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Varga R.S. — Scientific Computations on Mathematical Problems and Conjectures
Varga R.S. — Scientific Computations on Mathematical Problems and Conjectures



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Название: Scientific Computations on Mathematical Problems and Conjectures

Автор: Varga R.S.

Аннотация:

Studies the use of scientific computation as a tool in attacking a number of mathematical problems and conjectures. In this case, scientific computation refers primarily to computations that are carried out with a large number of significant digits, for calculations associated with a variety of numerical techniques such as the (second) Remez algorithm in polynomial and rational approximation theory, Richardson extrapolation of sequences of numbers, the accurate finding of zeros of polynomials of large degree, and the numerical approximation of integrals by quadrature techniques.
The goal of this book is not to delve into the specialized field dealing with the creation of robust and reliable software needed to implement these high-precision calculations, but rather to emphasize the enormous power that existing software brings to the mathematician's arsenal of weapons for attacking mathematical problems and conjectures.
Scientific Computation on Mathematical Problems and Conjectures includes studies of the Bernstein Conjecture of 1913 in polynomial approximation theory, the "1/9" Conjecture of 1977 in rational approximation theory, the famous Riemann Hypothesis of 1859, and the Polya Conjecture of 1927. The emphasis of this monograph rests strongly on the interplay between hard analysis and high precision calculations.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 128

Добавлена в каталог: 31.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Admissible kernel      48—49
Alternation set      82 83 86 88 101
Asymptotic series      69
Asymptotic series expansion      16—17
Beauzamy and Enflo      103
Beauzamy — Enflo extension      107
Bernstein conjecture      3—4 13 18
Bernstein constant      3—4 6—7 9 13 15 17
Best uniform approximation      1 4—5 6 11 81—83 86 98-100
Best uniform rational approximation      18
Binet function      69
Caratheodory — Fej$\acute{e}$r method      31 38
Central limit theorem      113
Chebyshev constants      24 26—27 32
Chebyshev polynomial      5 88 90
Chebyshev rational approximations      24 37
Chebyshev semi-discrete approximations      24—25
Chebyshev semi-discrete method      25—26
Concentration $d$ at degree $k$      103 116
Concentration at low degrees      103 119
Conjecture      17 115
Critical line      40 61
Critical strip      39—40 62
de Bruijn — Newman constant      52—56 58—60 62 63
De la Vall$\acute{e}$e — Poussin theorem      86
Enestrom — Kakeya theorem      65 79
Entire function      33 41—43 48—49 51 53—54 58 61 115 119
Equioscillation      10 91 96
Extremal polynomials      106—107 114 117
Fourier cosine transform      52
Fourier transforms      41 48 62
Gaussian elimination      11 25
Geometric convergence      26—27 33—34 38 59
Geometric convergence rate      32
Halphen constant      36
Hankel matrix      32
Hardy space      115 119
Hermite — Biehler theorem      61 62
Hurwitz polynomials      107
Hurwitz’s theorem      71
Jackson theorem      1
Jensen polynomial      54—57
Jensen’s formula      104 106—107 119
Jensen’s Inequality      103—104 105 107 119—120
Laguerre inequalities      41—42
Laguerre — P$\acute{o}$lya class      42 54
Length of the alternation set      82 101
Logarithmic concavity      47—49
Mobius transformation      93 104
Modulus of continuity      1
Moments      42 45—46 53 56—60 62
Near-circularity      91
Non — Pad$\acute{e}$ rational approximation      35
Normalized partial sums      65 68
Operator norm      25
Optimization routine      15
oscillations      87—88 97
Outer function      116
P$\acute{o}$lya Conjecture      41 44—49
Pad$\acute{e}$ rational approximations      24 26 34 38 80
Partial fraction decomposition      94
Partial sums      65 69 80
Prime number theorem      40
Remez algorithm      5 9—12 19 26—27
Richardson extrapolation      4 15—16 20 27 30—31
Riemann $\xi$-function      41 52
Riemann $\zeta$-function      39 41 61—63
Riemann hypothesis      39—44 49 53 60—61 62
Riemann’s conjecture      62
Romberg integration method      55
Semi-discrete approximation      24
Singular value      32
Spikes      94 96
Star-shaped      69 79
Stirlings asymptotic series formula      68
Szeg$\ddot{o}$ curve      65 75 77 79
Trapezoidal rule      59
Trivial zeros      39
Tur$\acute{a}$n differences      43 45 47
Tur$\acute{a}$n inequalities      41 43 45 61
Uniform norm      1
Univalent      72
Universal factors      48 51 53
Vector norms      25
Weierstrass approximation theorem      1
“1/9” Conjecture      23 26—27 32 35 37
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