Book contents
- Frontmatter
- Contents
- Preface
- 1 Some highlights of Harald Niederreiter's work
- 2 Partially bent functions and their properties
- 3 Applications of geometric discrepancy in numerical analysis and statistics
- 4 Discrepancy bounds for low-dimensional point sets
- 5 On the linear complexity and lattice test of nonlinear pseudorandom number generators
- 6 A heuristic formula estimating the keystream length for the general combination generator with respect to a correlation attack
- 7 Point sets of minimal energy
- 8 The cross-correlation measure for families of binary sequences
- 9 On an important family of inequalities of Niederreiter involving exponential sums
- 10 Controlling the shape of generating matrices in global function field constructions of digital sequences
- 11 Periodic structure of the exponential pseudorandom number generator
- 12 Construction of a rank-1 lattice sequence based on primitive polynomials
- 13 A quasi-Monte Carlo method for the coagulation equation
- 14 Asymptotic formulas for partitions with bounded multiplicity
- 15 A trigonometric approach for Chebyshev polynomials over finite fields
- 16 Index bounds for value sets of polynomials over finite fields
- 17 Rational points of the curve over
- 18 On the linear complexity of multisequences, bijections between ℤahlen and ℕumber tuples, and partitions
- Plate section
- References
3 - Applications of geometric discrepancy in numerical analysis and statistics
Published online by Cambridge University Press: 18 December 2014
Book contents
- Frontmatter
- Contents
- Preface
- 1 Some highlights of Harald Niederreiter's work
- 2 Partially bent functions and their properties
- 3 Applications of geometric discrepancy in numerical analysis and statistics
- 4 Discrepancy bounds for low-dimensional point sets
- 5 On the linear complexity and lattice test of nonlinear pseudorandom number generators
- 6 A heuristic formula estimating the keystream length for the general combination generator with respect to a correlation attack
- 7 Point sets of minimal energy
- 8 The cross-correlation measure for families of binary sequences
- 9 On an important family of inequalities of Niederreiter involving exponential sums
- 10 Controlling the shape of generating matrices in global function field constructions of digital sequences
- 11 Periodic structure of the exponential pseudorandom number generator
- 12 Construction of a rank-1 lattice sequence based on primitive polynomials
- 13 A quasi-Monte Carlo method for the coagulation equation
- 14 Asymptotic formulas for partitions with bounded multiplicity
- 15 A trigonometric approach for Chebyshev polynomials over finite fields
- 16 Index bounds for value sets of polynomials over finite fields
- 17 Rational points of the curve over
- 18 On the linear complexity of multisequences, bijections between ℤahlen and ℕumber tuples, and partitions
- Plate section
- References
Summary
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
- Type
- Chapter
- Information
- Applied Algebra and Number Theory , pp. 39 - 57Publisher: Cambridge University PressPrint publication year: 2014
References
[1] and , Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality. Submitted 2014, available at http://arxiv.org/pdf/1406.0230.pdf.
[2] and , Probabilistic star discrepancy bounds for double infinite random matrices. In: , , and (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2012, pp. 271–287. Springer-Verlag, Heidelberg, 2013.
[3] , and , Point sets on the sphere S2 with small spherical cap discrepancy. Discrete Comput. Geom. 48, 990–1024, 2012.Google Scholar
[4] and , Simulation with fluctuating and singular rates. Preprint, available at http://arxiv.org/pdf/1310.4555.pdf.
[5] , Sums of distances between points on a sphere an application of the theory of irregularities of distribution to discrete geometry. Mathematika 31, 33–41, 1984.Google Scholar
[7] and , Irregularities of Distribution. Cambridge Tracts in Mathematics, volume 89. Cambridge University Press, Cambridge, 2008.
[8] and , On the small ball inequality in three dimensions. Duke Math. J. 143, 81–115, 2008.Google Scholar
[9] , and , On the small ball inequality in all dimensions. J. Funct. Anal. 254, 2470–2502, 2008.Google Scholar
[10] , , and , Directional discrepancy in two dimensions. Bull. London Math. Soc. 43, 1151–1166, 2011.Google Scholar
[11] and , A simple proof of Stolarsky's invariance principle. Proc. Am. Math. Soc. 141, 2085–2096, 2013.Google Scholar
[12] , Quasi-random techniques for Monte Carlo methods. PhD Dissertation, Claremont Graduate School, Claremont, CA, 1976.
[13] and , Explicit constructions in the classical mean squares problem in irregularities of point distribution. J. Reine Angew. Math. 545, 67–95, 2002.Google Scholar
[14] , and , Consistency of Markov chain quasi-Monte Carlo on continuous state spaces. Ann. Stat. 39, 673–701, 2011.Google Scholar
[15] , Non-Uniform Random Variate Generation. Springer, New York, 1986.
[16] , Discrepancy bounds for infinite-dimensional order two digital sequences over F2. J. Number Theory 136, 204–232, 2014.Google Scholar
[17] and , Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010.
[18] and , Optimal L2 discrepancy bounds for higher order digital sequences over the finite field F2. Acta Arith. 162, 65–99, 2014.Google Scholar
[19] , and , Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte , 2013.
[20] , Irregularities of distribution and convex sets. Österreichisch-Ungarisch-Slowakisches Kolloquium über Zahlentheorie (Maria Trost, 1992). Grazer Math. Ber., 318, pp. 9–16. Karl-Franzens-Univ. Graz, Graz, 1993.
[21] and , Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, volume 1651. Springer-Verlag, Berlin, 1997.
[22] , Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41, 337–351, 1982.Google Scholar
[23] and , Spherical designs, discrepancy and numerical integration. Math. Comp. 60, 327–336, 1993.Google Scholar
[24] , On Roth's method in the theory of irregularities of point distributions. Recent Progress in Analytic Number Theory (Durham, 1979), volume 2, pp. 79–94. Academic Press, London, 1981.Google Scholar
[25] , On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2, 84–90, 1960.Google Scholar
[26] , Monte Carlo methods for solving multivariable problems. Ann. N. Y. Acad. Sci. 86, 844–874, 1960.Google Scholar
[27] , A generalized discrepancy and quadrature error bound. Math. Comp. 67, 299–322, 1998.Google Scholar
[28] , Funktionen von beschränkter Variation in der Theorie der Gleichverteilung (German). Ann. Mat. Pura Appl. 54, 325–333, 1961.Google Scholar
[29] , and , Automatic nonuniform random variate generation. Statistics and Computing. Springer-Verlag, Berlin, 2004.
[30] , Een algemeene stelling uit de theorie der gelijkmatige verdeeling modulo 1, Mathematica. 11, 7–11, 1942.Google Scholar
[31] and , Uniform Distribution of Sequences. Dover, New York, 1974.
[32] , , , and , Quasi-Monte Carlo for highly structured generalised response models. Methodol. Comput. Appl. Probab. 10, 239–275, 2008.Google Scholar
[33] , Discrepancy estimates for sets with small boundary. Stud. Sci. Math. Hung. 30, 105–109, 1995.Google Scholar
[34] , and , On array-RQMC for Markov chains: mapping alternatives and convergence rates. In: and (eds.), Monte Carlo and quasi-Monte Carlo Methods 2008, pp. 485–500. Springer, Berlin, 2009.
[35] , Discrepancy estimates of completely uniformly distributed and pseudorandom number sequences. Int. Math. Res. Notices, 1231–1251, 1999.Google Scholar
[36] , and , Hecke operators and distributing points on the sphere. I. Frontiers of the mathematical sciences 1985 (New York, 1985). Commum. Pure Appl. Math. 39, S149–S186, 1986.
[37] , and , Hecke operators and distributing points on S2. II. Commum. Pure Appl. Math. 40, 401–420, 1987.Google Scholar
[38] , Geometric Discrepancy. An illustrated guide. Algorithms and Combinatorics, volume 18. Springer-Verlag, Berlin, 1999.
[40] and , Smoothness and dimension reduction in quasi-Monte Carlo methods. Math. Comput. Modell. 23, 37–54, 1996.Google Scholar
[41] and , Distances of probability measures and uniform distribution mod 1. Math. Z. 142, 195–202, 1975.Google Scholar
[43] , Methods for estimating discrepancy. Applications of Number Theory to Numerical Analysis (Proc. Symp., Univ. Montreal, Montreal, 1971), pp. 203–236. Academic Press, New York, 1972.
[44] , Low-discrepancy and low-dispersion sequences. J. Number Theory 30, 51–70, 1988.Google Scholar
[45] , Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics, volume 63. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
[46] and , Diskrepanz und Distanz von Maßen bezüglich konvexer und Jordanscher Mengen (German). Math. Z. 144, 125–134, 1975.Google Scholar
[47] and , Low-discrepancy sequences and global function fields with many rational places. Finite Fields Appl. 2, 241–273, 1996.Google Scholar
[48] and , Rational Points on Curves over Finite Fields: Theory and Applications. London Mathematical Society Lecture Note Series, volume 285. Cambridge University Press, Cambridge, 2001.
[49] and , Monte Carlo Statistical Methods, second edition. Springer-Verlag, New York, 2004.
[54] , Irregularities of distribution X. Number Theory and Algebra, pp. 311–329. Academic Press, New York, 1977.
[55] , On a completely uniform distribution. Comput. Math. Math. Phys. 19, 249–253, 1979.Google Scholar
[56] , Harmonic analysis on totally disconnected groups and irregularities of point distributions. J. Reine Angew. Math. 600, 25–49, 2006.Google Scholar
[57] and , When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?J. Complexity 14, 1–33, 1998.Google Scholar
[58] , Distribution of points in a cube and approximate evaluation of integrals (Russian). Ž. Vyčisl. Mat. Mat. Fiz. 7, 784–802, 1967.Google Scholar
[59] and , Quasi-random methods for estimating integrals using relatively small samples. SIAM Rev. 36, 18–44, 1994.Google Scholar
[60] , Sums of distances between points on a sphere II. Proc. Am. Math. Soc. 41, 575–582, 1973.Google Scholar
[62] , Improving the rejection sampling method in quasi-Monte Carlo methods. J. Comput. Appl. Math. 114, 231–246, 2000.Google Scholar
[63] and , A construction of low-discrepancy sequences using global function fields. Acta Arith. 73, 87–102, 1995.Google Scholar
[64] , La discrépance isotrope et l'intégration numérique. Ann. Mat. Pura Appl. 87, 125–135, 1970.Google Scholar
[65] and , A discrepancy bound for a deterministic acceptance-rejection sampler. Electron. J. Stat. 8, 678–707, 2014.Google Scholar