### Citations

58 |
A generalized Fibonacci sequence,
- Horadam
- 1961
(Show Context)
Citation Context ...f L is formed by the vectors V7 = < F M F I 1 ^ , - F -iFl 1 - , > and T2 = <F ^ F l * , -F F l * ->. 1 ^ 2 j L t + l ' - / i - 1 2jLX+l L jLl+1 2 j L t + l ' -jll 2 / X + l Indeed, from (3) and from =-=(2)-=- with n = -2fx - 1 and with m = jit - 1 and m = jLi, respectively, we deduce F F 0 = -F - (mod F 9 , - ) [X 2jLt - / L l - 1 2jLt+l and jll+1 2j[i -jU 2jLt+l -so that F V = VA (mod 1) , and1970] GENE... |

49 |
Functionen von beschränkter variation in der theori der gleichverteilung (in german),”
- Hlawka
- 1961
(Show Context)
Citation Context ...lue of the error not exceeding VD(S), where V is the sum of the two-dimensional variation of f over Q 2 in the sense of Vitali and of the (one-dimensional) variations of f(x,l) and f(l,y) over [0,1] (=-=[3]-=-; for a slight sharpening of this result, see [9]). Thus sets of points with low extreme discrepancies provide us with a method of numerical integration over Q 2 even when the integrand cannot be expa... |

43 |
The approximate computation of multiple integrals.
- Korobov
- 1959
(Show Context)
Citation Context ...da. JAs a result of a misprint, the exponent of 8 log p appears to be -s in Hlawka's paper, but his proof applies to lattice points satisfying (1). Thus his results are sharper than those of Korobov (=-=[7]-=-, [8]). 185186 A REMARKABLE LATTICE [March what follows, by a multiple modulo 1 of any vector, we understand the result of reducing modulo 1 each coordinate of the multiple of the given vector. In th... |

10 |
Good lattice points, discrepancy, and numerical integration
- Zaremba
- 1966
(Show Context)
Citation Context ...ly to exist. However, in two dimensions, the best lattice points in the sense of maximizing the ratio (g):p are obtained by putting P = F n , gi = 1, g2 = Fn__1 , where <F > are the Fibonacci numbers =-=[9]-=-. One finds, then, p(g) = F 0 , j n— & which is of a better order of magnitude than (1). The case when the integrand has not the required properties of periodicity can be reduced to the periodic case.... |

5 |
Uniform distribution modulo 1 and numerical analysis
- Hlawka
- 1964
(Show Context)
Citation Context ...s s and denote by p(g) the minimum of R(h) for all the vectors having integral coordinates not all zero, and satisfying g • h = 0 (mod 1) , where the dot denotes, as usual, the scalar product* Hlawka =-=[5]-=- describes pg as a good lattice point modulo p if (1) p(g) > pCSlogp) 1 " 8 i because upper bounds for the error of integration can be expressed as rapidly decreasing functions of (g), and he proves t... |

4 |
On a General Fibonacci Identity
- Halton
- 1965
(Show Context)
Citation Context ...f L is formed by the vectors V7 = < F M F I 1 ^ , - F -iFl 1 - , > and T2 = <F ^ F l * , -F F l * ->. 1 ^ 2 j L t + l ' - / i - 1 2jLX+l L jLl+1 2 j L t + l ' -jll 2 / X + l Indeed, from (3) and from =-=(2)-=- with n = -2fx - 1 and with m = jit - 1 and m = jLi, respectively, we deduce F F 0 = -F - (mod F 9 , - ) [X 2jLt - / L l - 1 2jLt+l and jll+1 2j[i -jU 2jLt+l -so that F V = VA (mod 1) , and1970] GENE... |

3 |
On second order recurrences
- Wyler
- 1965
(Show Context)
Citation Context ...cases of M = 2 and M = 3 being trivial, assume JU > 3. In our search for the required vectors, we can dismiss those which have a coordinate equal to, or bigger than, F XF~ in absolute value, since by =-=(4)-=-, their length exceeds F~'* . All linear combinations o>V\ + (N\ in which p ^ 0 are thereby excluded because of their abscissa if «jSZo and because of their ordinate if afi < 0. There remain the multi... |

1 |
n The Extreme and L
- ton, Zaremba
(Show Context)
Citation Context ...* . All linear combinations o>V\ + (N\ in which p ^ 0 are thereby excluded because of their abscissa if «jSZo and because of their ordinate if afi < 0. There remain the multiples of "vf. But ((78) in =-=[1]-=-) (7) F Q = F 2 + 2F F ,, ; 2jLt jLt fX j t l + 1 ' This identity can also be deduced from (4) noting that F 2 - F 2 = F 2 + 2F F It follows from (7) that when JU > 3, we have FQ > 2F 2 , so that Vj i... |

1 |
ff Zur angerfaherten Berechnung mehrfacher Integrale
- Hlawka
- 1962
(Show Context)
Citation Context ...cases of M = 2 and M = 3 being trivial, assume JU > 3. In our search for the required vectors, we can dismiss those which have a coordinate equal to, or bigger than, F XF~ in absolute value, since by =-=(4)-=-, their length exceeds F~'* . All linear combinations o>V\ + (N\ in which p ^ 0 are thereby excluded because of their abscissa if «jSZo and because of their ordinate if afi < 0. There remain the multi... |

1 |
Teoretikocislovye Metody v riblizhennom analize" (Numbertheoretical methods in numerical analysis
- Korobov
(Show Context)
Citation Context ...As a result of a misprint, the exponent of 8 log p appears to be -s in Hlawka's paper, but his proof applies to lattice points satisfying (1). Thus his results are sharper than those of Korobov ([7], =-=[8]-=-). 185186 A REMARKABLE LATTICE [March what follows, by a multiple modulo 1 of any vector, we understand the result of reducing modulo 1 each coordinate of the multiple of the given vector. In the cas... |

1 |
Good Lattice Points in the Sense of Klawka and MonteCarlo Integration
- Zaremba
- 1968
(Show Context)
Citation Context ...ons, and the phenomenon illustrated by the lattice L even more accentuated as the number of dimensions increases. in two dimensions becomes The present author, impressed by the pattern of L n, proved =-=[10]-=- that if a set of p points of the s-dimensional unit interval Q s is generated by a good lattice point, then there exists an s-1-dimensional linear variety (or hyperplane, to use a rather old-fashione... |

1 |
The Mathematical Basis of Monte-Carlo and Quasi-MonteCarlo Methods
- Zaremba
- 1968
(Show Context)
Citation Context ...e regarded as describing the equidistribution of S oyer Q 2 . If a single number is required to characterize this equidistribution, it can be obtained by taking any of the plausible norms of g. ([1], =-=[11]-=-) to call D(S) = sup Jg(x,y)| <x,yxEQ 2 In particular, it has been proposed the extreme discrepancy of S in order to distinguish it from other possible norms of g; the previously used term is simply d... |

1 |
The Fibonacci Numbers, translated from the Russian by Normal
- Vorobyov
- 1963
(Show Context)
Citation Context ...lue of the error not exceeding VD(S), where V is the sum of the two-dimensional variation of f over Q 2 in the sense of Vitali and of the (one-dimensional) variations of f(x,l) and f(l,y) over [0,1] (=-=[3]-=-; for a slight sharpening of this result, see [9]). Thus sets of points with low extreme discrepancies provide us with a method of numerical integration over Q 2 even when the integrand cannot be expa... |