Concept# Gamma function

Summary

In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, \Gamma(n) = (n-1)!,.
Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral:
\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\text{ d}t, \ \qquad \Re(z) > 0,.
The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.
The gamma function has no zeros, so the reciprocal gamma function 1/Γ(''z'') is an entire function. In fact, the gamma function cor

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related publications (38)

Related people (2)

Loading

Loading

Loading

Related units (2)

Related concepts (86)

Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_{n

Euler's constant

Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the ha

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers le

Related courses (81)

MATH-313: Introduction to analytic number theory

The aim of this course is to present the basic techniques of analytic number theory.

MATH-201: Analysis III

Calcul différentiel et intégral: Eléments d'analyse vectorielle, intégration par partie, intégrale curviligne, intégrale de surface, théorèmes de Stokes, Green, Gauss, fonctions harmoniques;
Eléments d'analyse complexe: fonctions holomorphes ou analytiques, théorème des résidus et applications

PHYS-216: Mathematical methods for physicists

This course complements the Analysis and Linear Algebra courses by providing further mathematical background and practice required for 3rd year physics courses, in particular electrodynamics and quantum mechanics.

Related lectures (149)

The electrical discharge machining process (EDM) was discovered in the 1950s, and was then used essentially to destroy unrecoverable damaged screws. Since then, huge progress has been achieved in making this process reliable and able to perform the most complex machining operations on the most sophisticated materials. Two main processes use electrical discharge machining. First, Die Sinking EDM (DSEDM) in which an electrode, moved along a usually vertical axis, makes an imprint into a mechanical element ; second, Wire EDM (WEDM), uses a wire as an electrode and makes it possible to perform cut operations. Two specific aspects of the EDM process make it particularly challenging for optimization. First, the process evolves with the machining position. In the DSEDM process, where the electrode sinks deeply into the material, the fragments spawn by erosion (contamination) are trapped, thus modifying the sparking conditions. In the WEDM process, the main factors that drive the evolution of the process are the machining operations. Second, the measurements are very noisy, which is due to underlying, mainly random, physical phenomenon ; this is particularly true to spark triggering. The process evolution influences the single criterion to be minimized : total machining time. However, the latter is only known once the operations are completed. As a result, the whole history of manipulated variables influences the final criterion to minimize in this case of dynamic optimization. A major contribution of this work is a proof that a first-order model of the DSEDM process, with the machining position as a state variable, makes it possible to transform a dynamic optimization problem into a static optimization problem. The tools used in this demonstration are Pontryagin's Minimum Principle as well as Parametric Programming. The conclusion is that in order to achieve minimum machining time, maximum speed must be sought all along the trajectory. The online search for maximum speed is another important contribution of this thesis. Noisy efficiency functions are, indeed, known to be a significant challenge to the reliability of optimization algorithms. To address this issue the Golden Ratio Search and the Nelder-Mead Simplex algorithms were chosen as the starting point, as they do not rely explicitly on the gradient of the efficiency function. The addition of a further dilation condition makes these algorithms more effective in stochastic mode. This condition is based on the detection of contradictory measurement samples as compared to the shape of the efficiency function, which is assumed to be unimodal. As a result of this modification, the density of the final optimization points is well centred relative to the theoretical optimum, and dispersion is small. Moreover, the size of the search region for both algorithms never approaches zero. Consequently, as the machining conditions evolve, the optimizer can target a new optimum. This adaptability proves to be a significant improvement over existing algorithms. In the case of the DSEDM process, a simple model of the efficiency surface as a function of the control variables, which has been calibrated on sample measures, has allowed for a validation of the static optimization. For the WEDM process, conclusive results for the modified algorithms have been obtained both by simulation and during machine-based tests.

João Miguel De Oliveira Durães Alves Martins

Safety assessments of road bridges to braking events combine the braking force, acting along the longitudinal axis of the deck, with a vertical load that accounts for the vertical component of the traffic action. In modern design standards the vertical load models result from probabilistic calibration procedures targeting predefined return periods. On the contrary, the braking force was derived from a deterministic characterization of the vehicle configurations and of the braking process. Therefore, the return period of the braking force is unclear and may not be consistent with that of the vertical load model. Significant deviations from the target return period might lead to either uneconomical decisions, e.g. uncalled-for retrofitting interventions, or to inaccurate structural safety verifications. This thesis presents an original stochastic model to compute site-specific values of the braking force as a function of the return period. The developed stochastic model takes into account the length of the bridge deck and its dynamic properties for vibrations in the longitudinal direction, as well as different sources of randomness related to braking events, all of which comply with real-world measurements, including: - vehicle configurations, resorting to a time-history of crossing vehicles; - driver response times, randomly generated from probability distributions defined in the scope of this project; - deceleration profiles of the vehicles, resampled from catalogues of realistic deceleration profiles. The stochastic model uses Monte Carlo simulation of braking events and computes the maximum of the dynamic response of the bridge to each event. The computed maxima are collected in an empirical distribution function of the braking force. In the end, the model returns the quantile of this distribution that is suitable for safety assessments. This value of braking force is specific to the bridge given properties, to the traffic characteristics, and to the target return period. An additional novelty of this research work is the estimation of a rate of occurrence on motorways of braking events per vehicle-distance travelled. This parameter enables the estimation of the period of time covered by the simulations of braking events as a function of traffic flow and of the total number of braking events simulated. This step is fundamental to determine the value of the braking force that has a given return period. The braking forces returned by the stochastic model show significant dependence on the bridge length, the natural vibration period of the deck in the longitudinal direction, and the number of directions of traffic on the deck. On the contrary, damping ratio, traffic on the fast-lane or on weekends, and an augmentation of traffic in 20% show no substantial influence on the braking force. Moreover, the two motorway locations considered as sources of traffic data, Denges and Monte Ceneri, both in Switzerland, yielded braking forces with similar magnitudes, despite the significant differences in traffic characteristics. Finally, the results compiled served to calibrate an updated braking force that depends explicitly on the parameters found relevant, as well as on the return period so that it can be adopted by different standards even if they enforce different safety targets. This updated expression evidences that the braking forces of current codes tend to be conservative and, hence, can be improved based on the findings of this project.

Laurent Bultot, Caterina Collodet, Maria Deak, Kei Sakamoto

AMP-activated protein kinase (AMPK) plays diverse roles and coordinates complex metabolic pathways for maintenance of energy homeostasis. This could be explained by the fact that AMPK exists as multiple heterotrimer complexes comprising a catalytic alpha-subunit (alpha 1 and alpha 2) and regulatory beta (beta 1 and beta 2)- and gamma (gamma 1, gamma 2, gamma 3)-subunits, which are uniquely distributed across different cell types. There has been keen interest in developing specific and isoform-selective AMPK-activating drugs for therapeutic use and also as research tools. Moreover, establishing ways of enhancing cellular AMPK activity would be beneficial for both purposes. Here, we investigated if a recently described potent AMPK activator called 991, in combination with the commonly used activator 5-aminoimidazole-4-carboxamide riboside or contraction, further enhances AMPK activity and glucose transport in mouse skeletal muscle ex vivo. Given that the gamma 3-subunit is exclusively expressed in skeletal muscle and has been implicated in contraction-induced glucose transport, we measured the activity of AMPK gamma 3 as well as ubiquitously expressed gamma 1-containing complexes. We initially validated the specificity of the antibodies for the assessment of isoform-specific AMPK activity using AMPK-deficient mouse models. We observed that a low dose of 991 (5 mu M) stimulated a modest or negligible activity of both gamma 1- and gamma 3-containing AMPK complexes. Strikingly, dual treatment with 991 and 5-aminoimidazole-4-carboxamide riboside or 991 and contraction profoundly enhanced AMPK gamma 1/gamma 3 complex activation and glucose transport compared with any of the single treatments. The study demonstrates the utility of a dual activator approach to achieve a greater activation of AMPK and downstream physiological responses in various cell types, including skeletal muscle.