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Encyclopedia results for Arithmetic geometric mean

Arithmetic geometric mean





Encyclopedia results for Arithmetic geometric mean

  1. Arithmetic-geometric mean

    In mathematics , the arithmetic geometric mean AGM of two positive real number s math x and math y is defined as follows First compute the arithmetic mean of math x and math y and call it math a sub 1 sub . Next compute the geometric mean of math x and math y and call it math g sub 1 sub this is the square ... to the same number, which is the arithmetic geometric mean of math x and math y it is denoted ... method . Example To find the arithmetic geometric mean of math a sub 0 sub 24 and math g sub 0 sub 6 , first calculate their arithmetic mean and geometric mean, thus math a 1 tfrac12 24 6 15, math .... Properties The geometric mean of two positive numbers is never bigger than the arithmetic mean see inequality of arithmetic and geometric means as a consequence, math g sub n sub is an increasing ... math Indeed, since the arithmetic geometric process converges so quickly, it provides an effective way to compute elliptic integrals via this formula. The reciprocal of the arithmetic geometric mean ... , math which completes the proof. See also Generalized mean Inequality of arithmetic and geometric ... Arithmetic geometric mean process urlname a a130280 mathworld urlname Arithmetic GeometricMean title Arithmetic Geometric mean DEFAULTSORT Arithmetic Geometric Mean Category Means Category Special ...... 3 13.45820393250... 13.45813903099... 4 13.45817148175... 13.45817148171... The arithmetic geometric ... and arithmetic mean of math x and math y in particular it is between math x and math y . If math r 0 ... math named after Carl Friedrich Gauss . The geometric harmonic mean can be calculated by an analogous method, using sequences of geometric and harmonic mean harmonic means. The arithmetic harmonic mean can be similarly defined, but takes the same value as the geometric mean . Proof of existence From inequality of arithmetic and geometric means we can conclude that math g i leqslant a i math and thus ... of math x and math y which follows from the fact that both arithmetic and geometric means of two numbers ...   more details



  1. Geometric mean

    for their financial viability. If an arithmetic mean was used instead of a geometric mean, the financial ... of a geometric mean normalizes the ranges being averaged, so that no range dominates the weighting, and a given percentage change in any of the properties has the same effect on the geometric mean. So, a 20 change in environmental sustainability from 4 to 4.8 has the same effect on the geometric mean as a 20 change in financial viability from 60 to 72. The geometric mean is similar to the arithmetic ... expression states that the log of the geometric mean is the arithmetic mean of the logs of the numbers ... and the geometric mean is always in between see Inequality of arithmetic and geometric means ... i 1 n a i bigg 1 n sqrt n a 1 a 2 cdots a n . math The geometric mean of a data set inequality of arithmetic and geometric means is less than the data set s arithmetic mean unless all members of the data ... of the arithmetic geometric mean , a mixture of the two which always lies in between. The geometric ... 55 20 Geometric mean 31.622... 31.622... 20 The arithmetic and geometric means agree that computer ... B Computer C Program 1 1 10 20 Program 2 1 0.1 0.02 Arithmetic mean 1 5.05 10.01 Geometric mean ... Arithmetic mean 5.05 1 1.1 Geometric mean 1 1 0.632 In all cases, the ranking given by the geometric ... See Compound annual growth rate The geometric mean is more appropriate than the arithmetic mean for describing ... the arithmetic mean of 16 9 and 4 3 12 9, since 14 is the average of 16 and 12, while the precise geometric ... a spectrum is, is defined as the ratio of the geometric mean of the power spectrum to its arithmetic ... div style moz column count 3 column count 3 Arithmetic mean Arithmetic geometric mean Average Generalized ...The geometric mean , in mathematics , is a type of mean or average , which indicates the central tendency or typical value of a set of numbers. A geometric mean is often used when comparing different items ... 2012 ref . For example, the geometric mean can give a meaningful average to compare two companies ...   more details



  1. Arithmetic mean

    of an Experiment probability theory experiment . The term arithmetic mean is preferred in mathematics and statistics because it helps distinguish it from other average mean s such as the geometric mean geometric and harmonic mean . In addition to mathematics and statistics, the arithmetic mean is used ... forecasts Empirical measure Fr chet mean Generalized mean Geometric mean Harmonic mean Inequality of arithmetic and geometric means Ky Fan inequality Mean multicol break Median mode statistics Mode ... geommean.htm Calculations and comparisons between arithmetic and geometric mean of two numbers MathWorld urlname ArithmeticMean title Arithmetic Mean Statistics descriptive Use dmy dates date September 2010 DEFAULTSORT Arithmetic Mean Category Means interwiki ar az d di orta bg ...More footnotes date May 2010 In mathematics and statistics , the arithmetic mean , or simply the mean ... every academic field to some extent. For example, per capita GDP gives an approximation of the arithmetic average income of a nation s population. While the arithmetic mean is often used to report central ... by outlier s. Notably, for skewed distribution s, the arithmetic mean may not accord with one s notion .... Definition Suppose we have sample space math a 1, ldots,a n math . Then the arithmetic mean ... population , then the mean of that population is called a population mean . If the list is a sampling statistics statistical sample , we call the resulting statistic a sample mean . The arithmetic mean of a variable is often denoted by a bar, for example math bar x math read x bar would be the mean of some sample space math X math . Motivating properties The arithmetic mean has several properties ... distribution , the arithmetic mean is equal to both the median and the mode, other measures of central ... s. Na vely taking the arithmetic mean of 1 and 359 yields a result of 180 . This is incorrect for two ... x 1, ldots,x n math have mean X, then math x 1 X ldots x n X 0 math . Since math x i X math is the distance ...   more details



  1. Inequality of arithmetic and geometric means

    There is a similar inequality for the weighted arithmetic mean and weighted geometric mean . Specifically ... their arithmetic mean is x sub 1 sub and their product is x sub 1 sub sup n sup , so their geometric mean is x sub 1 sub therefore, the arithmetic mean and geometric mean are equal, as desired. The case ... math x 2 k 1 1 x 2 k 1 2 cdots x 2 k math in which case the first arithmetic mean and first geometric mean are both equal to x sub 1 sub , and similarly with the second arithmetic mean and second geometric mean and in the second inequality, the two sides are only equal if the two geometric means are equal ... the weighted arithmetic mean and the weighted geometric mean stated above. Since an x sub k sub with weight ...In mathematics , the inequality of arithmetic and geometric means , or more briefly the AM GM inequality , states that the arithmetic mean of a list of non negative real number s is greater than or equal to the geometric mean of the same list and further, that the two means are equal if and only if every number in the list is the same. Background The arithmetic mean , or less precisely the average ... of the numbers divided by n math frac x 1 x 2 cdots x n n . math The geometric mean is similar ... function exponential of the arithmetic mean of the natural logarithm s of the numbers math ... for first reading. With the arithmetic mean math mu frac x 1 cdots x n n math of the non negative ... Consider n     1 non negative real numbers. Their arithmetic mean satisfies math n 1 mu ... n mu x 1 cdots x n 1 underbrace x n x n 1 mu ,x n , math is also the arithmetic mean of math x 1 ... negative numbers math a 1, a 2, dots , a n math with arithmetic mean . By repeated application of the above ... mean is greater than the geometric mean. Clearly, this is only possible when n     ... power of 2 that is greater than n . So, if we have n terms, then let us denote their arithmetic mean ..., too. If at least one x sub k sub is zero but not all , then the weighted geometric mean is zero, while ...   more details



  1. Geometric-harmonic mean

    In mathematics , the geometric harmonic mean M x , y of two positive real number s x and y is defined as follows we form the geometric mean of g sub 0 sub x and h sub 0 sub y and call it g sub 1 sub , i.e. g sub 1 sub is the square root of xy . We also form the harmonic mean of x and y and call it h sub 1 sub , i.e. h sub 1 sub is the Multiplicative inverse reciprocal of the arithmetic mean of the reciprocals of x and y . These may be done sequentially in any order or simultaneously. Now we can iterate this operation with g sub 1 sub taking the place of x and h sub 1 sub taking the place of y . In this way, two sequence s g sub n sub and h sub n sub are defined math g n 1 sqrt g n h n math and math h n 1 frac 2 frac 1 g n frac 1 h n math Both of these sequences limit mathematics converge to the same number, which we call the geometric harmonic mean M x ,  y of x and  y . The geometric harmonic mean is also designated as the harmonic geometric mean .  cf. Wolfram MathWorld below. The existence of the limit can be proved by the means of Bolzano&ndash Weierstrass theorem in a manner almost identical to the proof of existence of arithmetic geometric mean . Properties M x ,  y is a number between the geometric and harmonic mean of x and y in particular it is between x and y . M x ,  y is also Homogeneous function homogeneous , i.e. if r     0, then M rx ,  ry   r M x ,  y . If AG x , y is the arithmetic geometric mean , then we also have math M x,y frac 1 AG frac 1 x , frac 1 y math Inequalities We have the following inequality involving the Pythagorean ... y is the harmonic mean, HG x ,  y is the harmonic geometric mean, G x ,  y   HA x ,  y is the geometric mean which is also the harmonic arithmetic mean , GA x ,  y is the geometric arithmetic mean, A x ,  y is the arithmetic mean. See also Arithmetic geometric mean Arithmetic harmonic mean Mean External links MathWorld title Harmonic Geometric Mean urlname Harmonic GeometricMean ...   more details



  1. Geometric-Arithmetic Parallel Processor

    The GAPP Geometric Arithmetic parallel processing Parallel Processor , invented by Poland Polish mathematics mathematician W odzimierz Holszty ski in 1981, was patented by Martin Marietta ref http patft.uspto.gov netacgi nph Parser?Sect1 PTO1&Sect2 HITOFF&d PALL&p 1&u 2Fnetahtml 2FPTO 2Fsrchnum.htm&r 1&f G&l 50&s1 4739474.PN.&OS PN 4739474&RS PN 4739474 Geometric arithmetic parallel processor US Patent 4,739,474, April 19, 1988 ref and is now owned by Silicon Optix , Inc. In terms of network topology , the GAPP is a mesh connected array of single bit SIMD processing elements PEs , where each PE can communicate with its neighbor to the north, east, south, and west. Each cell has its own memory. The space of addresses is the same for all cells. The data travels from the cell memories to the cells registers, and in the opposite direction, in parallel. Characteristically, the cell s ALU i.e. its PE in the early versions of GAPP was nothing but a full 1 bit adder subtractor, which efficiently served both the complex arithmetic as well as logical functions, and with the help of shifts it served also the geometric transformations in short, it was doing all three types of the tasks while other designs used three separate hardware special purpose units instead . In its most recent incarnation as of 2004 , the systems by Teranex utilize GAPP arrays of up to 294,912 processing elements. References references Refimprove date August 2007 Category SIMD computing Category Digital signal processing compu hardware stub ...   more details



  1. Arithmetic and geometric Frobenius

    In mathematics , the Frobenius endomorphism is defined in any commutative ring R that has characteristic algebra characteristic p , where p is a prime number . Namely, the mapping that takes r in R to r sup p sup is a ring endomorphism of R . The image of is then R sup p sup , the subring of R consisting of p th powers. In some important cases, for example finite field s, is surjective . Otherwise is an endomorphism but not a ring automorphism . The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to . This gives a mapping Spec R sup p sup Spec R of affine scheme s. Even in cases where R sup p sup R this is not the identity, unless R is the prime field . Mappings created by fibre product with , i.e. Grothendieck s relative point of view base change s, tend in scheme theory to be called geometric Frobenius . The reason for a careful terminology is that the Frobenius automorphism in Galois group s, or defined by transport of structure , is often the inverse mapping of the geometric Frobenius. As in the case of a cyclic group in which a generator is also the inverse of a generator, there are in many situations two possible definitions of Frobenius, and without a consistent convention some problem of a minus sign may appear. References Citation last1 Freitag first1 Eberhard last2 Kiehl first2 Reinhardt title tale cohomology and the Weil conjecture 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 12175 6 mr 926276 year 1988 volume 13 , p.  5 DEFAULTSORT Arithmetic And Geometric Frobenius Category Mathematical terminology Category Algebraic geometry Category Algebraic number theory ...   more details



  1. Weighted geometric mean

    In statistics , given a set of data, math X x 1,x 2 dots,x n math and corresponding weight function weights , math W w 1, w 2, dots,w n math the weighted geometric mean is calculated as math bar x left prod i 1 n x i w i right 1 sum i 1 n w i quad exp left frac sum i 1 n w i ln x i sum i 1 n w i quad right math Note that if all the weights are equal, the weighted geometric mean is the same as the geometric mean . Weighted versions of other means can also be calculated. Probably the best known weighted mean is the weighted arithmetic mean, usually simply called the weighted mean . Another example of a weighted mean is the weighted harmonic mean . The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values. See also average central tendency summary statistics Weighted mean Category Means Category Mathematical analysis statistics stub cs V en geometrick pr m r eo La peza geometria meznombro fr Moyenne g om trique pond r e ru ta ...   more details



  1. Quasi-arithmetic mean

    In mathematics and statistics , the quasi arithmetic mean or generalised f mean is one generalisation of the more familiar mean s such as the arithmetic mean and the geometric mean , using a function math f math . It is also called Kolmogorov mean after Russian scientist Andrey Kolmogorov . Definition ... any linear function math x mapsto a cdot x b math , math a math not equal to 0 then the f mean corresponds to the arithmetic mean . If we take math I math to be the set of positive real numbers and math f x log x math , then the f mean corresponds to the geometric mean . According to the f mean properties ... k text times ,x k 1 , dots,x n math The quasi arithmetic mean is invariant with respect to offsets .... Any quasi arithmetic mean math M math of two variables has the mediality property math M M x,y ,M z ..., any of those properties is essentially sufficient to characterize quasi arithmetic means see Acz l&ndash Dhombres, Chapter 17. Any quasi arithmetic mean math M math of two variables has the balancing ... functions math f math , the f mean is not. Indeed, the only homogeneous quasi arithmetic means are the power mean s and the geometric mean see Hardy&ndash Littlewood&ndash P lya, page 68. The homogeneity ... mean Jensen s inequality DEFAULTSORT Quasi Arithmetic Mean Category Means es Media f generalizada ko ... continuous function continuous and injective function injective then we can define the f mean ... For math n math numbers math x 1, dots, x n in I math , the f mean is math M f x 1, dots, x n f 1 left ..., it follows that f is a strictly monotonic function , and therefore that the f mean is neither ... mean corresponds to the harmonic mean . If we take math I math to be the set of positive real numbers and math f x x p math , then the f mean corresponds to the power mean with exponent math p math . Properties Partition of a set Partitioning The computation of the mean can be split into computations ... a priori, without altering the mean, given that the multiplicity of elements is maintained. With math ...   more details



  1. Arithmetic

    coding Arithmetic mean Arithmetic progression Arithmetic properties Associativity Commutativity Distributivity ...Image Tables generales aritmetique MG 2108.jpg thumb Arithmetic tables for children, Lausanne, 1835 Arithmetic ... of numbers. Professional mathematician s sometimes use the term higher arithmetic ref Harold Davenport Davenport, Harold , The Higher Arithmetic An Introduction to the Theory of Numbers 7th ed. , Cambridge ... results related to number theory , but this should not be confused with elementary arithmetic . History The prehistory of arithmetic is limited to a small number of artifacts which may indicate conception ... used all the elementary arithmetic operations as early as 2000 BC. These artifacts do not always reveal ... methods of calculation. The continuous historical development of modern arithmetic starts with the Hellenistic ... to each other, in his Introduction to Arithmetic . Greek numerals , derived from the hieratic Egyptian ... of arithmetic. For example, the ancient mathematician Archimedes devoted his entire work The Sand ... perform all four arithmetic operations. Although the Codex Vigilanus described an early form ... in comparison. In the Middle Ages , arithmetic was one of the seven liberal arts taught in universities ..., and trigonometry and nomogram nomographs in addition to the electrical calculator . Decimal arithmetic ... place and, with a radix point , using those same symbols to represent Arithmetic fraction fractions ... arithmetic computations using this type of written numeral. For example, addition produces the sum ... of the uses of number theory . Arithmetic operations The basic arithmetic operations are addition ... functions . Arithmetic is performed according to an order of operations . Any set of objects upon which all four arithmetic operations except division by zero can be performed, and where these four ... is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends ... is the second basic operation of arithmetic. Multiplication also combines two numbers into a single ...   more details



  1. Arithmetic IF

    The arithmetic IF statement has been for several decades a three way arithmetic Conditional programming conditional statement , starting from the very early version 1957 of Fortran , and including FORTRAN IV, FORTRAN 66 and FORTRAN 77. Unlike the Conditional programming logical IF statements seen in other languages, the Fortran statement defines three different branches depending on whether the result of an expression was negative, zero, or positive, in said order, written as IF expression negative,zero,positive While it was originally the only kind of IF statement provided in Fortran, the feature was used less and less frequently after the more powerful Conditional programming logical IF statements were introduced, and was finally labeled obsolescence obsolescent in Fortran 90. The arithmetic IF was also used in FOCAL programming language FOCAL . See also Sign function Three way comparison Conditional programming References http www.everything2.com index.pl?node arithmetic IF arithmetic IF everything2.com http www.liv.ac.uk HPC HTMLF90Course HTMLF90CourseNotesnode34.html Modular Programming with Fortran 90 Obsolescent Features Category Conditional constructs ru IF ...   more details



  1. Mean

    known as the power mean or H lder mean, is an abstraction of the quadratic, arithmetic, geometric and harmonic ... math f x x math arithmetic mean , math f x frac 1 x math harmonic mean , math f x x m math power mean , math f x ln x math geometric mean . Weighted arithmetic mean The weighted mean weighted arithmetic ... f. math This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric ... means div style moz column count 3 column count 3 Arithmetic geometric mean Arithmetic harmonic mean ...About the statistical concept In statistics , mean has two related meanings the arithmetic mean and is distinguished from the geometric mean or harmonic mean . the expected value of a random variable , which ... bar x math , pronounced x bar . This mean is a type of arithmetic mean. If the data set were based ... below. Examples of means Arithmetic mean AM Main Arithmetic mean The arithmetic mean is the standard ..., the arithmetic mean of five values 4, 36, 45, 50, 75 is math frac 4 36 45 50 75 5 frac 210 5 42. math Image Comparison mean median mode.svg thumb Comparison of the arithmetic mean, median and mode statistics ... with the median , Mode statistics mode or range. The mean is the arithmetic average of a set ... exponential and Poisson distribution s. Geometric mean GM Main Geometric mean The geometric mean ... and not their sum as is the case with the arithmetic mean e.g. rates of growth. math bar x left prod i 1 n x i right tfrac1n math For example, the geometric mean of five values 4, 36, 45, 50 ... 1 3 15. math Relationship between AM, GM, and HM Main Inequality of arithmetic and geometric means ... we get all means math m rightarrow infty math maximum math m 2 math quadratic mean math m 1 math arithmetic mean math m rightarrow0 math geometric mean math m 1 math harmonic mean math m rightarrow infty ... end, typically an equal amount at each end, and then taking the arithmetic mean of the remaining .... Interquartile mean The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic ...   more details



  1. Geometric progression

    of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression. An interesting ... since all the other terms cancel. Because r 1 for geometric series r 1 would give us an arithmetic progression ... the geometric mean of the progression s first and last term, and raising that mean to the power given ... sequence take the arithmetic mean of the first and last term and multiply with the number of terms .... A geometric progression is given this name because each term is the geometric mean of the two adjacent ...File Geometric sequences.svg thumb right 300px Diagram illustrating three basic geometric sequences of the pattern ... line represents the Infinite geometric series infinite sum of the sequence, a number that it will forever ... , a geometric progression , also known as a geometric sequence , is a sequence of number ... called the common ratio . For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1 2. The sum of the terms of a geometric progression, or of an initial segment of a geometric progression, is known as a geometric series . Thus, the general form of a geometric sequence is math a, ar, ar 2, ar 3, ar 4, ldots math and that of a geometric series is math a ar ar 2 ar 3 ar 4 cdots math where ... properties The n th term of a geometric sequence with initial value a and common ratio r is given by math a n a ,r n 1 . math Such a geometric sequence also follows the recursive relation math a n r ,a n 1 math for every integer math n geq 1. math Generally, to check whether a given sequence is geometric ... ratio of a geometric series may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance 1, 3, 9, 27, 81, 243, ... is a geometric sequence with common ratio 3. The behaviour of a geometric sequence depends on the value of the common ...   more details



  1. Outline of arithmetic

    The following outline is provided as an overview of and topical guide to arithmetic Arithmetic &ndash oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day to day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of operations that combine numbers. In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. Essence of arithmetic main Arithmetic Elementary arithmetic Decimal arithmetic Decimal point numeral system Numeral Place value History of arithmetic main Arithmetic History l1 History of arithmetic Arithmetic operations and related concepts seealso Operation mathematics Order of operations Addition Sum Additive inverse Subtraction Multiplication Multiplicative inverse Multiples Common multiple s Least common multiple Division mathematics Division Quotient ... Ratio Common denominator Lowest common denominator Factorization Factoring Fundamental theorem of arithmetic ... Odd number Even number Positive number Negative number Elementary statistics Mean Weighted mean ... mathematics Proportion Rounding Scientific notation Modern arithmetic Riemann zeta function L functions ... symbols External links sisterlinks Arithmetic http www.cut the knot.org WhatIs WhatIsArithmetic.shtml What is arithmetic? http mathworld.wolfram.com Arithmetic.html MathWorld article about arithmetic http www.aaamath.com Interactive Arithmetic Lessons and Practice http www.quiz tree.com math games level 1 windows.html Talking Math Game for kids s The New Student s Reference Work Arithmetic The New Student s Reference Work Arithmetic historical http zetamac.com arithmetic Arithmetic Game http www.quiz ... western work on arithmetic at http mathdl.maa.org convergence 1 Convergence outline footer Category Outlines Arithmetic Category Arithmetic Category Mathematics related lists Arithmetic ...   more details



  1. Geometric median

    dablink For another use of the term median in geometry, see Median geometry . The geometric median of a discrete ... nearest center. ref The geometric median is an important estimator of location parameter location ... x 1, x 2, dots, x m , math with each math x i in mathbb R n math , the geometric median is defined as Geometric Median math underset y in mathbb R n operatorname arg ,min sum i 1 m left x i y right 2 ... s is minimum. Properties For the 1 dimensional case, the geometric median coincides with the median . This is because the univariate median also minimizes the sum of distances from the points. The geometric median is unique whenever the points are not Line geometry collinear . The geometric median ... either by transforming the geometric median, or by applying the same transformation to the sample data and finding the geometric median of the transformed data. This property follows from the fact that the geometric median is defined only from pairwise distances, and doesn t depend on the system ... of the choice of coordinates. The geometric median has a breakdown point of 0.5. ref Lopuha and Rousseeuw .... Special cases For 3 points, if any angle of the triangle is more than 120 then the geometric median is the point making that angle. If all the angles are less than 120 , the geometric median is the point ... of the four points is inside the triangle formed by the other three points, then the geometric median is that point. Otherwise, the points form a convex quadrilateral and the geometric median is the crossing point of the diagonals of the quadrilateral. The geometric median of four coplanar points is the same ... concept, computing the geometric median poses a challenge. The centroid or center of mass , defined similarly to the geometric median as minimizing the sum of the squares of the distances to each ... but no such formula is known for the geometric median, and it has been shown that no explicit formula , nor an exact algorithm involving only arithmetic operations and k th roots can exist in general ...   more details



  1. Heinz mean

    In mathematics, the Heinz mean named after E. Heinz ref E. Heinz 1951 , Beitr ge zur St rungstheorie der Spektralzerlegung , Math. Ann. , 123 , pp. 415 438. ref of two non negative real number s A and B , was defined by Bhatia ref citation first R. last Bhatia title Interpolating the arithmetic geometric mean inequality and its operator version journal Linear Algebra and its Applications volume 413 issue 2 3 pages 355 363 year 2006 doi 10.1016 j.laa.2005.03.005 . ref as math H x A, B frac A x B 1 x A 1 x B x 2 . math with  0    x     1 2. For different values of x , this Heinz mean interpolates between the arithmetic mean arithmetic x     0 and geometric mean geometric x     1 2 means such that for 0    x     1 2 math sqrt A B H 1 2 A, B H x A, B H 0 A, B frac A B 2 . math The Heinz mean may also be defined in the same way for positive semidefinite matrix positive semidefinite matrices , and satisfies a similar interpolation formula. ref citation first1 R. last1 Bhatia first2 C. last2 Davis authorlink2 Chandler Davis title More matrix forms of the arithmetic geometric mean inequality journal SIAM Journal on Matrix Analysis and Applications volume 14 issue 1 pages 132 136 year 1993 doi 10.1137 0614012 . ref ref citation first Koenraad M.R. last Audenaert title A singular value inequality for Heinz means arxiv math 0609130 journal Linear Algebra and its Applications volume 422 issue 1 pages 279 283 year 2007 doi 10.1016 j.laa.2006.10.006 . ref See also Mean Muirhead s inequality Inequality of arithmetic and geometric means References reflist Category Means Mathapplied stub km ...   more details



  1. Chisini mean

    In mathematics , a function f of n variables x sub 1 sub , ..., x sub n sub leads to a Chisini mean M if for every vector < x sub 1 sub ... x sub n sub > , there exists a unique M such that f M , M , ..., M f x sub 1 sub , x sub 2 sub , ..., x sub n sub . The arithmetic mean arithmetic , harmonic mean harmonic , geometric mean geometric , generalised mean generalised , Heronian mean Heronian and quadratic mean quadratic means are all Chisini means, as are their weighted variants. References They were introduced by Oscar Chisini , in the paper Chisini, O. Sul concetto di media. Periodico di Matematiche 4, 106&ndash 116, 1929. Category Mathematical analysis Category Means it Media Chisini pl rednia Chisinego ...   more details



  1. Contraharmonic mean

    mean, G is geometric mean , L is the logarithmic mean , A is the arithmetic mean, R is the root ... 2 variable means. First, the geometric mean of the arithmetic and harmonic means is equal to the geometric ... over 2 cdot 2ab over a b sqrt ab G a,b math The second relationship is that the geometric mean of the arithmetic ...In mathematics, a contraharmonic mean is a function complementary to the harmonic mean . The contraharmonic mean is a special case of the Lehmer mean , math L p math , where p 2. Definition The contraharmonic mean of a set of positive numbers is defined as the arithmetic mean of the squares of the numbers divided by the arithmetic mean of the numbers math C x 1, x 2, dots , x n left x 1 2 x 2 2 cdots ... taking the mean of only two variables, the contraharmonic mean is as high above the arithmetic mean as the arithmetic mean is above the harmonic mean i.e., the arithmetic mean of the two variables is equal to the arithmetic mean of their harmonic and contraharmonic means . Two variable formulae From the formulas for the arithmetic mean and harmonic mean of two variables we have math A a,b a b ... to the sum of the arithmetic mean and the variance mean. ref name Kingley1989 Kingley MSC 1989 ... ref Since the variance is always 0 the contraharmonic mean is always greater than the arithmetic mean ... sample mean arithmetic mean is a biased estimator of the true mean. To see this consider math g x ... Eudoxus of Cnidus Eudoxos in the 4th century before Christ. See also Arithmetic mean Arithmetic geometric mean Arithmetic harmonic mean Average Ces ro mean Chisini mean Elementary symmetric mean Fr chet mean Generalized mean Geometric mean Gini mean Harmonic mean Heinz mean Heronian mean Identric ... the characteristic properties of a mean math C x 1, x 2, dots , x n in min x 1, x ... k fixed point property . The contraharmonic mean is higher in value than the average and also higher than the root mean square math min mathbf x leq H mathbf x leq G mathbf x leq L mathbf x leq ...   more details



  1. Arithmetic progression

    98 . math See Also Geometric progression Generalized arithmetic progression is a set of integers constructed as an arithmetic progression is, but allowing several possible differences. Harmonic progression ...In mathematics , an arithmetic progression AP or arithmetic sequence is a sequence of number s such that the difference between the consecutive terms is constant. For instance, the sequence 3, 5, 7, 9, 11, 13, is an arithmetic progression with common difference of 2. If the initial term of an arithmetic progression is math a 1 math and the common difference of successive members is d , then the n th term of the sequence is given by math a n a 1 n 1 d, math and in general math a n a m n m d. math A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series . The behavior of the arithmetic progression depends on the common difference d . If the common difference is Positive, the members terms will grow towards positive infinity . Negative, the members terms will grow towards negative infinity. Sum Other uses section Finite arithmetic series Infinite arithmetic series Infinite arithmetic series The Summation sum of the members of a finite arithmetic progression is called an arithmetic series . Expressing the arithmetic series in two different ways math S n a 1 a 1 d a 1 2d cdots a 1 n 2 d a 1 n 1 d math math S n a n n 1 d a n n 2 d ... So, for example, the sum of the terms of the arithmetic progression given by a sub n sub 3 n 1 5 ... of the members of a finite arithmetic progression with an initial element a sub 1 sub , common differences ... of the terms of the arithmetic progression given by a sub n sub 3 n 1 5 up to the 50th term ... 260 External links MathWorld urlname ArithmeticProgression title Arithmetic progression MathWorld urlname ArithmeticSeries title Arithmetic series DEFAULTSORT Arithmetic Progression Category Sequences ...   more details



  1. Geometric flow

    In mathematics , specifically differential geometry , a geometric flow is the gradient flow associated to a functional on a manifold which has a geometric interpretation, usually associated with some curvature extrinsic or intrinsic curvature . They can be interpreted as flows on a moduli space for intrinsic flows or a parameter space for extrinsic flows . These are of fundamental interest in the calculus of variations , and include several famous problems and theories. Particularly interesting are their critical point mathematics critical point s. A geometric flow is also called a geometric evolution equation . Examples Extrinsic Extrinsic geometric flows are flows on embedded submanifold s, or more generally immersed submanifold s. In general they change both the Riemannian metric and the immersion. Mean curvature flow , as in soap film s critical points are minimal surface s Willmore flow , as in minimax eversion s of spheres Inverse mean curvature flow Intrinsic Intrinsic geometric flows are flows on the Riemannian metric , independent of any embedding or immersion. Ricci flow , as in the Solution of the Poincar conjecture , and Richard Hamilton professor Richard Hamilton s proof of the Uniformization theorem Calabi flow Yamabe flow Classes of flows Important classes of flows are curvature flows , variational flows which extremelize some functional , and flows arising as solutions to parabolic partial differential equation s. A given flow frequently admits all of these interpretations, as follows. Given an elliptic operator L , the parabolic PDE math u t Lu math yields ... of the flow correspond to critical points of the functional. In the context of geometric flows, the functional ... Bakas, I. title The algebraic structure of geometric flows in two dimensions year 2005 arxiv hep th 0507284 cite journal author Bakas, I. title Renormalization group equations and geometric flows year 2007 arxiv hep th 0702034 DEFAULTSORT Geometric Flow Category Geometric flow ...   more details



  1. Arithmetic surface

    In mathematics, an arithmetic surface over a Dedekind domain R with Field of fractions fraction field math K math is a geometric object having one conventional dimension, and one other dimension provided ... ideal spectrum Spec Z being seen as analogous to a line. Arithmetic surfaces arise naturally ... point special fibers . Formal definition In more detail, an arithmetic surface math S math ... Topics in the Arithmetic of Elliptic Curves . Springer, 1994, p. 311. ref Over a Dedekind Scheme In even more generality, arithmetic surfaces can be defined over Dedekind schemes, a typical example of which is the spectrum of the ring of integers of a number field which is the case above . An arithmetic .... Algebraic geometry and arithmetic curves . Oxford University Press, 2002, chapter 8. ref This generalisation ... fields, which is important in positive characteristic. What makes them arithmetic? Arithmetic surfaces are the arithmetic analogue of fibred surfaces with the spectrum of a Dedekind domain replacing the base curve. ref Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves . Springer ... may also consider arithmetic schemes. ref Eisenbud, D. and Harris, J. The Geometry of Schemes . Springer Verlag, 1998, p. 81. ref Properties Dimension Arithmetic surfaces have dimension 2 and relative dimension 1 over their base. ref Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves ... divisors on arithmetic surfaces since every local ring of dimension one is regular. This is briefly stated as arithmetic surfaces are regular in codimension one. ref Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves . Springer, 1994, p. 311. ref The theory is developed in Hartshorne ... of scheme theory smooth , Glossary of scheme theory proper arithmetic surface over math R math ... R mathfrak m . math ref Silverman, J.H. Advanced Topics in the Arithmetic of Elliptic Curves . Springer ... over a global field , are examples of this construction, and are much studied examples of arithmetic ...   more details



  1. Heronian mean

    In mathematics, the Heronian mean H of two non negative real number s A and B is given by the formula math H frac 1 3 left A sqrt A B B right . math It is named after Hero of Alexandria , and used in finding the volume of a frustum of a pyramid or cone geometry cone . The volume is equal to the product of the height of the frustum and the Heronian mean of the areas of the opposing parallel faces. The Heronian mean of the numbers A and B is a weighted mean of their arithmetic mean arithmetic and geometric mean s math H frac 2 3 cdot frac A B 2 frac 1 3 cdot sqrt A B . math References Citation last1 Bullen first1 P.S. title Handbook of Means and Their Inequalities publisher Springer Science Business Media location Berlin, New York edition 2nd series Mathematics and Its Applications isbn 978 1 4020 1522 9 year 2003 Citation last1 Eves first1 Howard Whitley author1 link Howard Eves title Great Moments in Mathematics Before 1650 publisher Mathematical Association of America isbn 978 0 88385 310 8 year 1980 External links http jwilson.coe.uga.edu EMT668 EMAT6680.2000 Umberger EMAT6690smu Essay3smu Essay3smu.html Mean Trapezoids Geometric comparison of some mathematical means DEFAULTSORT Heronian Mean Category Means geometry stub es Media heroniana eu Batezbesteko herondar ko km zh ...   more details



  1. Geometric genus

    In algebraic geometry , the geometric genus is a basic birational invariant p sub g sub of algebraic varieties and complex manifold s. Definition The geometric genus can be defined for non singular complex projective varieties and more generally for complex manifold s as the Hodge number h sup n ,0 sup equal to h sup 0, n sup by Serre duality , that is, the dimension of the Canonical bundle General case canonical linear system . In other words for a variety V of complex dimension n it is the number of linearly independent holomorphic n differential form forms to be found on V . ref Danilov & Shokurov 1998 , Google books quote id mU6ciaFCC1IC page 53 text geometric genus p. 53 ref This definition, as the dimension of H sup 0 sup V ,&Omega sup n sup then carries over to any base field mathematics field , when &Omega is taken to be the sheaf of K hler differential s and the power is the top exterior power , the canonical bundle canonical line bundle . The geometric genus is the first invariant p sub g sub P sub 1 sub of a sequence of invariants P sub n sub called the plurigenera . The case of curves In the case of complex varieties, the complex loci of non singular curves are Riemann surfaces . The algebraic definition of genus agrees with the genus of a surface topological notion . On a nonsingular curve, the canonical line bundle has degree 2g 2 . The notion of genus features prominently in the statement of the Riemann Roch theorem see also Riemann Roch theorem for algebraic curves and of the Riemann Hurwitz formula . If C is an irreducible and smooth hypersurface in the Algebraic geometry of projective spaces projective plane cut out by a polynomial equation of degree d , then its ... O d C mathcal O d 3 C math . Genus of singular varieties The definition of geometric genus is carried over classically to singular curves C , by decreeing that p sub g sub C is the geometric genus of the normalization ... , the definition is extended by birational invariance. See also Genus mathematics Arithmetic ...   more details



  1. Affine arithmetic

    hull and affine arithmetic methods for algebraic curve drawing . Proc. Uncertainty in Geometric Computations ...Affine arithmetic AA is a model for self validated computation self validated numerical analysis . In AA .... Affine arithmetic is meant to be an improvement on interval arithmetic IA , and is similar to generalized interval arithmetic , first order Taylor arithmetic , the center slope model , and ellipsoid ... approximations to general formulas. Affine arithmetic is potentially useful in every numeric ... control , worst case analysis of electric circuit s, and more. Definition In affine arithmetic ... subset of the rectangle 2,18 13,27 . Affine arithmetic operations Affine forms can be combined with the standard arithmetic operations or elementary functions, to obtain guaranteed approximations ... range. One simply replaces each arithmetic operation or elementary function call in the formula .... For this reason, affine arithmetic will often yield much tighter bounds than standard interval arithmetic whose errors are proportional to h . Roundoff errors In order to provide guaranteed enclosure, affine arithmetic operations must account for the roundoff errors in the computation of the resulting ... that does not implement roundoff error control. Affine projection model Affine arithmetic can be viewed ... that AA is a zonotope arithmetic . Each step of AA usually entails adding one more row and one more .... Implementation Matrix implementation Affine arithmetic can be implemented by a global array A and a global ... 2004 Affine arithmetic concepts and applications. Numerical Algorithms 37 1&ndash 4 , 147&ndash 158. J. L. D. Comba and J. Stolfi 1993 , Affine arithmetic and its applications to computer graphics . Proc ... surfaces with affine arithmetic . Computer Graphics Forum , 15 5 , 287&ndash 296. fig sto 96 imp W. Heidrich 1997 , A compilation of affine arithmetic versions of common math library functions ... solution algorithm using affine arithmetic . NOLTA 98 &mdash 1998 International Symposium on Nonlinear ...   more details



  1. Geometric probability

    Problems of the following type, and their solution techniques, were first studied in the 19th century, and the general topic became known as geometric probability . Buffon s needle What is the chance that a needle dropped randomly onto a floor marked with equally spaced parallel lines will cross one of the lines? What is the mean length of a random chord of a unit circle? cf. Bertrand s paradox probability Bertrand s paradox . What is the chance that three random points in the plane form an acute rather than obtuse triangle? What is the mean area of the polygonal regions formed when randomly oriented lines are spread over the plane? For mathematical development see the concise monograph Solomon. ref cite book author Herbert Solomon title Geometric Probability year 1978 publisher Society for Industrial and Applied Mathematics location Philadelphia, PA ref Since the late 20th century the topic has split into two topics with different emphases. Integral geometry sprang from the principle that the mathematically natural probability models are those that are invariant under certain transformation groups. This topic emphasises systematic development of formulas for calculating expected values associated with the geometric objects derived from random points, and can in part be viewed as a sophisticated branch of multivariate calculus. Stochastic geometry emphasises the random geometrical objects themselves. For instance different models for random lines or for random tessalations of the plane random sets formed by making points of a Poisson process spatial Poisson process be say centers of discs. See also Wendel s theorem References references DEFAULTSORT Geometric Probability Category Geometry Category Probability theory eu Probabilitate geometriko uk ...   more details




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