Appendix

APPENDIX A

Further Reading

Calculus. Both books Calculus [39] and Calculus on Manifolds [38] byMichael Spivak are among the most brilliantly written books on moderncalculus, the latter being a concise and substantial introduction to theessential theorems of calculus. Advanced Calculus [23] by Lynn Loomisand Shlomo Sternberg is a comprehensive treatment of calculus, ODEsand calculus of manifolds, including classical mechanics.

Numerics. The book on numerical mathematics [36] by Hans RudolfSchwarz is a comprehensive and computer oriented reference. OrdinaryDifferential Equations by Fred Brauer and John Nohel is not only a reli-able reference to the theory of ODEs, but also includes some chapters onnumerics of ODEs, such as Runge-Kutta methods.

Categories. Saunders Mac Lane’s Categories for the Working Mathemati-cian [26] is a standard reference written by one of the fathers of categorytheory. Benjamin Pierce’s little book Basic Category Theory for ComputerScientists [30] is especially tailored to computer scientists.

Splines in all variants are a workhorse in computer graphics, espe-cially CAD (computer aided design) applications. Many books on numer-ical mathematics include a discussion of splines. The encyclopedic vol-ume Computer Graphics: Principles and Practice by James Foley and col-leagues [11] provides many details on theoretical and implementationissues.

Fourier Theory. Discussions of Fourier theory is generally found in anybook on numerical mathematics (see above). Since especially the Fast

336 Further Reading

Fourier Transform appears throughout digital signal processing, a bookon DSP like Richard Lyons’ [25] will be useful for the technical details.

Wavelets. Wavelet Transforms [32] by Raghuveer Rao and Ajit Bopar-dikar is a good reference to the theory of wavelets and includes manyalgorithms and examples. The article Fourier Analysis and Wavelet Anal-ysis [41] by James Walker provides an excellent comparison of Fourieragainst Wavelets algorithms.

Fractals. The approach used in this book is comprehensively developedin Michael Barnsley’s Fractals Everywhere [5]. It is mathematically com-plete and beautifully presented. For a wider scope see [29].

Neural Nets. Introduction to the Theory of Neural Computation [13] byJohn Hertz, Andreas Krogh and Richard Palmer offers a thorough treat-ment of a large variety of neural networks.

Probability Theory. The book on discrete structures [35] by ThomasSchickinger and Angelika Steger is a good reference focused on discreteprobability and statistics, including many useful exercises. For a thor-oughly mathematical treatment, refer to Kai Lai Chung’s A Course in Prob-ability Theory [8].

Lambda Calculus. Any text on functional programming worth its sub-ject features an introduction to the λ-calculus, for example the one byField and Harrison [10]. An accessible dedicated work is Hindley’s andSeldin’s Introduction to combinators and λ-calculus [16]. The classicaltreatise is Barendregt’s comprehensive The Lambda Calculus [4].

APPENDIX B

Bibliography

[1] Abelson, Harold & Sussman, Gerald Jay. Structure and Interpretationof Computer Programs. MIT Press, Cambridge 1996.

[2] Abraham, Ralph & Marsden, Jerrold E. Foundations of Modern Me-chanics. Addison Wesley, Reading 1967.

[3] Amann, Herbert. Analysis III. Birkhäuser, Basel 2001.

[4] Barendregt, Henk P. The Lambda Calculus. North Holland, Amster-dam 1984.

[5] Barnsley, Michael F. Fractals Everywhere. Morgan Kaufmann, SanFrancisco et al. 1993.

[6] Brauer, Fred & Nohel, John A. Ordinary Differential Equations. Ben-jamin, New York 1967.

[7] Bronstein, Ilya N. & Semendyayev, Konstantin A. Handbook of Math-ematics, Thomson Learning, 1991.

[8] Chung, Kai Lai. A Course in Probability Theory. Academic Press, Lon-don 2000.

[9] Cybenko, George. “Approximation by Superpositions of a SigmoidalFunction.” Mathematics of Control, Signals, and Systems, 2 (1989)pp. 303–314.

[10] Field, Anthony J. & Harrison, Peter. Functional Programming. Addi-son Wesley, Reading 1988.

[11] Foley, James D. et al. Computer Graphics: Principles and Practice inC. Addison Wesley, Reading 1995.

338 Bibliography

[12] Harary, Frank. Graph Theory. Addison Wesley, Reading 1972.

[13] Hertz, John, Krogh, Andreas & Palmer, Richard. Introduction to theTheory of Neural Computation. Addison-Wesley, Reading 1991.

[14] Hildebrandt, Stefan. Analysis I. Springer, Heidelberg et al. 2002.

[15] Hildebrandt, Stefan. Analysis II. Springer, Heidelberg et al. 2003.

[16] Hindley, J. Roger & Seldin, Jonathan P. Introduction to combinatorsand λ-calculus. Cambridge University Press, New York 1986.

[17] John, Fritz. Partial Differential Equations. Springer, Heidelberg et al.1978.

[18] Johnstone, Peter. Sketches of an Elephant: A Topos Theory Com-pendium. Oxford University Press, Oxford 2002.

[19] Jones, Huw. Computer Graphics through Key Mathematics. Springer,Heidelberg et al. 2001.

[20] Kolmogorov, Andrei Nikolaevich. Grundbegriffe der Wahrschein-lichkeitsrechnung. English Translation: Foundations of the Theory ofProbability. Chelsea, New York 1950.

[21] Krivine, Jean-Louis. Lambda-calcul, types et modèles. Masson, Paris1990.

[22] Lehn, Jürgen & Wegmann, Helmut. Einführung in die Statistik. Teub-ner, Stuttgart 1992.

[23] Loomis, Lynn H. & Sternberg, Shlomo. Advanced Calculus. AddisonWesley, Reading 1968.

[24] Louis, Alfred K. et al. Wavelets. Teubner, Stuttgart 1994.

[25] Lyons, Richard G. Understanding Digital Signal Processing. PrenticeHall, Upper Saddle River 1996.

[26] Mac Lane, Saunders. Categories for the Working Mathematician.Springer, Heidelberg et al. 1971.

[27] Mandelbrot, Benoit B. The Fractal Geometry of Nature. W H Freeman& Co., New York 1982.

[28] Mazzola, Guerino. The Topos of Music. Birkhäuser, Basel 2002.

[29] Peitgen, Heinz-Otto, Jürgens, Hartmut & Saupe, Dietmar. Chaos andFractals: New Frontiers of Science. Springer, Heidelberg et al. 1992.

Bibliography 339

[30] Pierce, Benjamin C. Basic Category Theory for Computer Scientists.MIT Press, Cambridge 1991.

[31] Pierce, Benjamin C. Types and Programming Languages. MIT Press,Cambridge 2002.

[32] Rao, Raghuveer M. & Bopardikar, Ajit S. Wavelet Transforms. AddisonWesley, Reading 1998.

[33] Rosenblatt, Frank. Principles of Neurodynamics. Spartan Books,Washington DC 1962.

[34] Rudin, Walter. Functional Analysis. McGraw-Hill, New York 1991.

[35] Schickinger, Thomas & Steger, Angelika. Diskrete Strukturen II.Springer, Heidelberg et al. 2002.

[36] Schwarz, Hans Rudolf. Numerische Mathematik. Teubner, Stuttgart1977.

[37] Scott, Dana. “Symbolic Computation and Teaching.” Artificial Intelli-gence and Symbolic Mathematical Computation, AISMC-3, vol. 1138pp. 1–20, 1996.

[38] Spivak, Michael. Calculus. Publish or Perish, Houston 1994.

[39] Spivak, Michael. Calculus on Manifolds. Westview Press, Boulder1971.

[40] Stein, Elias M. & Shakarchi, Rami. Fourier Analysis. Princeton Univer-sity Press, Princeton & Oxford 2003.

[41] Walker, James S. “Fourier Analysis and Wavelet Analysis.” Notices ofthe AMS, Volume 44, Number 6, pp. 658–670, 1997.

[42] Zerz, Eva & Helmke, Uwe & Prätzel-Wolters. Mathematical Theory ofNeural Networks. Berichte der AG Technomathematik 238, Kaiser-slautern 2001.

Index

Symbols! 153(?, ?) 4, 256(NM) 314(λx.M) 314(ψ,a0, b0) 221(f , g) 90fX ∗ fY 307U/F 1200 1531 153A(F,ζ, η, I, U) 115Acceptors 146AffR 146Aut(c) 143Automata 146Bdi (X) 171Bε(x) 13Bn 281B 281B(F, ζ, η, I, U) 116B(P0, P1, . . . Pd) 172B(n,p) 293BS p 181BXε (x) 15Bε(x) 5C0 146C 141C∞(U,V) 49Copp 147Cr 147C(?, y) 149C(R, n) 21C(r , l) 142

CN 147, 264CSwN 261CSw,+MCPercn,θ

261

CSw,−MCPercn,θ261

C0(X) 16Cat 148Cocone(D) 156Cone(D) 156Contra(X) 236Contra(X, Y) 236Conv(D) 163Djf(a) 51Dvf(a) 51DF0 40Drf 49∆d 161∆ 21∆(P0, . . . Pd) 162∆[P0, . . . Pd] 162Der(C∞(U)) 101Df 41Dfx 41Diff 147Diff (U,V) 41Digraph 144ET 273E 264E(X) 301EN 264End(c) 143Ex(λ) 298F(−∞) 291F(∞) 291F(x + 0) 291

342 Index

F(x − 0) 291FI(a0, b0) 221FV (t) 315F0 39FX 290FP,X 290Fern 250Fix(k) 107Frac(X) 238Func(C,D) 152Gasket 248Gr 145Graph 145H (X) 233H (f ) 234H 150Hm,n 223H+ 263H− 263HomC(r , l) 142Hom(r , l) 142I(P) 74Kε(x) 13K(a., b.) 75KXε (x) 15K5 33Kε(x) 5K3,3 33Koch 240LF 101Lψf 220L(V,C) 314L(f , P) 76L(x., δ) 310Li 169Λ 319LinR 146LinR(?, X) 149M(x0, f , δ) 83MCPercn,θ 261MK(f ) 76Metr 146Mon 145N⊥ 330N 264Nn(K) 245

Nθ 267Nw 259N (K, ε) 244N(µ,σ 2) 296NC 84NF 328Nat(F,G) 152O(R, n) 21Open(X,d) 106P(A|B) 286PF 328PRF 330P → Q 74Pλ 294Part(a, b) 74Part(a., b.) 74Percn,θ,σ 261R(a., b.) 79RK 130Rn 54Rand(A) 283Rings 145SD 265SwN 259Sequ(R, n) 21Sets 143Sierpinski 240Σ 21Sponge 248TU 42T 8Tf 42Tm(K) 165T rf 49T ba • f 220Tζ,η,F(y) 116Taylorx0

f(x) 55Taylornx0

f(x) 54

Thruk 267Thru 261Top(X, Y) 16U(f , P) 76V r (U) 99V(f) 65V(t) 315

Index 343

VX 282Vn 223Var(X) 305Wn 223X+ 261X− 262Y o 84[D, E] 104[d] 148[f ] 39[t/x]u 316[u] 120Erf 297�β 321�1β 321(nk

)24

�wN 259�N 259�f 265• 220f ◦ g 141x→G(y)F(x)→y 159c � d 237∫f 88∫f(x)dx 88∫

38∫ ba f 79∫ ba f(x)dx 79∫u F 120∫(a.,b.) f 79〈?, e.〉 218⊗

i Ri 178Rn1⊗n2⊗...nk 177∆d1⊗d2⊗...dk 177⊗ 177Y 84x 299∂Y 84∂fi/∂xj(x) 41u ∼ v 120≈ 157x̃ 300C×D 147f : r → l 142�o 259

�D 265�Sw.D� 266�SD 265k 327f ◦opp g 147f−1 143λ? 255

ihmj 274

ivmj 274ax 29actN 258arccos 63arcsin 63β(t) 324β−NF 322χn,k 222χ 222codom(f ) 142colim(D) 156cos 27cot 64df 100dimN 258dim(K) 244dim(K, ε) 244dom(f ) 142dx 38e 27etA(η) 118expa 29exp 24f(y)→ z 39false 326h(A,B) 233iX 234inf(A) 19iszero 327l(t) 314l2 218limi→∞ ci 9limy→x f(y) 39lim(D) 156loga 29log 27m(x0, f , δ) 83mK(f ) 76

344 Index

o(f ,x0) 83outN 258π 26plus 327plus 329pred 327σ 305sin 27succ 327supp(ψ) 215tan 64test 326true 326vn 223vol(I) 74vol(K) 75vol(S) 85w(u,p,w0) 268‖f‖ 90, 217‖x‖ 40 326

Aabsolutely convergent 23abstraction 314Acceptors 146Ackermann function 330actN 258activation function 254, 258adjoint

left - 159right - 159

adjunction 159AffR 146affine simplex 162algebra

Heyting - 8logical - 8

α-congruent 319α-conversion 319alternating series 22AND 262ANN 254application 314applicative order reduction 323arccos 63

Archimedes 73arcsin 63arithmetic mean 283, 299artificial neural network (ANN) 254atlas 70atom 314attractor 238Aut(c) 143auto 143Automata 146automorphism 143

group 143average 283axon 258

BBdi (X) 171Bε(x) 5Bε(x) 13B 281Bn 281BXε (x) 15B(n,p) 293B-spline 179

curve 181uniform - 181

Béziercovariance 172curve 171, 172spline 176surface 178, 179

Bézier, Pierre 161back-propagation algorithm 272ball, closed 13Banach space 116Barnsley’s fern 249basis

orthonormal - 219Schauder - 217

Bayes formula 286Bernoulli, Johann 73Bernstein

curve 171polynomial 171

β(t) 324β-contraction 321

Index 345

β−NF 322β-normal form 322

for a term 324β-redex 321β-reduction 321binomial

coefficient 24distribution 293

Bolzano, Bernhard 19Bolzano-Weierstrass theorem 11Borel set 281n-dimensional - 281

Borg spaceship 249bottom 330bound variable 315

change of - 319boundary 5, 84bounded

function 76sequence 10

box counting theorem 245BS p 181

CC 323C0 146C0(X) 16C 141C∞(U,V) 49Copp 147Cr 147Cr (U,V) 49C(?, y) 149C(r , l) 142C(R, n) 21calculus 38

integralis 73summatorius 73

car parking 123cardioid 66Cartesian product 153

category 147Cat 148category 14, 139, 141

Cartesian product - 147cocomplete - 156

cocone - 156complete - 156cone - 156finitely cocomplete- 156finitely complete - 156neural - 264of σ -algebras 281of categories 148of contractions 237opposite - 147

Cauchypolygon method 130product formula 26sequence 9, 107

Cauchy, Augustin 73central limit theorem 308certain event 280chain rule 43, 44change

of a bound variable 319of variables 89

chart 69Church numeral 327Church, Alonzo 313Church-Rosser theorem for β-

reduction 324Church-Turing thesis 313classical fourth-order Runge-Kutta

method 132classification of fractals 242closedε-ball 106ball 13cube 13, 75set 7, 106term 315

closure 84CN 147, 264cocone 154

category 156Cocone(D) 156codom(f ) 142codomain 142colim(D) 156colimit 156commutative diagram 150

346 Index

compact 13complete

category 156metric space 107

composition 141, 329, 331of functors 148

computability 125conditional probability 286cone 154

category 156Cone(D) 156conservative vector field 115constant 314

functor 148stream 255

continuous 16in x 16map between metric spaces 232

continuously differentiable 49continuously distributed random

variable 295Contra(X) 236Contra(X, Y) 236contractum 321contravariant 148control

point 169, 172theory 104, 121

Conv(D) 163convergent

absolutely - 23sequence 9, 106

convex 163hull 163

convolution 307coproduct 155cos 27cosine function 27cot 64cotangent function 64covariant 148CSwN 261CSw,+MCPercn,θ

261

CSw,−MCPercn,θ261

cube, closed 13, 75

cubic spline 165cumulative truncation error 128Cybenko approximation theorem

277Cybenko, George 277cycle 121

DDfx 41Djf(a) 51Drf 49Dvf(a) 51d’Alembert, Jean Le Rond 38data vector 299

deterministic 300de Casteljau algorithm 175de Faget de Casteljau, Paul 161decadic logarithm 29decreasing, monotonously 23definite integral 88deformation 219dendrite 258dense 221density 295Der(C∞(U)) 101derivation 98, 101derivative 41

Lie - 101, 104deterministic data vector 300Df 41df 100DF0 40diagram 150

commutative - 150process - 258scheme 150

dice space 280Diff 147Diff (U,V) 41diffeomorphism 63differentiable 41

continuously - 49in x 40manifold 70

differential 38, 100equation 97

Index 347

ordinary - 114partial - 114

differentiation 37Digraph 144dimN 258dim(K) 244dim(K, ε) 244dimension

fractal - 244fractional - 244geometric - 243of Koch curve 246of Sierpinski carpet 247

Dirac, Gabriel Andrew 35discrete

distribution 292random variable 282time 255time points 127

distance 4function 105

distribution 290binomial - 293discrete - 292exponential - 298function, n-dimensional - 307geometric - 292normal - 296Poisson - 294rectangular - 295standard normal - 296uniform - 295

divergent sequence 9dom(f ) 142domain 142dx 38dynamical system 97

Ee 27E 264ET 273E(X) 301Eilenberg, Samuel 139element 141, 143–145

elementaryevent 280function 331morphism 264

IO-shape of - 265empirical variance 300EN 264End(c) 143endo 143endomorphism 143

monoid 143epi 142epimorphism 142ε-ball 5, 106ε-cube 5equivalent metrics 242Erf 297error function 297error propagation 125Euler

formula 27number 27

Euler’sformula for polyhedra 31method 126, 129

event 280certain - 280elementary - 280impossible - 280space 280

eventually constant n-stream 255Ex(λ) 298exp 24expa 29expectation 301exponential

distribution 298function 24

for basis a 29exterior face 31

FF0 39FP,X 290FX 290face 31

348 Index

exterior 31interior 31

faithful functor 148false 326fan-in dimension 258feed-forward neural network 271Ferguson, James 161Fern 250FI(a0, b0) 221final object 153finitely cocomplete category 156finitely complete category 156Fix(k) 107fixpoint 107fluxion method 38form

Hermitian - 217uniform - 117

formal neuron 258Frac(X) 238fractal 238

deterministic - 238dimension 244

fractional dimension 244frame 219

theory 221free

input 266variable 315

Fubini’s Theorem 92full functor 148fully faithful functor 148Func(C,D) 152functor 140, 148

constant - 148faithful - 148full - 148fully faithful 148language - 149tangent - 150

functorial 43fundamental theorem of calculus 88fuzzy logic 9FV (t) 315

GGalilei, Galileo 37Galton’s board 308Gasket 248generator 144geometric

dimension 243distribution 292series 22

geometry of nature 231germ 39global integral curve 120Goupillaud, Pierre 215Gr 145gradient 101Graph 145Grossman, Alex 215Grothendieck, Alexander 139group action 220

HH 150H (X) 233H (f ) 234Hm,n 223h(A,B) 233Haar wavelet 215Hamiltonian mechanics 121harmonic series 22Haskell 313, 323Hausdorff metric 234Hausdorff-metric space 234Heaviside function 261Hebbian learning 254, 257

function 258Hermitian form 217Heyting algebra 8hidden layer 271Hilbert

space 217subspace 218

HomC(r , l) 142Hom(r , l) 142homeomorphism 16Hopfield network 271hyperplane 163

Index 349

IiX 234I(P) 74idempotent 141identity 141

functor 148left - 141right - 141

IMPLIES 262impossible event 280incompatible events 280indefinite integral 88independent

events 289random variables 307sequence of events 290

industrial shape design 161inf(A) 19inferential statistics 310initial object 153initially constant n-stream 255input 258

layer 271neuron 259sensorial - 259vector 259

input-output shape (IO-shape) 259instantaneous velocity 38, 40integrable function 79integral

curveglobal - 120local - 114

definite - 88indefinite - 88iterated - 92Riemann - 79sign 38, 73

integration by parts 90interior 84

face 31intermediate value theorem 19interval set 74invariant, numerical 243inverse 143

function theorem 61

left - 142right - 143

IO-shape 259of a morphism 265of a neural network 265of an elementary morphism 265

iso 143isomorphic categories 148isomorphism 143iszero 327iterated integral 92

JJacobian 52

matrix 41

KK3,3 33K5 33Kε(x) 5Kε(x) 13KXε (x) 15K(a., b.) 75Kepler, Johannes 73knot 169Koch 240Koch curve 240

dimension 246Kolmogorov axioms 283Kolmogorov, Andrei Nikolaevich 283Kuratowski’s theorem 33

Ll2 218LF 101Li 169Lψf 220L(f , P) 76l(t) 314L(V,C) 314L(x., δ) 310Lagrange

interpolation 168polynomial 169

Λ 319Λd 170

350 Index

lambda-calculusλ-calculus

pure - 314λ-calculus 313λ-compatible relation 318λ-term 314language functor 149Laplace principle 285Lawvere, William 139layer 270

hidden - 271input - 271output - 271

lazy evaluation 323learning 275

rate 274supervised - 276unsupervised - 276

Lebesgue, Henri Léon 74left

adjoint 159identity 141inverse 142

leftmostinnermost 323outermost 323

Leibniz criterion 22Lie

bracket 104, 123derivative 101, 104product 98

limi→∞ ci 9limy→x f(y) 39lim(D) 156limit 38, 156

of a function 39one-sided - 88

LinR 146LinR(?, X) 149linear

ODE 117spline 165

Lipschitz condition 116LISP 313local

fundamental theorem of ODEs 116

information flow 113integral curve 114truncation error 128

locally Lipschitz 116log 27loga 29logarithm

decadic - 29for basis a 29natural - 27

logicfuzzy - 9spatial - 4three-valued - 8two-valued - 8

logicalalgebra 8function 262

lower sum 76

MMK(f ) 76mK(f ) 76M(x0, f , δ) 83m(x0, f , δ) 83Mac Lane, Saunders 139Mallat, Stéphane 226Mandelbrot, Benoit 231manifold, differentiable 70maximum likelihood estimate 310Maxwell, James Clerk 37McCulloch-Pitts neuron 261MCPercn,θ 261mean value theorem 49

of integral calculus 80measure zero 81median 300Menger sponge 248methodus fluxionum 73Metr 146metric 4Mexican hat wavelet 216Meyer wavelet 216Meyer, Yves 226minimization 331Mon 145

Index 351

mono 142monomorphism 142monotonously decreasing 23Monty Hall problem 287Morlet wavelet 216Morlet, Jean 215morphism 141

elementary - 264of σ -algebras 281of cocones 156of cones 156of contractions 236

motion planning 121MSA 226multiplication formula 289multiscale analysis 226

NN⊥ 330N 264Nn(K) 245Nθ 267Nw 259N (K, ε) 244N(µ,σ 2) 296n-layered

digraph 270perceptron 271

n-stream 255domain 255eventually constant - 255initially constant - 255periodic - 255shifted - 255

Nat(F,G) 152natural 152

logarithm 27transformation 152

naturality of Hausdorff metric 235NC 84network, neural 253neural

category 264network 253

feed-forward - 271recurrent - 271

neuron 253formal - 258input - 259McCulloch-Pitts - 261output - 265

Newton’s method 111Newton, Isaac 37NF 328norm 4, 90, 217normal distribution 296normal order reduction 323NOT 262numerical

invariant 243mathematics 125solutions 125

NURBS 180

Oo(f ,x0) 83O(R, n) 21object 141

final - 153initial - 153universal - 153

occurrence of a variable 315ODE 114one-sided limit 88open set 7, 106Open(X,d) 106opposite category 147OR 262ordinary differential equation 114Oresme, Nicholas 37orthonormal 218

basis 219wavelet 222

oscillation 83outN 258outcome 280output 258

function 258layer 271neuron 265

352 Index

PPλ 294P(A|B) 286Part(a, b) 74Part(a., b.) 74partial

derivative 51differential equation 114

partial recursive function 330partition 74Pascal triangle 25path category 144PDE 114Peano axioms 326Percn,θ,σ 261perceptron 261n-layered - 271convergence theorem 269learning algorithm 268

periodic n-stream 255PF 328phase portrait 120π 26Picard-Lindelöf iteration procedure

117, 126plus 329plus 327Poisson distribution 294polar coordinate representation 27positive definiteness 105power series 24pred 327PRF 330primitive

function 88recursion 329, 331recursive function 328

PRNG 296probability

measure 283space 284

process diagram 258projection 329pseudo-random number generator

(PRNG) 296punctured 38

pure λ-calculus 314pushout 324

Qquadratic spline 165quincunx 308

RRn 54R(a., b.) 79Rand(A) 283random variable 282n-dimensional - 282, 306continuously distributed - 295discrete - 282symmetric - 302

range of sample series 300rectangular distribution 295recurrent neural network 271recursion theory 110refinement

common - 74relation 74

relative topology 15remainder, n-th - 54representation of partial recursive

functions 332retraction 143Riemann integral 79Riemann, Bernhard 70, 73right

adjoint 159identity 141inverse 143

Rings 145RK 130robot motion 122robotics 121roots of a function 111Rosenblatt, Frank 272round-off error 127Runge-Kutta method 126, 130

classical fourth-order - 132

SSD 265

Index 353

SwN 259saddle point 53sample series 299saturated 271scalar product 4, 90Schauder basis 217Scheme 313scheme, diagram 150Schuster, Seymour 35Schwarz inequality 90scope of a variable 315section 142self-organization 276semigroup 237semiring 237sensorial input 259Sequ(R, n) 21sequence

bounded - 10Cauchy - 9, 107convergent - 9, 106divergent - 9

series 21alternating - 22geometric - 22harmonic - 22power - 24sample - 299

setBorel - 281closed - 7interval - 74of cubes 75of values 282open - 7, 106

Sets 143shift operator 255shifted n-stream 255Sierpinski 240Sierpinski

carpet 240dimension 247

gasket 248σ 305σ -Alg 281σ -algebra 279

σ -algebragenerated by E 281

sigmoid function 261simplex

affine - 162standard - 161

sin 27sine function 27SML 313, 323space

Banach 116dice - 280event - 280Hilbert 217probability - 284state - 258

spline 161cubic - 165curve 165function 165linear - 165quadratic - 165surface 165

Sponge 248standard

deviation 300, 305normal distribution 296simplex 161

state 259space 258

of a neural network 265statistics, inferential 310stereographic projection 30stream 255

constant - 255domain 255

strict evaluation 323subsequence 11subspace, Hilbert 218substitution 316

canceling 317commutation 317composition 317inversion 317

succ 327successor function 329

354 Index

sumlower - 76map 237upper - 76

supervised learning 276supp(ψ) 215support 215surface, Bézier 178, 179symmetric random variable 302synaptic weight 254

TT 8Tf 42Tm(K) 165T rf 49TU 42tan 64tangent 42

bundle 42function 64functor 150map 42

Taylorapproximation 126polynomial 54

Taylorx0f(x) 55

Taylornx0f(x) 54

Taylor’s formula 55tensor product 176

sign 177spline 176

test 326theoretical computer science 139three-valued logic 8threshold 261Thru 261Thruk 267time delay 274time-dependent

data stream 254vector field 115

Top(X, Y) 16topology 8, 106

relative - 15topos 139

torus 70transformation, natural 152triangle inequality 90, 106true 326truncation error 128Turing adjunction 159Turing machine 313two-scale

equation 223relation 223

two-valued logic 8type of functions 165

UU(f , P) 76unconditionally convergent 26uniformB-spline 181distribution 295form 117

universalconstruction 140object 153

unsupervised learning 276upper sum 76

VVn 223vn 223V r (U) 99VX 282V(f) 65V(t) 315Var(X) 305variable 314

bound - 315free - 315occurrence of - 315scope of - 315

variance 305empirical - 300

vectordata - 299field 97, 98

velocity field 113vol(I) 74

Index 355

vol(K) 75vol(S) 85volume 74, 85

of a cube 75

WWn 223wavelet 215, 220

coefficient 219, 220Haar - 215Mexican hat - 216Meyer - 216Morlet - 216orthonormal - 222

weight matrix 271

XXOR 262, 269

YY -combinator 328

Zzero

fiber 65function 329

Zwischenwertsatz 19