and they give directly expressions used in the theory of causality and dispersion relations. It is possible to formulate field theory in a form in which the fields obey free field equations and the effects of interaction are contained in their commutation relations.

We investigate the functional quantization problem for stochastic processes with respect to Lp(IRd, μ)-norms, where μ is a fractal measure namely, μ is self-similar or a homogeneous Cantor measure. The derived functional quantization upper rate bounds are universal depending only on the mean-regularity index of the process and the quantization dimension of μ and as universal rates they are optimal. Furthermore, for arbitrary Borel probability measures μ we establish a (nonconstructive) link between the quantization errors of μ and the functional quantization errors of the process in the space Lp(IRd, μ).

The paper studies the motion of an electro-conductive non-viscous incompressible fluid in a generator of rectangular section having finite electrodes and the rest of its walls being insulating. It is assumed that in the electrodes region a homogeneous magnetic field is applied, whose orientation is normal to the plane of the motion. If the induced magnetic field is neglected, the current function ψ(x, y) and the electrical potential ϕ(x, y) satisfy a system of equations with partial derivatives of the second order, with distributions. Since the system depends upon the parameter λ = RM RH a method of successive approximations is developed, which permits one to obtain the exact solution of the problem. In the last paragraph the results are extended to the case of a generator with N pairs of symmetrical electrodes.

The aim of this paper is to answer a question of Coates and Greenberg: let $F$ be a commutative $m$-dimensional formal group over the ring of integers of a local field $k$, and let $K$ be an algebraic extension of $k$ with infinite ramification index. Denote by ${\cal M}_{\Mbar}$ the maximal ideal in the ring of integers of the separable closure of $K$. Suppose that the height of $F$ is greater than $m$. Does $H^1 (K, F ({\cal M}{}^m_{\Mbar}))=0$ imply that $K$ is deeply ramified? The answer is positive.

We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.

The problem of the transmission of power over a transmission line from an alternating current generator at one end to a synchronous motor at the other is easily solved when the line is uniform or made up of a small number of uniform line segments. The object of the present note is to apply the theory of integral equations to the general case of the non-uniform line. The solution obtained for the current at any point on the line in the form of a rational function of the frequency of the two alternators is of advantage when resonance effects are being considered; the latter effects are hard to trace in the case of the uniform line by the usual methods. It seems probable that numerical methods and computing devices can be developed for the solution of the integral equations given. The effect of armature inductance on the power limit of the system may then be more easily studied than at present.

It is a well known fact that every group G has a presentation of the form G=F/R, where F is a free group and R the kernel of the natural epimorphism from F onto G. Driven by the desire to obtain a similar presentation of the group of automorphisms Aut (G), we consider the subgroup Stab (R) ⊆ Aut (F) of those automorphisms of F that stabilize R and ask whether the natural homomorphism Stab (R) → Aut (G) is onto; if it is, we can try to determine its kernel.

Both parts of this task are usually quite hard. The former part received considerable attention in the past, whereas the more difficult part (determining the kernel) seemed unapproachable. Here we approach this problem for a class of one-relator groups with a special kind of small cancellation condition. Then, we address a somewhat easier case of 2-generator (not necessarily one-relator) groups and determine the kernel of the above-mentioned homomorphism for a rather general class of those groups.

A statistical table is in effect a classification of a finite number, N, of objects in respect of a finite number of different classes, A, B, C, … It is assumed that unambiguous rules have been laid down by which it is possible to determine whether any particular one of the objects does or does not belong to any particular one of the classes. The application of these rules to a particular object does not depend on the fact that that object is one of a finite number of N objects, so that the process of classification may be started on a non-finite collection of objects. When the collection of objects is non-finite the process can never end. On the other hand when the collection is finite the process must end; and when completed it will determine how many of the objects belong to any particular class. If in this way it is found that N1 (≤ N) of the objects belong to class A, the proper fraction N1/N is spoken of as the frequency of class A in the collection. In particular cases it may be zero or unity. In general it is a rational proper fraction. For its determination in complicated cases it may be convenient to suppose the collection to be arranged in some special way, but its value is absolutely independent of any such particular arrangement.

1. Consider the Cauchy problem for the classical wave equation in ℝn:

with initial conditions u(0, x) = uo(x) and ∂tu(0, x)= u1(x), where uj (j = 0,1) are compactly supported C∞ functions on ℝn. In this paper we shall establish the following result.

1. Let {μn} be a sequence of positive numbers increasing to infinity. If the series

converges for s > 0, and if f(s) → l as s → 0, then ∑an is said to be summable (A, μn) to l; this generalization of Abel summability, first introduced by Hardy, has long been familiar. Suppose that, for x ≥ μ0, ø(x)is a positive increasing function of x, which tends to infinity as x → ∞. We write throughout λn = ø(μn), and

if this exists.

We prove that the set of all endpoints of the Julia set of $f(z)=\exp\!(z)-1$ which escape to infinity under iteration of f is not homeomorphic to the rational Hilbert space $\mathfrak E$ . As a corollary, we show that the set of all points $z\in \mathbb C$ whose orbits either escape to $\infty$ or attract to 0 is path-connected. We extend these results to many other functions in the exponential family.

A manifold which admits a reducible genus-2 Heegaard splitting is one of the 3-sphere, S2 × S1, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the 3-sphere, S2 × S1 or a connected sum whose summands are lens spaces or S2 × S1, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-2 Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.

For stochastic processes which are induced by a signed measure, the Andersen-Jessen theorem asserts almost sure convergence and yields the identification of the limit. This result has been extended to real and vector-valued stochastic processes which are induced by a finitely additive set function or a set function process. In the present paper, we study the structure of such induced stochastic processes in order to locate the Andersen-Jessen theorem and its extensions in the family of convergence theorems for martingales and their generalizations. As an application of these results, we also show that the Andersen-Jessen theorem and its extensions can be deduced from the convergence theorems for conditional expectations and positive supermartingales.

Das Schwarzsche Lemma der klassischen Funktionentheorie einer Veränderlichen kann als vollständige Beschreibung der inneren Abbildungen des Einheitskreises (mit Fixpunkt O) aufgefaßt werden. Ist nämlich f(z) eine in |z| ≤ 1 reguläre Funktion mit |f(z)| ≤ 1 und f(0) = 0, so muß |f′(0)| ≤ 1 sein, und es stellt diese Ungleichung bekanntlich den ersten Schritt zur Lösung des Koeffizientenproblems der im Einheitskreis gleichmäßig beschränkten Funktionen dar. Die Aussage

kann andererseits so gedeutet werden, daß jede innere Abbildung des Einheitskreises mit Fixpunkt O auch innere Abbildung jedes konzentrischen Kreises |z| ≤ |ρ| < 1 ist.

During the last thirty years an immense amount of research has been done on differential equations of the form

where ε > 0 is small. It is usually assumed that the perturbing term on the right-hand side is a ‘good’ function of its arguments and that its dependence on t is purely trigonometric; this means that there is an expansion of the form

where the ωn are constants, and that one may impose any conditions on the rate of convergence of the series which turn out to be convenient. Without loss of generality we can assume

and for convenience we shall sometimes write ω0 = 0. Often f is assumed to be periodic in t, in which case it is implicit that the period is independent of x and ẋ (We can also allow f to depend on ε, provided it does so in a sensible manner.)

Let k be an algebraically closed field of ch(k) = 0 and G be a simple simply connected algebraic group G over k. By using results about cohomological invariants, we compute the coniveau spectral sequence for classifying spaces BG.

Let P(x) denote a polynomial with degree n and integer coefficients. By the height h of P we mean the maximum of the absolute values of the coefficients.

Let p be a prime and let Pi denote the Steenrod p-power if p ≠ 2 and the Steenrod square Sqi if p = 2. If p ≠ 2 let β denote the Bockstein co-boundary (defined so as to anticommute with suspension).

We are concerned in this work with the following question:

Suppose that i is a continuous algebraic isomorphism from the topological group H onto a subgroup of the topological group G and suppose that the image i(H) is not closed in G; then what can we say about the cardinal number

We observe two easy results.