Academic literature on the topic 'Gaussian Primes numbers'

Power laws and scaling are two features that have been known for some time in the distribution of returns (i.e., price fluctuations), and, more recently, in the distribution of volumes (i.e., numbers of shares traded) of financial assets. As in numerous examples in physics, these power laws can be understood as the asymptotic behavior of distributions that derive from nonextensive thermostatistics. Recent applications of the (Q-Gaussian distribution to returns of exchange rates and stock indices are extended here for individual U.S. stocks over very small time intervals and explained in terms of a feedback mechanism in the dynamics of price formation. In addition, we discuss some new empirical findings for the probability density of low volumes and show how the overall volume distribution is described by a function derived from q-exponentials. In March 1900 at the Sorbonne, a 30-year-old student—who had studied under Poincaré—submitted a doctoral thesis [2] that demonstrated an intimate knowledge of trading operations in the Paris Bourse. He proposed a probabilistic method to value some options on rentes, which were then the standard French government bonds. His work was based on the idea that rente prices evolved according to a random-walk process that resulted in a Gaussian distribution of price differences with a dispersion proportional to the square root of time. Although the importance of Louis Bachelier's accomplishment was not recognized by his contemporaries [24], it preceded by five years Einstein's famous independent, but mathematically equivalent, description of diffusion under Brownian motion. The idea of a Gaussian random-walk process (later preferably applied to logarithmic prices) eventually became one of the basic tenets of most twentieth-century quantitative works in finance, including the Black-Scholes [3] complete solution to the option-valuation problem—of which a special case had been solved by Bachelier in his thesis. In the times of the celebrated Black-Scholes solution, however, a change in perspective was already under way. Starting with the groundbreaking works of Mandelbrot [18] and Fama [11], it gradually became apparent that probability distribution functions of price changes of assets (including commodities, stocks, and bonds), indices, and exchange rates do not follow Bachelier's principle of Gaussian (or "normal") behavior.

Abstract This research studies the use of predetermined experimental plans in a live setting with a finite implementation horizon. In this context, we seek to determine the optimal experimental budget in different environments using a Bayesian framework. We derive theoretical results on the optimal allocation of resources to treatments with the objective of minimizing cumulative regret, a metric commonly used in online statistical learning. Our base case studies a setting with two treatments assuming Gaussian priors for the treatment means and noise distributions. We extend our study through a

Design optimization under uncertainty is notoriously difficult when the objective function is expensive to evaluate. State-of-the-art techniques, e.g., stochastic optimization or sampling average approximation, fail to learn exploitable patterns from collected data and, as a result, they tend to require an excessive number of objective function evaluations. There is a need for techniques that alleviate the high cost of information acquisition and select sequential simulations in an optimal way. In the field of deterministic single-objective unconstrained global optimization, the Bayesian globa

Abstract The reliability and safety of aero-engine are often the decisive factors for the safe and reliable flight of commercial aircraft. Hence, the vibration source location and fault diagnosis of aero-engine are of prime importance to detect faults and carry out fast and effective maintenance in time. However, the vibration signals collected by the sensors arranged on the casing of the aero-engine are generally the mixed signals of the main vibration sources inside the engine, and the components are extremely complicated. Therefore, the vibration source identification is a big challenge for