Unreferenced date February 2011 Biologicalexponentialgrowth is the exponentialgrowth of biological organisms. When the resources availability is unlimited in the habitat , the population of an organism living in the habitat grows in an Exponentialgrowthexponential or geometric fashion. Resource availability is obviously essential for the unimpeded growth of a population. Ideally, when resources in the habitat are unlimited, each species has the ability to realise fully its innate potential to grow in number, as Charles Darwin observed while developing his theory of natural selection . If, in a hypothetical population of size N , the birth rates per capita are represented as b and death rates per capita as d , then the increase or decrease in N during a time period t will be math dN dt b d x N math b d is called the intrinsic rate of natural increase and is a very important parameter chosen for assessing the impacts of any biotic or Abiotic components abiotic factor on population growth. Any species growing exponentially under unlimited resource conditions can reach enormous Population density population densities in a short time. Darwin showed how even a slow growing animal like the elephant could reach an enormous population if there were unlimited resources for its growth in its habitat. If birth giving takes two parents we get Nurgaliev s law . Category Biology Category Biology articles needing attention Category Population ecology ... more details
File Exponential.svg thumb 300px right The graph illustrates how exponentialgrowth green surpasses both linear red and cubic blue growth. legend green Exponentialgrowth legend red Linear growth legend blue Cubic growthExponentialgrowth including exponential decay when the growth rate is negative ... progression . The formula for exponentialgrowth of a variable x at the positive or negative growth ... to be 1.05 times i.e., 5 larger than what it was at the previous time. The exponentialgrowth model ... to grasp ramifications of exponentialgrowth, stating The greatest shortcoming of the human race ... can result in the exponentialgrowth of the amplified signal, although resonance effects may ... whose best fit line are exponential decay curves. Economics Economic growth is expressed in percentage terms, implying exponentialgrowth. For example, U.S. GDP per capita has grown at an exponential ... s law and technological singularity under exponentialgrowth, there are no singularities. The singularity ... by a factor of b math x t tau x t cdot b , . math If 0 and b 1, then x has exponentialgrowth ... positive number b . Thus the law of exponentialgrowth can be written in different but mathematically ... growth If a variable x exhibits exponentialgrowth according to math x t x 0 1 r t math , then the log ... sides of the exponentialgrowth equation math log x t log x 0 t cdot log 1 r . math This allows ... exponentialgrowth. Other growth rates In the long run, exponentialgrowth of any kind will overtake ... of conceivable growth rates that are slower than exponential and faster than linear in the long run ... exponential decay . Limitations of models Exponentialgrowth models of physical phenomena only apply ... assumptions of the exponentialgrowth model, such as continuity or instantaneous feedback, break down. further Limits to Growth , Malthusian catastrophe , Apparent infection rate Exponential stories ... returns Logistic curve Malthusian growth model Exponential algorithm Asymptotic notation EXPSPACE ... more details
Unreferenced date February 2007 The Growth of Biological Thought is a book written by Ernst Mayr , first published in 1982 in literature 1982 . It is subtitled Diversity, Evolution, and Inheritance , and is as much a book of philosophy and history as it is of biology. It is a sweeping, academic study of the first 2,400 years of the science of biology . It focuses largely on how the philosophical assumptions of biologists influenced and limited their understanding. It includes many important general observations about the role of philosophy in scientific inquiry and the place of biology amongst the sciences. External links http books.google.com books?id pHThtE2R0UQC&dq The Growth of Biological Thought Google Books Category Books about the history of science Category Biology books Category History of biology Category Philosophy of biology Category 1982 books DEFAULTSORT Growth of Biological Thought, The science book stub ... more details
wiktionary exponential exponentially Exponential may refer to any of several mathematical topics related to exponentiation , including Exponential function , also Matrix exponential , the matrix analogue to the above Exponential decay , decrease at a rate proportional to value Exponential discounting , a specific form of the discount function, used in the analysis of choice over time Exponentialgrowth , where the growth rate of a mathematical function is proportional to the function s current value Exponential map , in differential geometry Exponential notation , also known as scientific notation, or standard form Exponential object , in category theory Exponential time , in complexity theory in probability and statistics Exponential distribution , a family of continuous probability distributions Exponential family , sometimes used in place of exponential family Exponential smoothing , a technique that can be applied to time series data Function type Exponential type or function type, in type theory Topics listed at list of exponential topics Exponential may also refer to Exponential Technology , a vendor of PowerPC microprocessors disambiguation Category Mathematical disambiguation Category Exponentials ar eo Eksponento fr Exponentielle ... more details
Wiktionary growthGrowth refers to an increase in some quantity over time. The quantity can be Physical e.g., growth in height, growth in an amount of money Abstract e.g., a system becoming more complex, an organism becoming more mature . It can also refer to the mode of growth, i.e. numeric models for describing how much a particular quantity grows over time. Biology Cell growth A tumor is sometimes referred to as a growth Bacterial growth Human development biology deliberately deleting this item, but if included it would go here Human development biology Growth spurt , a short period of rapid growth Auxology , the study of all aspects of human physical growthGrowth hormone Social science Human development humanity Developmental psychology Personal development Personal growth Erikson s stages of psychosocial development , stages of individual growth Population growth Economy Economic growth For financial growth due to simple interest or compound interest see Interest Growth investing Mathematical models Linear function Linear growth Logistic function Logistic growth , characterized as an S curve Exponentialgrowth , also called geometric growth Hyperbolic growth Films Growth film Growth film , a 2010 American horror film disambig ar ca Creixement de Wuchs es Crecimiento desambiguaci n eu Hazkunde argipena fr Croissance ms Pertumbuhan nl Groei pl Wzrost ru simple Growth sk Rast sv Tillv xt ... more details
unreferenced date October 2011 Image Exponential.png thumb 300px right The graph illustrates how an exponentialgrowth surpasses both linear and cubic growths. Notice how quickly and substantially an error can be compounded over time. Exponential error is an idea expressing how a very small error can compound itself over time. It can be characterized as the exponentialgrowth of an error or the application of exponentialgrowth in terms of an error. See also Exponentialgrowth Computational complexity theory Scalability of algorithms Theory of computation Computer science Analysis of algorithms Math stub Category Exponentials ar ... more details
orphan date October 2009 A growth rate is said to be infra exponential if it is dominated by all exponentialgrowth rates, however great the doubling time . A continuous function with infra exponentialgrowth rate will have a Fourier transform that is a Fourier hyperfunction . References http eom.springer.de F f120110.htm Springer Online Mathematics Encyclopedia Category Exponentials mathanalysis stub ... more details
metabolize d see clearance medicine clearance according to exponential decay patterns. The biological ... also Exponential function Exponentialgrowth Radioactive decay for the mathematics of chains of exponential ...Image Plot exponential decay.svg thumb 400px A quantity undergoing exponential decay. Larger decay constants ..., 5, 1, 1 5, and 1 25 for x from 0 to 5. A quantity is said to be subject to exponential decay if it decreases ... equation derivation below is Exponential rate of change math N t N 0 e lambda t . , math Here ... the exponential time constant can be looked at as a scaling time , because we can write the exponential ... of the exponential is chosen to be 2, rather than e. In that case the scaling time is the half life . Half life main Half life A more intuitive characteristic of exponential decay for many people is the time ... this expression is inserted for math tau math in the exponential equation above, and ln 2 is absorbed ... The equation that describes exponential decay is math frac dN t dt lambda N t math or, by rearranging ... commonly used to describe exponential decay. Any one of decay constant, mean lifetime, or half ..., math c frac lambda N 0 . math We see that exponential decay is a scalar multiplication scalar multiple of the exponential distribution i.e. the individual lifetime of each object is exponentially distributed , which has a Exponential distribution Properties well known expected value . We can compute ... exponential processes the total half life can be computed, as above, as the harmonic mean of separate ... t 3 t 1 t 2 t 1 t 3 t 2 t 3 . , math Applications and examples Exponential decay occurs in a wide variety ... complexity of human behavior. However, a few roughly exponential phenomena have been identified there as well. Many decay processes that are often treated as exponential, are really only exponential ... exponential decay as long as the remaining number of atoms is large. The decay product is termed a radiogenic ..., the temperature difference between the object and the medium follows exponential decay in the limit ... more details
e . See exponentialgrowth for this usage. In general, the variable mathematics variable x can be any ... definition formal definition below . E mathematical constant Overview The exponential function arises whenever a quantity exponentialgrowth grows or exponential decay decays at a rate Proportionality ... decay Characterizations of the exponential function Exponential field Exponentialgrowth Exponentiation ...Image exp.svg thumb 200px right The natural exponential function math y e x math In mathematics , the exponential ..., 2006. ref ref The natural exponential function is identical with its derivative. This is really the source of all the properties of the exponential function, and the basic reason for its importance ... and methods edited by Stewart , 2nd revised edition, Oxford Univ. Press, 1996. ref The exponential ... as a superscript. class infobox width 200px colspan 2 align center Exponential Function Representation ... and spherical trigonometry , C.E. Merrill co., 1911. ref refer to the exponential function as the antilogarithm. Sometimes the term exponential function is used more generally for functions of the form ... . Later, in 1697, Johann Bernoulli studied the calculus of the exponential function. ref name mactutor ... without bound leads to the limit of a function limit definition of the exponential function, math ... of a Number , p.156. ref This is one of a number of characterizations of the exponential function ... be shown that the exponential function obeys the basic exponentiation identity, math exp x y exp ... of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself rather than equal to it is expressible in terms of the exponential function. This function property leads to exponentialgrowth and exponential decay. The exponential function extends to an entire function on the complex plane . Euler s formula relates its values at purely imaginary arguments to trigonometric functions . The exponential function also ... more details
Double exponential may refer to A double exponential function Double exponential time, a task with time complexity roughly proportional to such a function Double exponential distribution, which may refer to Laplace distribution , a bilateral exponential distribution Gumbel distribution , an iterated exponential distribution Double exponential integration, most commonly tanh sinh quadrature Double exponential smoothing mathdab ... more details
Distinguish2 the exponential family of probability distributions EDITORS Please see Wikipedia WikiProject ... such as this one. Probability distribution name Exponential type continuous pdf image Image exponential pdf.svg 325px Probability density function cdf image Image exponential cdf.svg 325px Cumulative ... log left lambda x right 1, math In probability theory and statistics , the exponential distribution a.k.a. negative exponential distribution is a family of continuous probability distribution s. It describes ... and memorylessness independently at a constant average rate. Note that the exponential distribution is not the same as the class of exponential family exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its ... density function pdf of an exponential distribution is math f x lambda begin cases lambda ..., we write X Exp . The exponential distribution exhibits infinite divisibility probability ... the probability density function pdf of an exponential distribution as math f x beta begin cases ... is the duration of time that a given biological or mechanical system manages to survive and X Exponential then E X . That is to say, the expected duration of survival of the system ... of events arriving at a rate , when the time between events which might be modelled using an exponential ... is being used if an author writes X Exponential , since either the notation in the previous ... median mean inequality . Memorylessness An important property of the exponential distribution is that it is memorylessness ... T 40 . , math That would be independence. These two events are not independent. The exponential distributions and the geometric distribution s are the only memoryless probability distributions. The exponential ... function for Exponential is math F 1 p lambda frac ln 1 p lambda , math for 0 p 1. The quartile s are therefore ... nowiki 0, and mean , nowiki the exponential distribution with 1 has the largest entropy. Alternatively ... more details
Exponential backoff is an algorithm that uses feedback to multiplicatively decrease the rate of some process, in order to gradually find an acceptable rate. Binary exponential backoff truncated exponential backoff In a variety of computer networks , binary exponential backoff or truncated binary exponential backoff refers to an algorithm used to space out repeated retransmission data networks retransmissions of the same block of data , often as part of network congestion avoidance . Examples are the retransmission of data frame frames in carrier sense multiple access with collision avoidance CSMA CA and carrier sense multiple access with collision detection CSMA CD networks, where this algorithm is part of the Media access control channel access method used to send data on these network. In Ethernet networks, the algorithm is commonly used to schedule retransmissions after collisions. The retransmission is delayed by an amount of time derived from the slot time and the number of attempts to retransmit. After c collisions, a random number of slot times between 0 and 2 sup c sup 1 is chosen. For the first collision, each sender will wait 0 or 1 slot times. After the second collision, the senders will wait anywhere from 0 to 3 slot times Interval mathematics inclusive . After the third collision, the senders will wait anywhere from 0 to 7 slot times inclusive , and so forth. As the number of retransmission attempts increases, the number of possibilities for delay exponentialgrowth increases exponentially . The truncated simply means that after a certain number of increases, the exponentiation stops i.e. the retransmission timeout reaches a ceiling, and thereafter does not increase .... Citation needed date September 2010 An example of an exponential backoff algorithm This example ... value within an acceptable range to ensure that this situation doesn t happen. An exponential ... also Control theory References Reflist FS1037C Use dmy dates date September 2010 DEFAULTSORT Exponential ... more details
See Lyapunov stability , which gives a definition of asymptotic stability for more general dynamical systems . All exponentially stable systems are also asymptotically stable. In control theory , a continuous LTI system theory linear time invariant system is exponentially stable if and only if the system has eigenvalue s i.e., the pole complex analysis pole s of input to output systems with strictly negative real parts. i.e., in the left half of the complex plane . ref David N. Cheban 2004 , Global Attractors Of Non autonomous Dissipative Dynamical Systems . p. 47 ref A discrete time input to output LTI system is exponentially stable if and only if the poles of its transfer function lie strictly within the unit circle centered on the origin of the complex plane. Exponential stability is a form of asymptotic stability . Systems that are not LTI are exponentially stable if their convergence is bounded function bounded by exponential growth exponential decay . Practical consequences An exponentially stable LTI system is one that will not blow up i.e., give an unbounded output when given a finite input or non zero initial condition. Moreover, if the system is given a fixed, finite input i.e., a Heaviside step function step , then any resulting oscillations in the output will decay at an exponential growth exponential rate , and the output will tend asymptote asymptotically to a new final, steady state value. If the system is instead given a Dirac delta function Dirac delta impulse as input, then induced oscillations will die away and the system will return to its previous value. If oscillations do not die away, or the system does not return to its original output when an impulse is applied, the system is instead marginal stability marginally stable . Example exponentially stable LTI systems Image AsymptoticStabilityImpulseScilab.png thumb 320px The impulse responses of two exponentially stable systems The graph on the right shows the impulse response of two similar ... more details
otheruses4 polynomials in variables and exponential functions the polynomials involving Stirling numbers Touchard polynomials In mathematics , exponential polynomials are functions on Field mathematics ... and an exponential function . Definition In fields There is no single definition of what an exponential ... kind of exponential function E x . In the complex numbers there is already a canonical exponential ... exponential polynomial is often used to mean polynomials of the form P x , e sup x sup where P C x , y is a polynomial in two variables. ref C. J. Moreno, The zeros of exponential polynomials ... particularly special about C here, exponential polynomials may also refer to such a polynomial on any exponential field or exponential ring with its exponential function taking the place of e sup ... to have one variable, and an exponential polynomial in n variables would be of the form ... in 2 n variables. In abelian groups A more general framework where the term exponential polynomial may be found is that of exponential functions on abelian groups. Similarly to how exponential functions on exponential fields are defined, given a topological abelian group G a homomorphism ... to the multiplicative group of nonzero complex numbers is called an exponential function, or simply an exponential. A product of additive functions and exponentials is called an exponential monomial, and a linear combination of these is then an exponential polynomial on G . ref L szl Sz kelyhidi ... ref P. G. Laird, On characterizations of exponential polynomials , Pacific Journal of Mathematics 80 1979 , pp.503&ndash 507. ref Uses Exponential polynomials on R and C often appear in transcendence theory , where they appear as auxiliary function s in proofs involving the exponential function. They also act as a link between model theory and analytic geometry . If one defines an exponential variety to be the set of points in R sup n sup where some finite collection of exponential polynomials ... more details
unreferenced date November 2007 In economics exponential discounting is a specific form of the discount function , used in the analysis of Intertemporal choice choice over time with or without uncertainty . Formally, exponential discounting occurs when total utility is given by math U c t t t 1 t 2 sum t t 1 t 2 delta t t 1 u c t , math where c sub t sub is Consumption economics consumption at time t , math delta math is the exponential discount factor , and u is the instantaneous utility function . In continuous time , exponential discounting is given by math U c t t t 1 t 2 int t 1 t 2 e rho t t 1 u c t ,dt, math Exponential discounting implies that the marginal rate of substitution between consumption at any pair of points in time depends only on how far apart those two points are. Exponential discounting is not dynamically inconsistent . For its simplicity, the exponential discounting assumption is the most commonly used in economics. However, alternatives like hyperbolic discounting have more empirical support. See also temporal discounting hyperbolic discounting intertemporal choice Category Intertemporal economics ... more details
An exponential factorial is a positive integer n exponentiation raised to the power of n &minus 1, which in turn is raised to the power of n &minus 2, and so on and so forth, that is, math n n 1 n 2 cdots . , math The exponential factorial can also be defined with the recurrence relation math a 0 1, quad a n n a n 1 . , math The first few exponential factorials are 1 number 1 , 1 number 1 , 2 number 2 , 9 number 9 , 262144, etc. OEIS id A049384 . So, for example, 262144 is an exponential factorial since math 262144 4 3 2 1 . , math The exponential factorials grow much more quickly than regular factorial s or even hyperfactorial s. The exponential factorial of 5 is 5 sup 262144 sup which is approximately 6.206069878660874 × 10 sup 183230 sup . The sum of the reciprocals of the exponential factorials from 1 onwards is the irrational number 1.6111149258083767361111... OEIS2C id A080219 . Like tetration , there is currently no accepted method of extension of the exponential factorial function to real number real and complex number complex values of its argument, unlike the factorial function, for which such an extension is provided by the gamma function . Numtheory stub References Jonathan Sondow, http mathworld.wolfram.com ExponentialFactorial.html Exponential Factorial From Mathworld , a Wolfram Web resource Category Integer sequences Category Factorial and binomial topics Category Large integers es Factorial exponencial he uk ... more details
In mathematics , an exponential field is a Field mathematics field that has an extra operation on its ... then F is called an exponential field, and the function E is called an exponential function on F . ref Helmut Wolter, Some results about exponential fields survey , M moires de la S.M.F. 2 sup e sup s rie, 16 , 1984 , pp.85&ndash 94. ref Thus an exponential function on a field is a homomorphism from the additive group of F to its multiplicative group. Examples There is a trivial exponential function ... of the field under multiplication. Thus every field is trivially also an exponential field, so the cases of interest to mathematicians occur when the exponential function is non trivial. If a field has Characteristic algebra characteristic p 0 then it can be shown that the only exponential function on the field is the trivial one. ref name Dries Lou van den Dries, Exponential rings, exponential polynomials and exponential functions , Pacific Journal of Mathematics, 113 , no.1 1984 , pp.51 ..., exponential fields are sometimes required to have characteristic zero. ref Martin Bays ... 0810.4457 ref When the field does have characteristic zero then there can be non trivial exponential ... and one, has infinitely many exponential functions. One such function is the usual exponential function ... field R equipped with this function gives the ordered real exponential field, denoted R sub exp sub R , , , ,0,1,exp . In fact any real number a > 0 gives an exponential function ... to the real exponential field, there is the Complex number complex exponential field, C sub exp sub C , , ,0,1,exp . Boris Zilber constructed an exponential field K sub exp sub that, crucially, satisfies the equivalent formulation of Schanuel s conjecture with the field s exponential ... zero , Ann. Pure Appl. Logic, 132 , no.1 2005 , pp.67&ndash 95. ref It is conjectured that this exponential ... mathematics ring , R , and concurrently the exponential function is relaxed to be a homomorphism ... more details
Unreferenced date December 2009 In the mathematics mathematical theory of dynamical systems , an exponential dichotomy is a property of an equilibrium point that extends the idea of hyperbolic equilibrium point hyperbolicity to non autonomous system s. Definition If math dot mathbf x A t mathbf x math is a linear system linear non autonomous dynamical system in R sup n sup with fundamental solution matrix t , 0 I , then the equilibrium point 0 is said to have an exponential dichotomy if there exists a constant matrix mathematics matrix P such that P sup 2 sup P and positive constants K , L , , and such that math Phi t P Phi 1 s le Ke alpha t s mbox for s le t infty math and math Phi t I P Phi 1 s le Le beta s t mbox for s ge t infty. math If furthermore, L 1 K and , then 0 is said to have a uniform exponential dichotomy . The constants and allow us to define the spectral window of the equilibrium point, &minus , . Explanation The matrix P is a projection onto the stable subspace and I &minus P is a projection onto the unstable subspace. What the exponential dichotomy says is that the norm of the projection onto the stable subspace of any orbit in the system exponential decay decays exponentially as t and the norm of the projection onto the unstable subspace of any orbit decays exponentially as t &minus , and furthermore that the stable and unstable subspaces are conjugate because math scriptstyle P oplus I P mathbb R n math . An equilibrium point with an exponential dichotomy has many of the properties of a hyperbolic equilibrium point in autonomous system mathematics autonomous system s. In fact, it can be shown that a hyperbolic point has an exponential dichotomy. DEFAULTSORT Exponential Dichotomy Category Dynamical systems Category Dichotomies ... more details
Distinguish2 the Tsallis Tsallis statistics q exponential q exponential Unreferenced date December 2009 Lowercase In combinatorics combinatorial mathematics , the q exponential is a q analog of the exponential function . Definition The q exponential math e q z math is defined as math e q z sum n 0 infty frac z n n q sum n 0 infty frac z n 1 q n q q n sum n 0 infty z n frac 1 q n 1 q n 1 q n 1 cdots 1 q math where math n q math is the q factorial and math q q n 1 q n 1 q n 1 cdots 1 q math is the q Pochhammer symbol . That this is the q analog of the exponential follows from the property math left frac d dz right q e q z e q z math where the derivative on the left is the q derivative . The above is easily verified by considering the q derivative of the monomial math left frac d dz right q z n z n 1 frac 1 q n 1 q n q z n 1 . math Here, math n q math is the q bracket . Properties For real math q 1 math , the function math e q z math is an entire function of z . For math q 1 math , math e q z math is regular in the disk math z 1 1 q math . Relations For math q 1 math , a function that is closely related is math e q z E q z 1 q math Here, math E q t math is a special case of the basic hypergeometric series math E q z 1 phi 0 0 q,z prod n 0 infty frac 1 1 q n z math DEFAULTSORT Q Exponential Category Q analogs Category Exponentials eo Q eksponenta funkcio pl Funkcja q wyk adnicza it Funzione q esponenziale ... more details
to be of exponential type with respect to math K math if for every math varepsilon 0 math there exists ... title Functions of exponential type journal Ann. of Math. 2 volume 65 year 1957 pages 582&ndash ... more details
about the exponential map in differential geometry discrete dynamical systems Exponential map discrete ... right The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography. In differential geometry , the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection . Two important special cases of this are the exponential map for a manifold with a Riemannian metric , and the exponential map from a Lie algebra to a Lie group . Definition Let ... exponential map is defined by exp sub p sub v sub v sub 1 . In general, the exponential map .... An affine connection is called complete if the exponential map is well defined at every point of the tangent bundle . Lie theory Lie groups In the theory of Lie group s the exponential map is a map ... structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras. The ordinary exponential function of mathematical analysis is a special case of the exponential map when G is the multiplicative group of non zero real number s whose Lie algebra is the additive group of all real numbers . The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function ... of math G math . The exponential map is a map math exp colon mathfrak g to G math which can be defined in several different ways as follows It is the exponential map of a canonical left invariant affine connection on G , such that parallel transport is given by left translation. It is the exponential ... subgroups acting by left or right multiplication so give the same exponential map. It is given by math ... near zero. If math G math is a matrix Lie group , then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion math exp X sum k 0 infty frac X k k I X ... more details
Exponential smoothing is a technique that can be applied to time series data, either to produce smoothed ... observations are weighted equally, exponential smoothing assigns exponentially decreasing weights over time. Exponential smoothing is commonly applied to financial market and economic data, but it can ... by x sub t sub , and the output of the exponential smoothing algorithm is commonly written as s sub ... of observations begins at time t 0, the simplest form of exponential smoothing .... The exponential moving average Exponential smoothing was first suggested by Charles C. Holt in 1957 ... is the one commonly used, is attributed to Brown and is known as Brown s simple exponential smoothing ... of exponential smoothing is given by the formulae math begin align s 1& x 0 s t& alpha x t 1 1 alpha ... the same as the original series with lag of one time unit . Simple exponential smoothing is easily applied ... the original signal without information loss all stages of the exponential moving average ... those to be skipped. This simple form of exponential smoothing is also known as an Moving average Exponential moving average exponentially weighted moving average EWMA . Technically it can also be classified .... ref cite web url http www.duke.edu rnau 411avg.htm title Averaging and Exponential Smoothing Models accessdate 26 July 2010 ref Why is it exponential ? By direct substitution of the defining equation for simple exponential smoothing back into itself we find that math begin align s t& alpha x t 1 1 ... progression is the discrete version of an exponential function , so this is where the name for this smoothing method originated. Comparison with moving average Exponential smoothing and moving average ... 2 k 1 . They differ in that exponential smoothing takes into account all past data, whereas moving ... average requires that the past k data points be kept, whereas exponential smoothing only ... Double exponential smoothing Simple exponential smoothing does not do well when there is a Trend estimation ... more details
An exponential tree is almost identical to a binary search tree , with the exception that the dimension of the tree is not the same at all levels. In a normal binary search tree, each node has a dimension d of 1, and has 2 sup d sup children. In an exponential tree, the dimension equals the depth of the node, with the root node having a d 1. So the second level can hold two nodes, the third can hold eight nodes, the fourth 64 nodes, and so on. Layout Exponential Tree can also refer to a method of laying out the nodes of a tree structure in n typically 2 dimensional space. Nodes are placed closer to a baseline than their parent node, by a factor equal to the number of child nodes of that parent node or by some sort of weighting , and scaled according to how close they are. Thus, no matter how deep the tree may be, there is always room for more nodes, and the geometry of a subtree is unrelated to its position in the whole tree. The whole has a fractal structure. In fact, this method of laying out a tree can be viewed as an application of the upper half plane model of hyperbolic geometry , with isometries limited to translations only. Deleted image removed Image ExponentialTree2.PNG See also http www.parc.xerox.com csl groups sda publications papers Lamping UIST94 for web.pdf link not working CS Trees Category Exponentials Category Trees structure datastructure stub ... more details
In computational complexity theory , the exponential hierarchy is a hierarchy of complexity class es, starting with EXPTIME math rm EXPTIME bigcup k in mathbb N mbox DTIME left 2 n k right math and continuing with math mbox 2 EXPTIME bigcup k in mathbb N mbox DTIME left 2 2 n k right math math mbox 3 EXPTIME bigcup k in mathbb N mbox DTIME left 2 2 2 n k right math and so on. We have P complexity P EXPTIME 2 EXPTIME 3 EXPTIME . Unlike the analogous case for the polynomial hierarchy , the time hierarchy theorem guarantees that these inclusions are proper that is, there are languages in EXPTIME but not in P, in 2 EXPTIME but not in EXPTIME and so on. The union of all the classes in the exponential hierarchy is the class ELEMENTARY . References Computational Complexity . Addison Wesley, 1994. pp 497 498 ComplexityClasses DEFAULTSORT Exponential Hierarchy Category Complexity classes it Gerarchia esponenziale zh ... more details
Unreferenced date March 2008 The ordered exponential also called the path ordered exponential is a mathematics mathematical object, defined in non commutative algebra s, which is equivalent to the exponential function of the integral in the commutative algebras. Therefore it is a function mathematics function , defined by means of a function from real number s to a real or complex associative algebra . In practice the values lie in matrix math matrix and Operator mathematics operator algebras. For the element A t from the algebra math g, math set g with the non commutative product , where t is the time parameter , the ordered exponential math OE A t equiv left e int 0 t dt A t right math of A can be defined via one of several equivalent approaches As the Limit mathematics limit of the ordered product of the infinitesimal exponentials math OE A t lim N rightarrow infty left e epsilon A t N e epsilon A t N 1 cdots e epsilon A t 1 e epsilon A t 0 right math where the time moments math t 0, t 1, ... t N math are defined as math t j j epsilon math for math j 0, ... N math , and math epsilon t N math . Via the initial value problem , where the OE A t is the unique solution of the system of equations math frac partial OE A t partial t A t OE A t , math math OE A 0 1. math Via an integral equation math OE A t 1 int 0 t dt A t OE A t . math Via Taylor series expansion math OE A t 1 int 0 t dt 1 A t 1 int 0 t dt 1 int 0 t 1 dt 2 A t 1 A t 2 math math int 0 t dt 1 int 0 t 1 dt 2 int 0 t 2 dt 3 A t 1 A t 2 A t 3 cdots math See also Related Path ordering describes essentially the same concept. Product integral Category Abstract algebra Category Ordinary differential equations ... more details
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