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ON THE EXPONENTIAL FUNCTION

by Robert Gove, Jan Rychtář
"... Abstract. The natural exponential function is one of the most important functions students should learn in calculus classes. The applications range from mathematics, statistics, natural sciences, and economics. Despite its wide use and importance, instructors often struggle with the proper definitio ..."
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Abstract. The natural exponential function is one of the most important functions students should learn in calculus classes. The applications range from mathematics, statistics, natural sciences, and economics. Despite its wide use and importance, instructors often struggle with the proper

Exponential functionals of Lévy processes

by Jean Bertoin, Marc Yor - Probabilty Surveys , 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0). 0

Comment on the history of the stretched exponential function

by Manuel Cardona, Ralph Chamberlin, M. Cardona, R. V. Chamberlin, W. Marx
"... The history of the stretched exponential function ..."
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The history of the stretched exponential function

AIMD algorithms and exponential functionals

by Fabrice Guillemin, Philippe Robert, Bert Zwart - Ann. Appl. Probab , 2002
"... ABSTRACT. The behavior of connection transmitting packets into a network according to a general additive-increase multiplicative-decrease (AIMD) algorithm is investigated. It is assumed that loss of packets occurs in clumps. When a packet is lost, a certain number of subsequent packets are also lost ..."
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lost (correlated losses). The stationary behavior of this algorithm is analyzed when the rate of occurrence of clumps becomes arbitrarily small. From a probabilistic point of view, it is shown that exponential functionals associated to compound Poisson processes play a key role. A formula

Quantum Exponential Function

by S. L. Woronowicz - Rev. Math. Phys , 2000
"... A special function playing an essential role in the construction of quantum \ax + b"-group is introduced and investigated. The function is denoted by F~(r; %), where ~ is a constant such that the deformation parameter q2 = e−i~. The rst variable r runs over non-zero real numbers; the range of t ..."
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of the second one depends on the sign of r: % = 0 for r> 0 and % = 1 for r < 0. After the holomorphic continuation the function satises the functional equation F~(ei~r; %) = (1 + ei~=2r)F~(r;−%): The name \exponential function " is justied by the formula: F~(R; )F~(S; ) = F~([R + S]; e); where R

Inequalities For Power-Exponential Functions

by Feng Qi, Lokenath Debnath , 2000
"... The following inequalities for power-exponential functions are proved y x y x y x > y x > y x x y , # y x # xy > y y x x , where 0 < x < y < 1 or 1 < x < y. Key words and phrases: Inequality, power-exponential function, revised Cauchy's mean-value theor ..."
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The following inequalities for power-exponential functions are proved y x y x y x > y x > y x x y , # y x # xy > y y x x , where 0 < x < y < 1 or 1 < x < y. Key words and phrases: Inequality, power-exponential function, revised Cauchy's mean

On exponential functionals of Lévy processes

by Anita Behme , Alexander Lindner - J. Theor , 2013
"... Abstract Exponential functionals of Lévy processes appear as stationary distributions of generalized Ornstein-Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further we use these results to investigate propertie ..."
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Abstract Exponential functionals of Lévy processes appear as stationary distributions of generalized Ornstein-Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further we use these results to investigate

Image Smoothing with Exponential Functions

by Mitra Basu, Min Su - International Journal of Pattern Recognition and Artificial Intelligence , 2001
"... Noise reduction in images, also known as image smoothing, is an essential and first step before further processings are done on the image. The key to image smoothing is to preserve important features while removing noise from the image. Gaussian function is widely used in image smoothing. Recently i ..."
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it has been reported that exponential functions (value of the exponent is not equal to 2) perform substantially better than Gaussian functions in modeling and preserving image features. In this paper we propose family of exponential functions, that include Gaussian when the value of the exponent is 2

The Exponential Function Expository Paper

by Shawn A. Mousel, Jim Lewis Advisor , 2006
"... One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range ..."
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(Lial & Miller, 1975). An exponential function is a function with the basic form f (x) = ax, where a (a fixed base that is a real, positive number) is greater than zero and not equal to 1. The exponential function is not to be confused with the polynomial functions, such as x 2. One way

AN EXCEPTIONAL EXPONENTIAL FUNCTION

by unknown authors
"... There are at least two surprising results in this article. The first is that there is actually a link between the standard calculus problem of finding the best view of a painting and graphs of exponential functions. The second is that the exponential function with the “best view ” is not the one wit ..."
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There are at least two surprising results in this article. The first is that there is actually a link between the standard calculus problem of finding the best view of a painting and graphs of exponential functions. The second is that the exponential function with the “best view ” is not the one
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