Let H_k = 1 + 1/2 + 1/3 + ⋯ + 1/k denote the kth harmonic number. We present an easy-to-implement algorithm for the computation of explicit closed-form evaluations, in terms of the digamma and polygamma functions, for Euler sums of the form ∑_k=1^∞ R(k) H_k, where R(k) is a rational function (quotient of two polynomials) whose denominator degree is at least two larger than the numerator degree. We apply the same method to show how the computation of a general formula for Euler sums of the form ∑_k=1^∞H_k/(m_1 k + n_1)^p_1 (m_2 k + n_2)^p_2⋯ (m_r k + n_r)^p_r reduces to partial fraction decomposition. We present explicit formulae for sums with one or two terms in the denominator, with powers p_i ranging up to 3, and with multipliers m_i ranging up to 4. We also include results for related Euler sums such as ∑_k=1^∞k^q H_k/(m k + n)^p. Computation of Euler sums directly to very high precision enables us to rigorously check the above-mentioned formulas in many specific cases.

Several features of financial research make it particularly prone to the occurrence of false discoveries. First, the probability of finding a positive (profitable investment strategy) is very low, due to intense competition. Second, true findings are mostly short-lived, as a result of the non-stationary nature of financial systems. Third, unlike in the natural sciences, it is rarely possible to verify statistical findings through controlled experiments. Finance’s inability to conduct controlled experiments makes it virtually impossible to debunk a false claim. One would hope that, in such a field, researchers would be particularly careful when conducting statistical inference. Sadly, the opposite is true.Tenure-seeking researchers publish thousands of academic articles that promote dubious investment strategies, without controlling for multiple testing. Some of those articles are written for, funded, or promoted by investment firms with a commercial interest. As a consequence, today’s academic finance exhibits some resemblance with medicine’s predicament during the 1950-2000 period, when Big Tobacco paid for thousands of studies in support of their bottom line. Unlike finance, medical journals today impose strict controls for multiple testing. Academic finance’s denial of its replication crisis risks its branding as a pseudoscience.

This paper presents a catalogue of mathematical formulas and iterative algorithms for evaluating the mathematical constant \(\pi\), ranging from Archimedes' 2200-year-old iteration to some formulas that were discovered only in the past few decades. Computer implementations and timing results for these formulas and algorithms are also included. In particular, timings are presented for evaluations of various infinite series formulas to approximately 10,000-digit precision, for evaluations of various integral formulas to approximately 4,000-digit precision, and for evaluations of several iterative algorithms to approximately 100,000-digit precision, all based on carefully designed comparative computer runs.

Recently some scientific computing users have discovered that they can replace 64-bit with 32-bit operations for carefully selected portions of the computation, and still retain acceptable accuracy in the final results. In addition, developers of some emerging applications such as machine learning have discovered that they can achieve acceptable results with only 16-bit precision in certain portions of the code. At the other end of the precision spectrum, some users have explored using 128-bit arithmetic in some particularly demanding applications, while others have done computations using much higher precision—hundreds or even thousands of digits. Such work has underscored the need to develop new mathematical and software frameworks to support a dynamically variable level of precision, and, more generally, to rethink what “reproducibility” means in a variable precision environment. This article summarizes some of the work being done in this arena, and lists research problems that need to be solved.

Many investors rely on market experts and forecasters when making investment decisions, such as when to buy or sell securities. Ranking and grading market forecasters provide investors with metrics on which they may choose forecasters with the best record of accuracy for their particular market exposure. This study develops a novel ranking methodology to rank the market forecaster. In particular, we distinguish forecasts by their specificity, rather than considering all predictions and forecasts equally important, and we also analyze the impact of the number of forecasts made by a particular forecaster. We have applied our methodology on a dataset including 6,627 forecasts made by 68 forecasters.

Glioblastoma (GBM) remains one of the most intransigent of cancers, with a median overall survival of only 15 months after diagnosis. Drug treatments have largely proven ineffective; it is thought that this is related to the heterogeneous nature and plasticity of GBM-initiating stem cell lineages. Although many combination drug therapies are being positioned to address tumour heterogeneity, the most promising therapeutic approaches for GBM to date appear to be those targeting GBM by vaccination or antibody- and cell-based immunotherapy. We review the most recent clinical trials for GBM and discuss the role of adaptive clinical trials in developing personalised treatment strategies to address intra- and inter-tumoral heterogeneity.

The Gordon Bell Prize is awarded each year by the Association for Computing Machinery to recognize outstanding achievement in high-performance computing (HPC). The purpose of the award is to track the progress of parallel computing with particular emphasis on rewarding innovation in applying HPC to applications in science, engineering, and large-scale data analytics. Prizes may be awarded for peak performance or special achievements in scalability and time-to-solution on important science and engineering problems. Financial support for the US$10,000 award is provided through an endowment by Gordon Bell, a pioneer in high-performance and parallel computing. This article examines the evolution of the Gordon Bell Prize and the impact it has had on the field.

This is an Accepted Manuscript of an article published by Taylor & Francis in Regional Studies on 19 January 2017, available online: http://www.tandfonline.com/10.1080/00343404.2016.1255720.

This article briefly summarizes the extraordinary career of Jonathan Borwein, with a focus on his contributions to the field of experimental mathematics.

Computer-based tools for mathematics are changing how mathematics is researched, taught and communicated to society. Future technology trends point to ever-more powerful tools in the future. Computation in mathematics is thus giving rise to a new mode of mathematical research, where algorithms, datasets and public databases are as significant as the resulting theorems, and even the definition of what constitutes secure mathematical knowledge is seen in a new light.

While tremendously useful, automated techniques for tuning the precision of floating-point programs face important scalability challenges. We present Blame Analysis, a novel dynamic approach that speeds up precision tuning. Blame Analysis performs floating-point instructions using different levels of accuracy for their operands. The analysis determines the precision of all operands such that a given precision is achieved in the final result of the program. Our evaluation on ten scientific programs shows that Blame Analysis is successful in lowering operand precision. As it executes the program only once, the analysis is particularly useful when targeting reductions in execution time. In such case, the analysis needs to be combined with search-based tools such as Precimonious. Our experiments show that combining Blame Analysis with Precimonious leads to obtaining better results with significant reduction in analysis time: the optimized programs execute faster (in three cases, we observe as high as 39.9% program speedup) and the combined analysis time is 9 x faster on average, and up to 38 x faster than Precimonious alone.

Mathematical research is undergoing a transformation from a mostly theoretical enterprise to one that involves a significant amount of experimentation. Indeed, computational and experimental mathematics is now a full-fledged discipline with mathematics, and the larger field of computational science is now taking its place as an experimental discipline on a par with traditional experimental fields. In this new realm, reproducibility comes to the forefront as an essential part of the computational research enterprise, and establishing procedures to ensure and facilitate reproducibility is now a central focus of researchers in the field. In this study, we describe our attempts to reproduce the results of a recently published article by Reinhard Ganz, who concluded that the decimal expansion of pi is not statistically random, based on an analysis of several trillion decimal digits provided by Yee and Kondo. While we are able to reproduce the specific findings of Ganz, additional statistical analysis leads us to reject his overall conclusion.

This note presents a short history mathematical formulas involving the mathematical constant , and how they have been used in mathematical research through the ages.

Data, code, and workflows should be available and cited

The foundation of scientic research is theory and experiment, carefully documented in open publications, in part so that other researchers can reproduce and validate the claimed