1. Regular Data Array/Lattice “Holographic” Principle
GENERALIZED (N-D) CROSS FOOTING
Algorithmic Cross Footing
Regular Data Array/Lattice “Holographic” Principle
THE START
325 Goldstein AC PVL
&
416A WP GC
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

2. Analogy to Holography: Objects in a volume of space
Regular Data Array/Lattice “Holographic” Principle
Analogy to Holography:
Objects in a volume of space
are mapped 1-1 onto
a lower dimensional (2-D) Screen.
Proves N-D Cross-footing
is Possible.
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

3. The holographic principle is a mathematical principle that
Regular Data Array/Lattice “Holographic” Principle
Also, this is in analogy to the Holographic Principle
The holographic principle is a mathematical principle that
the total information contained in a volume of space corresponds
to an equal amount of information contained on the boundary
of that space. This dependence of information on surface area,
rather than volume, is one of the key principles of black hole
thermodynamics.
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

4. Regular Data Array/Lattice “Holographic” Principle
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

5. Regular Data Array/Lattice “Holographic” Principle
An Array is the dual of a Lattice which is the dual of the Array.
They have identical structure.
Galactic Outline
Shape List Definition of a Regular Array (Lattice) (RAL).
RALSACP (based on the shape list)
“Regular Array Lattice Structure Characteristic Polynomial”:
Counts all full-span Sub RALs of a given RAL.
1898 High School Algebra Formula
Apply 3. to 2. & get result.
Notice the relationship to cross footing.
Algorithm for generating sub RALs
Algorithm for placing sub RAL in the semi shell
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

6. Regular Data Array/Lattice “Holographic” Principle
1. & 2.
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

7. Regular Data Array/Lattice “Holographic” Principle
RALSCAP = Regular Array/Lattice Structure Associated
Characteristic Polynomial, which, for an N-D RAL
Counts ALL Full Span sub RALs by Dimension (j)
Derivation of the RALSACP
Frank, Ronald I. (2002) “Regular Array Expansions in Null Arrays with Applications," Dissertation Pace University. ProQuest Digital Dissertations (AAT )
[A heuristic-based long tedious algebraic derivation of the RALSACP.]
Frank, Ronald I. (2003). A Generating Function that Counts the Combinatorial Full-Span Sub Array Structure of a Regular Array. Proceedings of the ACM SIGAPL Federated Computer Research Conference (October 2003). ISBN
[Still heuristic-based but a shortened derivation of the RALSACP.]
Frank, Ronald I. (2003). An Algorithm to Compute All Full-span Sub Arrays of a Regular Array. Proceedings of the ACM APL (SIGAPL) Federated Computer Research Conference (October 2003). ISBN [An executable computation based on the heuristic.]
Frank, Ronald I. (2008) A Computing Research Example: Following a Question. In The Proceedings of the Conference on Information Systems Applied Research 2008, V 1 (Phoenix): §
[Related results but not the Holographic Principle.]
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

8. Regular Data Array/Lattice “Holographic” Principle
Algorithm for COUNTING sub RALs.
Algorithm to Count sub arrays.
Set dimension j = 0.
Set N dimension counters D[k]=0 for k e [0…N]
Chose a j-element subset of the Shape List. [see next slide]
Form the product of the complimentary Shape List Elements.
Sum to the D[j] counter.
Repeat to 3 so long as there is a new j-element subset of the Shape List.
Increase j by 1 so long as (N ³j) else exit.
Repeat to 3.
The D[k] are the coefficients of the RALSACP.
D[k] is the number of k-Dimension sub RALs.
There will be a total of j-element subsets.
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

9. Regular Data Array/Lattice “Holographic” Principle
Algorithm for Generating j-Element Subsets
Computational trick for generating the j-element subsets o f N shapes numbers.
The N-bit binary number can represent 2N bit configurations.
All possible combinations of N bits will be generated
Generate all 2N configurations by forming the N-bit binary numbers 0 – (2N-1)
Sort the 2N configurations by number of bits
The number of j-bit numbers is
Use the j-bit numbers as masks to extract the j-Element subsets of the Shape List
Or use the (N-j)-bit numbers as masks to get the complimentary sets
Recall that
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

10. Regular Data Array/Lattice “Holographic” Principle
Reminder
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

11. Regular Data Array/Lattice “Holographic” Principle
Our basis is the combinatorial formula from 1898:
Chrystal, G., Textbook of Algebra.
Vol. II., 6th Ed. Chelsea Publishing Co., NY 1952.
[Multi-variable 'binomial'-like combinatorial expansion on page 21.]
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

12. Regular Data Array/Lattice “Holographic” Principle
Galactic Outline
1898 High School Algebra Formula
4. Apply 3. to 2. & get result.
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

13. Total cells in expanded Array semi shell [k=0, to N-1.] Total cells
Regular Data Array/Lattice “Holographic” Principle
Total cells
in expanded
array.
Total cells in expanded Array semi shell [k=0, to N-1.]
Total cells
in original
array.
4. Apply 3. to 2. & get result.
All
non (0-D)
sub arrays
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

14. Regular Data Array/Lattice “Holographic” Principle
The 1898 left hand side gives the count of all of the cells of the
Expanded-by-1 RAL The right hand side (RHS)
gives the counts of the sub RALS of the given RAL. The last
term gives the count of the cells of the given RAL (j=0, k=N).
These are common to both RALs.
Therefore, the terms (j = 1 to N) (k = N-1 to 0) count the cells
of the semi shell of the expanded RAL. But the terms
(j = 1 to N) (k = N-1 to 0) also count all of the non-(0-D)
sub RALS of the given RAL. QED.
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

15. Example: This is generalized cross footing.
Regular Data Array/Lattice “Holographic” Principle
Notice the relationship to cross footing.
Since every sub array maps 1-1 onto a semi shell cell, if it contains data, we can sum over the cells of any sub array and insert the sum in the corresponding semi shell cell.
Example:
Sum every [j-D] sub array in a [(j+1)-D] sub array into their cells in the semi shell. Sum all of these semi shell cells ( = S1). E.g. j=1 vectors in a j=2 plane.
Sum all cells in the [(j+1)-D] sub array and put it into its semi shell cell ( = S2). E.g., the plane.
The two values are equal (S1 = S2) since the cells used are the same. This is cross footing.
This is generalized cross footing.
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

16. Regular Data Array/Lattice “Holographic” Principle
Algorithm for generating sub RALs.
Algorithm to Generate ALL sub arrays [j e 0...N]:
From the Shape List, create the general Index List.
E.g., {2, 3, 4, 2}®[i, k, m, n].
i e (1-2), k e (1-3), m e (1-4), and n e (1-2).
Choose a (j-element) sub set of the general index list. E.g., [i, 3, m. 2]
Odometer the complementary (N-j) index set. [i, k, m, n]
For each odometered value set of the complementary index list, odometer the j indices. This generates 1 (j-D) array.
Therefore, the product of the lengths of the complementary index set is the total number of (j-D) sub RALs the chosen (j-element) sub set can generate.
Do 2 & 3 for all j-element sub sets. This generates ALL (j-element) sub arrays.
Do 2-4 and generate all sub arrays (j=0 … N).
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

17. Regular Data Array/Lattice “Holographic” Principle
Algorithm placing a sub RAL into the semi shell.
For any index set defining sub arrays, extend its defining indices to the semi shell (maximize them).
Set the complimentary index values to their general indices for odometering.
Odometer the complimentary set to define the set of semi shell cells.
Example for {2, 3, 4, 2} ® {3, 4, 5, 3}:
[i, 3, 4, n] ® [3, k, m, 3] which defines 12 (kxm) semi shell cells mapping the planes defined by i & n.
Null case example for {2, 3, 4, 2} ® {3, 4, 5, 3}: [-, -, -, -] ®[3,4,5, 3] which places the single (4-D) array in the remoter corner of the 3rd brick.
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

18. Regular Data Array/Lattice “Holographic” Principle
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

19. Regular Data Array/Lattice “Holographic” Principle
RALSCAP = Regular Array/Lattice Structure Associated
Characteristic Polynomial
Counts ALL Full Span sub RALs by Dimension
2-D Example {2, 3}
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

20. Regular Data Array/Lattice “Holographic” Principle
2-D Example {2, 3} ® {3, 4}
1898 High School Algebra Formula
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

21. Regular Data Array/Lattice “Holographic” Principle
Engulfing Semi shell
Also Models Cross-footing.
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

22. Regular Data Array/Lattice “Holographic” Principle
3-D Example {3, 3, 2} ® {4, 4, 3}
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

23. Regular Data Array/Lattice “Holographic” Principle
3-D Example {3, 3, 2} ® {4, 4, 3}
1898 High School Algebra Formula
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

24. Regular Data Array/Lattice “Holographic” Principle
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

25. {3,3,2} ® {4,4,3} Cross-Footing Example:
Plane 1
Plane 2
Plane 3 1 1 1
Clear is the count of a plane = 8.
Light Blue is the one sum over all data items; the 3-D array.
There are 18 gray cells (here containing values).
Yellow is the row & column counts in a plane.
There are 9 more vectors across planes 1 & 2 = 21 total.
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

26. {3,3,2} ® {4,4,3} Cross-Footing Example:
Plane 1 {4,4,3}
Plane 2 {4,4,3}
Plane 3 {4,4,3} 1 2 3 6 5 4 15 12 10 11 33 4 5 7 16 3 6 13 12 10 18 41 5 7 10 22 9 28 6 8 24 20 30 74
Gray is the original {3,3,2}
Light Blue is the cross-footing in the plane verifying the plane {4,4,3} cross-footing. It is the map of the whole {3,3,2}.
Yellow is the vector sum of {3,3,2} rows & columns.
Clear is the cross-footing in the plane verifying the {3,3} cross-footing.
In plane 1 & 2 yellow shows the vector maps. Clear shows the planar maps. In plane 3, the added plane, yellow shows the cross plane vector maps (sums). Clear is the plane maps.
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

27. {3,3,2} ® {4,4,3} Cross-Footing Example:
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

28. [See the 3-D example above.
{3,3,2} ® {4,4,3} Cross-Footing Example:
Cross-footing is applied on arrays that contain values. We have been studying only structure.
Cross-footing follows the Holographic Principle.
Every Sum is over a sub RAL and is placed into a cell of the expanded array.
Some of the sums are the sum of smaller sub RALs. Thus corresponding to the cross footing of accounting.
[See the 3-D example above.
and a 4-D example below]
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

29. 4-D Example: {2, 3, 4, 2} ® {3, 4, 5, 3} Cells (48)
{2, 3, 4, 2} ® (2x3x4x2)=48
Vectors (76)
Front to Back Vector {2, 3, m, 2}®[i, j, 5, n] (2x3x2)=12
Left to Right Vector {2, j, 4, 2} ®[i, 4, m, n] (2x4x2)=16
Top to Bottom Vector {i, 3, 4, 2} ®[3, j, m, n] (3x4x2)=24
Top & Bottom Vector {2, 3, 4, n) ®[i, j, m, 3] (2x3x4)=24
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

30. 4-D Example: {2, 3, 4, 2} ® {3, 4, 5, 3} Planes (44)
Front to Back Vertical Plane {i, 3, m, 2} ®[3, k, 5, n] (3x2)=6
Left to Right Vertical Plane {i, k, 4, 2} ®[3, 4, m, n] (4x2)=8
Horizontal Planes {2, k, m, 2} ®[i, 4, 5, n] (2x2)=4
Top & Bottom Column/Planes {i, 3, 4, n} ®[3, k, m, 3] (3x4)=12
Top & Bottom Row/Planes {2, k, 4, n} ®[i, 4, m, 3] (2x4)=8
Top & Bottom Planar/Planes {2, 3, m, n} ®[i, k, 5, 3] (2x3)=6
Left to Right Planar-Brick {i, 3, m, n} ®[3, k, 5 3] (3)=3
Top & Bottom Horizontal Planar-Brick {2, k, m, n) ®[i, 4, 5 3] (2)=2
Bricks (11)
Top & Bottom Brick {i, k, m, 2} ®[3, 4, 5, n] (2)=2
Top & Bottom Planar-Brick {i, k, 4, n} ®[3, 4, m, 3] (4)=4
4-D Object (1, 4-D Brick)
{-, -, -, -) ®[3 4, 5 3] (1)
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

31. Consider A Cross-Footing Example: with 5 values per factor has:
Data in a {5, 5, 5, 5, 5} ® {6, 6, 6, 6, 6}
This 5 factor data table
with 5 values per factor has:
D cells
D rows & columns to sum over
D planes to sum over
D cubes to sum over
25 4-D hyper cubes to sum over
1 5-D table to sum over
4651 Sums This is a lot of data.
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Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014

32. Regular Data Array/Lattice “Holographic” Principle
THE END
325 Goldstein AC PVL
&
416A WP GC
GENERALIZED (N-D) CROSS FOOTING
Algorithmic Cross Footing
11/29/2018
Regular Data Array/Lattice “Holographic” Principle V14. © Ronald I. Frank 2014