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# ANALYSIS OF ALGORITHMS CSCE 750

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This 5 page Class Notes was uploaded by Trace Mante MD on Monday October 26, 2015. The Class Notes belongs to CSCE 750 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 23 views. For similar materials see /class/229598/csce-750-university-of-south-carolina-columbia in Computer Science and Engineering at University of South Carolina - Columbia.

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Date Created: 10/26/15

A Primer for Communicating Mathematics Through Email You will need to communicate your quiz answers to me via text based email Your answer may include one or more mathematical expressions There are informal conventions for communicating mathematics via normal US ASCII text These conventions are used throughout the research community7 and are loosely based on the LaTeX typesetting system with some programming language constructs thrown in This brief primer will explain these conventions7 allowing you to communicate math via email quickly and easily These conventions have evolved informally over the years and obey no written hard and fast rules They are meant for human to human communication Some sloppynesscreativity is tolerated7 and even encouraged7 if the existing conventions are inadequate Easy Constructs Easiest are expressions involving Roman letters7 the arithmetic oper ations of addition7 subtraction7 multiplication7 and division7 as well as equals7 less than7 greater than7 and parentheses These expressions can be written practally verbatim For ex ample7 2X 677 can be written as 2X6 or 2X 6 or 2 X 6 Multiplication is usually just juxtaposition with optional whitespace in between you could write 2X67 but you needn7t For another example7 z lt x2y 7 4w77 is also writable verbatim as z lt X2y 4 w Here are some other expressions that can be written entirely verbatim o absolute value of m o n 7 2 n 7 2 factorial o x gt 0 l x is not an integer the set of all real positive x that are not integers o x z 1 let x equal 1 o xyz gt 0 x7 y and z are all positive 0A z j the 2397jth entry of the array A o my the pair Ly write the angle brackets using lt and gt7 thus getting ltXygt o logx lt siny expz z transcendental functions can usually be written verbatim7 but extra parentheses may be needed to clarify the grouping The nonstrict inequalities S and 2 can be written as lt and gt7 respectively7 but 1e or leq are also possible for g and ge or geq for 2 Thus7 z 2 0 can be written as either X gt 0 or X ge O The set membership relation 6 is usually written as in Thus7 z E Y becomes X in Y Here is a table of standard mathematical symbols and possibilities for their ASCII text equivalents Note that many symbols set relations7 for example can be written in more than one way I prefer the rst of each alternative rendition Symbol Possible renditions ltleleq gtgegeq neneq congtocongcongruenc equiv in notin subsetofsubsetsubseteq notsubsetofnot subset ofetc supersetofsupersetsupseteq unioncup intersectintersectioncap emptysetnullset orvee andwedge notneg gt gt mapsto gtimplies ltgtiff forallfor eachetc existsthere isetc infinityinfty floor X the largest integer S m ceilingx the smallest integer 2 m sqrtx X vectorvectorxvecx 8mlt llligtlt63cwmmmmmwwwm 8282 4 2 Standard Number Systems Number systems with standard notation include N natural numbers Z integers Q rational numbers R real numbers C complex numbers and H quaternions You may use these letters verbatim provided it is clear what you mean For example if you use Z for the integers don7t use the letter Z to mean anything else in the message To be sure to avoid confusion you can just write out a set long hand ie naturalsintegersrationalsrealscomplexesquaternions For example 71 E N n gt 2 and n is prime Q n E N n is odd may be written as n in N ngt2 and n is prime subsetof n in N n is odd Greek Letters Write Greek letters out long hand Thus 04 is written alpha etc Lower case Greek letters are written all lower case Upper case Greek letters such as P are written with initial capitals eg Gamma For example sin0 2P cosa 7T can be written as sin theta 2 Gamma cosalpha pi Superscripts and Subscripts Use the character uppercase 6 to signify that the next thing is a superscript Thus X 2 rneans x2 If there is more than one token in the superscript such as 2 then you should surround the entire superscript with some kind of delimiter Curly brackets braces are often used Thus 2 is written as 2 n1 instead of 2 n1 the latter means 2 1 Subscripts are similar but use the underscore character instead Thus x2 is written as X2 The delirniter requirements are as with superscripts thus AM is written as Aij You may combine both superscripts and subscripts after the same object but it may help clarity to insert extra parentheses Thus could be written as Xi 2 but it is clearer to say Xi 2 instead Some people dont bother with the underscore for simple subscripts saying X2 to mean 2 for example This is a ghastly practice in my opinion Sums Products Integrals Etc To render a sum like write Sumi0 n 1 2 1 2 n 1 For products use prod or product instead of sum Note that what appears below the surn rnation sign is written as if it were a subscript and what appears above is like a superscript Don7t say Sigma for a sum sum77 is clearer You can do the same with other surn like con structs For de nite integrals use integral you could use int but it may cause confusion for pararneterized unions use union or UNION etc Some more examples 00 2 67m Zdz 00 as integral infty infty e X 22 dX Hz39 S nlSlk 13965 0 Write 0 Write as sumS subsetof n Sk prodi in S i 0 Write lim1 em as limn gtinfty 1 xn n e x You may use limit77 instead of lim7 0 Write b Va b30 a g c g b mm fcb 7 1 as forall altb exists C altcltb and integrala b fxdx fc b a Binomial Coe icients The binomial coef cient 71717 1dnik1 1 is the number of different ways of choosing k things from a set of 71 things without regard to order There are several other ways of expressing this value mathematically Cmk7 Bn7 k7 Ck Bk etc I prefer the notation in Equation 17 which is commonly used in the United States7 and is pronounced7 n choose k In text7 render this as 11 Choose k or 11 Choose k or some such The Choose77 operator has pretty low precedence even lower that plus or minus7 so if you say7 eg7 n k Choose 11 k l7ll interpret it as rather than7 say7 n 7 k Fractions Fractions can be a bit problematic You could render a complicated fraction like 6n2a2 b2 i CZ2 1 sinz y using ASCII art e n2 a 2 b 2 C 2 2 1 sqrtn sin 2 gamma The problem with this is that it may show up crooked if your recipient7s email reader uses a font that is pitched differently from yours This is especially true with a series of several fractions in the same expressionithe crookedness compounds so that the fractions on the right of the expression may be unrecognizable Although it can look run on7 the lesser of two evils is just to put the numerator and denominator on the same line7 separated by the division operator the forward slash Surround the numerator and denominator each with delimiters if there7s any doubt about what constitutes the fraction The fraction above could then be written7 e n2a 2 b 2 6 2 2 1 sqrtn sin 2 gamma Exercises Do these exercises for your own practice I don7t require you to submit them Render the following mathematical expressions into ASCII text 1 f log 2 login 3 6M 71 4 fA B 5 1n CT z y 5 f96 fy 7 P0 a Pn a VnPn 8 p 1 V on n V x4 9 V8 gt 07 n5 9lgn 10 Xil IgT UXlET 13961 11 n i Z jg 1239 P A m B PrA l B W 13 n 22 nkH 14 n z gt zk 15 00 cm 2 g I the symbol C is the Greek letter zeta

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