Very Special Functions — Unbeknownst To Mathematica And .
Z ENTRUM M ATHEMATIKT ECHNISCHE U NIVERSITÄT M ÜNCHENVery Special Functions — unbeknownst to Mathematica and kinshipnumerical explorations of random matrix distributions6operator determinants(b)(a)1000500Ku( x ) 0-500-1000-1000-5000500(d)(c)1000(µm)det( I zK ) Z ba n 02000K ( x, y)u(y) dyznn!Zndet K (ti , t j ) dt[ a,b]n i,j 115001000500[w,x] QuadratureRule(a,b,m);w sqrt(w); [xi,xj] ndgrid(x,x);Fig. 3 Growing DSM2 cluster with a circular (a,b) and flat (c,d) interface. (a,c) Raw images. Indicated below each image is theelapsed time after the emission of laser pulses. (b,d) Snapshots of the interfaces at t 3, 8, . . . , 28 s for the circular case (b) andt 10, 20, . . . , 60 s for the flat case (d). The gray dashed lines indicate the mean radius (height) of all the circular (flat) interfacesd det(eye(m) z*(w’*w).*K(xi,xj));fluctuations of turbulent liquid crystals00100020003000(µm)recorded at t 28 s (t 60 s). See also Supplementary Movies of Ref. [96].Crecent study showed that overhangs are irrelevant for the scaling of the interfaces [79], here, for the sake ofsimplicity and direct comparison to theoretical predictions, we take the mean of all the detected heights at agiven coordinate x to define a single-valued function h(x,t) for each interface. The spatial profile h(x,t) is statistically equivalent at any point x because of the isotropic and homogeneous growth of the interfaces, which,together with the large numbers of the realizations, provides accurate statistics for the interface fluctuationsanalyzed below.Before presenting the results of the analysis, it is worth noting different characters of the “system size”L, or the total lateral length, of the circular and flat interfaces in general. While the system size of the flatHALLENGESENTURYXP thatATHOMPULYinterfaces is chosenINa priori STand fixedduring the evolution,of the circularinterfaces is the circumference21CE .M.C., ICERM J2014Folkmar BornemannF OLKMAR B ORNEMANN1
S CALING L IMITS OF R ANDOM M ATRICESWigner semicircle law (1955)example: GUE0.5Gaussian unitary ensembleA randn(n) i*randn(n);A (A A’)/2;probability density0.4spectrum as n ? fluctuations?0.20.1010.50.80.40.60.40.200 2 10 eigenvalues/ n12edge: Tracy–Widom distribution F2probability densityprobability densitybulk: Gaudin–Mehta distribution E20.30.30.20.10.5211.5normalized eigenvalue spacing2.530 5 40 3 2 1normalized maximum eigenvalue12universality: fluctuation statistics does only depend on symmetry class (Soshnikov; Tao; L. Erdős, . . . )C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN2
U NIVERSALITY WITHIN M ATHEMATICSMontgomery Odlyzko lawMontgomery–Odlzyko “law” (’73/’87)120.9 iγn of Riemann ζ-functionlarge n statistics of the spacings ofγnγnlog2π2π0.8Probability densitynontrivial zeros10.70.60.50.40.30.20.1 Gaudin–Mehta distribution E2000.511.522.53Normalized consecutive spacingsfluctuations in Ulam’s problem (Baik/Deift/Johansson ’99)ln length of longest increasing subsequence of a random permutation of order n ln 2 n Tracy–Widom distribution F21/6nC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN3
U NIVERSALITY: A V ERY S HORT I NTRODUCTIONB./Ferrari/Prähofer '08H. Spohn: Random Matrices and the KPZ Equation (June 1, ’12, HCM, Bonn)Universality: The macroscopic statistics depend on the models, but the microscopic statistics areindependent of the details of the systems except the symmetries.— L. Erdős ’10C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN4
V ERY S PECIAL F UNCTIONS , I NDEEDspacing distributions of GUE @ bulk( 1)n nP(exactly n eigenvalues in (0, s)) E2 (s; z)nn! zJimbo/Miwa/Môri/Sato ’80Gaudin ’61 E2 (s; z) det I zK L2 (0,s)z 1 w/ kernelK ( x, y) sinc(π ( x y))can be expressed in terms ofradial prolate spheroidal wave functions E2 (s; z) exp w/ σ-form of Painlevé VZ πsσ( x; z)0xdx ( xσ00 )2 4(σ xσ0 )( xσ0 σ σ0 2 )zz2 2σ( x; z) ' x 2 xππC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014( x 0)F OLKMAR B ORNEMANN5
V ERY S PECIAL F UNCTIONS , V ERY M UCH S O I NDEEDspacing distributions of GUE @ soft edge( 1)n nF2 (s; z)P(exactly n eigenvalues in (s, )) nn! zForrester ’93 F2 (s; z) det I z K L2 (s, )w/ kernelK ( x, y) 0z 1Tracy/Widom ’93 F2 (s; z) exp 0Ai( x ) Ai (y) Ai ( x ) Ai(y)x yZ s( x s)u( x; z)2 dx w/ Painlevé IIu00 2u3 xu u( x; z) ' z Ai( x )( x )Without the Painlevé representations, the numerical evaluation of the Fredholm determinants isquite involved.— Tracy/Widom ’00Recently a numerical analyst has shown that the most efficient way to compute spacing distributionsin classical RMT is to use Fredholm determinant formulas.— Forrester ’10C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN6
F REDHOLM D ETERMINANTS VS . PAINLEVÉ T RANSCENDENTSvery special functions: nonlinear ode, but integrableIvar Fredholm (1866–1927)Paul Painlevé (1863–1933)determinant of integral operator (1899)six families of irreducible transcendental functions (1895)Ku( x ) Z baK ( x, y)u(y) dyu00 6u2 xu00 2u3 xu αu00 u 1 u0 2 x 1 u0 x 1 (αu2 β) γu3 δu 1det( I zK ) n 0znn!Zndet K (ti , t j ) dt[ a,b]n i,j 1.C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN7
C OMPARING D IFFERENT N UMERICAL A PPROACHESnumerical evaluation of the Tracy–Widom distribution F2 via Painlevé II as IVP (backwards)Painlevé II by IVPF (x) by IVP0102Prähofer (’04): 16 digits (1500 internally!)Painlevé II by BVPF2(x) by BVP10absolute errorBejan (’05): 3 digitsF2(x) by Fredholm 5machine precisionEdelman/Persson (’05): 8 digits @ 8.9 sec via Painlevé II as BVP 1010Tracy/Widom (’94): 10 digits (75 internally!)Dieng (’05): 9 digits @ 3.7 sec 1510Driscoll/B./Trefethen (’08): 13 digits @ 1.3 sec 2010 8 6 4 202468x via Fredholm determinantB. (’10): 15 digits @ 0.69 secabsolute error using IEEE double precisionC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN8
T HE N EED FOR C ONNECTION F ORMULAEinstabilitysolution of Painlevé II,3u xx 2u xu,separatrix for z 1u( x ) ' ( x ),z Ai( x )IVP highly -8-6-4-2u( x ) with 2-24-2z 1 10 8 , 1, 1 10 8u( x ) with z 1 10 16 , 1, 1 10 16consequence calculate F2 via a BVP solution u( x ) ' z Ai( x ) ( x )connection formula needed: C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014u( x ) ' ?( x )F OLKMAR B ORNEMANN9
T HE N EED FOR C ONNECTION F ORMULAEinstabilitysolution of Painlevé II,3u xx 2u xu,separatrix for z 1u( x ) ' ( x ),z Ai( x )IVP highly -8-6-4-2u( x ) with 2-24-2z 1 10 8 , 1, 1 10 8u( x ) with z 1 10 16 , 1, 1 10 16consequence calculate F2 via a BVP solutionu( x ) 'Ai( x )( x )connection formula needed: u( x ) ' x/2C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014( x )F OLKMAR B ORNEMANN9
Q UADRATURE M ETHODNyström (1930)solved a Fredholm equation ( I zK )u f of the 2nd kind,u( x ) zZ baK ( x, y)u(y) dy f ( x )( x ( a, b)),using an m-point quadrature formula w/ weights w j & nodes x ju ( xi ) ui :mui z w j K ( xi , x j ) u j f ( xi )Evert Nyström (1895–1960)(i 1, . . . , m)j 1straightforward idea (B. ’08)approximate det( I zK ) simply by the corresponding n n determinant mdet I z (w j K ( xi , x j ))i,j 1C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014 F OLKMAR B ORNEMANN10
C ONVERGENCE R ATE OF THE Q UADRATURE M ETHODMatlab code[w,x] QuadratureRule(a,b,m);w sqrt(w); [xi,xj] ndgrid(x,x);d det(eye(m) z*(w’*w).*K(xi,xj));Theorem (B. ’10)for quadrature formula of order ν w/ positive weights: if kernel is C k 1,1 ([ a, b]2 ),error O(ν k ); if kernel is bounded analytic, there is ρ 1 w/error O(ρ ν ).C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN11
P ROOFidea of proof (Hilbert 1904, B. ’10)m-point quadrature formulaZ baf (t) dt m wk f ( xk )k 1yieldsdet( I zK )Fredholm 1903Hadamard 1893v. Koch 1892w/ the m m-matrix zn n!n 0 zn n!n 0Z bam k 1 1Z bdt1 · · ·wk1 · · ·am k n 1ndtn det K (ti , t j )i,j 1nwkn det K ( xki , xk j )i,j 1det( I zKm ) Km wi1/2 K ( xi , x j )w1/2jC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014 mi,j 1F OLKMAR B ORNEMANN12
E XAMPLE 1Gaudin–Mehta distribution E2 E2 (0; s) det I K L2 (0,s) ,K ( x, y) sinc(π ( x y))0approximation error for E2(0;2)10 510 1010 1510 20100510152025dimensionstars: Gaudin’s method, dots: Gauss–Legendre, circles: Clenshaw–CurtisC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN13
E XAMPLE 2Tracy–Widom distribution F2 F2 (s) det I K L2 (s, ) ,Ai( x ) Ai0 (y) Ai0 ( x ) Ai(y)K ( x, y) x y0approximation error for F2( 2)10 510 1010 1510 201005101520253035404550dimensiondots: Gauss–Legendre, circles: Clenshaw–Curtisperturbation bound for m-dimensional determinants: (B. ’10) round-off error 6 m kKm k F · emachineC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN14
DYSON B ROWNIAN M OTIONAn (t) n n hermitean-matrix valued process, coefficients Ornstein–Uhlenbeck 1/3λmax An (nt) 2nA2 (t) limn 2 1/2 n 1/6Prähofer/Spohn ’02relation to the PNG droplet model (universality!) P(A2 (t) 6 x, A2 (0) 6 y) det I w/ kernelKt (u, v) Z Z 00 K0KtK tK0 L2 ( x, ) L2 (y, ) e ξt Ai(u ξ ) Ai(v ξ ) dξt 0e ξt Ai(u ξ ) Ai(v ξ ) dξt 0C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN15
T HERE IS A PDE . . .Adler/van Moerbeke ’05G (t, x, y) log P(A2 (t) 6 x, A2 (0) 6 y) satiesfies nonlinear 3rd order PDEt t 2 x2 2 y2 G 3 G 2 2 G 2 G 2 G x y t2 x y x2 y y2 x2 2 232 G G G G 2 x y t22 2 2 x y y x x y 33 G G G 33 x y x y y xw/ asymptotic boundary conditionsaside: useful for numerical calculations? most probably not . . .C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN16
T HE FATE OF A C ONJECTURE?edge scaling limit of GOE matrix diffusion Airy1 processMn (t) n n symmetric-matrix valued process, coefficients Ornstein–Uhlenbeck 1/3λmax Mn (2nt) nA1 (t) limn n 1/6Sasamoto ’05, Borodin/Ferrari/Prähofer/Sasamoto ’07conjectured relation to the flat PNG model (universality!) K0 K t? L2 (s , ) L2 (s , ) P(A1 (t) 6 s1 , A1 (0) 6 s2 ) det I 21K t K0w/ kernel exp( ( x y)2 /(4t))3 /3 2t(x y) 2t Ai( x y t )e 4πtKt ( x, y) Ai( x y t2 )et( x y) 2t3 /3C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANNt 0t6017
A C OMPUTER E XPERIMENT: T HE T WO -P OINT C ORRELATION F UNCTIONSTwo Point Correlation Function of the Airy2 ),A1(0))Two Point Correlation Function of the Airy1 red: Monte–Carlo matrix diffusion;22.500green: asymptotics;246t81012blue: numerical Fredholm determinantsconclusion (B./Ferrari/Prähofer ’08)limitof GOE {zmatrix diffusion} 6 Airy1 process identification is an open problemC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN18
A P HYSICAL E XPERIMENT — I NTERFACE G ROWTH Ievidence for geometry-dependent universal fluctuationsTakeuchi/Sano/Sasamoto/Spohn ’11 (Nature)6866(a)(c)K.A. Takeuchi, M. SanoFig. 8 Histogram(b) of the rescaled1000heightlocalq (h v t)/(Γ t)1/3 for the500circular (solid symbols) and flat(open0 symbols) interfaces. Theblue circles and red diamonds-500 the histograms for thedisplaycircular interfaces at t 10 s and30-1000s,-1000respectively,-5000 while500 the1000(µm)turquoise up-trianglesandpurple(d)down-triangles are for the flat2000interfacesat t 20 s and 60 s,1500respectively.The dashed anddottedcurvesshow the GUE and1000GOE TW distributions,500respectively, defined by the0randomχ20000 variables10003000GUE and(µm)χGOE . (Color figure online)Fig. 3 Growing DSM2 cluster with a circular (a,b) and flat (c,d) interface. (a,c) Raw images. Indicated below each image is theelapsed time after the emission of laser pulses. (b,d) Snapshots of the interfaces at t 3, 8, . . . , 28 s for the circular case (b) andt 10, 20, . . . , 60 s for the flat case (d). The gray dashed lines indicate the mean radius (height) of all the circular (flat) interfacesrecorded at t 28 s (t 60 s). See also Supplementary Movies of Ref. [96].rescaled height χturbulent liquid crystal (DSM2 cluster)Fortheinterfacesflat interfaces,in principleneeds to impose a tilt u to measure v (u) [53], butrecent study showed that overhangs are irrelevant for the scalingof the[79], here, foronethe sakeofsimplicity and direct comparison to theoretical predictions, we supposingtake the mean ofallthedetectedheightsatathe rotational invariance of the system, one again finds v (u) 1 u2 v andgiven coordinate x to define a single-valued function h(x,t) for each interface. The spatial profile h(x,t) is stathus Eq. (13).justifiedby practically the same values of our estimates v in Eq. (10).tistically equivalent at any point x because of the isotropic and homogeneousgrowthThisof theisinterfaces,which,together with the large numbers of the realizations, provides accurate statistics for the interface fluctuationsIn passing, the rotational invariance also implies v0 0 in the KPZ equation (5), without theanalyzed below.Before presenting the results of the analysis, it is worth notingdifferentcharactersthe “system size”needto invoketheofcomovingframe.L, or the total lateral length, of the circular and flat interfaces in general. While the system size of the flatinterfaces is chosen a priori and fixed during the evolution, that of the circular interfaces is the circumferencewhich grows linearly with time and is therefore not independentThis mattersmost when one3.4of dynamics.DistributionFunctiontakes an average of a stochastic variable, e.g., the interface height. For the flat interfaces, the spatial andensemble averages are equivalent provided that the system size L is much larger than the correlation lengthUsingtwo experimentallydetermined parameter values (10) and (11), we shall directlyl t 1/z . In contrast, for the circular interfaces, the two averagesmake athesignificantdifference, because thesystem size is inevitably finite and the influence of finite-size effectsvaries in thetime.interfaceTo avoid thisfluctuationscomplication, χ with the mathematically defined random variables χGUEcomparewe take below the ensemble average denoted by h· · ·i unless otherwise stipulated, which turns out to be theχGOEexponent. This isright choice when one measures characteristic quantities such asandthe growthβ . achieved by defining the rescaled local heightKPZ theory & universalityht h xx h2x space-time white noiseKardar/Parisi/Zhang ’86 (4000 citations)h(t, x ) ' λt (Γt)1/3 χscaling lawrigorous solution concept: M. Hairer ’13h v tq C HALLENGESIN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 20143 Experimental results1/3χ(14)F OLKMAR B ORNEMANN19
A P HYSICAL E XPERIMENT — I NTERFACE G ROWTH IIone-parameter fitΓ 21 A2 λ 2.2 103 µm3 /sλ 33µm/s,universal spatial two point correlation (Takeuchi/Sano ’12)Cs (l; t) hh( x l, t)h( x, t)i hh( x l, t)ihh( x, t)i' (Γt)2/3 2/3/2 15(b) 1.2g j two point correlation function of Airy j process A j (t)0.8B./Ferrari/Prähofer '080.6Circular, Airy20.4B./Ferrari/Prähofer '080.20g j Al (Γt)Csint 0 Cs '(ζ) dζCs' (ζ) Cs (l) / ( Γt)2/3(a) Flat,Airy1012-2/3ζ (Al/2)(Γt)3g2int1010g2int – Csint (t)slope -1/3Nov 13, ’13. Tim Halpin-Healy @ Columbia University:0.8t (s)-1100.60.40.2021010the 10You have becomemeccato whichall pilgrims journey for Airy 1&2covariance. I apologize,but.g1intCould20 I troubleyou4060 for the same?01t (s)Fig. 10 Spatial correlation function Cs (l;t). (a) Rescaled correlation function Cs0 (ζ ;t) Cs (l;t)/(Γ t)2/3 against rescaled lengthζ (Al/2)(Γ t) 2/3 . The symbols indicate the experimental data for the circular and flat interfaces (top and bottom pairs ofsymbols,respectively),obtainedat t 10 Es XPand30ATHs for. theformerand t 20s and60 s for the latter (frombottom toBtop).C HALLENGESIN 21ST C ENTURY.MC OMP., ICERMJ ULY2014F OLKMARORNEMANN20
H IGHER -O RDER S PACING D ISTRIBUTIONSgenerating function ( 1)n ndet I zK L2 ( J )P(exactly n eigenvalues in J ) n! znnumerical method0.80.7f (z) det( I zK ) is entire of order 00.6F2(n,s)0.5f (n) ( z )1 n!2πr n0.40.30.2Z 2π0e inθ f (z reiθ ) dθ trapezoidal rule exponentially convergent0.10z 1 10 8 6 4s 202probability density of n-th largest eigenvalue in GUE stability: careful choice of r 0C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN21
E XAMPLE 1what is the right radius?f (z) exp(z), order of differentiation n 100 @ z 0rIEEE hardware arithmetic19.93042 57391 0998 1014325100400 3.60245 53557 5186 1013software arithmetic @ # digits1.0000 00000 0000 @ 159 digits1.0000 00000 0000 @ 32 digits1.0000 00000 00001.0000 00000 0000 @ 16 digits5.11842 41787 3295 10551.0000 00000 0000 @ 75 digitstheory (B. ’11) log-convexityunique optimal radius ropt (n) f (z) entire of order ρ 0 and type τ 0 (Borel classification) 1/ρnropt (n) 'τρC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN22
E XAMPLE 2generating function of probabilities absolute error: r 1 appropriate (Lyness/Sande ’71) relative error ( accurate tails):ropt argmin r n f (1 r )r 0unique solution of log-convex optimization problem (B. ’11)15010 1010 20101010510radius rabsolute error10 30 40 51010 5010201010 1046810121416s18102E2 (9; s): absolute error vs. sC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 20144681012141618sE2 (9; s): optimal radius ropt vs. sF OLKMAR B ORNEMANN23
E XAMPLE : H IGH - DIMENSIONAL M ULTIVARIATE S TATISTICS (PCA)Apr 29, ’10. Nick Patterson @ Broad Institute (MIT/Harvard)Some things I would like: A table of the mean of F1(k,s) for as largek as is practical. I’d certainly like to get this for k 1, ., 50.Dec 19, ’13. Edoardo Saccenti @ Systems and Synthetic Biology Lab (Netherlands)I would be interested in having a list of the 95th and 99thpercentiles for, let’s say, the first 50 eigenvalues?B. '10: n 50.6Dieng '05: n 2,3,40.5Tracy/Widom '96: n 10.31F (n,s)0.40.20.10 10 8 6 4s 2024pdf of n-th largest level in edge scaled GOEC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN24
PCA: n- TH L ARGEST E IGENVALUE OF GOE @ S OFT E DGE I( 1)n nP(exactly n eigenvalues in (s, )) F (s; z)n! zn 1z 1Forrester/Nagao/Honner ’99, Tracy/Widom ’05 F1 (s; z) det I z S( x, y) KAi ( x, y) 12 SD ( x, y) y KAi ( x, y) 1 12IS( x, y) 21 sgn( x y) 12S( x, y)SD ( x, y)IS( x, y)S(y, x )Ai( x )Z yAi(η ) dη 1/2 X (s, ) X (s, ) 21 Ai( x ) Ai(y)Z xKAi (ξ, y) dξ 12 ZxyAi(ξ ) dξ Z xAi(ξ ) dξZ yAi(η ) dη Ai( x ) Ai0 (y) Ai0 ( x ) Ai(y)KAi ( x, y) x yregularized Hilbert–Carleman determinantvery slow convergenceC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN25
PCA: n- TH L ARGEST E IGENVALUE OF GOE @ S OFT E DGE IIFredholm determinant reformulation (B. ’10, Forrester ’06) 1F1 (s; z) 1 2 rz2 z det I qz(2 z) K L2 (s/2, ) w/ kernel K ( x, y) Ai( x y)idea of proof (B. ’10) statistical decomposition (Forrester/Rains ’01)GSEm even(GOE2m 1 ),GUEm even(GOEm GOEm 1 ) determinantal decomposition of GUE/GSEF2 (s; z) F (s; z) · F (s; z),F4 (s; z) 12 ( F ( s; z ) F ( s; z )) F (s; z) det I z K L2 (s/2, )(at first by computer experiments)C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN 26
C OMBINATORICS : R ANDOM P ERMUTATIONS Ilength of longest increasing subsequenceσ {3, 7, 10, 5, 9, 6, 8, 1, 4, 2} S10 (σ) 4 egf of probability distributionP(σ S N : (σ ) 6 n) Toeplitz determinant (Gessel ’90)n 1 φn (z) det( Tn ( a)) det Ij k (2 z)j,k 0w/ symbol a(t) exp( z(t t 1 )) NzN!φn (z)Fredholm determinant (Baik/Deift/Rains ’01)n z φn (z) 2 e det I Kn L2 (S1 )w/ kernel 1 tn ã(t)/sn ã(s)Kn (t, s) 2πi (t s) ã(t) exp( z(t t 1 ))C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN27
TOEPLITZ VS . F REDHOLMstrong Szegő limit theoremexampleoptimal radius(n )φn ( x ) ' exp( x )ropt argminx 0 ψ( x )evaluating φn well posed w.r.t. xw/ log-convex functionψ( x ) N!x N φn ( x )numerical instability of Toeplitz10200φ̂n ( x ) φn ( x )(1 δ) det Ij k (2 x )(1 e jk )w/ e jk 6 emach ; perturbation bound n 1j,k 0 min a(eiθ ) 101001050100101 δ κ2 ( Tn ( a))emachmax a(eiθ ) ψ( x ) 10150102x103n 120, N 200 (red: Toeplitz; green: Fredholm)emach e 4 xcomplete loss of digits at about x w/emachC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014e 4 x emach 1F OLKMAR B ORNEMANN28
C OMBINATORICS : R ANDOM P ERMUTATIONS IInumerical examplep P(σ S100 : (σ ) 6 15) z100100! φ15 (z)rp @Toeplitzp @Fredholm1 2.70942775790365 10143 6.78309621528621 10142108.750.2290 63853 920770.22769 70405 77202cpu times: 1.7s optimal radius; 35s accurate value of pfun with high-precision arithmetic (@ 166 digits)p q/100! w/q 212 50103 06485 64864 48137 80469 11384 43791 46971 06696 69185 28019 00140 85873 81972 3724514519 98898 81271 08844 81409 58002 42098 38222 02649 76840 02234 15476 61239 51004 32207 27880cf. Odlyzko/Rains ’99 (Schensted corresp., hook formula, generating the 190 569 292 partitions of 100)C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN29
R ELIABILITY: T HE S ETTINGcomputation of “very special functions”: how accurate?special functions 7 integrals 7 operator determinants 7 derivatives 7 integralsexponential convergence each step accurate to the level of “numerical noise” functions represented by polynomial interpolation in Chebyshev pointswhat is the “total noise”?model (prior)f num f e · noise,knoisek 1estimation of e (inverse problem) universal scaling robust statisticsC HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN30
E XAMPLElow frequency modesAiry function1000.60.410 50high frequency modes ck Ai( x )0.210 10numerical noise 0.2 0.410 15 0.6 0.8 50 40 30 20x 10010 200100200300400kChebyshev coefficients for n 400 40 000library Airy function: how much noise?noise level estimator based on numerical Chebyshev expansion (FFT)f num ( x ) n ck Tk (x),eest n1/2k 0yieldseest 1.33 · 10 14(n 400),C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014max ck k : noisy taileest 1.52 · 10 14(n 40 000)F OLKMAR B ORNEMANN31
U NIVERSAL S CALINGChebyshev coefficients of interpolated noise10010 110 110 210 210 310 3 ck ck 10010 410 410 510 510 6050100k15020010 6050100k/100150200robust statisticsmaxk 1:n ck '4 1/2·n3C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014(uniform noise)F OLKMAR B ORNEMANN32
N OISE E STIMATE IN ACTION9-level bulk-spacing distribution of GUE ( 1)k dkdet I zKsin L2 (0,s)E2 (k; s) k! dzk0.71000.610 20.510 40.410 6 ck p2 (9; s)d2 9p2 (9; s) 2 (10 k) E2 (k; s),ds k 00.310 80.210 100.110 1206810s1210 1414z 000000.412548685C HALLENGES IN 21 ST C ENTURY E XP. M ATH . C OMP., ICERM J ULY 2014F OLKMAR B ORNEMANN33
Very Special Functions — unbeknownst to Mathematica and kinship numerical explorations of random matrix distributions 6 (a) (c)-1000 -500 0 500 1000-1000-500 0 500 1000 0 1000 2000 3000 0 500 1000 1500 2000 (b) (d) (mm) (mm) Fig. 3 Growing DSM2 cluster with a circular (a,b) and at (c,d) in
work/products (Beading, Candles, Carving, Food Products, Soap, Weaving, etc.) ⃝I understand that if my work contains Indigenous visual representation that it is a reflection of the Indigenous culture of my native region. ⃝To the best of my knowledge, my work/products fall within Craft Council standards and expectations with respect to
Aug 13, 2020 · exponential functions. Unit 5.1 –Exponential Functions & Their Graphs So far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions. In this chapter, you will study two types of nonalgebraic functions –exponential funct
Unit 1 - Chapter 2 Oracle Built in Functions There are two types of functions in Oracle. 1) Single Row Functions: Single row or Scalar functions return a value for every row that is processed in a query. 2) Group Functions: These functions group the rows of
Functions are useful while writing SQL queries also. Functions can be applied to work on single or multiple records (rows) of a table. Depending on their application in one or multiple rows, SQL functions are categorised as Single row functions and Aggregate functions. 1.2.1 Single Row Functions These are al
29 Functions and their Graphs The concept of a function was introduced and studied in Section 7 of these notes. In this section we explore the graphs of functions. Of particular in-terest, we consider the graphs of linear functions, quadratic functions, cubic functions, square root functions,
These are functions that are not linear. Their graph is not a straight line. The degree of these functions is not 1. Non-linear functions can be: Quadratic functions – a polynomial of degree 2 Cubic functions – a polynomial of degree 3 Other higher order functions Exponential functions
Lesson 7-7B Comparing Functions Lesson Comparing Functions Chapter 7 7-7B BIG IDEA Different descriptions of functions make it possible to compare functions in a variety of ways. You have seen functions described verbally, in tables, by equations, and using graphs. With any of these descriptions, you can compare functions. Example 1
when using chords in your riffs and solos. mattwarnockguitar.com 14 After working this line in a few keys, experiment by adding bends to other chords in your vocabulary. Audio Example 14 The last line uses A7#9 chord shapes to create tension in this Stevie Ray Vaughan style line. Remember, you can use 7#9 chords anywhere in a blues, though they fit most naturally on the V7 chord. If you use .
