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 Exponential growth , 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
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
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
orphan date October 2009 A growth rate is said to be infra exponential if it is dominated by all exponential growth rates, however great the doubling time . A continuous function with infra exponential growth 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
unreferenced date October 2011 Image Exponential.png thumb 300px right The graph illustrates how an exponential growth 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 exponential growth of an error or the application of exponential growth in terms of an error. See also Exponential growth Computational complexity theory Scalability of algorithms Theory of computation Computer science Analysis of algorithms Math stub Category Exponentials ar ... 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. Trivial exponential function There is a trivial exponential function on any field, namely the map that sends every element to the identity element 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. Exponential fields are sometimes required to have Characteristic algebra characteristic zero as the only exponential ... Dries, Exponential rings, exponential polynomials and exponential functions , Pacific Journal of Mathematics ... infinitely many exponential functions. One such function is the usual exponential function , that is E ... 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 on R , specifically ... 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 function .... Pure Appl. Logic, 132 , no.1 2005 , pp.67&ndash 95. ref It is conjectured that this exponential field is actually C sub exp sub , and a proof of this fact would thus prove Schanuel s conjecture. Exponential ... be a Ring mathematics ring , R , and concurrently the exponential function is relaxed to be a homomorphism ... object is called an exponential ring . ref name Dries An example of an exponential ring ... 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
Exponential growth Image exp.svg thumb 200px right The natural exponential function math y e x math In mathematics , the exponential function is the function mathematics function e sup x sup , where ... , 11th ed., Prentice Hall, 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 .... ref The exponential function is used to model a relationship in which a constant change in the independent ... the independent variable as a superscript. class infobox width 200px colspan 2 align center Exponential ... and Durrel, Plane 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 ... , not necessarily e . See exponential growth for this usage. In general, the variable mathematics ... object see the Formal definition formal definition below . E mathematical constant Overview The exponential function arises whenever a quantity exponential growth grows or exponential decay decays at a rate ... n, math now known as e . Later, in 1697, Johann Bernoulli studied the calculus of the exponential ... intervals per year grow without bound leads to the limit of a function limit definition of the exponential ... Eli Maor , e the Story of a Number , p.156. ref This is one of a number of characterizations of the exponential ... of these definitions it can be shown that the exponential function obeys the basic exponentiation identity ... rate of change of the exponential function is the exponential function itself. More generally ... in terms of the exponential function. This function property leads to exponential growth 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 has analogues for which the argument is a matrix exponential matrix , or even ... 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
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
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 ... distribution . 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 members, but also includes the normal ... others. Characterization Probability density function The probability density function pdf of an exponential ... nowiki nowiki . If a random variable X has this distribution, we write X Exp . The exponential ... pdf of an exponential distribution as math f x beta begin cases frac 1 beta e x beta , & x ge 0, 0 ... biological or mechanical system manages to survive and X Exponential then E X ... , when the time between events which might be modelled using an exponential distribution has a mean ... X Exponential , since either the notation in the previous using or the notation ... between the mean and the median median mean inequality . Memorylessness An important property of the exponential .... These two events are not independent. The exponential distributions and the geometric distribution s are the only memoryless probability distributions. The exponential distribution is consequently also ... The quantile function inverse cumulative distribution function for Exponential is math F 1 p ... theory support nowiki 0, and mean , nowiki the exponential distribution with 1 has the largest ... accessdate 2011 06 02 ref Distribution of the minimum of exponential random variables ... 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
File Exponential.svg thumb 300px right The graph illustrates how exponential growth green surpasses both linear red and cubic blue growth. legend green Exponential growth legend red Linear growth legend blue Cubic growth Exponential growth including exponential decay when the growth rate is negative ... progression . The formula for exponential growth 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 exponential growth model ... of exponential responses the loudness and frequency of sound are perceived logarithmically ... increase, rather than an exponential increase. This has survival value . Generally it is important ... exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material .... Due to the exponential rate of increase, at any point in the chain reaction 99 of the energy will have ... amplification can result in the exponential growth of the amplified signal, although resonance ... results whose best fit line are exponential decay curves. Economics Economic growth is expressed in percentage terms, implying exponential growth. For example, U.S. GDP per capita has grown at an exponential rate of approximately two percent per year for two centuries. Multi level marketing . Exponential ... exponential growth of the capital. See also rule of 72 . Pyramid scheme s or Ponzi scheme s also ... also Moore s law and technological singularity under exponential growth, there are no singularities. The singularity here is a metaphor. . In computational complexity theory , computer algorithms of exponential ... with an exponential algorithm . Also, the effects of Moore s Law do not help the situation much because ... for x to increase by a factor of b math x t tau x t cdot b , . math If 0 and b 1, then x has exponential growth. If 0 and b 1, or 0 and 0 b 1, then x has exponential decay . Example If a species ... positive number b . Thus the law of exponential growth can be written in different ... more details
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 ... when the base 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 ... ln 2. math When this expression is inserted for math tau math in the exponential equation above, and ln ... equation The equation that describes exponential decay is math frac dN dt lambda N math or, by rearranging ... that is most commonly used to describe exponential decay. Any one of decay constant, mean lifetime ..., 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 ... t 2 t 3 t 1 t 2 t 1 t 3 t 2 t 3 . , math Applications and examples Exponential decay occurs in a wide ... that are often treated as exponential, are really only exponential so long as the sample is large ... to a different state, the number of atoms in the original state follows exponential decay as long as the remaining ... difference between the object and the medium follows exponential decay in the limit of slow processes ... order reactions first order reactions consequently follow exponential decay. For instance, many ... external load resistance R . The exponential time constant for the process is R C , and the half ... clearance medicine clearance according to exponential decay patterns. The biological half life biological ... 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 ... commonly used, is attributed to Brown and is known as Brown s simple exponential smoothing . ref cite ... year 1963 publisher Prentice Hall location Englewood Cliffs, NJ ref The simplest form of exponential ... series with lag of one time unit . Simple exponential smoothing is easily applied, and it produces ... signal without information loss all stages of the exponential moving average must also be available ... to be skipped. This simple form of exponential smoothing is also known as an Moving average Exponential ... url http www.duke.edu rnau 411avg.htm title Averaging and Exponential Smoothing Models accessdate ... . Why is it exponential ? By direct substitution of the defining equation for simple exponential smoothing ... 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 are similar in that they both ... in that exponential smoothing takes into account all past data, whereas moving average only takes into account ... that the past k data points be kept, whereas exponential smoothing only needs the most recent forecast ... edition 6th edition ISBN 0073377856 Page needed date September 2011 ref Double exponential smoothing Simple exponential smoothing does not do well when there is a Trend estimation trend in the data ... were devised under the name double exponential smoothing . One method, sometimes referred to as Holt ... 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
Exponential Technology was a vendor of PowerPC microprocessor s. The company was founded by George Taylor and Jim Blomgren in 1993. The company s plan was to use BiCMOS technology to produce very fast processors for the Apple Computer market. Logic used 3 level ECL circuits single ended for control logic, and differential for datapaths while RAM structures used CMOS. The company was originally named Renaissance Microsystems . Rick Shriner was the CEO . Their chips were manufactured by Hitachi, Ltd. Hitachi . Their product, the Exponential X704 , was advertised to run at 533 MHz, but the first version of the device only ran at about 400 MHz. This lower frequency along with small level one CPU cache caches , produced systems which had good but not stellar performance. This allowed Motorola Apple s traditional processor vendor , to convince the computer maker that Motorola s future roadmap would produce processors with similar performance, hence making it less attractive for Apple to rely on the small startup company for critical technology. Due to Apple s financial problems at the time, Exponential starting marketing the device to Apple Macintosh clone makers such as Power Computing and UMAX . In order to diversify into other markets, a second design team was started under Paul Nixon engineer Paul Nixon in Austin, TX to build a BiCMOS Intel x86 processor. Due to Apple s decision to close off the Macintosh clone market, Exponential ran out of possible customers for their chips. The company closed in 1997, though the Texas design team run by Paul Nixon continued on as EVSX. EVSX changed its name to Intrinsity Intrinsity, Inc. in 2000. ref EVSX Announces Name Change And Technology Focus, http linuxpr.com releases 1885.html Linux PR , May 24, 2000 ref External links http www.cs.utexas.edu users karu papers exponential.html Fast Company article on the company http ieeexplore.ieee.org iel3 4 13972 00641683.pdf?arnumber 641683 JSSC paper on the processor http www.intrinsity.com ... 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 exponential growth increases exponentially . The truncated simply means that after a certain number of increases, the exponentiation .... 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 Backoff Category Networking algorithms Category Ethernet de Binary Exponential Backoff hu Exponenci lis ... more details
unreferenced date September 2009 In mathematics , an exponential sum may be a finite Fourier series i.e. a trigonometric polynomial , or other finite sum formed using the exponential function , usually expressed by means of the function math e x exp 2 pi ix . , math Therefore a typical exponential sum may take the form math sum e x n , math summed over a finite sequence of real number s x sub n sub . Formulation If we allow some real coefficients a sub n sub , to get the form math sum a n e x n math it is the same as allowing exponents that are complex number s. Both forms are certainly useful in applications. A large part of twentieth century analytic number theory was devoted to finding good estimates for these sums, a trend started by basic work of Hermann Weyl in diophantine approximation . Estimates The main thrust of the subject is that a sum math S sum e x n math is trivially estimated by the number N of terms. That is, the absolute value math S le N , math by the triangle inequality , since each summand has absolute value 1. In applications one would like to do better. That involves proving some cancellation takes place, or in other words that this sum of complex numbers on the unit circle is not of numbers all with the same Parameter argument . The best that is reasonable to hope for is an estimate of the form math S O sqrt N , math which signifies, up to the implied constant ... of Weyl differencing investigated by Weyl involving a generating exponential sum math G tau ... general application. Types of exponential sum Many types of sums are used in formulating particular ... is between a complete exponential sum , which is typically a sum over all residue class es modular arithmetic modulo some integer N or more general finite ring , and an incomplete exponential sum ... exponential sums are Gauss sum s and Kloosterman sum s these are in some sense finite field ... i.e., along an algebraic variety over a finite field . One of the most general types of exponential ... more details
Infobox data structure name Exponential tree type tree invented by Arne Andersson invented year 1995 space avg O n log n space worst O n log n search avg O min log n , log n log w , log log n , log w log log n search worst O min log n , log n log w , log log n , log w log log n insert avg O min log n , log n log w , log log n , log w log log n insert worst O min log n , log n log w , log log n , log w log log n delete avg O min log n , log n log w , log log n , log w log log n delete worst O min log n , log n log w , log log n , log w log log n 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 ... number3 pxc3873876.pdf Implementation and Performance Analysis of Exponential Tree Sorting CS Trees ... more details
orphan date April 2012 In mathematics , the Carlitz exponential is a characteristic p analogue to the usual exponential function studied in real analysis real and complex analysis . It is used in the definition of the Carlitz module an example of a Drinfel d module . Definition We work over the polynomial ring F sub q sub T of one variable over a finite field F sub q sub with q elements. The Completion metric space completion C sub sub of an algebraic closure of the field F sub q sub T sup &minus 1 sup of formal Laurent series in T sup &minus 1 sup will be needed. It is a complete and algebraically closed field. First we need analogues to the factorials , which appear in the definition of the usual exponential function. For i 0 we define math i T q i T, , math math D i prod 1 le j le i j q i j math and D sub 0 sub 1. Note that that the usual factorial is inappropriate here, since n vanishes in F sub q sub T unless n is smaller than the Characteristic algebra characteristic of F sub q sub T . Using this we define the Carlitz exponential e sub C sub C sub sub C sub sub by the convergent sum math e C x sum j 0 infty frac x q j D i . math Relation to the Carlitz module The Calitz exponential satisfies the functional equation math e C Tx Te C x left e C x right q T tau e C x , , math where we may view as the power of q map or as an element of the ring F sub q sub T of noncommutative polynomials . By the universal property of polynomial rings in one variable this extends to a ringhomomorphism F sub q sub T C sub sub , defining a Drinfel d F sub q sub T module over C sub sub . It is called the Carlitz module. References reflist Citation last1 Goss first1 D. authorlink David Goss title Basic structures of function field arithmetic publisher Springer Verlag location Berlin, New York series Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Results in Mathematics and Related Areas 3 isbn 978 3 540 61087 8 mr 1423131 year 1996 volume 35 Citation ... more details
In mathematics , the matrix exponential is a matrix function on square matrix square matrices analogous to the ordinary exponential function . Abstractly, the matrix exponential gives the connection between ... or complex number complex matrix mathematics matrix . The exponential of X , denoted by e sup X sup ... k mathbf X k. math The above series always converges, so the exponential of X is well defined. Note that if X is a 1× 1 matrix the matrix exponential of X is a 1× 1 matrix consisting of the ordinary Exponential function exponential of the single element of X . Properties Let X and Y be n × ... matrix by I and the zero matrix by 0. The matrix exponential satisfies the following properties ... for the importance of the matrix exponential is that it can be used to solve systems of linear ... where A is a constant matrix, is given by math y t e At y 0. , math The matrix exponential can also ..., but the Magnus series gives the solution as an infinite sum. The exponential of sums We know that the exponential ... of E. Lieb journal Commun Math. Phys. volume 31 page 317 325 year 1973 ref The exponential map Note that the exponential of a matrix is always an invertible matrix . The inverse matrix of e sup X sup is given by e sup &minus X sup . This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map math exp colon M n mathbb C to mathrm ... is surjective which means that every invertible matrix can be written as the exponential of some other ... matrix norm . It follows that the exponential map is continuity mathematics continuous and Lipschitz ... d dt X t frac 1 3 X t , X t , frac d dt X t cdots math The determinant of the matrix exponential ... A . math In addition to providing a computational tool, this formula shows that a matrix exponential .... Computing the matrix exponential Finding reliable and accurate methods to compute the matrix exponential is difficult, and this is still a topic of considerable current research in mathematics ... more details
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