Array Research: A Research Example

Transcription

1 Array Research: A Research Example THE START Pace University 35 Goldstein AC PVL & 416A WP GC pace.edu 1

2 Table of Contents 1/3 TOC 7 (7 Head, 43 text) 3[-4] Research Tree Flow Diagram 3[5-7] Observation & Binomial Theorem 3 [8-1] Scalars & 1-D Vectors 3 [11-13] Outer Products & -D Null 3 [14-16] Master Equation 3[17-] Mystery Solved 3[1-3] Catenation (Try 1) 3[4-5] 5

3 Table of Contents /3 Analysis of Catenation 3 [6-8] The Law of Catenation 3 [9-31] Continuous 3 [3-34] Negative Length & Dual View 4[35-38] Complex (& Imaginary) Length 3 [39-41] Fractional Dimensions 3 [4-44] P(F(A))= F(P(A)) 4[45-48] Given P(A), Find A. (Roots) 3[49-51] 6 3

4 Table of Contents 3/3 Structure Topology Questions 3 [5-54] Catenation and the Big Bang 3[55-57] Alternative Summary [58-59] Appendix I: The Binomial Theorem [6-61] Appendix II: The Binomial Coefficients 3[6-64] Appendix III: Some Coefficients of (1+X) 1/ [65-66] Appendix IV: Detail of Big Bang 4[67-7] 4 19

5 The Tree Flow of the Research 5

6 Discovered Array Structure Characteristic Polynomial Polynomial == Polynomial Form 6

7 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length Topological Structure? 7 Dual View Functions of Polys P[F(A)] = F[P(A)] 8?-6 Develop fractional dimension?-8 Work out the details.?-7 What do the edges of the egg crate mean? The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 VisioDocument 7

8 The Observation, & Binomial Theorem 8

9 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 9

10 Basic Representation Idea N N N nnnn n n ( n) T{,,,,..., } T for short. { nnnn,,,,..., n} is the "Shape Vector" T {} and T {} and T {,, } k ( ) ( ) k ( ) Basic Binomial Theorem Idea N N N Tn ( n) = ( n 1+ 1) = ( n 1) + 1 = n 1 1 k= k N N N N k N k Tn ( n 1 ) Tn 1 all smaller arrays. k= k k= k N ( ) ( ) ( ) N N k N k 1 k

11 All Dimension Scalars Are = & The Canonical 1-D 1-Vector Array 11

12 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 1

13 Derivation of [ ] T = T n n T n n n T n ( ) = ( 1 + 1) = ( 1) (1) (1) n 1(1) n n 1 n n n [ ] T = (1) T (1) = (1) T (1) = = (1) T (1) = T n Derivation of a 1-D, N-vector polynomial form [ ] 1 1 T = ( nt ) + T n n 1 1 T n n n n n ( ) = ( 1 + 1) = ( 1) (1) + ( 1) (1) RECURSION RECURSION 1 1 (1) T n 1(1) + (1) Tn 1(1) = (1) T + Tn 1 = usi ngtn 1 = T n = () T + T = ( n) T + T = = ( n) T + T = T QED n n n n 13

14 Outer Product, Null, {, 3} Example are Made by Outer Product of 1-D k-vectors (The k i are the elements of the shape vector) The Algebra of Null & A {, 3} Array Structure Example 14

15 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 15

16 CREATING A {, 3} ARRAY BY THE OUTER PRODUCT OF VECTORS Polynomial Multiplication T T = ( T + T ) (3 T + T ) = 6T + 5T + T 1 1 1,3 This means 6 Cells (-D Scalars), 5 1-D Vectors, 1 (-D) Thingie. i j ( ) and T ( ) i ( ) ( ) ( ) i j i j i+ j i+ j T T = T [ integer i, j ] This corresponds to null shape vector catenation. {,, },{,, }={,, } 1 i 1 j 1 i+ j j T 1 T T 3 3 T, T T,,

17 The Master Equation The Master Equation 17

18 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 18

19 T N = ( nt 1 + T) ( nt + T) ( n 1T + T) ( n T + T,,,,,, ) { n n n n n n } T N N N N N N N j T { } = T 1,, 3,,, 1, [( n ) ( ) ( ) ( )] n n n nn nn nn i x n 1 i x x n i x n N j 1 in j j= N N j {, 3, 4} Example N Sum over all combinations of N-j factors { } = ( T + T) ( 3 T + T) ( 4 T + T) = ( 4T + 6T + 9 T + T,3,4 ) 4 = x3x4 the coefficient of T ( j = ) 1 6 = x3 + x4 + 3x4 the coefficient of T ( j = 1) The sum of products of 3 things taken (3 = N- =3-) at a time. The sum of product of 3 things taken ( = N-1 = 3-1) at a time. 9 = the coefficient of T ( j = ) The sum of product of 3 things taken (1 = N- = 3-) at a time. 1 = 3 N 1 the coefficient of T ( j = 3) C is defined as 1 The sum of product of 3 things taken ( = N-3 = 3-3) at a time. 19

20 THE MASTER EQUATION Example x3x4 Full Decompostion to Null There are 4 scalar cells. [-D] There are 6 vectors: [1-D] 6 vectors front to back 1 vectors top to bottom 8 vectors right to left (x3x4) "Dual" View There are 9 matrices (planes): [-D] horizontal planes top to bottom 3 vertical planes left to right 4 vertical planes front to back There is one 3-D Entity. [3-D] There are no higher D Entities. [4...-D] T 3 (, 3, 4) = T (4) + T 1 (6) + T (9) + T 3 (1) What is Research Like? V.44. Ronald I. Frank 3, 4, 5, 6, 7, 8 (C)

21 Mystery Solved Mystery Solved 1

22 { }, {1}, {1, 1}, {1, 1, 1} Contents Counts Are The Same = 1 ( ) 1 k 1 T T + for k =,1,, All Structure Polynomials Start With ( ) 1 T The structure of an array is independent of the mapping to the co-domain that determines the cell contents.

23 First Try at Catenation First Try at Catenation 3

24 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 4

25 ( ) ( ) ( ) ( T 3 ) ( 3 ) ( 3 ), 1, 4, T,,4 = T, 3, 4 T ( T + T ) (1 T + T ) (4 T + T ) = 8T + 14T + 7T + T ,1,4 T ( T + T ) ( T + T ) (4 T + T ) = 16T + T + 8T + T ,,4 T ( T + T ) (3 T + T ) (4 T + T ) = 4T + 6T + 9T + T , 3,4 ( ) T 8T + 14T + 7T + T 3 1 3, 1,4 ( ) + T + 16T + T + 8T + T 3 1 3,,4 ( ) T 4T + 34T + 15T + T 3 1 3, 3,4 ( ) T 4T + 6T + 9T + T 3 1 3, 3,4 Difference = T + 8T + 6T + 1T ( ) 1 3 T T + 8T + 6T + 1T 3 1 3,,4 5

26 Analysis of a Catenation Example Analysis of a Catenation Example 6

27 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 VisioDocument?-8 Work out the details. 7

28 ( ) T ( T + T ) ( T + T ) (4 T + T ) = T + 8T + 6T + T ,,4 ( T 3 ) ( 3 ) ( 3 ), 1, 4, T,,4 = T, 3, 4 ( T 3 ),( T 3 ) = ( T 3 ) ( T 3 ), 1, 4,,4, 3, 4,, 4 ( 3 T ),, 4 is the shape of the catenation interface, called the "anti-glue". The anti glue must be subtracted from catenations And added (glued) into decatenations. 8

29 The Law of Catenation The Law of Catenation 9

30 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 3

31 ( T ),( T ) = ( T ) ( T ) 1, 1 1,1,1, 1 + = + ( T 3 ),( T 3 ) = ( T 3 ) ( T 3 ), 1, 4,,4, 3, 4,, 4 The anti glue must be subtracted from catenations And added (glued) into decatenations. 31

32 Continuous Continuous 3

33 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 33

34 x y { Lx, Ly} L = dx & L = dy x y ( ) ( ) T = ( L T + T ) ( L T + T ) = L L T + L + L T + T L, L x y x y x y x x 34

35 Negative Length & the Dual View Negative Length 35

36 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 36

37 N N N N N N k T n ( n) = ( n 1+ 1) = (( n 1) + 1) = ( n 1) ( 1) k k = = N N k k N N N ( n + 1) ( 1) ( 1) or T n ( 1) ( n ) k= k T = ( T + T ) ( 3 T + T ) = 6T 1T + T 1 1 1, 3 37

38 T = ( T + T ) (3 T + T ) = 6T + 1T + T 1 1 1, 3 T = ( T + T ) (3 T + T ) = 6T + 5T + T 1 1 1, 3 ( T + T ),(3 T + T ) = T + T T = 1T + T

39 Complex (or Imaginary) Length Array Complex (or Imaginary) Length 39

40 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 4

41 ( ) ( ) T = a T + T 1 1 a T = ib T + T 1 1 ib ( ) T + = a + ib T + T 1 1 a ib ( ) ( ) a, ib ( ) ( ) T T = a T + T + ib T + T T ( ) ( ) a, ib ( ) ( ) ( ) T T = a + ib T + T = a + ib T + T ( ),( ) T T = T a ib a ib 41

42 Fractional Dimension (The Square Root of a 1-Vector) Fractional Dimension (The Square Root of a 1-Vector) [New results on shape vectors adds to this] 4

43 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 43

44 1 1 ( 1T + T ) ( 1T + T ) ( 1T + T ) = ( 1T + T ) 1 α α α αα ( 1) ( α+ 1 k) k= k k= k k= kk ( 1) (1) k= k= k= 1 1 k k k ( 1T + T ) = ( T ) = ( T ) = ( T ) ( ) ( ) Infinite dimensionality nt + T = n T + T n n T T n T T n T T n T n n n ( ) ( ) + + = + = ( + T ) 44

45 P(F(A)) = F(P(A)) Array Structure of Functions of P(F(A)) = F(P(A)) Array Structure of Functions of [New results on shape vectors adds to this] 45

46 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? P[F(A)] Topological Functions Dual View = Structure? of Polys F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 46

47 α k= k k k α 1 1 α α = 1 1 α α n k= k n k= k n k ( ) n T + T = n T = n T = k= = α αα ( 1) ( α+ 1 k) 1 αα ( 1) ( α+ 1 k) n T n T k= kk ( 1) (1) n k= nkk ( 1) (1) k k α k ( ) = k ( ) PFA ( ( )) = FPA ( ( )) k This is based on the polynomial. There is a parallel but very different definition of fractional dimension array based on the shape vector. 47

48 F N ( T ) 1 N ( n1, n, n3,, nn, nn 1, n ) T N ( n1, n, n3,, nn, nn 1, nn) ) ) N 1 ) N ) N ( ( T ( n ) 1, n, n3,, nn, nn 1, nn) ) = = ( nt + T ( nt + T ( n T + T ( n T + T = ) ) N 1 ) ( nt N T) F P = ( nt + T ( nt + T ( n T + T N ( ( T ) ) 1 N P T = ( n1, n, n3,, n, 1, ) ( 1,, 3,,, 1, )... N nn n P F N n n n nn nn n QED N orfpa ( ( )) = PFA ( ( )) + = ( T T ( T T T T T ( T T T T T ) P( ) ({1,1} ) = ( 1 + ) ( 1 + ) ( 1 + ) ( 1 + ) = ( ) ({1,1} ) = ({1,1} )... {1,1} = 1 + ) 1 + ) = = = {1,1} P T T T T T T T T T T T T T or P P Q E D ( T T (T T T T T ( T T T T T ) P( ) ({1, } ) = ( 1 + ) ( + ) ( 1 + ) ( + ) = ( ) ({1, } ) = ({1, } )... {1, } = 1 + ) + ) = = = {1, } P T T T T T T T T T T T T T or P P Q E D 48

49 Given P(A) Find A. Observation on Defective Given P(A) Find A. Observation on Defective 49

50 b ± b 4ac ( ) = 1+ = = a PA x x ± x = 4(1)(1) (1) x = ± i = ± i so A = T i, i A = 1+ x Observation: "Defective" arrays in ordinary space often have polynomials that lie in more complicated spaces. 5

51 PA ( ) 1 x x x b b 4ac ± = + + = = a 1 1 4(1)(1) 1 3 ± x = = ± = (1) 1 3 x = ± i so A = T i, i A= 1+ x+ x 51

52 Questions About the Meaning of Array Representations The Topology of Array Structure Questions About the Meaning of Array Representations The Topology of Array Structure 5

53 ?-1 Alternate data structure access using hierarchical array structure?- Relationship to alternate big bang??-3 Develop continuous length?-4 Develop negative length Catenation Original Observation (n) N Canonical n-vector Array Shape Characteristic Polynomial Binomial Theorem Expansion in Shorter Length Nulls 1 Outer Product Generic? Apply to OLAP manipulations Fractional Dimensions? Law of Catenation & Anti-glue Continuous 3 Negative Complex Fractional Length Length Dimensions 4 5 6?-5 Develop Complex length?-6 Develop fractional dimension?-7 What do the edges of the egg crate mean? Topological Structure? Dual View Functions of Polys P[F(A)] = F[P(A)] 7 8 The Array Shape Characteristic Polynomial Research Tree and Open Questions Ronald I. Frank 8 ArrayResearchLectureV9.pptx?-8 Work out the details. 53

54 {, 3} Array on the surface of a torus Same Array!! 54

55 Catenation and the Big Bang (Alternative) Catenation and the Big Bang (Alternative) Catenation applied to Steinhardt & Turok idea. 55

56 T, T = T T = { xyzt,,, } { xyzt,,, } { xyzt,,, + t' } { xyz,,,} ( ) xyz ( t + t ') T + ( )( ') ( ) yz + xz + xy t + t + xyz T + ( + + )( + ') + ( + + ) + T x y z t t xy yz xz T ( ) ( ) 4 ' 3 + x+ y+ z + t+ t T 1 56

57 T 4 { xyz,,,} = 1 { x, y, z,} = ( xyz) T + ( yz+ xz+ xy+ xyz) T + ( ) ( ) xy + xz + x + yz + y + z T + x + y + z + T + T 3 4 ( ) ( ) ( ) ( ) = T + xyz T + xy + xz + yz T + x + y + z T + T ( ) T T ( x) T T ( y) T ( ) = + + T + z T + T = T T 1 3 { } { xyz,, } This is a full "instantaneous" 3-D Space. 57

58 Appendix: The Binomial Coefficients SUMMARY 58

59 Summary Example of following two ideas Discover new things Uncover unexpected connections Get new ideas on related connections Don t limit your view to a discipline Keep notebooks Write something every day Use your unconscious Keep at it in spite of the experts (DeBroglie) 59

60 Appendix: The Binomial Coefficients Appendix I The Binomial Theorem 6

61 Binomial Theorem i= N N N N i i ( x+ y) = x y [ N ] i= i x = 1 i= N N N (1 + y) = y i= i x= 1 = y i= N N N N ( x+ y) = () = i= i i 61

62 Appendix: The Binomial Coefficients Appendix II The Binomial Coefficients 6

63 N N N! Ck k k! ( N k)! The combinations of N things taken k at a time. For N k: ( )( ) ( )( ) ( )( ) ( ) N! = N N 1 N 1 ( )( ( )) N! = N N 1 N N N N N 1 (N) terms. ( )( ) ( ) N! = N N 1 N N ( N k) + 1 k! (N-k) initial terms. ( )( ) ( ) ( ) N! = N N 1 N N ( k 1) N k! (k) initial terms. N N N! C = k k k! ( N k)! N( N 1)( N ) ( )( 1) ( 1)( ) ( )( 1 ) ( )(( ) 1 )(( ) ) ( )( 1) k k k N k N k N k ( )( ( )) N( N 1)( N ) N ( N ) N N 1 ( 1)( ) ( )( 1 ) ( )(( ) 1 )(( ) ) ( )( 1) k k k N k N k N k = 63

64 OR N( N 1)( N ) ( N ( N k) + 1 ) k! k( k 1)( k ) ( )( 1 ) ( N k)( ( N k) 1 )(( N k) ) ( )( 1) N( N 1)( N ) ( N ( N k) + 1) = ( N k)( ( N k) 1 )(( N k) ) ( )( 1) N! N( N 1)( N ) ( k 1 N N + ) Ck k = k! ( N k)! ( N k)! OR N( N 1)( N ) ( N ( k 1) ) ( N k)! ( 1)( ) ( )( 1 ) ( )(( k) 1 )(( N k) ) ( )( 1) ( 1)( ) ( ( 1) ) k( k 1)( k ) ( )( 1)! N( N 1)( N ) ( N k 1 N N + ) Ck =!( )! [!] k k k N k N N N N N k = N k k N k k = = 64

65 Appendix: The Binomial Coefficients Appendix III Some coefficients of (1+X) 1/ 65

66 αα ( 1) ( α+ 1 k) αα ( 1) ( α ( k 1)) BC.. = kk ( 1) (1) = kk ( 1) (1) k BC.. α [ ] By definition 1 = α 1 [ 1/ ] By definition = α /(1/ 1) 1 1 = = 3 (1) 8 ( ) 1/( 1/)(1/ ) 1/( 1/)( 3/) 3 1 = = = 4 4 ( 3)( )(1) ( 3)( )(1) ( )( 3) ( ) 1/( 1/)(1/ )(1/ 3) 1/( 1/)( 3/)( 5/) 15 5 = = = ( 3)( )(1) 4 ( 3)( )(1) ( )( 3) ( ) 1/( 1/)( 3/)( 5/)(1/ 4) 1/( 1/)( 3/)( 5/)( 7 / ) ( 1)( 7) 7 = = = 8 8 5( 4)( 3)( )(1) 5( 4)( 3)( )(1) ( ) ( ) 7 (1 / 5) = 8 = = ( ) 6 ( ) ( ) 8 ( ) ( ) ( ) ( )( ) (n 3) n ( BC..) n 1 Where n= 1,, 3... ( n) 7 1 ( 7 3) 7 (11) (7)(11) (7) = ( = ) ( ) 66

67 Appendix: The Binomial Coefficients Appendix IV Details of Big Bang 67

68 {,,, } ( ) ( ) ( ) ( ) x y z t = xyzt T + yzt + xzt + xyt + xyz T + xy + xz + xt + yz + yt + zt T + x + y + z + t T + T {,,, } ( ) ( ) ( ) ( ) x y z t = xyzt T + yzt + xzt + xyt + xyz T + xy + xz + xt + yz + yt + zt T + x + y + z + t T + T ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 1 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 3 4 T { }, T ' ' ' ' =,,, ' ' ' ' ' ' ' {,,, } T xyzt x y z t { x, y, z, t t } T + { x, y, z,} Assuming x=x', y=y', z=z' : conformability. ' ' ' ' { xyzt,,, },{ x, y, z, t} ' ' ' ' ( xyzt) + ( x y z t ) T = ' ' ' ' ' ' ' ' ' ' ' ' 1 ( yzt xzt xyt xyz) ( y z t x z t x y t x y z ) T ( xy xz xt yz y + ) + ( ) ' ' ' ' 3 4 ( x y z t) ( x y z t ) T T ( )( + ') + ( + + )( + ') + ( ) ( ) ( ) ( ) ( ) ' ' ' ' ' ' ' ' ' ' ' 1 + x + y + z ( t + t ') + xy + yz + xz + T ' x y z t t T + T = t zt x y x z xt y z y t z t T = xyz t t T yz xz xy t t xyz T {,,,} ( ) ( ) ( ) ( ) = ( ) T + ( xyz) T + ( xy + xz + yz) T + ( x + y + z) T + T = ( ) T + T ( x) T + T ( y) T + T ( z) T + T x y z = xyz T + yz + xz + xy + xyz T + xy + xz + x + yz + y + z T + x + y + z + T + T

69 ' ' ' ' { xyzt,,, },{ x, y, z, t} = ( ) + ( ) ( ) xyzt x y z t T T ' ' ' ' ( ) ( ) ( ) + yzt + xzt + xyt + xyz + y z t + x z t + x y t + x y z T xyz T ' ' ' ' ' ' ' ' ' ' ' ' 1 1 ( ) ( ) ( ) ' ' ' ' 3 ( x y z t) ( x y z t ) T x y+ z T + xy + xz + xt + yz + yt + zt + x y + x z + xt + y z + y t + z t T xy + xz + yz T ( ) 3 + T T 4 4 ' ' ' ' { xyzt,,, },{ x, y, z, t} ' ' ' ' ( xyzt) + ( x y z t ) T ' ' ' ' ' ' ' ' ' ' ' ( ) ( ) ( ) ( ) ' ' ' ' 3 ( t) ( x y z t ) T + yzt + xzt + xyt + y z t + x z t + x y t + x y z T 4 ' ' ' ' ' ' ' ' ' ' ' ' 1 + xt yt zt x y x z xt y z y t z t T T = ' ' ' ' ' ' ' ' ' ' ' 69

70 T { }, T ' ' ' ' =,,, ' ' ' ' ' ' ' {,,, } T xyzt x y z t { x, y, z, t t } T + { x, y, z,} ( )( + ') + ( + + )( + ') + ( ) + ( + + )( + ') + ( + + ) xyz t t T yz xz xy t t xyz T x y z t t xy yz xz T ( ) ( ) 1 ' x y z t t T + T ( ) ( ) ( ) ( ) T + xyz T + xy + xz + yz T + x + y + z T + T = ( )( ') ( ) ( )( ') ( ) ( ) ( )( ') ( ) ( ) xyz t + t T T yz + xz + xy t + t + xyz T xyz T + x + y + z t + t + xy + yz + xz + T xy + xz + yz T + ( ) ( ) ( ) ' 3 3 x+ y+ z + t+ t T x+ y+ z T + + T T = 4 4 7

71 Array Research: A Research Example THE END 71