## Studies in Mathematical Physics: Lectures Presented at the NATO Advanced Study Institute on Mathematical Physics held in Istanbul, August, 1970Mathematical physics has become, in recent years, an inde pendent and important branch of science. It is being increasingly recognized that a better knowledge and a more effective channeling of modern mathematics is of great value in solving the problems of pure and applied sciences, and in recognizing the general unifying principles in science. Conversely, mathematical developments are greatly influenced by new physical concepts and ideas. In the last century there were very close links between mathematics and theo retical physics. It must be taken as an encouraging sign that today, after a long communication gap, mathematicians and physicists have common interests and can talk to each other. There is an unmistak able trend of rapprochement when both groups turn towards the com mon source of their science-Nature. To this end the meetings and conferences addres sed to mathematicians and phYSicists and the publication of the studies collected in this Volume are based on lec tures presented at the NATO Advanced Study Institute on Mathemati cal Physics held in Istanbul in August 1970. They contain review papers and didactic material as well as original results. Some of the studies will be helpful for physicists to learn the language and methods of modern mathematical analysis-others for mathematicians to learn physics. All subjects are among the most interesting re search areas of mathematical physics. |

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### Contents

BASIC NOTIONS IN TOPOLOGICAL VECTOR SPACES | 1 |

LECTURES ON MATHEMATICAL ASPECTS OF PHASE TRANSITIONS | 51 |

THE MATHEMATICAL STRUCTURE OF THE BCSMODEL AND RELATED MODELS | 81 |

A SHORT SURVEY OF MODERN STATISTICAL MECHANICS | 113 |

GENERALIZED EIGENVECTORS IN GROUP REPRESENTATIONS | 135 |

MACKEYWIGNER AND COVARIANT GROUP REPRESENTATIONS | 155 |

DERIVATION OF THE LORENTZ AND GALILEI GROUPS FROM ROTATIONAL INVARIANCE | 179 |

GENERALIZED EIGENVECTORS AND GROUP REPRESENTATIONS THE CONNECTION BETWEEN REPRESENTATIONS OF SO41 AND T... | 197 |

THE INVERSE DECAY PROBLEM | 247 |

ON OPERATOR EQUATIONS IN PARTICLE PHYSICS | 253 |

LECTURES IN NONLINEAR ALGEBRAIC TRANSFORMATIONS | 263 |

THE NEW MATHEMATICAL AND PHYSICAL PROBLEMS OF SUPERSTRONG COUPLING OF MAGNETIC MONOPOLES AND COMPOSITE... | 315 |

### Common terms and phrases

0-neighbourhood algebra application Banach space barreled base basis bounded bounded sets called circled closed commuting compact complete condition consider contained continuous convergence convex corresponding course defined Definition denote dense direct sum discuss dual eigenvalues elements equation equicontinuous equivalent example exists extension fact field finite fixed point follows functions give given hence Hilbert space identity implies inductive infinite integral introduce invariant iterates Lecture Lemma limit sets linear Math mathematical matrix means measure metrizable norm Note nuclear obtain operator particular Phys physical positive problem projective Proof properties prove reduced Reference relations Remark representation respect restriction result satisfy semi-norms separated sequence solution Spec spectrum subset subspace tensor theorem theory tion topology transformation unitary usual vector vector space