Ten Problems in Experimental Mathematics

Bailey, David H.; Borwein, Jonathan M.; Kapoor, Vishaal; Weisstein, Eric W.

doi:10.1080/00029890.2006.11920330

Cited by 32 publications

(22 citation statements)

References 21 publications

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“…Such sums are called alternating Euler sums. Only the first two cases of (5.20) when n = 1 and n = 2 have been proved thus far (see [3,4,6]). For example,…”

Section: Wwwmn-journalcommentioning

confidence: 99%

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

Choi

Srivástava

2009

Mathematische Nachrichten

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The Gamma function and its nth logarithmic derivatives (that is, the polygamma or the psi-functions) have found many interesting and useful applications in a variety of subjects in pure and applied mathematics. Here we mainly apply these functions to treat convolutions of the Rayleigh functions by recalling a general identity expressing a certain class of series as psi-functions and to evaluate a class of log-sine integrals in an algorithmic way. We also evaluate some Euler sums and give much simpler psi-function expressions for some known parameterized multiple sums.

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“…Such sums are called alternating Euler sums. Only the first two cases of (5.20) when n = 1 and n = 2 have been proved thus far (see [3,4,6]). For example,…”

Section: Wwwmn-journalcommentioning

confidence: 99%

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

Choi

Srivástava

2009

Mathematische Nachrichten

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“…Here an ideal (or totally asymptotic) tetrahedron in H 3 is a hyperbolic tetrahedron with all four vertices at infinity. In 1998 Borwein and Broadhurst conjectured that = L −7 (2) ∼ = 1.151925470544491 (2) where L −7 (s) is the primitive Dirichlet L-series modulo seven [1,[6][7][8][9][10]. The sign ?…”

Section: Introductionmentioning

confidence: 99%

A dilogarithmic integral arising in quantum field theory

Cvijović¹

2009

Journal of Mathematical Physics

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Recently, an interesting dilogarithmic integral arising in quantum field theory has been closed-form evaluated in terms of the Clausen function $\text{Cl}_2(\theta)$ by Coffey [J. Math. Phys.} 49 (2008), 093508]. It represents the volume of an ideal tetrahedron in hyperbolic space and is involved in two intriguing equivalent conjectures of Borwein and Broadhurst. It is shown here, by simple and direct arguments, that this integral can be expressed by the triplet of the Clausen function values which are involved in one of the two above-mentioned conjectures.Comment: 6 page

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“…Examples of such problems are factoring very large numbers (RSA depends on this problem's computational difficulty) [5], finding the solution to partial differential equations [6], and deciding whether a knot in 3-dimensional Euclidean space is unknotted (topological problem) [7]. More of such problems can be found in [4][5][8][9][10]. Complex problems in science and engineering, including the ones above, are computationally intensive in nature [11].…”

Section: Introductionmentioning

confidence: 99%

Reconfigurable Hardware Implementation of the Successive Overrelaxation Method

Kasbah

Haraty

Damaj

2008

Lecture Notes in Electrical Engineering

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2Reconfigurable Hardware Implementation of the SOR Method the system at hand. There are two basic approaches for solving linear systems: Direct Methods and Iterative Methods. In the first approach, a finite number of operations are performed to find the exact solution. In the second approach, an initial approximate of the solution is generated, then this initial guess is used to generate another approximate solution, which is more accurate than the previous one [12] The robustness of applying iterative methods over direct methods is shown in different areas including: circuit analysis and design, weather forecasting and analyzing financial market trends.The well-known iterative methods are: Gauss-Seidel, Multigrid, Jacobi and Successive Over-Relaxation (SOR) which is of a interest in this chapter. SOR has been devised to accelerate the convergence of Gauss-Seidel and Jacobi [13], by introducing a new parameter, , referred to as the relaxation factor. The SOR rate of convergence is highly dependent on the relaxation factor. The main difficulty of using SOR is finding a good estimate of the relaxation factor [12]. Several techniques have been proposed for determining the exact value of which accelerates the rate of convergence of the method [12,13].All available iterative methods packages, including SOR, are done in software. Examples are the: ITPACK 3A, ITPACK 3B, ITPACK 2C, ITPACK 2D, and the ELLPACK package [14,15]. Several sequential and parallel techniques were used in these packages to accelerate the method [16].The emergence of the new computing paradigm, Reconfigurable Computing (RC), introduces novel techniques for accelerating certain classes of applications including signal processing (e.g., weather forecasting, seismic data processing, Magnetic Resonance Imaging (MRI), adaptive filters), cryptography and DNA matching [17]. RC-systems combine the flexibility offered by software and the performance offered by hardware [18]. It requires a reconfigurable hardware, such as an FPGA, and a software design environment that aids in the creation of configurations for the reconfigurable hardware [17].In [19], the first hardware implementation of an iterative method-the Multigridis presented. The speedup achieved demonstrates that hardware design can be suited for such computationally intensive applications. Toward proving the hypothesis that accelerated versions of the iterative methods can be realized in hardware, we undertook the first hardware implementation of the SOR method; using the same FPGAs that were used in [19][20][21].In this chapter, we study the feasibility of implementing SOR in reconfigurable hardware. We use Handel-C, a higher-level design tool to code our design, which is analyzed, synthesized, and placed and routed using the FPGAs proprietary software (DK Design Suite, Xilinx ISE 8.1i and Quartus II 5.1). We target Virtex II Pro, Altera Stratix and Spartan3L which is embedded in the RC10 FPGA-based system from Celoxica. We report our timing results when targeting Virtex II Pro and compare them ...

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Ten Problems in Experimental Mathematics

Cited by 32 publications

References 21 publications

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

A dilogarithmic integral arising in quantum field theory

Reconfigurable Hardware Implementation of the Successive Overrelaxation Method

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