M
MacDonald function, 509 Maclaurin formula, 415 series expansion, 418 Macsyma (computer algebra system), 953 majorant series, 413 manifold center manifold theorem local, 835 mappings, 835 of solutions, 890 stable, 806, 812 unstable, 806, 812 manipulation algebraic expressions, 979 non-polynomial expressions, 981 mantissa, 10, 939 manyness set, 298 Maple (computer algebra system), 953 Maple addendum to syntax, 978 algebraic expressions, 971 manipulation, 981 multiplication, 981 array, 973 attribute, 978 basic structure elements, 968 context, 978 conversion of numbers, different bases, 970 Maple (continued) differential equations, 997 differential operators, 977 differentiation, 995 eigenvalues, eigenvectors, 992 elements of linear algebra, 989 environment variable, 978 equation one unknown, 986 transcendental, 987 factorization of polynomials, 982 floating-point number, conversion, 970 formula manipulation, introduction, 953 functions, 975 graphics, 1005 introduction, 955 three-dimensional, 1008 two-dimensional, 1005 help and information, 978 input, output, 949, 969 input—output, 953 integral definite, 997 indefinite, 996 multiple, 997 lists, 972 manipulation, general expressions, 983 matrices, 973 numerical calculations, introduction, 954 numerical mathematics, 949 differential equations, 952 equations, 950 expressions and functions, 949 integration, 951 object classes, 968 objects, 968 operations important, 986 polynomials, 983 operators functional, 976 important, 971 partial fraction decomposition, 983 procedures, 975 programming, 977 sequences, 972 short characteristc, 953 special package plots, 1007 system description, 968 system of equations, 987 systems of equations, linear, 990 table structures, 973 types, 968 types of numbers, 970 vectors, 973 mapping, 294, 296 between groups, 302 bijective, 296, 600 complex number plane, 684 conformal, 674 circular transformation, 676 exponential function, 679 fixed-point, 675, 676 inversion, 676 linear fractional, 676 linear function, 675 logarithm, 678 quadratic function, 677 square root, 677 sum of linear and fractional linear functions, 678 contracting, 608 equivalent, 843 function, 48, 294 Henon mapping, 812 horseshoe, 827 injective, 296, 600 inverse mapping, 296 kernel, 302 lifted, 843 linear, 317, 319, 600 modulo, 816 one-to-one, 296 Poincare, 827 Poincare mapping, 808 reduced, 835 regular, 318 rotation mapping, 817 shift, 820 surjective, 296, 600 tent, 816 topological conjugate, 813 unit circle, 843 marginal distribution, 754 mass double integral, 474 line integral, first type, 464 triple integral, 480 matching, 356 maximal, 356 perfect, 356 saturated, 356 Mathcad (computer algebra system), 953 Mathematica (computer algebra system), 953 Mathematica 3D graphics, 1004 algebraic expressions, manipulation, 979 algebraic expressions, multiplication, 979 apply, 965 attribute, 966 basic structure element, 956 context, 966 curves parametric representation, 1003 two-dimensional, 1002 differential and integral calculus, 992 differential equation, 994 differential quotient, 992 differentiation, 965 eigenvalues, eigenvectors, 989 elements, 956 elements of linear algebra, 988 equation, 985 manipulation, 984 transcendent, 985 expression, 956 factorization, polynomials, 980 Mathematica (continued) FixedPoint, 965 FixedPointList, 965 floating-point number, conversion, 958 formula manipulation, introduction, 953 function, inverse, 965 functional operations, 964 functions, 963 graphics, 998 functions, 1001 introduction. 955 options, 1000 primitives, 998, 999 head,956 input and output, 946 input-output, 953, 957 integral definite, 994 indefinite, 993 multiple, 994 lists, 959 manipulation with matrices. 962 manipulation with vectors. 962 Map, 965 matrices as lists, 961 messages, 967 Nest, 965 NestList, 965 numerical calculations, introduction, 954 numerical mathematics, 946 curve fitting, 946 differential equations, 948 integration, 948 interpolation, 947 polynomial equations, 947 objects, three-dimensional, 1004 operations, polynomials, 980 operators, important, 959 partial fraction decomposition, 980 pattern, 964 programming, 966 short characteristc, 953 surfaces, 1004 surfaces and space curves, 1003 syntax, additional information, 966 system description, 956 system of equations, 986 general case, 988 special case, 988 types of numbers, 957 vectors as lists, 961 mathematics, discrete, 286 matrices arithmetical operations, 254 associative law, 254 commutative law, 254, 255 distributive law, 254 division, 254 eigenvalue, 278 eigenvalue problem, 278 eigenvectors, 278 equality, 254 multiplication, 254 powers, 258 rules of calculation, 257 matrix, 251 adjacency, 350 adjoint, 251, 260 admittance, 355 anti-Hermitian, 253 antisymmetric, 252 block tridiagonal, 912 complex, 251 conjugate, 251 deflation, 284 degree, 355 diagonal, 252 distance, 351 extended coefficient, 892 full rank, 277 fundamental, 802 Hermitian, 253 Hessian, 869 identity, 253 incidence, 350 inverse, 256, 260 inverse calculation, 272 Jacobian, 897 Jordan,282 lower triangular, 892 main diagonal element, 252 monodromy, 803, 812 normal, 252 of coefficients, 272 augmented, 273 orthogonal, 257 rank, 255 real, 251 rectangular, 251 regular, 256 rotation, 257 scalar, 252 self-adjoint, 253 singular, 256 size, 251 skew-symmetric, 252 spanning tree theorem, 355 sparse, 912 spur, 252 square, 251, 252 stochastic, 766 symmetric, 252 trace, 252 transposed, 251 triangle decomposition, 890 triangular, 253 unitary, 257 upper triangular, 892 zero, 251 matrix exponential function, 802 matrix product, disappearing, 257 max-min composition, 371 maximum absolute, 60, 123, 390 global, 390 point, 847 maximum (continued) relative, 390 maximum-criterion method, 373 Maxwell, diagonal method, 684 mean arithmetic, 19, 753, 790 estimate, 771 geometric, 20 golden, 845 harmonic, 20 quadratic, 20 sample function, 771 statistics, 773 value, 19 weighted, 753, 790, 793 mean squares problem different versions, 403 non-linear, 922 mean squares value problem linear, 277 rank deficiency case, 278 mean value formula, 899 mean value function, integral calculus, 444 mean value method, 107 mean value theorem differential calculus, 389 generalized, 390 integral calculus, 444 generalized, 445 measure concentrated on a set, 816 counting, 635 degrees, 130 dimension, 824 Dirac, 636, 816 ergodic, 817 function, convergence theorems. 638 Hausdorff, 822 invariant, 816 Lebesgue, 636, 816 natural, 816 physical, 817 probability, 638 invariant, 816 SBR (Sinai-Bowen-Ruelle), 817 a-algebra, 635 (T-finite, 638 support, 816 measured value, 787 measurement error, 787 error density, 788 error, characterization, 788 protocol, 788 measuring error, distribution, 788 median, 142 sample function, 772 statistics, 774 triangle, 132 Mellin transformation, 707 Melnikov method, 841 membership degree, 360 function, 360, 361 bell-shaped, 362 trapezoidal, 361 relation, 290 meridian, 160, 244 convergence, 169 tangential, 176 method Bairstow, 889 barrier, 876 Bernoulli, 889 bisection, 283 broken line, Euler, 904 center of area, 374 center of gravity, 373 generalized, 374 parametrized, 374 Cholesky, 278, 893 collocation boundary value problem, 909 integral equation, 576 partial integral equation, 913 cryptoanalysis, 337 cryptography, 335 cutting plane, 877 descent, 868 difference ordinary differential equations, 908 partial differential equations, 911 extrapolation, 905 Fibonacci, 868 finite difference, 534 finite element, 534, 561. 913 Galerkin, 909 Gauss elimination, 890 Gauss-Newton, 897 derivative free, 898 Gauss-Seidel, 895 gradient. 561 Graeffe, 889 Green, 531, 532 Hildreth-d'Esopo, 867 Householder, 278, 921 Hungarian, 860 inner digits of squares, 783 integrating factor, 490 iteration, 283, 884, 890, 895 ordinary, 884, 896 Jacobi, 283, 895 Kelley's, optimization, 877 Lagrange multiplier, 862 Lanczos, 283 largest area. 374 least squares, 107, 890, 909, 913, 919, 921 Mamdani, 374 maximum-criterion, 373 mean-of-maximum, 373 Monte Carlo, 783 multi-step, 905 multiple-target, 910 Newton, 885, 897 modified, 885 non-linear operators, 632 method (continued) operator, 709 orthogonalization, 278, 893, 894 parametrized center of area, 374 pivoting, 271 polygonal, Euler, 904 predictor-corrector, 906 public key, 338 regula falsi, 886 relaxation, 896 Riemann, 530 Ritz, 560, 909 Romberg, 901 RSA encryption, 339 Runge-Kutta, 904 separation of variables, 489 shooting, 910 single-target, 910 SOR (successive overrelaxation), 896 statistical experiment, 787 steepest descent, 869 Steffensen, 886 successive approximation Banach space, 621 Picard, 496 Sugeno, 375 transformation, 283 undetermined coefficients, 16, 497 variation of constants, 501 Vigenere cypher, 336 Wolfe, 865 metric, 604 Euclidean, 604 surface, 247 Meusnier, theorem, 247 midpoint line segment plane, 192 space, 213 midpoint rule, 906 minimal surface, 249 minimum absolute, 60. 123, 390 global, 390 point, 862 global, 862 local, 862 problem, 847 relative, 390 Minkowski inequality, 32 integrals, 32 series, 32 mixed product, 187 modal value, statistics, 774 mode, statistics, 774 model Hopf-Landau, turbulence, 841 urn model, 755 module of an element, 603 modulo congruence, 297, 327 mapping, 816 modulus, analytic function, 672 de Moivre formula complex number, 37 hyperbolic functions, 90 trigonometric functions, 79 Mollweide equations, 142 moment of inertia, 263, 452 double integral, 474 line integral, first type, 464 triple integral, 480 moment, order n, 753 monodromy matrix, 803, 804, 812 monotonicity, 635 monotony, 388 function, 49 sequence, numbers, 404 Monte Carlo method, 783 application in numerical mathematics, 785 usual, 785 Monte Carlo simulation, 783 example, 784 Morse-Smale system, 815 multi-index, 613 multi-scale analysis, 743 multi-step method, 905 multiple integral, 471 transformation, 709 multiple-target method, 910 multiplication complex numbers, 36 computer calculation, 940 polynomials, 11 rational numbers, 1 multiplicity, of divisors, 320 multiplier,490, 803, 804, 812 Lagrange method, 403 mutation, 870