_cP!_ COMPREHENIIVE vvvvOfvvvv [)ICTI()~Al?"
MATHEMATICS Chief Editor & Compiler: -~
Roger Thompson
~ ABHISHEK
All rights reserved. No part of this book may be reproduced in any form, electronically or otherwise, in print, photoprint, micro film or by any other means without written permission from the publisher.
ISBN Copyright
Revised Edition
978-81-8247-341-6 Publisher
2010
Published by ABHISHEK PUBLICATIONS, S.C.O. 57-59, Sector 17-C, CHANDIGARH-1600 17 (India) Ph.-2707562,Fax-OI72-2704668 Email:
[email protected] Preface Mathematicians, scientists, engineers, and students will look this dictionary for defmitive coverage of all branches of mathematics, both pure and applied. Featuring more than 1500 terms, the book defmes terms and expressions in algebra, number theory, operator theory, logic, complex numbers, fmite mathematics, topology, and other areas----each with definition along with pictorial representations of many terms. This dictionary is authoritative, comprehensive comprising latest terms and is carefully reviewed to ensure its accuracy, clarity, and completeness. This Dictionary of Mathematics puts a wealth of essential information at your fingertips. Whether you're a professional, a student, a writer, or a general reader with science curiosity, this comprehensive resource defines the current language of pure and applied Mathematics and gives you a better understanding of the ideas and concepts you need to know. It has, for the first time, brought together in one easily accessible form the best-expressed thoughts that are especially illuminating and pertinent to the discipline of Mathematics. This dictionary will be a handy reference for the mathematician or scientific reader and the wider public interested in who has said what on mathematics.
· The overall aim of the dictionary is to provide an accessible description of what one judges to be the core material for damn good dictionary. The subject has a reputation for being disagreeably difficult. I have tried to alter that perception by showing that key ideas can be presented simply judiciously but not overwhelmingly deployed, clarifies and provides power.
11 2-by-2 table I abeliangroup
5
.2-by-2 table 1. this is a two-way table where the numbers of levels of the ro·.v and column-classifications are each 2. If the row and column classifications each divide the observational units into subsets, then it is likely that it will be useful to analyse the data using the Fisher Test.
. ; .
• 3-D figure a set of points in space; examples: box, cone, cylinder, parallelpiped, prism, pyramid, regular pyramid, right cone, right cylinder, right prism, sphere.
; 'prime'; designates an image : corresponding to the preimage ~ using the same variable.
A~
T
I I
45'
0.
I
~ .8
I
.• aRb I a is an element in b
I.
abacus ~ a Japanese counting device and : calculator.
~ • abelian group I
a group in which the binary
.'1...
°',;'" rIO;'.. .._'1 ~.if..... . •. .if. ...... (~, '=)
f
Wi'Z"
• 45-45-90 triangle an isoscoles right triangle
w;;g,
========!!!!!!!!*
sbseisss IIIJlinetnlnsj'tmnllritm
!!!!!6="
?peratio~
II
is commutative, that for all elements a abd b m the group. • abscissa the x-coordinate of a point in a 2-dimensional coordinate systern . • absolute value the positive value for a real n~ber, disregarding the sign. Wntten 1x I. For example, 131 =3, 1-41 =4, and 101 =0. • abundant number a positive integer that is smaller than the sum of its proper divisors.
~ • additive identity property ; the sum .pf any number and : zero is the original number· ~ zero is the identity element of ~ addition. ; • adjacent angles . .g ht and nonzero ; two non-straJ. : angles that have a common side ~ in the interior of the angle ; f?rmed by the non-s:ommon : SIdes
• acceleration ~e rate of change of velocity With respect to time. • "'......-...hl I els f --1'~ e eVi 0 accuracy ~e p~eClSlon determined by the sItuation or the given numbers stude?ts should help develo~ what IS acceptable according to the situation.
. ; • ad)a~t interior angle : the mtenor angle that forms a ~ linear pair with a given exterior I angle of a triangle. : . ." adjacent SIde ~ (of an angle.in a triangle~, one : of the two SIdes of the mangle ~ that form the sides of the angle.
IS,. ab= ba
~
lnIeriorangl ....nlhe ....... side oflhe InirIlIversaI A
~ ... ~" Be 0 DE
IntedonI on the ......
::~
1IIC3o_18D
; ~
.
I·
• accuracy ; • affine superimposition the closeness of a measurement ; a su~rimposition for which the or estimate to its true value. : assocIated transformations are I all affine. • acute angle : an angle whose measure is ~ • affine transforma,tion greater than 0 but less than 90 ; a transformation for which degrees. : parallel lines remain parallel.· I
~~~~~~~~~~==~~~M
llilleeimlie elJllllhon lilleeimlie
numb:.. =========~7
Mfine transformations of the plane take squares into parallelograms and take circles into ellipses of the same shape. Mfine transformations of a 3-dimensional space take cubes into parallelopipeds (sheared bricks) and spheres into ellipsoids all of the same shape. Similar results are produced in higher dimensional spaces . Equivalent to "uniform transformation" . As far as form is concerned' (that is, ignoring translation and rotation), any affine transformation can be diagrammed as a pure strain taking a square to a rectangle on the same axes. In studies of shape, where scale is ignored as well, the picture is the same but now the sum of the squares of the axes is unchanging. Still ignoring scale (that is, as far as shape is concerned), any affine transformation can be also diagrammed as a pure shear tak ing a square into a parallelogram of unchanged base segment and height. This diagram of shear came into morphometrics via an application to principal components analysis somewhat before it
~ was applied ; based shape . I
'g
I
:or
I
'. ---Y 'II
to
landmark-
!r-(.~-s!:i'"
I
I
uses the words if and only if. • bijection a one-to-one onto function.
J I
-_.. -
.
...... "'.add ...""" I;'
:. :. 32
_
. . .2
....
• tIII
OIl
.... • .•• .. 2
.
II hiM", operation I biseet
17
• binary operation ~ a binary operation is an opera- I tion that involves two operands. ; For example, addition and sub-. traction are binary operations. ; •. . : bmolD1~.
I
an expression that IS the sum of two terms. • binomial coefficient the coefficients of x in the expansion of (x+ 1)n.
: ~
• binomial distribution a random variable has a binomial distrib,:!:tion (with parameters n -and p) if it is the number of "successes" in a fixed number n of independent random trials, all of which have the same probability p of resulting in "success." Under these assumptions, the probability of k successes (and n _k failures) is nC pk(l-p)n-k, where nC is the k k number of combinations of n objects taken k at a time: nC = k n!j(k!(n-k)!). The expected value ~f a r~do~ v~iabl~ wi~ the Bmomial dIstnbutIon IS nxp, and the.stanruu:d error of a random vanable With the Binomial distribution is (nxpx(l p» V2. This page shows the
;
~ ; : I
probability histogram of the binomial distribution. • binomial test .. . . . this IS a stansnc~ test refemng to a repeated bmary process such as would be expected to generate outcomes with a binomial distribution. A value for the parameter 'p' is hypothesised (null hypothesis) and the difference of the actual value from this is assessed as a value of alpha.
~
• biplot a single diagram that represents two separate scatterplots on the same pair of axes. One scatter is of some pair of columns of ~ the matrix U of the singular I value decomposition of a ma: trix S, and the other scatter is ~ of the matching pair of columns I of V. When S is a centered data ~ ~atrix, the effect is to plot prin: Clpal component loadings and ~ scores on the same diagram. ~ • biquadratic equation ; a polynomial equation of the : 4th degree. : ~ ; :
I
•
~. bl~e~t
. ; to diVide mto two congruent I parts.
18
bisecWrpflJ~t I bootstmp
II
• bisector pf a segment ~ human subjects, it is usually any plane, point or two-dimen- ; necessary to administer a plasional figure containing the the cebo to the control group. midpoint of the segment and no I • bootstrap other points on that segment I this is a form of randomisation • bit I test which is one of the alternaa binary digit. tives to exhaustive re• bivariate randomisation. Th~ bootstrap having or having to do with two I scheme involves generating variables. For example, bivari- I subsets of the data on the basis ate data are data where we have of random sampling with retwo measurements of each "in- I placements as the data are dividual." These measurements I sampled. Such resampling promight be the heights and vides that each datum is equally in the weights of a group of people I represented ". di ·dual"· ) th I randomisation scheme; how(~ghID VI IS a person, e: ever, the bootstrap procedure hel ts 0 f fathers and sons (an I h Co hi h di· .h ".IIId··d . asf leatures IVI ua1"·IS a f at h er-son .. h w c d songws f .) th d I It rom t e proce ure 0 a pair, e pressure an tem-.. M earl0 test. The distm. onte.. perature of a fIXed volume of I ". di .dual"· th 1· gwshing features of the boot( gas an ID VId IS ~ vo - ; strap procedure are concerned lin th . ume 0 f gas un er a certam set. .th of experimental conditions), ; WI o~er-samp thg erebels nO · · constramt upon e num r 0 f etc. Scatterp1ots, the corre1anon·. tha d be coefficient, and regression ~ nmes d ~ a atum ~ay inglrepmake sense for bivariate data ; resente lingID genberaong a ~ e .. d . resamp su set; the SIZe 0 f but not uruvariate ata. ; the resampling subsets may be • blind experiment in a blind experiment, the subjects do not know whether they are in the treatment group or the control group. In order to have a blind experiment with
fixed arbitrarily independently : of the parameter values of the ~ experimental design and may ; even exceed the total number of : data. The positive motive for ~ bootstrap resampling is the gen-
1I--=======MAthemsties
II bootstrapestimateofstandarderror I: eral relative ease of devising an appropriate resampling algorithm when the experimental design is novel or complex. A negative aspect of the bootstrap is that the form of the resampling distribution with prolonged resampling converges to a form which depends not only upon the data and the test statistic, but also upon the bootstrap resampling subset size thus the resampling distribution should not be expected to converge to the gold standard form of the exact test as is the for Monte-Carlo case resampling. An effective necessity for the bootstrap procedure is a source of random codes or an effective pseudo-random generator.
model
19
~ the sample to estimate the SE ; of sampling from the popula: tion. For sampling from a box ~ of numbers, the SD of the ; sample is the bootstrap estimate : of the SD of the box from ~ which the sample is drawn. For I sample percentages, this takes ~ a particularly simple form: the : SE of the sample percentage of I n draws from a box, with re~ placement, is SD(box)/nY2, : where for a box that contains ~ only zeros and ones, SD(box) ; - ( (fraction of ones in : box) x (fraction of zeros in box) ~ )Y2. The bootstrap estimate of ; the SE of the sample percent: age consists of estimating ~ SD(box) by ((fraction of ones I in sample) x (fraction of zeros : in sample»'h. When the ~ sample size is large, this apI proximation is likely to be ~ good. : - box ~ a surface made up of rect~ angles; a rectangular parallel; epiped
- bootstrap estimate of standard error the name for this idea comes from the idiom "to pull oneself up by one's bootstraps," which connotes getting out of a hole without anything to stand on. The idea of the bootstrap is to ~ • box model assume, for the purposes of es: an analogy between an experitimating uncenainties, that the ~ ment and drawing numbered sample is the population, then ; tickets "at random" from a box use the SE for sampling from : with replacement. For example, I
Msthemsties=========~ II
20 =~~~~~~=*
, to evalu- I, suppose we are trymg M ate a cold remedy by giving it ; R or a placebo to a group of n in- : dividuals, randomly choosing ~ half the individuals to receive I Q p the remedy and half to receive : the placebo. Consider the me- ~ dian time to recovery for all the I individuals (we assume every- ~ N one recovers from the cold : eventually; to simplify things, ~ • box plot we also ~sume that no one ,re- ; a representation of data above covered m exactly the median: a numbered scale where the time, and that n is' even). By ~ "box" encloses all data between defmition, half the individuals ;~ the median of the lower half got better in less than, the me- : (quartile 1) and the median of dian time, and half 10 more ~ the upper half (quartile 3), with than the median time. The in- I a vertical line inside the box to dividuals who received the: indicate the median of the data; treatment are a random sample ~ a dot represents each of the high of size n/2 from the set of n I and low values of the data, and subjects, half of whom got bet- ~ a horizontal line called a whister in less than median time, and : ker connects each dot to the half in longer than median ~ box. time, If the remedy is ineffec-; b h d b d . • ranc tIve, the number 0 f sub'Jects : , -an - oun . , ' d the remedy and I exploratIon w h0 recelve " , 'of a randOlrusanon 1 th : distrlbunon m such a way as to ' ' , w ho recovered mess an me- I 'lik th f nl2 . anOclpate the effect' of the next ' dian orne lS e e sum 0 , "I th ' replacement firom a I, rannOffilSatIOn to ale draws Wlth d 're atIve . This box with two tickets in it: one ; present r~ OffilSanon. ,with a "1" on it and one with a : lows selecove search of ~aro~"0"" I lar zones of a randoffilsatIon on It. . the context 0 f a .. distrl'b' uoon; m ~ randomisation test such selec-
II =======MsthtmUJms
II hreaItdownpoint I canonicalcorreUJ";"nalysis
21
tive search may be concerned ~ relations. Each score (linear with the tail of the i combination) from either list is randomisation distribution : correlated with no other com~ bination from its list and with • breakdown point . the breakdown point of an esti- ~ ~nly one score from the other mator is the smallest fraction of i st. observations one must corrupt to make the estimator take any '------------value one wants. • byte the amount of memory needed to represent one character on a computer, typically 8 bits. • calculator notation ~ • canonical correlation the symbols used by a calcula- i analysis tor for scientific notation. : a multivariate method for as; sessing the associations be• caliban puzzle a logic puzzle in which one is I tween two sets of variables asked to infer one or more facts : within a data set. The analysis ~ focuses on pairs of linear comfrom a set of given facts. I binations of variables (one for • canonical ~ each set) ordered by the maga canonical description of any : nitude of their correlations with statistical situation is a descrip~ each other. The fIrst such pair tion in terms of extracted vec; is determined so as to have the tors that have especially simple : maximal correlation of any ordered relationships. For in~ such linear combinations. Substance, a canonical correlations ; sequent pairs have maximal coranalysis describes the relation : relation subject to the constraint between two lists of variables ~ of being orthogonal to those in terms of two lists of linear I previously determined. combinations that show a remarkable pattern of zero cor- I
Msthmulties======= II
22
canonical 11am; analysis I causation, causal relation
II
• canonical variates analysis ~. capacity a method of multivariate analy- ; the amount of liquid that can fill sis in which the variation among : an object. • I groups is expressed relative to I the pooled within-group cova- I • cartesian p ane . . . C . I . a rectangular coordinate svstem . . . J' nance matrIX. anoruca varIal . finds lin I COnSIStIng of a honzontal numates an. YSIS ear tr~s- ber line (x-axis) and a vertical fiormatlons of the data which I b lin ( .). maximise the among group ~um ethr e . ~-axIS , mtersect. . , I mg at e ongm (zero on each varIatIOn relatIve to the pooled . be lin) within-group variation. The ca- ; num r e. nonica! variates then may be I • categorical variable displayed as an ordination to : a variable whose value ranges show the group centroids and ~ over categories, such as {red, scatter within groups. This may ; green, blue}, {male, female}, be thought of as a "data reduc- : {Delhi, Calcutta, Mumabai}, tion" method in the sense that ~ {short, tall}, {Asian, Mricanone wants to describe among ; American, Caucasian, Hispanic, group differences in few dimen- : Native American, Polynesian}, sions. The canonical variates are ~ {straight, curly}, etc. Some cat'uncorrelated, however the vec- ; egorical variables are ordinal. tors of coefficients are not or- : The distinction between catthogonal as in Principal Com- ~ egorical variables and qualitaponent Analysis. The method is I tive variables is a bit blurry. closely related to multivariate .I • catenary analysis of variance : a curve whose equation is y = (MANOVA), multiple discrimi- ~ (a/2)(e +eA chain susnant analysis, and canonical cor- I pended from two points forms relation analysis. A critical as- : this curve sumption is that the within- ~ •. group variance-covariance;. causation, causal relation structure is similar otherwise : two variables are causally rethe pooling of th~ data over ~ lated if changes in the value of groups is not very sensible. i one cause the other to change. For example, if one heats a rigid X/ 3
X/ 3 ) .
II =======.M.thelulies
II
23
ceilingfunction I centroid
*================= container filled with a gas, that I • center of gravity causes the pressure of the gas I the mean of the coordinates of in the container to increase. points in a figure, whether one, Two variables can be associated I two, or three-dimensional without having any causal rela- I • central angle tion, and even if two variables I (of a chord or arc) An angle have a causal relation, their cor- : whose vertex is the center of a relation can be small or zero. I circle and whose sides pass I through the endpoints of a • ceiling function the ceiling function of x is the chord or arc. smallest integer greater than or • central angle of a circle equal to x. an angle whose vertex is the • center (of a circle or sphere) The point from which all points on the figure are the same distance. • center of a circle the point that all points in the circle are equidistant from .
I
I
I
I I
\
... \
''''
• center of a rotation the point where the two intersecting lines of a rotation meet
central limit theorem the central limit theorem states that the probability histograms of the sample mean and sample sum of n draws with replacement from a box of labeled tickets converge to a normal curve as the sample size n grows, in the following sense: As n grows, the area of the probability histogram for any range of values approaches the area tmder the normal curve for the same range of values, converted to standard units.
I •
I
--..
center of the circle
I
• centroid the point of concurrency of a triangle's three medians.
MlJthemR.tiu=======
II
",,24=========== ..centroid size I Chebychev'si1UlJUR1ity
II
_ centroid size - cevian centroid size is the square root I a line segment extending from of the sum of squared distances a vertex of a triangle to the opof a set of landmarks from their I posite side. centroid, or, equivalently, the I square r'?Ot of the sum of the ~ variances of the landmarks about that centroid in xand y- I directions. Centroid Size is used in geometric morphometries _ chance variation, chance because it is approximately I error uncorrelated with every shape a random variable can be devariable when landmarks are composed into a sum of its exdistributed around lTiean posi- ~ pected value and chance variations by independent noise of ; tion around its expected value. the same small variance at ev- The expected value of the ery landmark and in every di- I chance variation is zero; the rection. Centroid Size is the size I standard error of the chance measure used to scale a con- variation is the same as the stanfiguration of landmarks so they dard error of the random varican be plotted as a point in I able-the size of a "typical" difKendall"s shape space. The de- ference between the random nominator of the formula for variable and its expected value the Procrustes distance between I - Chebychev's inequality two sets of landmark configuI for lists: For every number rations is the product of their k>O, the fraction of elements Centroid Sizes. in a list that are k SD's or furI ther from the arithmetic mean - certain event an event is certain if ItS prob- of the list is at most Ijk2. For ability is 100%. Eve~ if an event random variables: For every is certain, it might not occur. I number k>O, the probability However, by the complement I that a random variable X is k rule, the chance that it does not SEs or further from its expected occur is 0%. I value is at most l/k2.
II chi-square curve I chi-squared d~tion • chi-square curve the chi-square curve is a family of curves that depend on a parameter called degrees of freedom (d.f.). The chisquare curve is an approximation to the probability histogram of the chi-square statistic for multinomial model if the expected number of outcomes in each category is large. The chi-square curve is positive, and its total area is 100%, so we can think of it as the probability histogram of a random variable. The balance point of the curve is d.f., so the expected value of the corresponding random variable would equal d.f.. The standard error of the corresponding random variable would be (2xd.f.)V2. As d.f. grows, the shape of the chi-square curve approaches the shape of the normal curve. • chi-square statistic the chi-square statistic is used to measure the agreement between categorical data and a multinomial model that prediets the relative frequency of outcomes in each possible category. Suppose there are n in-
25
~ dependent trials, each of ; which can result in one of k : possible outcomes. Suppose ~ that in each trial, the probabil; itythatoutcome i occurs is pi, : for i = 1, 2, ... , k, and that ~ these probabilities are the I same in every trial. The ex: pected number of times out~ come 1 occurs in the n trials I is n.xp1; more generally, the ~ expected number of times out: come i occurs is ~ • chi-squared distribution ~ where expected frequencies are ; sufficiently high, hypothesised : distributions of counts may be ~ approximated by a normal dis; tribution rather than an exact : binomial distribution. The cor~ responding distribution of the ; chi-squared statistic can be de: rived algebraically this is the ~ chi-squared distribution which I has been computed and pub~ lis~ed historically as extensive : prmted tables. Use of the tables ~ is notably simple, as the chi; squared distribution depends : upon only one parameter, the ~ degrees of freedom, defined as ; one less than the number of cat: egories. I
Mslthema.ties======= II
chi-SlJ.tl4red statistic I circumfoYence II
26
~============*~===========
• chi-squared statistic I this is a long-established test statistic for measuring the extent to which a set of categori- : cal outcomes depart from a. I hypothesised set of probabili- ~ ties. It is calculated as a sum of : terms over the available catego- ~ ries, where each term is of the i form: ((0-E)2)fE; '0' represents the observed frequency for I the category and 'E' represents the corresponding expec~ed ~e- ~ quency based upon multIplymg : the sample size by the I h thesised probability for the : being considered (therefore 'E' will generally not I be an integer value). In situa- : tions where the numbe~ of cat- ~ egories is 2 an alternatIve pro- ~ cedure is to use an exact i biniomial test.
c~gory
~
Q Chord
c
D
"A
E
F
A line segment that connects 2 points on a circle.
CD
and
EF
are chords of circle A.
• circle the set of points on a plane at a certain distance (radius) from a certain point (ceI1ter); a polygon with infinite sides •
• cl1'cular cone . . a cone whose base IS a crrcle. • circularity when on a search, circling back to a previous place visited (definition, web site, etc.), ustially unhelpful or redundant
i • circumcenter • chord : the circumcenter of a triangle 1. a line segment whose end- ~ is the center of the circum; scribed circle. points lie on a circle.. 2. the line joining two pomts on i • circumcircle a curve is called a chord. : "the circle circumscribed about I • chord of a circle : a figure. a segment whose end~oints ~ • circumference are on a circle i the distance around a circle, ~ given by the formula C = 2m;
II circumscribed I cluster analysis
27
*=============== where r is the radius of the circle. • circumscribed passing through each vertex of a figure, usually referring to circles circumscri bed around polygons or spheres circum _ scribed around polyhedrons. The figure inside is inscribed in the circumscribed figure. • cissoid a curve with equation y2(a_ X)=X3.
P2
~
: ~
• class interval in plotting a histogram, one starts by dividing the range of v~ue~ into a set of non-ov~rlap pmg ~tervals, called class mtervals, m such a way that every ~atum is contained in some class mterval. • class of functions family of functions such as linear, quadratic, power (polynomial), exponential, or logarith-
;
mic.
; :
~
~ .
~
I
I
~
~ • classes of numbers : family of numbers or number ~ systems such as natural, inte; ger, rational, irrational, real, or : complex. I
: • classify ~ to categorise something accord; ing to some chosen character: istics. I
: • clockwise ~ in orientation, the direction in ; which the points are named : when, if traveling along the line, ~ the interior of the polygon is on ~ the right. • class boundary a point that is the left endpoint I • cluster analysis of one class interval, and the ~ a method of analysis that repright endpoint of another class : resents multivariate variation in I data as a series of sets. In biolinterval. ~ ogy, the sets are often con-
II
corJJicient I combinatitms
28
structed in a hierarchical manner and shown in the form of a tree-like diagram called a dendrogram. • coefficient a coefficient, in general, is a number multiplying a function. In multivariate data analysis, usually the "function" is a variable measured over the cases of the analysis, and the coefficients multiply these variable values before we add them up to form a score. A coefficient is not the same as a loading. • coincide lying exactly on top of each other. Line segments that coincide are identical; they have all the same points.
~
; :
~ ~ ~
I
: I I
I
I
I I
: I : I
I
~
;
II
• coincidental lines lines that are identical (one and the same) • collinear lying on the same line. • combinations the number of combinations of n things taken k at a time is the number of ways of picking a subset ofk of the n things, without replacement, and without regard to the order in which the elements of the subset are pickc:d. The number of such combinations is nCk = n!j(k!(nk)!), where k! (pronounced "k factorial") is kx(k-l)x(k-2)x ... x 1. The numbers nCk are also called the Binomial coefficients. From a set that has n elements one can form a total of 2n subsets of all sizes. For example, from the set {a, b, c}, which has 3 elements, one can form the 23 = 8 subsets U, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}. Because the number of subsets with k elements one can form from a set with n elements is nck, and the total number of subsets of a set is the sum of the numbers of possible subsets of each size, it follows that nco +nC1 +nC2 + ... +nCn = 2n.
II ~m#n~~I~~n~~~~~~~~~~~2==9 The calculator has a button ~ - complementary angles (ncm ) that lets you compute the ; two angles whose measures number of combinations of m : have the sum 90°. things chosen from a set of n I things. To use the button, first I ArSfc. type the value of n, then push the nCm button, then type the value of m, then press the "=" I 8 () button. mLABC + mLCBD =90"
~
LABC and LCBD are complementary angles.
- commutative properties I properties about order of addi- : - complex numbers tion' a + b = b + a; of multi- ~ complex numbers are an algeplication, a x b = b x a" ; braic way of coding points in : the ordinary Euclidean plane - compass ~ so that translation (shift of a drawing tool used to draw ; position) corresponds to the circles at different radii :I addition of complex numbers . - compatible numbers : and both rescaling (enlargenumbers that can be easily ma- I ment or shrinking) and rotanipulated and operated on men- ~ tion correspond to multiplicatally. : tion of complex numbers. In _ complement rule ~ this system of notation, inthe probability of the comple- ; vented by Gauss, the x-axis is ment of an event is 100% mi- : identified with the "real numnus the probability of the event: ~ bers" (ordinary decimals P(Ac) = 100% peA). ; numbers) and the y-axis is : identified with "imaginary -thcompIement I f b f .I num b ers" (t h e square roots 0 f e.comp e~ent 0 a su .set 0 ; negative numbers). When you a gIven set IS the collecnon of" ul" I " thi" b all elements of the set that are ; ~ tip pomts on " s axiS hY not elements of the subset. : t emse ves accord mg to t e I rules, you get negative points : on the "real" axis just defined. ~ Many operations on data in
r
MJJ.th_tics========== II
~30~~~~~~~~~~~~~~mmW~l~i~p70b~i~ two dimensions can be proved valid more directly if they are written out as operations on complex numbers. • compose numbers put a set of numbers together to form a new number using addition or multiplication. • composite transformation the composite of a first transformation S and a second transformation T is the transformation mapping a point P onto T(S(P».
/I
concentric circles circles that share the same center, but have different radii
I •
I
I
I
I
• concrete materials objects to be manipulated (e.g., • composition (of transformations) The trans- I pattern blocks, snap cubes, formation that results when geoboards, tangrams, color one transformation is applied ~ tiles, base ten blocks). after another transformation. • concurrent I intersecting at a single point • compound eventS the point of two or more events in a prob- ( called I concurrency). ability situation such as flipping a coin and spinning a spinner. I • conditional I a statement that tells if one • concave thing happens, another will folcurved from the inside. low. • concave polygon a polygon having at least one • conditional probability diagonal lying outside the poly- I suppose we are interested in the probability that some event A gon; not convex. occurs, and we learn that the I event B occurred. How should I
II =======MRthem4ries
31
*================= we update the probability of A to reflect this new knowledge? This is what the conditional probability does: it says how the additional knowledge that B occurred should affect the probability that A occurred quantitatively. For example, suppose that A and B are mutually exclusive. Then if B occurred, A did not, so the conditional probability that A occurred given that B occurred is zero. At the other extreme, suppose that B is a subset of A, so that A must occur whenever B does. Then if we learn that B occurred, A must have occurred too, so the conditional probability that A occurred given that B occurred is 100%. For in-between cases, where A and B intersect, but B is not a subset of A, the conditional probability of A given B is a number between zero and 100%. Basically, one "restricts" the outcome space S to consider only the part of S that is in B, because we know that B occurred. For A to have happened given that B happened requires that AB happened, so we are interested in the event AB. To have a legitimate probability requires that P(S) = 100%, so
~
if we are restricting the out; come space to B, we need to : divide by the probability of B ~ to make the probability of this ; new S be 100%. On this scale, : the probability that AB hap~ pened is P(AB)fP(B). This is , the deftnition of the conditional : probability of A given B, pro~ vided P(B) is not zero (division ~ by zero is undefined). Note ; that the special cases AB = {} : (A and B are mutually exclu~ sive) and AB = B (B is a subset ; of A) agree with our intuition : as described at the top of this ~ paragraph. Conditional prob; abilities satisfy the axioms of : probability, just as ordinary ~ probabilities do. ~ - conditional proof ; a proof of a conditional state: ment.
,
: ~ , :
~ ~
; : ~
; :
- cone a solid whose surface consists of a circle and its interior, and all points on line segments that connect points on the circle to a single point (the cone's vertex) that is not coplanar with the circle. The circle and its interior form the base of the cone. The radius of a cone is the radius of the base. The altitude of a cone
Msthemtllti&s==========
II
~32~~~~~~~~~~~Mlmn~m~M is the line segment from the vertex to the plane of the base an~ perpendic~ar to it. The ?elgh~ of a cone 18 t.he length of Its aln~de. If the line segment c~nnecnng the vertex of a cone with th~ center of its base is perpendi~ar ~o the base, then the cone 18 a nght cone; otherwise it is oblique. • confidence interval for a given re-randomisation distribution, a family of related distributions may be defined according to a range of hypothe~cal values of the pattern whIch the test statistic measures. For instance, for the pitman permutation test to test for a scale shift between .two groups, a related distribution may be formed by shifting all the observations in one group by a common amount where this common shift is r~garded as a continuous variable, With fmite numbers of data the number of related distributions will be fmite, and typically considerably smaller than the number of points of the randomisation distribution. The likelihood of the outcome value may be calculated for each distribution in
II
~ the family, and these likelihoods ; may be then used to defme a ~ contiguous set of values which : occupy a certain proportion of I the total unit weight of the like: lihoods integrated over all val~ ues of the test statistic, the con~ fidence interval is defmed by ' the minimum and maximum ~ values of the range of values so ~ defined. The proportion of the ; total weight within the range of : values is regarded as an alpha ~ probabili~ ~at, the v~ue of the I test stanstic hes WIthin this ~ range, Generall~ the defmition : of a confidence mterval cannot ~ be unique wi~out imposing ' further constramts. Approaches ~ to providing suitable con~ ~traints, s~ch that a confidence ; mtet;al will be unique, include : d~g the confidence interval ~ : to mclude the whole of one tail ; of the ~istribution; or to be : centred m some sense upon the ~ outcome value; or to be centred ; bet,ween TAILS of equal ~ weIght" In, the ,cas,e of re: randomIsanon distributions, I these are discrete distributions : so there will generally be no ~ range of,values with weight corI responding exactly to an arbi: trary nominal alpha criterion
II ===:========Mstht:'l'/Ul.tU:s
II con.fideme level I congruentfiDU1'eS.
33
level, and the problem of non- ~ the treatment (if any). For exuniqueness is therefore not gen- ; ample, prominent statisticians eratly solvable. : questioned whether differences ~ between individuals that led ; some to smoke and others not : to (rather than the act of smok~ ing itself) were responsible for I the observed difference in the ~ frequencies with which smok: ers and non-smokers contract I various illnesses. If that were ~ the case, those factors would be - confidence level : confounded with the effect of the confidence level of a confi- ~ smoking. Confounding is quite dence interval is the chance that ; likely to affect observational the interval that will ~esult on~e : studies and experiments that data are collected wIll contam I are not randomised Confoundthe corresponding parame~er. If ~ ing tends to be d~creased by one computes confidence mter- : randomisation. vals again and again from inde- ~ pendent tiata the long-term . - congruent limit of the fr;ction of intervals ~ equilateral, equal, exactly the that contain the parameter is I same (size, shape, etc.) the confidence level. ; - congruent angles : two or more angles that have • confounding when the differences between ~ the same measure. • I the treatment and control I: - congruent clrc es groups other than the treat- I two or more circles that with the ment produce differences in re- : same radius. I sponse that are not distinguishable from the effect of the treat- : - congruent figures ~ two figures where one is the ment, those differences beI image of the other under a retween the groups are said to be ~ flection or composite of reflecconfounded with the effect of : tions.
MIIthemtlnes=======
\I
~34~~~~~~~~~~C~ongruen=.tpolygOllS I constantofRnetpmtUm • congruent polygons two or more polygons with the exact same size and shape. • congruent segments two or more segments that have the same measure or length. • conic section the cross section of a right circular cone cut by a plane. An ellipse, parabola, and hyperbola are conic sections. 1\
I
I
I
I I I
~ --
...... =:. :\ ,, .......
....... - ...
,~ - ............
I
=-'= I
;
\
-
:.,-l_tt.. I \
) '/ i
"
.'----- - - -- /
I
1\
l '
f,I .-------\\ (
~
I ..-
\
~ / \\ -
-
I \
'/
• conic solid the set of points between a point (the venex) and a non-coplan;:lI region (the base), including the point and the region. • conjecture a guess, usually made as a result of inductive reasoning. • consecutive angles (of a polygon), two angles that have a side of the polygon as a common side. • consecutive sides (of a polygon), two sides that have a common vertex.
• consecutive vertices (of a polygon or polyhedron), two vertices that are connected by a side or edge. consensus configuration a single set of landmarks intended to represent the central tendency of an observed sample for the production of superimpositions, of a weight matrix, or some other morphometric purpose. Often a consensus configuration is ·computed to optimize some measure of fit to the full sample : in particular, the Procrustes mean shape is computed to minimise the sum of squared Procrustes distances from the the consensus landmarks to those of the sample.
I •
/ \ !
•......:
II
I
I I I
• consequent the second or "then" part of a conditional statement. constant of an equation the term that has no variable in an equation; example: "0
I •
I I
1 (v - va) 2 x - Xo = vot + - - - t 2 t 1
1
x - Xo = Vat + -tit - -vat 2 2 1 1 2 2 1 x - Xo = -(tJo + tI)t 2
x - Xa = .,..-vat + -vt I
II =====.===MsJthmuJnes
II ~consts~~n~tffl,~te~oJi~'ch~'Q;~~~e~Iconti~~·n~Uf,~·ty=~ • constant rate of change set .of data or table of values in which ~e amount of the dependent vanable changes by a constant (fixed) value as the value of the independent variable changes by a constant value. • construct create a figure using only a straight edge and compass. • construction a precise way of drawing which allows only 2 tools: the straightedge and the compass • continuity correction in using the normal approximation to the binomial probability histogram, one can get more accurate answers by fmding the area under the normal curve corresponding to half-integers, transformed to standard units. This is clearest if we are seeking the chance of a particular number of successes. For example, suppose we seek to approximate the chance of 10 successes in 25 independent trials, each with probability p = 40% of success. The number of successes in this scenario has a binomial distribution with parameters n = 25 and p = 40%.
35
~ The expected number of suc-
; cesses is np = 10, and the stan~ dard error is (np( I-p» lh = 6lh . = 2.45. If we consider the area ; under the normal curve at the : point 10 successes transformed ~ to standard units,' we get zero: ~ the area under a point is always ; zero. We get a better approxi: mation by considering 10 suc~ cesses to be the range from 9 ; 1/2 to 101/2 successes. The only : possible number of successes ~ between 91/2 and 10 1/2 is 10, ; so this is exactly right for the : binomial distribution. Because ~ the normal curve is continuous ; and a binomial random variable : is discrete, we need to "smear ~ out" the binomial probability ' over an appropriate range. The ; lower endpoint of the range, 9 : 1/2 successes, is (9.5 10)/2.45 '. = -0.20 standard units. The ~ upper endpoint of the range, 10 : 1/2 successes, is (10.5 10)/2.45 ~ = +0.20 standard units. The ; area under the normal curve : between -0.20 and +0.20 is ~ about 15.8%. The true bino; mial probability IS : 25CI0x (0.4)10x (0.6)15 = ~ 16%. In a similar way, if we ' seek the normal approximation : to the probability that a bino-
,
36
cuntinUIIIIS tlistriburion I cuntmaitm I
II
..
mial random variable is • the ~ for any possible ~a!ue of ~e range from i successes to k suc- ; cumulative probab~ty there IS cesses, inclusive, we should find ~ an exact; c~n:espon~ value of the area under the normal curve : the statIStiC m question. from i-I/2 to k+ 1/2 successes, I • continuous variable transformed to standar~ .units. ~ a quantitative variable. is conIf we seek the probability of : tinuous if its set of pOSSible valmore than i successes and fewer ~ ues is uncountable. Examples than k successes, we should find ; inclUde temperature, exact the area under ¢e normal curve : height exact age (including corresponding to the range i + 1/ ~ parts of a second). In practice, 2 to k-I/2 successes, trans- lone can never measure a conformed to standard units. If we : tinuous variable to infinite preseek the probability of more ~ cision so continuous variables than i but no more than k suc- ; are so~etimes approximated by cesses, we should find the area: discrete variables. A random , I . unde.r the normal curve corre- : variable X is also called conttnusponding to the range i + 1/2 to I ous if its set of possible values k+ 1/2 successes, transformed ~ is uncountable, and the chance to standard units. If we se~~ the : that it takes any particular value probability of at least I but ~ is zero (in symbols, ifP(X = x) fewer than k successes, we ; = 0 for every real number x). should fmd the area under the : A random variable is continunormal curve corresponding to ~ ous if and only if its cumulative the range i-I/2 to k-I/2 suc- ; probability distributi?n function cesses, transformed to standard : is a continuous funCtIon (a funcunits. Including or excluding the ~ tion ,with no jumps). • half-integer ranges at the ends I of the interval in this manner is : • contrac~on . . called the continuity correction. ~ the oppo~lte of dilatlo~ a ~. ure resulttng from multlplymg • continuous distribution ~ all dimensions of a given figure a probability distribution of a ; by a number betwec:n zero and continuous statistic, based upon ~ one. an algebraic formula, such that : I
37 II contm,positi1le I cunwnien&e sample ================*================
- contrapositive if p and q are two logical propositions, then the contrapositive of the proposition (p IMPLIES q) is the proposition «NOT q) IMPLIES (NOT p) ). The contrapositive is logically equivalent to the original proposition. _ control for a variable to control for a variable is to try to separate its effect from the treatment effect, so it will not confound with the treatment. There are many methods that try to control for variables. Some are based on matching individuals between treatment and control; others use assumptions about the nature of the effects of the variables to ~ to model the effect mathematically, for example, using regressl0n. - control group the subjects in a controlled experiment who do not receive the treatment. -control there are at least three senses of "control" in statistics: a member of the control group, to whom no treatment is given;
~
a controlled experiment, and to ; control for a possible confound: ing variable. I
: - controlled experiment ~ an experiment that uses the ; method of comparison to evalu: ate the effect of a treatment by ~ comparing treated subjects I with a control group, who do ; not receive the treatment. :I - controlled, randomised • : expenment I a controlled experiment in ~ which the assignment of sub: jects to the treatment group or ~ control group is done at ran; dom, for example, by tossing a : com. I
: - convenience sample ~ a sample drawn because of its ; convenience; not a probability : sample. For example, I might ~ take a sample of opinions in ; Delhi (where I live) by just ask: ing my 10 nearest neighbors. I : That would be a sample of conI venience, and would be unlikely : to be representative of all of ~ Delhi. Samples of convenience ~ are not typically representative, ; and it is not typically possible : to quantify how unrepresentaI
MMIIenIaeics==--====____.11
38
..
~~~~WD~e,!!!!!·c~O'1¥I1i~m!!!!!~!!!!!mee~!!!!!1coo!!!!!~"~rJt~~na~te II.
tive results based on samples of ~ 2. if P and q are two logical i propositions, then the converse convenience will be. of the proposition (p IMPLIES -~~-=.:=.:~ q) is the proposition (q IM~ i i PLIES p). v.,...,
T_.
:
_of
__ of
_.....
.:..
1>..
. c.: a.
..... ."" .....
~,-#
>
I
1.8UJIa!i1C9
COl
-'Ii>!
00.
. . ,,'
.,.,
-
i-conversion factor : relationship between two umts ~ from different systems of meai surement used to convert from ; . one system ~o the other (e.g., : 2.~4 centimeters corresponds to ~ 1 mch). i_convex polygon : a polygon having no diagonal ~ lying outside the polygon.
--!
•
. ':" ~ : : : - ::
_001
_ converge convergence a sequence ~f numbers xl, x2, x3 . . . converges if there is a number x such that for any number E>O, there is a number k (which can depend on E) such that Ixj xl < E whenever j > k. I . . . . - convex If such a number x eXISts, It 18 f set hich all egcalled the limit of the sequence ~ a set 0 pomts om w fSth 1 x2 3 i ments connectmg pomts 0 e x, , x set lie entirely in the set; There - convergence in probability ~ are three things one can do to a sequence of random variables i see if a figure is convex look for Xl, X2, X3 converges in : "dents", extend the segments probability if there is a random ~ (they shouldn't enter the figvariable X such that for any jure), and connect any two number E>O, the sequence of : points within the figure with a numbers P( IXI XI < e), P( 1X2 ~ segment (if any part of the segXI < e), P( IX3 XI < e), . I ment lies outside the figure, it's converges to 100%. : concave). 0
0
0
0
0
0
•
0
0
:
0
0
0
0
I
- converse 1. (of a conditional statement), the statement formed by exchanging the "if" and "then" parts 0 f an "if.th" - en statement
: _ coordinate ~ a number that identifies (or I helps to identify) a point on a : number line (or on a plane, or . space) :I m I
39 ~ vector may be thought of as - coordinate geometry the study of geometrically rep- ; coordinates in a geometric resenting ordered pairs of num- : sense. I bers : - coordinatised. line ~ a line on which every point is ; identified with exactly 1 num: her and vice versa; a one-di~ mensional graph. The distance I between 2 parts on a : coordinatised line is the abso~ lute value of the difference of I their coordinates. - coordinate plane a plane in which every point is I _ coplanar identified with exactly 1 num- ~ lying in the same plane. ber and vice versa; a two-di- ; - coprime mensional graph : integers m and n are coprime if _ coordinate proof ~ gcd(m,n) =1. a proof using coordinate geom- ~ _ corollary to a theorem ; a theorem that is easily proved etry. _ coordinate system : from the first set of ordered pairs used to 10- I p cate an object or point on the I two-dimensional plane.
- coordinates a set of parameters that locate a point in some geometrical I space. Cartesian coordinates, for instance, locate a point on a plane or in physical space by I projection onto perpendicular . lines through one single point, :I - correI abon the origin. The elements of any I relation between two or more ; variables. Frequently the word
M/lthemR,riu==--======
II
correlation moment correlation which is I a measure of linear association the covariance divided by the between two (ordered) lists. product of the standard ~ Two variables can be strongly deviations,rxy=Sxy / Sx.Sy- This ; correlated without having any correlation coefficient IS + 1 or causal relationship, and two -1 when all values fall on a variables can have a causal restraight line, not parallel to ei- I lations hip and yet be ther axis. However, there are uncorrelated. also Kendall, Spearman, • corresponding angles tetrachoric, etc. correlations I if two parallel lines are cut by a which measure other aspects of ~ transversal, corresponding die relation between two vari- angles are translations of each ables. , other along the transversal . is used for Pearson's producr-
I •
• correlation coefficient • cosine the correlation coefficif!nt r is I (of an acute angle) The ratio of a measure of how nearly a the length of the adjacent side scatterplot falls on a straight I to the length of the hypotenuse line. The correlation coeffi- I in any right triangle containing cient is always between -1 and I the angle. + 1. To compute the correla- . tion coefficient of a list of ; • coterminal angles pairs of measurements (X,Y), : two angles that have the same first transform X and Y indi- .~" terminal side vidually into standard units. "I • countable set Multiply correspopding ele- ; a set is countable if its elements ments of the transformed : can be,put in one~to-one correpairs to get a single list of ~ spondence with a subset of the numbers. The correlation co- ; integers. For eXaniple the sets efficient is the mean of. that {O~ 7 -3} {red, gre~n, blue} list of , , "'-2 :1" 0" 1 2 ... }', . products. This page .~. { ... contams a .too~ tha~ lets y?U I {straiWtt, curly}, and the set of generate" bIVarIate data WIth : all fractions are countable. If a . YQU .I set is not countable ' · coeffiICIent any correIanon . , it is unwant.
II counterclockwise I criticalveUue
41
~~~======~*==~===========
countable. The set of all real ~ given transformation has only numbers is uncountable. lone direction of covariants, but : a full plane (four landmarks) or • counterclockwise in orientation, the direction in ~ hyperplane (five or more landwhich points are named when I marks) of invariants.
· travelling on the line, the in-' if terior of the figure is on the left side.
• counterexample a situation in a conditional for which the antecedent is true, but the conditional is false; aka contradiction • counting techniques a variety of methods used to determine the total possible outcomes, typically in a probability situation, including the multiplication principle, trees and lists. • covariant a covariant of a particular shape change is a shape variable whose gradient vector as a function of changes in any complete set of shape coordinates lies precisely along the change in question. For transformations of triangles, the relation between invariants and covariants is a rotation by 90 degrees in the shape-coordinate plane. For more than three landmarks, a
C
