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Real roots of algebraic equations (Kac, 1943)f (t) = ξ 0 + ξ 1 t + ξ 2 t 2 + · · · + ξ n−1 t n−1P{ξ = (ξ 0 , . . . , ξ n−1 ) ∈ A} =∫Ae −‖x‖2 /2(2π) n/2 dx (ξ ∼ N(0, In×n ) )
Real roots of algebraic equations (Kac, 1943)f (t) = ξ 0 + ξ 1 t + ξ 2 t 2 + · · · + ξ n−1 t n−1P{ξ = (ξ 0 , . . . , ξ n−1 ) ∈ A} =Theorem: N n = the number of real zeroes of f∫Ae −‖x‖2 /2(2π) n/2 dx (ξ ∼ N(0, In×n ) )E{N n } =4 π∫ 10[1 − n 2 [x 2 (1 − x 2 )/(1 − x 2n ] 2 ] 1/21 − x 2 dx
Page 4 and 5: Our heroesMarc KacStephen O. Rice19
Page 10 and 11: Zeroes of complex polynomialsf (z)
Page 12 and 13: Zeroes of complex polynomialsf (z)
Page 14 and 15: Thinking more generally:1: f : R N
Page 16 and 17: Thinking more generally:1: f : R N
Page 18 and 19: The original (non-specific) Rice fo
Page 30 and 31: Constructing Gaussian Processes◮
Page 32 and 33: Constructing Gaussian Processes◮
Page 34 and 35: Existence of Gaussian processesTheo
Page 36 and 37: Existence of Gaussian processesTheo
Page 38 and 39: Example: The Brownian sheet on R NE
Page 40 and 41: Example: The Brownian sheet on R NE
Page 42 and 43: Constant variance Gaussian processe
Page 50 and 51: Gaussian Kac-Rice with no simplific
Page 52 and 53: Gaussian Kac-Rice with no simplific
Page 54 and 55: Some pertinent thoughts◮ Real roo
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Some pertinent thoughts◮ Real roo
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The Kac-Rice “Metatheorem”
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The Kac-Rice “Metatheorem”◮ T
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The Kac-Rice “Metatheorem”◮ T
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The original (non-specific) Rice fo
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The original (non-specific) Rice fo
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The Kac-Rice Conditions (the fine p
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Higher (factorial) moments◮ Facto
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The Gaussian case: What can/can’t
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The Gaussian-related casef (t) = (f
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The Gaussian-related casef (t) = (f
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The Gaussian Kinematic Formula (GKF
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The perturbed-Gaussian case◮ A ph
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Applications I: Exceedence probabil
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Applications I: Exceedence probabil
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Applications II: Local maxima on th
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Applications III: Local maxima on M
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Applications III: Local maxima on M
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Applications IV: Longuet-Higgins an
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Applications V: Higher moments and
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Applications VI: Poisson limits and
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Applications VI: Poisson limits and
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Applications VII: Eigenvalues of ra
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Applications VII: Eigenvalues of ra
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Appendix I: The canonical Gaussian
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Exceedence probabilities for canoni
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◮ We needP { sup〈U, t〉 ≥ u/
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Appendix II: Stationary and isotrop
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Appendix II: Stationary and isotrop
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◮ Elementary considerations give{
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◮ Elementary considerations give{
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Appendix III: Regularity of Gaussia
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Appendix III: Regularity of Gaussia
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Special cases of the entropy result
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Special cases of the entropy result
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Appendix IV: Borell-Tsirelson inequ
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Appendix IV: Borell-Tsirelson inequ
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