Table of contents

On 21 August 1994 Academician Nikolai Nikolaevich Bogolyubov, an eminent mathematician and theoretical physicist of our time, would have been 85. The editorial board decided to publish in the present issue of the journal a series of survey papers covering a wide range of important directions of his scientific activities. These papers have been written by the colleagues and students of Nikolai Nikolaevich.

CONTENTS §1. Introduction §2. Theory of open systems §3. Kinetic theory and hydrodynamics §4. Quantum statistical mechanics References

CONTENTS §1. The first version of Bogolyubov's "edge of the wedge" theorem §2. Generalizations of Bogolyubov's "edge of the wedge" theorem §3. Bogolyubov's global "edge of the wedge" theorem §4. Microlocal versions of Bogolyubov's "edge of the wedge" theorem §5. Bogolyubov's "edge of the wedge" theorem for wedges with a "non-linear" edge §6. Some applications of Bogolyubov's "edge of the wedge" theorem References

CONTENTS §1. Preliminaries from the theory of almost periodic functions §2. Bogolyubov's contribution to the theory of almost periodic functions §3. The influence of Bogolyubov's research on further development in the theory of almost periodic functions References Appendix 1. Bogolyubov's proof of the approximation theorem Appendix 2. The proof of Bogolyubov's theorem on arithmetic properties of relatively dense sets

CONTENTS §1. Introduction 1.1. Quantum fields 1.2. The origin of Bogolyubov's renormalization group 1.3. Episode with an `illusory pole' §2. History of the renormalization group in quantum field theory 2.1. Renormalizations and renormalization invariance 2.2. The discovery of the renormalization group 2.3. The origin of the renormalization group method 2.4. Other early applications of the renormalization group §3. Further development of the renormalization group 3.1. Quantum field theory 3.2. Spin lattices 3.3. Turbulence 3.4. Ways of extending the renormalization group 3.5. Two faces of the renormalization group in quantum field theory 3.6. Functional self-similarity Conclusion References