## Selected Papers of Norman LevinsonJ.A. Nohel, D.H. Sattinger, Gian-Carlo Rota The deep and original ideas of Norman Levinson have had a lasting impact on fields as diverse as differential & integral equations, harmonic, complex & stochas tic analysis, and analytic number theory during more than half a century. Yet, the extent of his contributions has not always been fully recognized in the mathematics community. For example, the horseshoe mapping constructed by Stephen Smale in 1960 played a central role in the development of the modern theory of dynami cal systems and chaos. The horseshoe map was directly stimulated by Levinson's research on forced periodic oscillations of the Van der Pol oscillator, and specifi cally by his seminal work initiated by Cartwright and Littlewood. In other topics, Levinson provided the foundation for a rigorous theory of singularly perturbed dif ferential equations. He also made fundamental contributions to inverse scattering theory by showing the connection between scattering data and spectral data, thus relating the famous Gel'fand-Levitan method to the inverse scattering problem for the Schrodinger equation. He was the first to analyze and make explicit use of wave functions, now widely known as the Jost functions. Near the end of his life, Levinson returned to research in analytic number theory and made profound progress on the resolution of the Riemann Hypothesis. Levinson's papers are typically tightly crafted and masterpieces of brevity and clarity. It is our hope that the publication of these selected papers will bring his mathematical ideas to the attention of the larger mathematical community. |

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### Contents

XII | 11 |

XIII | 21 |

XIV | 28 |

XV | 34 |

XVI | 51 |

XVII | 58 |

XVIII | 65 |

XIX | 69 |

XXXV | 230 |

XXXVI | 234 |

XXXVII | 238 |

XXXVIII | 254 |

XXXIX | 266 |

XL | 267 |

XLI | 288 |

XLII | 325 |

XX | 91 |

XXI | 99 |

XXII | 114 |

XXIII | 141 |

XXIV | 152 |

XXV | 155 |

XXVI | 159 |

XXVII | 164 |

XXVIII | 192 |

XXIX | 193 |

XXX | 199 |

XXXI | 203 |

XXXII | 207 |

XXXIII | 211 |

XXXIV | 226 |

XLIII | 333 |

XLIV | 357 |

XLV | 366 |

XLVI | 376 |

XLVII | 384 |

XLVIII | 396 |

XLIX | 415 |

L | 423 |

LI | 439 |

LII | 449 |

LIII | 462 |

LIV | 467 |

LV | 494 |

LVI | 505 |

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Amer Anal analytic applied assume Asymptotic base interval behavior boundary condition boundary value problem bounded class C(A Clearly closed curve closed integral curve Coddington consider constant continuous function converges corresponding defined degenerate system denote domain dp(u Duke Math Eigenfunction eigenvalues elliptic exists fixed point follows easily formula given Hence holds hypothesis implies inequality initial values integral curve integral equation interior inverse J. E. Littlewood Lemma limit-point Mathematics matrix maximum principle Moreover nonlinear Norman Levinson obtain Ordinary Differential Equations oscillations paper periodic solution plane positive Proc proof of Theorem proves Theorem relaxation oscillations replaced result satisfies second order self-adjoint singular singularly perturbed solution of 1.1 solution x(t stable Sturm-Liouville problem sufficiently small Systems of Differential term Theorem 1.1 theory transformation uniformly vanishes variables vector Volterra Equations Wasow yields zero