von Foerster http www.robotwisdom.com science logarithmic.html Detailedlogarithmictimeline of the Universe http www.futuretimeline.net http www.futuretimeline.net a timeline of future history DEFAULTSORT DetailedLogarithmicTimeline Category Timelines ru ...Main logarithmictimeline This timeline allows one to see the whole history of the universe, the Earth, and mankind Humankind is a bastard word, half Latin, half Germanic. Its use is based on the mistaken idea that mankind is sexist. in one table. Each row is defined in years ago , that is, year s before the present calendar date date , with the earliest times at the top of the chart. In each table cell on the right, references to events or notable people are given, more or less in chronological order within the cell. Each row corresponds to a change in log time before present of about 0.1 using log base 10 , similar to Renard numbers . The table for recent events assumes a present of 1 January 2010. class wikitable style background silver abbr Interval Time interval, before the present time. a annum year style background silver abbr Period List of time periods Period style background silver abbr Event Event, Invention or Historical development align center 13.7 Gigaannum Ga &ndash 12.6 Ga Big Bang , Inflation cosmology Inflation , Galaxy formation and evolution Stars and galaxies , Earliest quasar s align center 12.6 Ga &ndash 10 Ga Omega Centauri star cluster forms align center 10 Ga &ndash 8 Ga Gliese ... . A logarithmictimeline can also be devised for events which should occur in the future, barring ... s fall or are flung away from their stars. See also Timeline of the far future List of timelines Timeline of the Big Bang Geologic time scale Timeline of evolution Orders of magnitude time World history Technological singularity Logarithmictimeline Future of an expanding universe References Reflist ... more details
to that of Sparks Histomap can be found at Detailedlogarithmictimeline . Example of a forward looking logarithmictimeline In this table each row is defined in seconds after the Big Bang , with earliest ... science logarithmic.html Detailedlogarithmictimeline of the Universe http urss.ru cgi ...Unreferenced date April 2009 A logarithmictimeline is a timeline laid out according to a logarithmic .... Example of a backward looking logarithmictimeline In this table each row is defined in years ... fade in an exponential manner. Logarithmic timelines have also been used in future studies to justify the idea of a technological singularity . A logarithmic scale enables events throughout time to be presented ... modern times, and the logarithmic scale fulfills just this condition without any break in the continuity ... Planck Epoch align center 10 sup 35 sup to 10 sup 30 sup Timeline of the Big Bang Epoch of Grand Unification ... of the chart. Each event is an occurrence of an observed or inferred process. Note that the logarithmic ... , Nanotechnology , Global warming , Timeline of historic inventions 2000s more... align center 10 sup 1 sup to 10 sup 2 sup 20th century Timeline of transportation technology 20th century ... war s, Nuclear energy , Timeline of historic inventions 20th century more... align center 10 sup ... press , Industrial Revolution , Colonialism , Firearms , Steam engine , Timeline of historic inventions ... , Wheel , Civilization , Major world religions Religions , Philosophy , Timeline of historic inventions ... species extinction, Ice age ends, Domestication Timeline of agriculture and food technology Agriculture ... sup 10 sup Precambrian , Timeline of cosmological eras Cosmology Timeline of the Big Bang Big Bang , Galaxy formation and evolution Stars and galaxies , Earth , Origin of Life Life See also Timeline of the Big Bang Geologic time scale Timeline of evolution Natural history Orders of magnitude time Technological singularity Graphical timeline from Big Bang to Heat Death Timeline External links http ... more details
Logarithmic can refer to Logarithm , a transcendental function in mathematics Logarithmic scale , the use of the logarithmic function to describe measurements Logarithmic growth Logarithmic distribution , a discrete probability distribution Natural logarithm mathdab ... more details
In mathematics and statistical mechanics , a Markov process is said to have detailed balance if the transition probability , P , between each pair of states i and j in the state space obey math pi i P ij pi j P ji ,, math where P is the Markov transition matrix transition probability , i.e. , P sub ij sub P X sub t sub j X sub t 1 sub i and sub i sub and sub j sub are the equilibrium probabilities of being in states i and j , respectively. ref name OHagan Cite book last1 O Hagan first1 Anthony authorlink1 last2 Forster first2 Jonathan authorlink2 title Kendall s Advanced Theory of Statistics, Volume 2B Bayesian Inference trans title url archiveurl archivedate format accessdate type edition series volume date year 2004 month origyear publisher Oxford University Press location New York isbn 0 340 807520 oclc doi id page 263 pages ... s P s,s ,. math A Markov process that has detailed balance is said to be a reversible Markov process or Markov chain Reversible Markov chain reversible Markov chain . ref name OHagan The detailed ... processes with stationary distributions that do not have detailed balance. Detailed balance ... flow condition implies detailed balance. A Markov process with detailed balance can be described ... P s , s are called doubly stochastic and they always have detailed balance. In these cases ... with detailed balance, it may be possible to continuously transform the coordinates until the equilibrium ... of sub states. Physical significance In a closed and isolated physical system, detailed ... classical physics is time reversible, but not quantum physics . Detailed balance holds because ... Hermitian . A consequence of detailed balance holding is the second law of thermodynamics . ref ... Detailed Balance Category Probability theory Category Statistical mechanics Category Markov models de Detailed Balance ja ... more details
Impact Hazard Scale Logarithmictimeline counting f stop s for ratios of photographic exposure rating ...Refimprove date May 2009 Cleanup date August 2007 File Logarithmic Scales.svg thumb 400px Various scales ... . A logarithmic scale is a scale measurement scale of measurement using the logarithm of a physical ... are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4. Each unit increase on the logarithmic scale thus ... 10, in this case . Presentation of data on a logarithmic scale can be helpful when the data cover a large ... reduces a wide range to a more manageable size. Some of our sense s operate in a logarithmic fashion Weber Fechner law , which makes logarithmic scales for these input quantities especially appropriate ... differences in pitch. In addition, studies of young children in an isolated tribe have shown logarithmic ... 10.1126 science.1156540 pmid 18511690 pmc 2610411 ref Definition and base Logarithmic scales are either ... units. Deviating from these units means that the logarithmic measure will change by an additive ... to be a dimensional quantity expressed in generic indefinite base logarithmic units. Example scales On most logarithmic scales, small values or ratios of the underlying quantity correspond to negative values of the logarithmic measure. Well known examples of such scales are Richter magnitude scale ... curves of soil Some logarithmic scales were designed such that large values or ratios of the underlying quantity correspond to small values of the logarithmic measure. Examples of such scales ... samples. Logarithmic units Logarithmic units are abstract mathematical units that can be used to express any quantities physical or mathematical that are defined on a logarithmic scale, that is, as being proportional to the value of a logarithm function. In this article, a given logarithmic ... here denotes the indefinite logarithm function Log . Examples Examples of logarithmic units include ... log e , and other logarithmic scale units such as the Richter scale point log 10 or more generally ... more details
Unreferenced date December 2009 Image Log.svg thumb A graph of logarithmic growth In mathematics , logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. y C log x . Note that any logarithm base can be used, since one can be converted to another by a fixed constant. Logarithmic growth is the inverse of exponential growth and is very slow. A familiar example of logarithmic growth is the number of digits needed to represent a number, N , in positional notation , which grows as log sub b sub N , where b is the base of the number system used, e.g. 10 for decimal arithmetic. Another example is in cryptography , where the key cryptography key size needed to protect against a brute force attack for a certain period of time grows logarithmically with the desired protection interval. In the design of computer algorithm s, logarithmic growth, and related variants, such as log linear, or linearithmic , growth are very desirable indications of efficiency. Logarithmic growth can lead to apparent paradoxes, as in the martingale roulette system martingale roulette system, where the potential winnings before bankruptcy grow as the logarithm of the gambler s bankroll. It also plays a role in the St. Petersburg paradox . In microbiology , the rapidly growing exponential growth phase of a cell culture is sometimes called logarithmic growth. During this bacterial growth phase, the number of new cells appearing are proportional to the population. See also Iterated logarithm an even slower growth model DEFAULTSORT Logarithmic Growth Category Logarithms ... more details
Unreferenced date December 2009 In mathematics , specifically in calculus and complex analysis , the logarithmic ... f or, the derivative of the natural logarithm of f . This follows directly from the chain rule . The logarithmic ... also apply to the logarithmic derivative, even when the function does not take values in the positive ..., we have math log uv log u log v log u log v . math So for positive real valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we ... u u frac v v . math Thus, it s true for any function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors when they are defined . Similarly in fact this is a consequence , the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function math frac 1 u 1 u frac u u 2 1 u frac u u , math just .... More generally, the logarithmic derivative of a quotient is the difference of the logarithmic .... Generalising in another direction, the logarithmic derivative of a power with constant real exponent is the product of the exponent and the logarithmic derivative of the base math frac ... , a reciprocal rule , a quotient rule , and a power rule compare the list of logarithmic identities each pair of rules is related through the logarithmic derivative. Computing ordinary derivatives using logarithmic derivatives Main Logarithmic differentiation Logarithmic derivatives can simplify ... it directly, we compute its logarithmic derivative. That is, we compute math frac f f frac u u frac ... makes it possible to compute Nowrap &fnof by computing the logarithmic derivative of each factor, summing, and multiplying by . Integrating factors The logarithmic derivative idea is closely connected ... operator by the logarithmic derivative G &prime G . In practice we are given an operator ... pole . Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily ... more details
Any formula written in terms of logarithm s may be said to be in logarithmic form . Logarithmic differential forms In contexts including complex manifold s and algebraic geometry , a logarithmic differential form is a 1 form that, locally at least, can be written math frac df f math for some meromorphic function resp. rational function f . That is, for some open covering , there are local representations of this differential form as a logarithmic derivative modified slightly with the exterior derivative d in place of the usual differential operator D . These forms are quite highly constrained in their behaviour. For example on a Riemann surface it follows that they have simple pole s, and everywhere integer Residue complex analysis residues at them. In higher dimension one needs the Poincar residue to formulate their distinctive behaviour at places where f takes the value 0 or &infin . Classically, for example in elliptic function theory, the logarithmic differential forms were recognised as complementary to the differentials of the first kind . They were sometimes called differentials of the second kind and, with an unfortunate inconsistency, also sometimes of the third kind . The classical theory has now been subsumed as an aspect of Hodge theory . For a Riemann surface S , for example, the differentials of the first kind account for the term H sup 0,1 sup in H sup 1 sup S , when by the Dolbeault isomorphism it is interpreted as the sheaf cohomology group H sup 0 sup S ,&Omega this is tautologous considering their definition. The H sup 1,0 sup direct summand in H sup 1 sup S , as well as being interpreted as H sup 1 sup S ,O where O is the sheaf of holomorphic function s on S , can be identified more concretely with a vector space of logarithmic differentials. Number theory See Linear forms in logarithms . External links http www.ucl.ac.uk Mathematics geomath level2 hyper hy8d.html The logarithmic form for inverse hyperbolics http www.intmath.com MethInt 2 BLog.php ... more details
In algebraic geometry , a logarithmic pair consists of a Algebraic variety variety , together with a divisor along which one allows mild logarithmic singularities. They were studied by harvtxt Iitaka 1976 . Definition A boundary Q divisor on a variety is a Q divisor D of the form &Sigma d sub i sub D sub i sub where the D sub i sub are the distinct irreducible components of D and all coefficients are rational numbers with 0&le d sub i sub &le 1. A logarithmic pair , or log pair for short, is a pair X , D consisting of a normal variety X and a boundary Q divisor D . The log canonical divisor of a log pair X , D is K D where K is the canonical divisor of X . A logarithmic 1 form on a log pair X , D is allowed to have logarithmic singularities of the form d log z d z z along components of the divisor given locally by z 0. References Citation authorlink Shigeru Iitaka last1 Iitaka first1 Shigeru title Logarithmic forms of algebraic varieties id MathSciNet id 0429884 year 1976 journal Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics issn 0040 8980 volume 23 issue 3 pages 525 544 Citation last1 Matsuki first1 Kenji title Introduction to the Mori program publisher Springer Verlag location Berlin, New York series Universitext isbn 978 0 387 98465 0 id MathSciNet id 1875410 year 2002 Category algebraic geometry ... more details
align right Image Logarithmic Spiral Pylab.svg 260px thumb Logarithmic spiral pitch 10 Image NautilusCutawayLogarithmicSpiral.jpg ... logarithmic spiral File Brassica romanesco.jpg 200px thumb Romanesco broccoli , which grows in a logarithmic spiral Image Mandel zoom 04 seehorse tail.jpg thumb 200px A section of the Mandelbrot set following a logarithmic spiral Image Low pressure system over Iceland.jpg thumb 200px A low pressure area over Iceland shows an approximately logarithmic spiral pattern File Messier51 sRGB.jpg thumb 200px The arms of Spiral galaxy spiral galaxies often have the shape of a logarithmic spiral, here the Whirlpool Galaxy Image Polygon spiral.svg thumb 300px A logarithmic spiral , equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic ... , Latin for miraculous spiral , is another name for the logarithmic spiral. Although this curve ... , Evolutes. p. 206 ref Properties The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression , while in an Archimedean spiral these distances are constant. Logarithmic spirals ... lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals ..., the mapping of all lines to all logarithmic spirals is onto . The pitch angle of the logarithmic ... to a logarithmic spiral in the complex plane. One can construct a golden spiral , a logarithmic spiral ... 17.03239 degrees , or approximate it using Fibonacci number s. Logarithmic spirals in nature In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows .... last Chin date 8 December 2000 title Organismal Biology Flying Along a Logarithmic Spiral journal ... ref Our own galaxy, the Milky Way , has several spiral arms, each of which is roughly a logarithmic ... of the cornea in a logarithmic spiral pattern . ref name Yu C. Q. Yu CQ and M. I. Rosenblatt, Transgenic ... more details
unreferenced date October 2010 The scale convolution of two functions math s t math and math r t math , also known as their logarithmic convolution is defined as the function br math s l r t r l s t int 0 infty s left frac t a right r a frac da a math when this quantity exists. Results The logarithmic convolution can be related to the ordinary convolution by changing the variable from math t math to math v log t math math s l r t int 0 infty s left frac t a right r a frac da a int infty infty s left frac t e u right r e u du math math int infty infty s left e log t u right r e u du. math Define math f v s e v math and math g v r e v math and let math v log t math , then math s l r v f g v g f v r l s v . , math planetmath id 5995 title logarithmic convolution Category Logarithms Category Mathematics ... more details
In mathematics, the logarithmic norm is a real valued Functional mathematics functional on Operator mathematics ..., or its induced operator norm . The logarithmic norm was independently introduced by Germund Dahlquist ... unbounded operators as well. ref Gustaf S derlind, The logarithmic norm. History and modern theory ... matrix norm. The associated logarithmic norm math mu math of math A math is defined math mu A lim limits ... equals math mu A math , and is in general different from the logarithmic norm math mu A math , as math ... 0 math , but the logarithmic norm math mu A math may also take negative values, e.g. when math A math is Positive definite matrix negative definite . Therefore, the logarithmic norm does not satisfy the axioms of a norm. The name logarithmic norm, which does not appear in the original reference, seems ... d mathrm dt math is the Dini derivative upper right Dini derivative . Using logarithmic differentiation ... If the vector norm is an inner product norm, as in a Hilbert space , then the logarithmic norm ... to be unbounded. Thus Differential operator differential operators too can have logarithmic norms, allowing the use of the logarithmic norm both in algebra and in analysis. The modern, extended theory ... norm and the logarithmic norm are then associated with extremal values of Quadratic form quadratic ... x neq 0 frac real langle x, Ax rangle langle x,x rangle math Properties Basic properties of the logarithmic ... e t mu A , math for math t geq 0 math math mu A 0 , Rightarrow , A 1 leq 1 mu A math Example logarithmic norms The logarithmic norm of a matrix can be calculated as follows for the three most common ... sum limits j, j neq i a ij math Applications in matrix theory and spectral theory The logarithmic norm ... A leq real , lambda k leq mu A math . More generally, the logarithmic norm is related to the numerical ... , R A leq 1. math Thus the matrix norm and logarithmic norms may be viewed as generalizing the modulus ... theory and numerical analysis The logarithmic norm plays an important role in the stability analysis ... more details
Cleanup date February 2008 Logarithmic decrement , , is used to find the damping ratio of an underdamped system in the time domain. The logarithmic decrement is the natural logarithm natural log of the amplitudes of any two successive peaks math delta frac 1 n ln frac x 0 x n , math where x sub 0 sub is the greater of the two amplitudes and x sub n sub is the amplitude of a peak n periods away. The damping ratio is then found from the logarithmic decrement math zeta frac 1 sqrt 1 frac 2 pi delta 2 . math The damping ratio can then be used to find the undamped natural frequency sub n sub of vibration of the system from the damped natural frequency sub d sub math omega d frac 2 pi T , math math omega n frac omega d sqrt 1 zeta 2 , math where T, the period of the waveform, is the time between two successive amplitude peaks. The damping ratio can also be found using a slightly simplified variation on these equations for two adjacent peaks. This method is identical to the above, but simplified for the case of n equal to 1 math zeta frac 1 sqrt 1 frac 2 pi ln x 0 x 1 2 , math where x sub 0 sub is the left peak and x sub 1 sub is the first peak to its right. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5 it does not apply at all for a damping ratio greater than 1.0 because the system is overdamped. The method of fractional overshoot can be useful for damping ratios between about 0.5 and 0.8. The fractional overshoot OS is math OS frac x p x f x f , math where x sub p sub is the amplitude of the first peak of the step response and x sub f sub is the settling amplitude. Then the damping ratio is math zeta frac 1 sqrt 1 frac pi ln OS 2 . math See also Damping Damping factor Q factor Category Logarithms de Logarithmisches Dekrement pl Logarytmiczny dekrement t umienia ru ... more details
Dablink Logarithmic derivative is a separate article. Calculus In calculus , logarithmic differentiation or differentiation by taking logarithms is a method used to derivative differentiate function mathematics function s by employing the logarithmic derivative of a function , ref cite book title Calculus demystified pages 170 first Steven G. last Krantz publisher McGraw Hill Professional year 2003 isbn 0071393080 ref math ln f frac f f math The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. Logarithmic differentiation relies on the chain rule as well as properties of logarithms in particular, the natural logarithm , or logarithmic to the base e mathematics e to transform products into sums and divisions into subtractions, and can also applied to functions raised to the power of variables or functions. ref cite book title Golden Differential Calculus pages 282 author N.P. Bali publisher Firewall Media year 2005 isbn 8170081521 ref ref name Bird cite book title Higher Engineering Mathematics first John last Bird pages 324 publisher Newnes year 2006 isbn 0750681527 ref However, the principle can be implemented, at least in part, in the differentiation of almost all differentiable function s, providing that these functions are non zero. Overview For a function math y f x , math logarithmic differentiation typically begins by taking the natural logarithm, or the logarithm to the base e constant ... Differentiation Rules Logarithmic differentiation extratext see for textbook examples of logarithmic ... Lie group s List of logarithm topics List of logarithmic identities Notes references External ... title Logarithmic differentiation accessdate 2009 03 10 cite web url http tutorial.math.lamar.edu Classes CalcI LogDiff.aspx title Calculus I Logarithmic differentiation accessdate 2009 03 10 DEFAULTSORT Logarithmic Differentiation Category Differential calculus et Logaritmiline diferentseerimine ... more details
y x ln y ln x end array math The area interpretation allows to easily derive basic properties of the logarithmic ... mean which is related to logarithms is the geometric mean . The logarithmic mean is a special case of the Stolarsky mean . References http www.everything2.com index.pl?node id 801020 Logarithmic mean Everything2.com http jipam old.vu.edu.au v4n4 088 03.html Oilfield Glossary Term logarithmic mean mathworld Arithmetic Logarithmic GeometricMeanInequality Arithmetic Logarithmic Geometric Mean Inequality ... 3E2.0.CO 3B2 6 Generalizations of the logarithmic mean , Mathematics Magazine, Vol. 48, No. 2, Mar., 1975 ... more details
Probability distribution name Logarithmic type mass pdf image Image Logarithmicpmf.svg 300px center Plot of the logarithmic PMF small The function is only defined at integer values. The connecting lines are merely guides for the eye. small cdf image Image Logarithmiccdf.svg 300px center Plot of the logarithmic CDF parameters math 0 p 1 math support math k in 1,2,3, dots math pdf math frac 1 ln 1 p frac p k k math cdf math 1 frac Beta p k 1,0 ln 1 p math mean math frac 1 ln 1 p frac p 1 p math median mode math 1 math variance math p frac p ln 1 p 1 p 2 , ln 2 1 p math skewness exists, but too complex kurtosis exists, but too complex entropy exists, but too complex mgf math frac ln 1 p , exp t ln 1 p text for t ln p , math char math frac ln 1 p , exp i ,t ln 1 p text for t in mathbb R math pgf math frac ln 1 pz ln 1 p text for z frac1p math In probability and statistics , the logarithmic distribution also known as the logarithmic series distribution or the log series distribution is a discrete probability distribution derived from the Maclaurin series expansion math ln 1 p p frac p 2 2 frac p 3 3 cdots. math From this we obtain the identity math sum k 1 infty frac 1 ln 1 p frac p k k 1. math This leads directly to the probability mass function of a Log p distributed random variable math f k frac 1 ln 1 p frac p k k math for k &ge 1, and where 0 p 1. Because of the identity above, the distribution is properly normalized. The cumulative distribution function is math F k 1 frac Beta p k 1,0 ln 1 p math where B is the incomplete beta function . A Poisson mixture of Log p distributed random variables has a negative binomial distribution . In other words, if N is a random variable with a Poisson distribution , and X sub i sub , i 1, 2, 3, ... is an infinite ... described the logarithmic distribution in a paper that used it to model relative species abundance ... chapter Chapter 7 Logarithmic and Lagrangian distributions isbn 9780471272465 MathWorld urlname Log ... more details
Portal box History Time List of timelines Timeline of world history Chronology Living graph LogarithmictimelineDetailedlogarithmictimeline Synchronoptic view SIMILE WikiTimeScale References Citation ... use a linear timescale, for very large or small timespans, logarithmictimeline s use a logarithmic ...Other uses Selfref For Wikipedia s timeline and related tools, see Wikipedia Timeline . A timeline is a project Artifact archaeology artifact . It is typically a graphic design showing a long bar labeled with dates alongside itself and usually events labeled on points where they would have happened. It is used ... biographies . Examples include Chronology of Shakespeare s plays Timeline of the African American Civil Rights Movement Timeline of European exploration Timeline of United States history 1930 1949 World War I timeline Natural sciences Timelines are also used in the natural world and sciences for subjects such as astronomy , biology and geology . 2009 flu pandemic timeline 2009 H1N1 Flu Pandemic Timeline Geologic time scale Timeline big bang Timeline of the Big Bang Timeline of evolution Timeline of Evolution Project management Another type of timeline is used for project management . In these cases ... what time schedule. For example in the case of establishing a project timeline in the implementation ... to a set amount of time. This time scale is dependent on the events in the timeline. A timeline of evolution can be over millions of years, whereas a timeline about the September 11, 2001 timeline ... s map 1861 of Napoleon s invasion of Russia is an example of a non standard timeline that also uses ... of the Timeline publisher Princeton Architectural Press year 2010 pages 272 isbn 978 1568987637 references External links Commons Timeline Wiktionary timeline http www.bl.uk timeline Timelines sources from history a British Library interactive history timeline that explores collection items chronologically ... czasu pt Linha do tempo th Timeline zh ... more details
wa Im dje Mape walonreye wi.jpg Detailed map of Wallonia, taken from Walloon Wikipedia. Some translation required. GFDL en migration relicense ... more details
Orphan date February 2009 A logarithmic spiral beach is a type of beach which develops in the direction under which it is sheltered by a headland, in an area called the shadow zone. It is characterized as a logarithmic spiral because if you look at it in plan view or aerially, it represents the same shape that is created from the logarithmic spiral function. These beaches are also commonly referred to as zeta cure bays , half heart or crenulate shaped bays, or headland bays . Logarithmic spiral function Image Logarithmic Spiral Pylab.svg thumb Logarithmic Spiral The logarithmic spiral can be determined using the equation written in polar coordinates r e sup cot sup where the angle of rotation, is located between two lines drawn from the origin to any two points on the spiral. r the ratio of the lengths between two lines that extend out from the origin. The two lines are given as R sub O sub and R. So r also equals the ratio R R sub O sub . the angle between any line R from the origin and the line tangent to the spiral which is at the point where line R intersects the spiral. is a constant for any given logarithmic spiral. Spiral development This type of beach forms due to the refraction of approaching waves and their diffraction by an upcoast headland . The approaching wave front curves as a result of wave diffraction at the headland, which in turn causes the shoreline to bend and yield a log spiral shape. Log spiral beaches are often on swell dominated coasts where waves .... Famous logarithmic spiral beaches Image Half Moon Bay State Beach 1.jpg thumb Half Moon Bay ... of Logarithmic Spiral Beach. Australian Geographer 14.1 1978 44 45. Kimberley, M. M. Fitting a Logarithmic Spiral to the Shoreline of a Headland Bay Beach Computers & Geoscience 15 No. 7 1989 1089 1108. LeBlond, Paul H. An Explanation of the Logarithmic Spiral Plan Shape of HeadlandBay Beaches .... Logarithmic Spiral Coastlines The Northern Zululand Coastline. The South African Geographical Journal ... more details
In theoretical physics , a logarithmic conformal field theory is a generalization of the concept of usually two dimensional conformal field theory in which the correlator s of the basic fields are allowed to be multiply valued and be functions of the logarithm of the separation of the operators. References http xstructure.inr.ac.ru x bin theme3.py?level 1&index1 59186 Logarithmic conformal field theory on arxiv.org V. Gurarie, http dx.doi.org 10.1016 0550 3213 93 90528 W Logarithmic operators in conformal field theory , Nucl. Phys. B410 1993 535 549. M. R. Gaberdiel, H. G. Kausch, http dx.doi.org 10.1016 0550 3213 96 00364 1 Indecomposable fusion products , Nucl. Phys. B477 1996 293 318. Category Conformal field theory quantum stub ... more details
In mathematics , the logarithmic integral function or integral logarithm li x is a special function . It occurs in problems of physics and has number theory number theoretic significance, occurring in the prime number theorem as an estimate of the number of prime number s less than a given value. Image Logarithmic integral.svg thumb right Logarithmic integral Integral representation The logarithmic integral has an integral representation defined for all positive real number s math x ne 1 math by the integral definite integral math rm li x int 0 x frac dt ln t . math Here, math ln math denotes the natural logarithm . The function math 1 ln t math has a mathematical singularity singularity at t 1, and the integral for x 1 has to be interpreted as a Cauchy principal value math rm li x lim varepsilon to 0 left int 0 1 varepsilon frac dt ln t int 1 varepsilon x frac dt ln t right . math Offset logarithmic integral The offset logarithmic integral or Eulerian logarithmic integral is defined as math rm Li x rm li x rm li 2 , math or math rm Li x int 2 x frac dt ln t , math As such, the integral representation has the advantage of avoiding the singularity in the domain of integration. This function is a very good approximation to the number of prime numbers less than x. Series representation The function li x is related to the exponential integral Ei x via the equation math hbox li x hbox Ei ln x , , math which is valid for x 1. This identity provides a series representation ... Logarithmic Integral ref is math rm li x gamma ln ln x sqrt x sum n 1 infty frac 1 n 1 ln x n n , 2 ... integral . Infinite logarithmic integral Clarify date November 2009 math int infty infty frac M t 1 t 2 dt math and discussed in Paul Koosis, The Logarithmic Integral , volumes I and II, Cambridge University Press, second edition, 1998. Number theoretic significance The logarithmic ... Pedersen Gram Skewes number References references AS ref 5 228 dlmf id 6 title Exponential, Logarithmic ... more details
In computer science a doubly logarithmic tree is a Tree data structure tree where each internal node of height 1 has two children, and each internal node of height math h 1 math has math 2 2 h 2 math children. Each child of the root contains math sqrt n math leaves. The number of children at a node as we go from leaf to root is 0,2,2,4,16, 256, 65536, ... i.e. 0,2, http www.research.att.com njas sequences A001146 OEIS A001146 A similar tree called a k merger is used in Prokop et al s cache oblivious Funnelsort to merge elements. File Double log tree.png Notes reflist References citation first1 Omer last1 Berkman first2 Baruch last2 Schieber first3 Uzi last3 Vishkin author3 link Uzi Vishkin title Optimal doubly logarithmic parallel algorithms based on finding all nearest smaller values journal Journal of Algorithms volume 14 pages 344 370 year 1993 issue 3 doi 10.1006 jagm.1993.1018 . Harald Prokop. http citeseer.ist.psu.edu prokop99cacheobliviou.html Cache Oblivious Algorithms . Masters thesis, MIT. 1999. M. Frigo, C.E. Leiserson, H. Prokop, and S. Ramachandran. Cache oblivious algorithms. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science FOCS 99 , p.285 297. 1999. http ieeexplore.ieee.org iel5 6604 17631 00814600.pdf?arnumber 814600 Extended abstract at IEEE , http citeseer.ist.psu.edu 307799.html at Citeseer . Erik Demaine . http courses.csail.mit.edu 6.897 spring03 scribe notes L17 lecture17.pdf Review of the Cache Oblivious Sorting . Notes for MIT Computer Science 6.897 Advanced Data Structures. Category Parallel computing ... more details
the precision of floating point math operations. History Logarithmic number systems have been independently ... introduced logarithmic arithmetic for digital signal processing in 1971. ref cite journal author N. G. Kingsburg and P. J. W. Rayner title Digital filtering using logarithmic arithmetic journal ... Logarithmic Microprocessor ELM . ref cite journal author J. N. Coleman, C. I. Softley, J. Kadlec, R. Matousek, M. Tichy, Z. Pohl, A. Hermanek, and N. F. Benschop title The European Logarithmic ... Comparing Floating point and Logarithmic Number Representations for Reconfigurable Acceleration author ... more details
In probability theory and statistics , the exponential logarithmic EL distribution is a family of lifetime probability distribution distributions with decreasing failure rate , defined on the interval 0, . This distribution is Parametric family parameterized by two parameters math p in 0,1 math and math beta 0 math . TABLE class infobox bordered wikitable style FONT SIZE 95 MARGIN BOTTOM 0.5em MARGIN LEFT 1em WIDTH 325px CAPTION Exponential Logarithmic distribution EL CAPTION TR style TEXT ALIGN center TD colSpan 2 Probability density function BR File Pdf EL.png TD TR TR style TEXT ALIGN center TD colSpan 2 Hazard function BR File Hazard EL.png TD TR TR vAlign top TH Parameters TH TD SPAN math p in 0,1 math SPAN BR SPAN math beta 0 math SPAN TD TR TR TH Support TH TD math x in 0, infty math TD TR TR TH Probability density function pdf TH TD math frac 1 ln p times frac beta 1 p e beta x 1 1 p e beta x math TD TR TR TH Cumulative distribution function cdf TH TD math 1 frac ln 1 1 p e beta x ln p math TD TR TR TH Mean TH TD math frac text polylog 2,1 p beta ln p math TD TR TR TH Median TH TD math frac ln 1 sqrt p beta math TD TR TR TH Mode TH TD 0 TD TR TR TH Variance TH TD math frac 2 text polylog 3,1 p beta 2 ln p frac text polylog 2 2,1 p beta 2 ln 2 p math TD TR TR TH Skewness TH TD TD TR TR TH Excess kurtosis TH TD TD TR TR TH Moment generating function mgf TH TD math frac beta 1 p ln p beta t math br math text hypergeom 2,1 1, frac beta t beta , frac 2 ... in biological terms . The exponential logarithmic model, together with its various properties ... distribution has been generalized to form the Weibull logarithmic distribution. ref Ciumara1,Roxana ... The Weibull logarithmic distribution in lifetime analysis and its properties . In L. Sakalauskas ... from a logarithmic distribution where the parameter p in the usual parameterisation is replaced by nowrap 1 1 &minus p , then X has the exponential logarithmic distribution in the parameterisation ... more details
The following is a list of integral s antiderivative functions of logarithmic function s. For a complete list of integral functions, see list of integrals . Note x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity. math int ln ax dx x ln ax x math math int ln ax b dx frac ax b ln ax b ax b a math math int ln x 2 dx x ln x 2 2x ln x 2x math math int ln x n dx x sum n k 0 1 n k frac n k ln x k math math int frac dx ln x ln ln x ln x sum infty k 2 frac ln x k k cdot k math math int frac dx ln x n frac x n 1 ln x n 1 frac 1 n 1 int frac dx ln x n 1 qquad mbox for n neq 1 mbox math math int x m ln x dx x m 1 left frac ln x m 1 frac 1 m 1 2 right qquad mbox for m neq 1 mbox math math int x m ln x n dx frac x m 1 ln x n m 1 frac n m 1 int x m ln x n 1 dx qquad mbox for m neq 1 mbox math math int frac ln x n dx x frac ln x n 1 n 1 qquad mbox for n neq 1 mbox math math int frac ln x n dx x frac ln x n 2 2n qquad mbox for n neq 0 mbox math math int frac ln x ,dx x m frac ln x m 1 x m 1 frac 1 m 1 2 x m 1 qquad mbox for m neq 1 mbox math math int frac ln x n dx x m frac ln x n m 1 x m 1 frac n m 1 int frac ln x n 1 dx x m qquad mbox for m neq 1 mbox math math int frac x m dx ln x n frac x m 1 n 1 ln x n 1 frac m 1 n 1 int frac x m dx ln x n 1 qquad mbox for n neq 1 mbox math math int frac dx x ln x ln left ln x right math math int frac dx x n ln x ln left ln x right sum infty k 1 1 k frac n 1 k ln x k k cdot k math math int frac dx x ln x n frac 1 n 1 ln x n 1 qquad mbox for n neq 1 mbox math math int ln x 2 a 2 dx x ln x 2 a 2 2x 2a tan 1 frac x a math math int frac x x 2 a 2 ln x 2 a 2 dx frac 1 4 ln 2 x 2 a 2 math math int sin ln x dx frac x 2 sin ln x cos ln x math math int cos ln x dx frac x 2 sin ln x cos ln x math math int e ... cbm aands page 69.htm page 69 . Lists of integrals Category Integrals Logarithmic functions Category Mathematics related lists Integrals of logarithmic functions ar ... more details
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