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Topic: Characterizations of the exponential function

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	Characterizations of the exponential function - Wikipedia, the free encyclopedia
		In mathematics, the exponential function can be characterized in many ways.
		For instance, when the value of the function is defined by a sequence or series, the convergence of this sequence or series needs to be established.
		First, the equivalence of characterizations 1 and 2 is established, and then the equivalence of characterizations 1 and 3 is established.
	en.wikipedia.org /wiki/Characterizations_of_the_exponential_function (670 words)

	Exponential function - Wikipedia, the free encyclopedia
		The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives.
		The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin.
		The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices (in which case the function is called the matrix exponential).
	en.wikipedia.org /wiki/Exponential_function (1889 words)

	Exponentiation Summary
		Exponentiation with various bases: red is to base e, green is to base 10, and purple is to base 1.7.
		Exponentiation is a basic mathematical tool that is used pervasively in many other fields as well, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.
		A special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of -1 indicates the inverse function.
	www.bookrags.com /Exponentiation (5500 words)

	Exponential function Summary
		Exponential functions have a wide range of applications in finance, economics, biology, ecology, physics, and other sciences.
		Scientists from various fields use exponential functions as models for growth and decay phenomena in which a quantity is assumed to grow or decay at a rate which is proportional to the amount of the quantity at any given time.
		Exponential functions and modifications of exponential functions are currently being used to study such phenomena as the growth of the internet, the spread of AIDS, the projected growth or decay of the national debt, and much more.
	www.bookrags.com /Exponential_function (1700 words)

	Descriptive Complexity
		Some of the results arising from this approach include characterizing polynomial time as the set of properties expressible in first-order logic plus a least fixed-point operator, and showing that parallel time on a Parallel Random Access Machine is linearly related to first-order inductive depth.
		Yuri Gurevich gave an alternate characterization of logspace as the set of boolean queries computable by the "finite, primitive-recursive" functions.
		Exponential time is the set of boolean queries computable in second-order logic extended with the capacity to define new relations by induction.
	www.cs.umass.edu /~immerman/descriptive_complexity.html (991 words)

	PlanetMath: Taylor series (Site not responding. Last check: 2007-10-12)
		Both are useful to linearize or otherwise reduce the analytical complexity of a function.
		In contrast with the complex case, it turns out that all holomorphic functions are infinitely differentiable and have Taylor series that converge to them.
		This of course makes the theory of analytic functions very nice, and many questions about real power series and real analytic functions are more easily answered by looking at the complex case.
	planetmath.org /encyclopedia/TaylorSeries.html (745 words)

	Growth Models, Part 2
		Another characterization of exponential growth is that the percentage or relative growth of Q, i.e.
		In other words, Malthus is claiming that, for a population undergoing exponential growth, the time it takes to double is independent of the size of the population.
		Superimpose the graph of your exponential function over the plot of the population.
	www.math.duke.edu /education/postcalc/growth/growth2.html (1424 words)

	AFRICAN DIASPORA JOURNAL OF MATHEMATICS
		ABSTRACT.We establish that necessary conditions on the initial functions are also sufficient conditions for the existence of the classical solution, weakened on the axis in Holder spaces of the third mixed problem with central symmetry for the tree-dimensional wave equation.
		Thus, by using the fundamental theorems of the geometry of alternate matrices, the characterization of φ is obtained.
		We obtain a characterization of multivariate natural exponential families with k-th degree polynomial variance functions via a notion of k-orthogonality of some associated polynomials.
	www.african-j-math.org /eprints.htm (4640 words)

	Characterizations of Multivariate Differences and Associated Exponential Splines - Kunkle (ResearchIndex) (Site not responding. Last check: 2007-10-12)
		Abstract: The subject of this investigation is the class of dierence functionalslinear combinations of nitely many function and/or derivative evaluationswhich annihilate the nullspace of a certain constant coecient dierential operator.
		Any such functional can be viewed as an integral-dierential operator whose Peano kernel is a compactly supported exponential spline.
		2 The exponentials in the span of the multiiinteger translates..
	citeseer.ist.psu.edu /435940.html (549 words)

	A performance methodology for commercial servers
		The fifth attribute is the degree of contention for shared data, which is characterized by the frequency of synchronization primitives such as load-reserve/store conditional [6].
		A set of key quantitative descriptors are needed to effectively characterize and model the performance of a server system running a complex workload such as TPC-C. These descriptors are typically broken down into processor core or on-chip attributes, memory subsystem or memory nest attributes, and system attributes for a given workload.
		The quality of the performance data is a function of the instrumentation mechanism itself and the care taken in its deployment.
	www.research.ibm.com /journal/rd/446/kunkel.html (13842 words)

	WWS509 - Lecture Notes
		An alternative characterization of the distribution of T is given by the hazard function, or instantaneous rate of occurrence of the event, defined as
		These results show that the survival and hazard functions provide alternative but equivalent characterizations of the distribution of T. Given the survival function, we can always differentiate to obtain the density and then calculate the hazard using Equation 7.3.
		It turns out that the conditional density, hazard and survivor function for those who experience the event are related to the unconditional density, hazard and survivor for the entire population.
	data.princeton.edu /wws509/notes/c7s1.html (1227 words)

	Summary of WWW Characterizations
		Characterizations of user behavior captured at the browser are the most informative yet are quite rare.
		The first study to characterize WWW client behavior was performed by [Catledge and Pitkow 1995] during the summer of 1994.
		One of the graver shortcomings of the characterizations to date is the focus on US education settings and the subsequent scarcity of client characterizations from commercial Internet Service Providers, corporate environments, and international users.
	www7.scu.edu.au /1877/com1877.htm (4126 words)

	\Large Normal Distribution \\ \large characterizations with applications (Site not responding. Last check: 2007-10-12)
		We have managed to avoid functional equations for non-differentiable functions; in many proofs in the literature lack of differentiability is a major technical difficulty.
		The book is primarily aimed at graduate students in mathematical statistics and probability theory who would like to expand their bag of tools, to understand the inner workings of the normal distribution, and to explore the connections with other fields.
		Fall as it may, its deviation from the mark is error, and the probability of that error is the unknown function of its square, ie.
	math.uc.edu /~brycw/probab/charakt/charakt.htm (5891 words)

	research
		Functional analysis is the study of spaces of functions and other Banach spaces, and is related to differential equations, linear algebra, topology and abstract algebra.
		However, it is a very useful function and is used extensively in mathematics.
		He is currently working on the normal form of the Navier-Stokes equations, the Langrangian averaged Navier-Stokes alpha model of turbulence, numerically determining modes, the Bouligand dimension and the bioremediation of contaminated soil.
	www.unr.edu /cos/math/research.html (1641 words)

	Abstract11 (Site not responding. Last check: 2007-10-12)
		Characterizations of the exponential distribution based on certain properties of its characteristic function
		Two characterizations of the exponential distribution among distributions with support the nonnegative real axis are presented.
		The characterizations are based on certain properties of the characteristic function of the exponential random variable.
	www.unipi.gr /faculty/geh/abstract11.htm (68 words)

	v2n7
		Both Riemann-Stieltjes and Riemann integrals are evaluated with a variety of assumptions about the integrand enabling the characterisation of the bound in terms of a variety of norms.
		From this, some monotonicity results and inequalities for the gamma or incomplete gamma functions are deduced or extended, a unified proof of some known results for the gamma function is given.
		Identities and inequalities are obtained involving n-time differentiable functions in terms of evaluations at an interior and at the end points.
	rgmia.vu.edu.au /v2n7.html (923 words)

	Measurements of Shape and Morphology
		This causes the surface brightness to be underestimated, and the inverse concentration index to be overestimated, for objects of size comparable to the PSF.
		A second measure of ellipticity is given by measuring the ellipticity of the 25 magnitudes per square arcsecond isophote (in all bands).
		measures the radius of a particular isophote as a function of angle and Fourier expands this function.
	www.sdss.org /dr2/products/general/edr_html/node42.html (947 words)

	Autocorrelation Function and Power Spectrum of Two-State Random Processes Used in Neurite Guidance -- Odde and Buettner ...
		In principle the value of r for the growth phase (A) could be different from that for the shortening phase (B); however, the present analysis only considers the case where they are the same.
		The similarity in microtubule and filopodial dynamics may be a direct reflection of a similarity in function: the growing
		Time series characterization of simulated microtubule dynamics in the nerve growth cone.
	www.biophysj.org /cgi/content/full/75/3/1189 (3478 words)

	Sensible Calculus Chapter IX: DIFFERENTIAL EQUATIONS AND POLYNOMIALS: TAYLOR'S THEOREM
		As we have seen there are many ways to characterize and define these numbers and use these characterizations to find an estimate.
		Thus the difference between the function's value at `x` and `a` can be measured using the value of the derivative of `f` at `a`..
		Note: Since the exponential function has a positive derivative for all `x`, the function is increasing for all `x`.
	www.humboldt.edu /~mef2/book/ch9/IXA.htm (1880 words)

	v7n3
		be a non-negative functional on the positive cone of the space of all real-valued locally bounded functions on
		It is shown here that the proposed fractal interpolation function and its first p derivatives are good approximations of the corresponding derivatives of the original function.
		In this paper, by using some properties of geometrically convex functions, some monotone functions and sequences are constructed.
	rgmia.vu.edu.au /v7n3.html (1143 words)

	Machines
		To use a TM as a computational device and thus to associate a function with its operation, we have to fix where its input and output are on the tape.
		Yet all these characterizations have been shown to be equivalent in terms of their ability to compute the same functions.
		We saw that certain functions which are computable on a TM are not computable on an FSM.
	www.rci.rutgers.edu /~cfs/472_html/TM/MachineChapter.html (5056 words)

	Book Review: Introduction to Computational Biology (Site not responding. Last check: 2007-10-12)
		After delineating these mathematical characterizations, Waterman presents several algorithms in use today for analyzing biological data.
		To give an example of problems treated in the book, consider the immense efforts that are being put into sequencing the genome of bacteria, viruses, yeasts, human beings and other animals (e.g.
		It may also allow us to deduce the function of one gene based on its similarity to another gene whose function is known.
	www.acm.org /crossroads/xrds5-1/introcb.html (475 words)

	AMCA: Some characterizations for the exponential stability of linear skew-product semiflows by Ciprian Preda (Site not responding. Last check: 2007-10-12)
		We analyze the exponential stability of linear skew-product semiflows on locally compact metric space with Banach fibers.
		This characterization is a very general one, it includes as particular cases many interesting situations among them we can mention some results due to Clark, Datko, Latushkin, van Minh, Montgomery-Smith, Randolph, Rabiger, Schnaubelt.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/n/e/89.htm (127 words)

	Content of the AAA_readme file at ftp.cs.wisc.edu/Approx (Site not responding. Last check: 2007-10-12)
		Items on: approximation orders, box splines and exponential box splines, dimensions of kernels of linear operators, frames, multivariate polynomial interpolation, polynomial ideals, commutative algebra and approximation theory, quasi-interpolation, radial basis function approximation, refinable functions, scattered data approximation, shift-invariant spaces, subdivision, surveys, univariate splines, wavelets, Weyl-Heisenberg systems, linear algebra, \TeX, splinebib, file mailing,...
		Characterizations of linear independence and stability of the shifts of a univariate refinable function in terms of its refinement mask
		The exponentials in the span of the multiinteger translates of a compactly supported function: quasiinterpolation and approximation order
	www.cs.wisc.edu /~deboor/ftpreadme.html (1631 words)

	Citebase - Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions (Site not responding. Last check: 2007-10-12)
		Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions
		Authors: Biane, P. Pitman, J. Yor, M. This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables.
		We present some characterizations of these probability laws, and some approximations of Riemann's zeta function which are related to these laws.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9912170 (157 words)

	Amazon.com: Advanced Probability Theory (Probability, Pure and Applied, Vol 10): Books: Janos Galambos (Site not responding. Last check: 2007-10-12)
		Key Phrases: distributed exponential variables, total probability rule, limiting distribution function, New York, Proof Let, Academic Press (more...
		Key Phrases - SIPs: distributed exponential variables, total probability rule, limiting distribution function, proper distribution function, simple random variable (more)
		New York, Proof Let, Academic Press, Definition Let, Lecture Notes, Lemma of Section, Marcel Dekker, Proof First, Theory Prob, Characterizations of Probability Distributions, Corollary Let, Foundations of the Theory of Probability, Measure Theory, Remark Note, Remark Notice, San Francisco, Sectio Math
	www.amazon.com /Advanced-Probability-Theory-Pure-Applied/dp/0824793323 (890 words)

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