Kneser— Hecke— operators in coding theory

The Kneser-Hecke-operator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of self-dual codes of fixed length. It maps a linear self-dual codeC over a finite field to the formal sum of the equivalence classes of those self-dual codes that intersectC in a codi-mension 1 subspace. The eigenspaces of this self-adjoint linear operator may be described in terms of a coding-theory analogue of the Siegel Φ-operator.     

Stochastic investment models—theory and applications

The paper discusses a stochastic model for investment variables, involving four series: the Retail Prices Index, an index of share dividend yields, an index of share yields, and the yield on ‘consols’. Section 2 describes the model and explains its derivation on the basis of historical data. Section 3 shows how the model can be used for forecasting the distributions of the variables. Section 4 discusses possible applications, and describes two in detail, relating to the expense charges of unit trusts and to guarantees incorporated in index linked life annuities.     

Progress in physical concepts of string and superstring theory—thirty years of string theory

In this paper we discuss the evolution of physical concepts which led to the generation and development of string theories. The paper is conceived with the intention of summarizing and extending with new aspects the specific characteristics of strings which refer to the physical intuition and experiment. We hope to present new insights into the physics of strings and make it understandable from the point of view of a non-string theorist. Even if there exist some opinions that the (super)string theory appertains to the twenty-first or twenty-second century or that there are no concrete new predictions of string theory at low energies, we believe that string theory presents a rich field of research and a source of physical intuition not only for mathematicians but also for theoretical and experimental physicists. We offer as an example an atomic electron cloud which can also be interpreted in terms of a fixed point in a string theory We propose also an experiment to verify the fundamental hypotheses. Finally we deduce that the number of dimensions of spacetime must be infinite by virtue of the axiom of universality of motion.     

Massive vector gauge theory and comparison with Higgs—Connes—Lott

A massive vector gauge theory constructed from the matrix derivative approach of non-commutative geometry is compared with the Higgs—Connes—Lott theory. In the massive vector gauge theory, a new extra shift-like symmetry which is due to the one form constant matrix derivative allows the theory to have a mass term while keeping the gauge symmetry intact. In the Higgs—Connes—Lott theory, the transformation of scalar field makes up the deficiency of symmetry due to the mass term. Thus, when the scalar field is absent there remains no gauge symmetry just like the Proca model. In the massive vector gauge theory, the shift-like symmetry makes up the deficiency of symmetry due to the mass term even in the absence of the scalar field.     

Left linear theories — A generalization of module theory

In this paper we introduce left linear theories of exponentN (a set) on the setL as mapsL ×L ^N (l, ) l · L such that for alll L and , L ^N the relation (l · ) =l( · ) holds, where · L ^N is given by ( · )(i) = (i),i N. We assume thatL has a unit, that is an element L ^N withl · =l, for alll L, and · = , for all L ^N . Next, left (resp. right)L-modules andL-M-bimodules and their homomorphisms are defined and lead to categoriesL-Mod, Mod-L, andL-M-Mod. These categories are algebraic categories and their free objects are described explicitly. Finally, Hom(X, Y) andX Y are introduced and their properties are investigated.Herrn Professor Dr. D. Pumplün zum 60. Geburtstag gewidmet     

Calculations in conformal theory needed for verifying the Alday—Gaiotto—Tachikawa hypothesis

Explicitly verifying the Alday—Gaiotto—Tachikawa (AGT) relation between the conformal blocks controlled by the WN symmetry and U(N) Nekrasov functions requires knowing the Shapovalov matrix and various triple correlators for W-algebra descendants. We collect the simplest expressions of this type for N = 3 and for the two lowest descendant levels together with the detailed derivations, which can now be computerized and used in more general studies of conformal blocks and AGT relations at higher levels.     

Unified theory for abstract parabolic boundary problems—a semigroup approach

This paper presents and abstract semigroup formulation ofparabolic boundary value problems. Smoothness of solutions, represented by a semigroup formula, is the primary object of discussion. The generality of our approach enables us to treat in a unified manner theregularity of solutions to parabolic equations for a large variety of nonhomogeneous boundary value problems. In particular, the approach presented here allows us to translate known regularity results of the elliptic theory directly into regularity results for the parabolic solutions. On the one hand, our theory recaptures known regularity results of the parabolic solutions over smooth spatial domains. On the other hand, however, our theory also covers the case of conical spatial domains, for which the standard assumption ofC -boundaries is violated by suitable application of recent relevant results of elliptic theory for such domains. In the concluding section, an application of our general theory to a boundary control problem with a quadratic performance index is presented.Research supported in part under Grant no. 78-3550, AFOSR, Applied Math Division, USAF.     

Conley pairs in geometry—Lusternik–Schnirelmann theory and more

Firstly, we wish to motivate that Conley pairs, realized via Salamon’s definition (Salamon, 1990), are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces (Weber, 2017). From this fell off quite naturally, firstly, an alternative proof (Weber, 2016) of the cell attachment theorem in Morse theory (Milnor, 1963) and, secondly, some ideas (Majer and Weber, 2015) how to try to organize the closures of the unstable manifolds of a Morse–Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use these Conley pairs to implement the gradient flow proof of the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934) proposed in Bott’s survey (Bott, 1982).Secondly, we shall use this opportunity to provide an exposition of Lusternik–Schnirelmann (LS) theory based on thickenings of unstable manifolds via Conley pairs. We shall cover the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934), cuplength, subordination, the LS refined minimax principle, and a variant of the LS category called ambient category.     

Reliability theory and applications — Proceedings of the China-Japan Reliability Symposium

Book Reviews

Reliability theory and applications — Proceedings of the China-Japan Reliability SymposiumShunji Osaki and Jinhua Cao (eds): World Scientific, Singapore, New Jersey, Hong Kong, 1987, xii+433 pp, £47.15     

A subjective Bayesian approach to the theory of queues I — Modeling

This is the first of an expository two-part paper which outlines a point of view different from that currently used in queueing theory. In both parts, the focus is on concepts. Here we adopt a personal probability point of view to all sources of uncertainty in the theory of queues and explore the consequences of our approach by comparing our results to those that are currently available. For ease of exposition, we confine attention to the M/M/1/ and the M/M/1/K queues. In Part I we outline the general strategy and focus on model development. In Part II we address the problem of inference in queues within the subjective Bayesian paradigm and introduce a use of Shannon's measure of information for assessing the amount of information conveyed by the various types of data from queues.     

Numerical calculations on the new approach to the cascade theory —I

In the new approach to the cascade theory suggested by Alladi Ramakrishnan and one of us (S. K. S.), we deal with the number of electrons produced between 0 andt in a soft cascade, each of these electrons having an energy greater than E at the point of its production as contrasted with the usual approach where we are interested in the number of electrons with energy greater than E att. We present here numerical calculations of the mean numbers on the basis of the new approach.     

Numerical calculations on the new approach to the cascade theory —II

Numerical results on the mean numbers of electrons produced in small thicknesses in a shower initiated by a single electron are presented on the basis of the new approach to the cascade theory.     

A mixed theory of information—X: Information functions and information measures

In the probabilistic theory of information, measures of information depend only upon the probabilities of the events, whereas in the nonprobabilistic theory these measures depend only upon the events. In a new, mixed theory of information the measures of information are assumed to depend on both the probabilities and the events. In this paper we consider measures depending upon a n-tuple of events and upon a finite number of n-ary complete, discrete probability distributions and characterize these measures of information only by two properties: by a recursivity condition, which states how the information changes by splitting one event of a system (one outcome of an experiment, market situation etc.) into two events, and by a weak symmetry condition (no regularity condition is assumed). Our result generalizes all recent results on this topic and especially we get from this one theorem a lot of characterization theorems for some well-known (purely probabilistic) information measures like the Shannon entropy, the entropy of degree α, the inaccuracy, the directed divergence, and the information improvement.     

Herman’s theory revisited

We prove that a C ^2+α -smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class D _δ , 0≤δ<α≤1, α−δ≠1, is C ^1+α−δ -smoothly conjugate to a rigid rotation. This is the first sharp result on the smoothness of the conjugacy. We also derive the most precise version of Denjoy’s inequality for such diffeomorphisms.     

Eddington’s statistical theory

The density-particle correlation, described in a previous paper of the series, is used to define the «uncertainty of the physical origin» considered by Eddington, and to relate it to the Hubble constant.