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111
On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
, 2010
"... Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and t ..."
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Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.
A survey of the merit factor problem for binary sequences, Sequences and Their
 Applications, Proceedings of SETA 2004, Lecture Notes in Computer Science 3486, 30–55
, 2005
"... A classical problem of digital sequence design, first studied in the 1950s but still not well understood, is to determine those binary sequences whose aperiodic autocorrelations are collectively small according to some suitable measure. The merit factor is an important such measure, and the problem ..."
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A classical problem of digital sequence design, first studied in the 1950s but still not well understood, is to determine those binary sequences whose aperiodic autocorrelations are collectively small according to some suitable measure. The merit factor is an important such measure, and the problem of determining the best value of the merit factor of long binary sequences has resisted decades of attack by mathematicians and communications engineers. In equivalent guise, the determination of the best asymptotic merit factor is an unsolved problem in complex analysis proposed by Littlewood in the 1960s that until recently was studied along largely independent lines. The same problem is also studied in theoretical physics and theoretical chemistry as a notoriously difficult combinatorial optimisation problem. The best known value for the asymptotic merit factor has remained unchanged since 1988. However recent experimental and theoretical results strongly suggest a possible improvement. This survey describes the development of our understanding of the merit factor problem by bringing together results from several disciplines, and places the recent results within their historical and scientific framework. 1
Elliptic integral evaluations of Bessel moments and applications
, 2008
"... We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. In particular, we develop formulae for integrals of products of six or fewer Bessel functions. ..."
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We record and substantially extend what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. In particular, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for cn,k: = � ∞ (t)dt with integers n = 1,2,3,4 and k ≥ 0, 0 tkKn 0 obtaining new results for the even moments c3,2k and c4,2k. We also derive new closed forms for the odd moments sn,2k+1: = � ∞ 0 t2k+1I0 (t)K n−1 0 (t)dt with n = 3,4 and for tn,2k+1: = � ∞ 0 t2k+1I2 0 (t)Kn−2 0 (t)dt with n = 5, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of s5,2k+1, make substantial progress on the evaluation of c5,2k+1, s6,2k+1 and t6,2k+1 and report more limited progress regarding c5,2k, c6,2k+1 and c6,2k. In the process, we obtain 8 conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in 4dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike.
Highprecision floatingpoint arithmetic in scientific computation
 Computing in Science and Engineering, May–June
, 2005
"... At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required: some of these applications require roughly twice ..."
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Cited by 17 (1 self)
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At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required: some of these applications require roughly twice this level; others require four times; while still others require hundreds or more digits to obtain numerically meaningful results. Such calculations have been facilitated by new highprecision software packages that include highlevel language translation modules to minimize the conversion effort. These activities have yielded a number of interesting new scientific results in fields as diverse as quantum theory, climate modeling and experimental mathematics, a few of which are described in this article. Such developments suggest that in the future, the numeric precision used for a scientific computation may be as important to the program design as are the algorithms and data structures.
HighPrecision Computation and Mathematical Physics
"... At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required. Such calculations are facilitated by highpreci ..."
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Cited by 16 (3 self)
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At the present time, IEEE 64bit floatingpoint arithmetic is sufficiently accurate for most scientific applications. However, for a rapidly growing body of important scientific computing applications, a higher level of numeric precision is required. Such calculations are facilitated by highprecision software packages that include highlevel language translation modules to minimize the conversion effort. This paper presents a survey of recent applications of these techniques and provides some analysis of their numerical requirements. These applications include supernova simulations, climate modeling, planetary orbit calculations, Coulomb nbody atomic systems, scattering amplitudes of quarks, gluons and bosons, nonlinear oscillator theory, Ising theory, quantum field theory and experimental mathematics. We conclude that highprecision arithmetic facilities are now an indispensable component of a modern largescale scientific computing environment.
A Strong Hot Spot Theorem
 Proceedings of the American Mathematical Society
"... Abstract. A real number α is said to be bnormal if every mlong string of digits appears in the baseb expansion of α with limiting frequency b −m. We prove that α is bnormal if and only if it possesses no baseb “hot spot. ” In other words, α is bnormal if and only if there is no real number y su ..."
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Cited by 16 (5 self)
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Abstract. A real number α is said to be bnormal if every mlong string of digits appears in the baseb expansion of α with limiting frequency b −m. We prove that α is bnormal if and only if it possesses no baseb “hot spot. ” In other words, α is bnormal if and only if there is no real number y such that smaller and smaller neighborhoods of y are visited by the successive shifts of the baseb expansion of α with larger and larger frequencies, relative to the lengths of these neighborhoods. 1.
A compendium of BBPtype formulas for mathematical constants
, 2000
"... A 1996 paper by the author, Peter Borwein and Simon Plouffe showed that any mathematical constant given by an infinite series of a certain type has the property that its nth digit in a particular number base could be calculated directly, without needing to compute any of the first n−1 digits, by me ..."
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Cited by 15 (2 self)
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A 1996 paper by the author, Peter Borwein and Simon Plouffe showed that any mathematical constant given by an infinite series of a certain type has the property that its nth digit in a particular number base could be calculated directly, without needing to compute any of the first n−1 digits, by means of a simple algorithm that does not require multipleprecision arithmetic. Several such formulas were presented in that paper, including formulas for the constants π and log 2. Since then, numerous other formulas of this type have been found. This paper presents a compendium of currently known results of this sort, both formal and experimental. Many of these results were found in the process of compiling this collection and have not previously appeared in the literature. Several conjectures suggested by these results are mentioned.
The SIAM 100digit challenge, a study in highaccuracy numerical computing. http://wwwm3.ma.tum.de/m3old/ bornemann/challengebook/index.html
, 2004
"... Digit Challenge ” [6]. In this note he presented ten easytostate but hardtosolve problems of numerical analysis, and challenged readers to find each answer to tendigit accuracy: � 1 1. What is limɛ→0 ɛ x−1 cos(x−1 log x) dx? 2. A photon moving at speed 1 in the xy plane starts at t = 0 at (x, ..."
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Digit Challenge ” [6]. In this note he presented ten easytostate but hardtosolve problems of numerical analysis, and challenged readers to find each answer to tendigit accuracy: � 1 1. What is limɛ→0 ɛ x−1 cos(x−1 log x) dx? 2. A photon moving at speed 1 in the xy plane starts at t = 0 at (x, y) = (1/2, 1/10) heading due east. Around every integer lattice point (i, j) in the plane, a circular mirror of radius 1/3 has been erected. How far from the origin is the photon at t = 10? 3. The infinite matrix A with entries a11 = 1, a12 = 1/2, a21 = 1/3, a13 = 1/4, a22 = 1/5, a31 = 1/6, etc., is a bounded operator on ℓ 2. What is A? 4. What is the global minimum of the function exp(sin(50x))+sin(60e y)+sin(70 sin x)+ sin(sin(80y)) − sin(10(x + y)) + (x 2 + y 2)/4? 5. Let f(z) = 1/Γ(z), where Γ(z) is the gamma function, and let p(z) be the cubic polynomial that best approximates f(z) on the unit disk in the supremum norm   · ∞. What is f − p∞? 6. A flea starts at (0, 0) on the infinite 2D integer lattice and executes a biased random walk: At each step it hops north or south with probability 1/4, east with probability 1/4 + ɛ, and west with probability 1/4 − ɛ. The probability that the flea returns to (0, 0) sometime during its wanderings is 1/2. What is ɛ? 7. Let A be the 20000 × 20000 matrix whose entries are zero everywhere except for the primes 2, 3, 5, 7, · · · , 224737 along the main diagonal and the number 1 in all the positions aij with i − j  = 1, 2, 4, 8, · · · , 16384. What is the (1, 1) entry of A −1.
Resolution of the QuinnRandStrogatz constant of nonlinear physics
 EXP. MATHEMATICS, TO APPEAR, HTTP://CRD.LBL.GOV/˜DHBAILEY/DHBPAPERS/QRS.PDF. COMPUTATION AND MATHEMATICAL PHYSICS DAVID H. BAILEY
, 2007
"... Herein we develop connections between zeta functions and some recent “mysterious” constants of nonlinear physics. In an important analysis of coupled Winfree oscillators, Quinn, Rand, and Strogatz [14] developed a certain Noscillator scenario whose bifurcation phase offset φ is implicitly defined, ..."
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Herein we develop connections between zeta functions and some recent “mysterious” constants of nonlinear physics. In an important analysis of coupled Winfree oscillators, Quinn, Rand, and Strogatz [14] developed a certain Noscillator scenario whose bifurcation phase offset φ is implicitly defined, with a conjectured asymptotic behavior: sin φ ∼ 1 − c1/N, with experimental estimate c1 = 0.605443657.... We are able to derive the exact theoretical value of this “QRS constant ” c1 as a real zero of a particular Hurwitz zeta function. This discovery enables, for example, the rapid resolution of c1 to extreme precision. Results and conjectures are provided in regard to higherorder terms of the sin φ asymptotic, and to yet more physics constants emerging from the original QRS work.
Experimental Determination of ApéryLike Identities for ζ(2n + 2)
, 2006
"... We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others. 1 ..."
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We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others. 1