PROOF #1: Regular matrix multiplication: [A]ij[B]jk = [C]ik . [A]Tij o [B]jk = Cjk First I will do a micro-proof for simplicity, then a regular proof. A11 A21 A31 B11 B12 B13 A12 A22 A32 o B21 B22 B23 = A13 A23 A33 B31 B32 B33 A14 A24 A34 B41 B42 B43 Separating each column in the [A]T matrix into a column matrix, we cross multiply and get the following Nested Array: A11 B11 B12 B13 A21 B11 B12 B13 A31 B11 B12 B13 A12 o B21 B22 B23 A22 o B21 B22 B23 A32 o B21 B22 B23 A13 B31 B32 B33 A23 B31 B32 B33 A33 B31 B32 B33 A14 B41 B42 B43 A24 B41 B42 B43 A34 B41 B42 B43 We take the first column in [A]T and multiply it straight across the [B]jk matrix. This operation is equivalent to diagonalizing the first column of [A]ij and multiplying across [B]. i.e. This is equal to: But this is a Nested Array, and I will show later that when we transpose a Nested Array, only the matrices are transposed and not their elements. Since in Half-Multiplying we originally transposed the Spreadsheet Matrix, we must now re-transpose the nested array back in it¡¦s un-transposed state. Then we will remove the inner brackets of the Nested Array using Partitioning and put the sub-Matrices into a single array. Now we will sum the columns for each individual matrix: ( we must remember that we are working with nested arrays, but computers are not programmed to handle these yet, so we must set up the pre-multiplier so that it operates on the individual Sub-Matrices) The answer to this is quite long on MathCad, but since we are working with Nested Arrays, we may also look at the multiplication in this manner: But this is equal to: micro-QED CLASSICAL PHYSICS C([A]Tij o [B]jk)= [C]jk Let c = m, and [B]jk = [a]jk (acceleration) and [C]jk = Fjk, then [C]jk = m([A]Tij o [B]jk ) but [A]ij = [I]jj and [B]jk = [a]jk and [C]jk = Fjk, so Fjk = m([I]jj o [a]jk) for m = to a constant. Fjk = m([a]jk) If [A]ij „j [I]jj , then we have Fjk = m([A]Tij o [a]jk ) EINSTEIN¡¦S FIELD EQUATION FOR GRAVITATION C([A]Tij o [B]jk)= [C]jk Let [C]jk = [G]jk , [B]jk = [T]ij and [A]ij = [I]jj and c = 8ƒàƒâ. Then we have: 8ƒàƒâ([A]Tij o[T]jk) = [G]jk but [A]ij = [I]jj so we have 8ƒàƒâ[T]jk = [G]jk which is Einstein¡¦s First Field Equation. This is the equation as we now understand it, but the true equation is 8ƒàƒâ([A]Tij o[T]jk) = [G]jk. GRAVITATIONAL WAVE EQUATION: [G]jk = 8ƒàƒâ ƒÉT[T]jkƒÉ 8ƒàƒâ ƒÉT([A]Tij o [T]jk)ƒÉ or [G]jk = ƒÉT ƒÉ ƒÉT ƒÉ STATISTICS There was, until now, no field equation describing the field of statistics. Going back to the Unified Field Equation: C([A]Tij o [B]jk)= [C]jk and for the simple case, letting [B]jk = [I]jk and c = 1/N, the equation becomes: 1/N([A]Tij o [B]jk )= [C]jk =1/N([A]Tij o [I]jk)= 1/N([A]Tij) = [C]ij Since [A][I] is a straight multiplication problem, we do not need to transpose it. But we do need to sum the columns of [A], so the basic statistical equation becomes: 1/N([1]1,I[A]ij) = [C]1j But to make this statistical, we must square the above expression so that it becomes: 1/N([1]1,i[A]ij)2 = [C]21j Where [C]21j = CCT. (this gives us a one by one matrix as a solution). Suppose [B]jk is not = to [I]jk. Then we have our basic complex statistical field equation, the simplest form of which is the Analysis of Variance. 1/N([1]1,i[A]ij[B]jk)2 = [C]21,k where [C]2 = CCTequation But we also need to subtract the correction factor(s), so the total statistical field equation becomes: 1/N([1]1,i[A]ij[B]jk)2 - correction factor(s) = C21,k - correction factor(s). There are actually two basic equations, they are 1/N([1]1,i[A]ij[B]jk)2 = [C]21k Between Subjects equation 1/N([A]ij[B]jk {1]k,i} 2 = [C]2k1 Within Subject There are only 3, perhaps 4 operators from which statistics (perhaps all of statistics) may be computed: 1/2„¸([A]Tij o [B]jk = i Cjk matrices (Half-Multiplier mode) ( where i = #rows in un-transposed matrix) C„¸ 1/N([A]Tij o [B]jk ) = 1/N[C]ik regular matrix multiplication. R„¸ 1/N([A]Tij o [B]jk ) = 1/N([B]jk[1]k,1o[A]Tij) M„¸ 1/N([A]Tij o [B]jk ) = 1/N([A]Tij[1]i,1o[B]jk) QUANTUM STATISTICS: [A]TMN[A]MN = [A]NM[A]MN = [A]2NN Then we do the following: 1/N[DB1]iN [A]2NN[DB1]TiN 1/N[DB1]iN [A]2NN[DB1]Ni 1/N[DB1]iN [A]2NN[DB1]Ni = = = [DB1]iN[DB1]TiN [DB1]iN[DB1]Ni [DB1]2ii 1/N[C]2ii [DB1]2ii QUANTUM MECHANICS FROM EINSTEIN¡¦S EQUATION: 8ƒàƒâ ƒÉT[T]jkƒÉ [G]jk = ƒÉT ƒÉ c ƒÉTjk[H]jkƒÉjk cƒÉT([A]Tij o [H]jk)ƒÉ Ejk or [E]jk = ƒÉT ƒÉ ƒÉT ƒÉ INVENTORY/ACCOUNTING SYSTEM: [A]Tij o [B]jk = [C]jk = [INV]ij[DB]jk = [SOL]ik This is the sum total of everything bought, sold, manufactured, etc. i[A]Tij o [B]jk = i[C]jk = i[INV]Tij o [DB]jk = i[AP]jk This keeps and individual item accounting of everything bought, sold, manufactured, etc. The superscript i just tells us wich row in the un-transposed matrix we have hollow-dotted onto the [DB] matrix. j[B]jk o [A]Tij = j[C]jk = j[DB]jk o [INV]Tij = j[IP]jk This takes an individual item from the database matrix (a column) and multiplies it across the inventory(or accounting page if we wish) giving us a slice of the total pie, so to speak. It tells us how much of that item was used, by whom or what machine or smokestack, and shows how it is distributed throughout the total inventory. [A]Tij o [B]jk = i, [C]jk sub-matrices . [INV]Tij o [DB]jk = i, [SOL]jk sub-matrices This is the pure half-multiplier operation giving all the accountpage matrices, i of them, in a single operation.

This is a handbook for scientists,engineers and those intersted inmathematics and its applications.It presents a brief history ofmathematics followed by a summaryof basic undergraduate methods,techniques and formulas togetherwith numerous fi gures and examples.Within the text is a summary ofthe basic formulas from algebra,trigonometry, plane geometry, solidgeometry, calculus, vector calculus,ordinary differential equations,probability and statistics along withdiagrams and illustrations showinghow formulas are to be applied. Thisis followed by a chapter on specialfunctions which arise in a variety ofmathematics applications. The fi nalchapter is a collection of selectedapplied mathematics applicationstaken from the subject areas ofmechanics, physics and chemistry.There are three appendices. Theappendix A contains units ofmeasurement from the SystèmeInternational d’ Unitès (designatedSI in all Languages). The appendix Bcontains tables of integrals for bothindefi nite and defi nite integrals, withover 850 integrals presented. Theappendix C contains miscellaneoustopics that students entering the fi eldsof science or engineering should beaware of. There is also an extensiveindex to aid in fi nding informationabout a specifi c topic. •
•
This handbook presents asummary of selected mathematicstopics from college/university level mathematics courses.Fundamental principles are reviewed and presented byway of examples, fi gures, tables and diagrams. It condensesand presents under one cover basic concepts from severaldifferent applied mathematics topics. The following topicsare examined:•
History of mathematics

Geometry

Algebra and Trigonometry

Calculus and Vector Calculus

Ordinary Differential Equations

Probability and Statistics

Selected topics from Physics and Chemistry

Selected applied mathematics topics

Numerous applications of mathematics

Special functions

Numerical Methods

Table of integrals

Syst`eme International d’Unit`es

This is a handbook for scientists,engineers and those intersted inmathematics and its applications.It presents a brief history ofmathematics followed by a summaryof basic undergraduate methods,techniques and formulas togetherwith numerous fi gures and examples.Within the text is a summary ofthe basic formulas from algebra,trigonometry, plane geometry, solidgeometry, calculus, vector calculus,ordinary differential equations,probability and statistics along withdiagrams and illustrations showinghow formulas are to be applied. Thisis followed by a chapter on specialfunctions which arise in a variety ofmathematics applications. The fi nalchapter is a collection of selectedapplied mathematics applicationstaken from the subject areas ofmechanics, physics and chemistry.There are three appendices. Theappendix A contains units ofmeasurement from the SystèmeInternational d’ Unitès (designatedSI in all Languages). The appendix Bcontains tables of integrals for bothindefi nite and defi nite integrals, withover 850 integrals presented. Theappendix C contains miscellaneoustopics that students entering the fi eldsof science or engineering should beaware of. There is also an extensiveindex to aid in fi nding informationabout a specifi c topic. •
•
This handbook presents asummary of selected mathematicstopics from college/university level mathematics courses.Fundamental principles are reviewed and presented byway of examples, fi gures, tables and diagrams. It condensesand presents under one cover basic concepts from severaldifferent applied mathematics topics. The following topicsare examined:•
History of mathematics

Geometry

Algebra and Trigonometry

Calculus and Vector Calculus

Ordinary Differential Equations

Probability and Statistics

Selected topics from Physics and Chemistry

Selected applied mathematics topics

Numerous applications of mathematics

Special functions

Numerical Methods

Table of integrals

Syst`eme International d’Unit`es

This quick reference guide is suitable for students in middle school, grades 6–8. The book will serve as a true companion by providing quick access to the metric system, basic algebra, geometry, exponents, probability, mathematical formulas, intro to statistics, and much more. This is truly the ultimate companion for students who wishes to excel in math. The Personal Math Companion will have all the answers to the tough math questions that are frequently asked by students in the sixth, seventh, or eighth grades. Do you want to know how to find the third angle of a triangle? How to use and understand a number line for better assistance when handling real numbers? Classifying the different types of polygons? Well, it has all these answers and more, plus helpful tips that will help your children to understand the different concepts of mathematics!

This quick reference guide is suitable for students in middle school, grades 6–8. The book will serve as a true companion by providing quick access to the metric system, basic algebra, geometry, exponents, probability, mathematical formulas, intro to statistics, and much more. This is truly the ultimate companion for students who wishes to excel in math. The Personal Math Companion will have all the answers to the tough math questions that are frequently asked by students in the sixth, seventh, or eighth grades. Do you want to know how to find the third angle of a triangle? How to use and understand a number line for better assistance when handling real numbers? Classifying the different types of polygons? Well, it has all these answers and more, plus helpful tips that will help your children to understand the different concepts of mathematics!

This quick reference guide is suitable for students in middle school, grades 6–8. The book will serve as a true companion by providing quick access to the metric system, basic algebra, geometry, exponents, probability, mathematical formulas, intro to statistics, and much more. This is truly the ultimate companion for students who wishes to excel in math. The Personal Math Companion will have all the answers to the tough math questions that are frequently asked by students in the sixth, seventh, or eighth grades. Do you want to know how to find the third angle of a triangle? How to use and understand a number line for better assistance when handling real numbers? Classifying the different types of polygons? Well, it has all these answers and more, plus helpful tips that will help your children to understand the different concepts of mathematics!