Periodic matrix set
Encyclopedia
In mathematics
, a periodic matrix set is a set of square matrices in which each square matrix
is of a different size, and such that each cell within each matrix within a set contains data associated with some type of periodic distribution.
Each matrix within a set has an identification number (a) and must contain a "root cell". A root cell must be located at any corner of a matrix. All root cells must be located at the same corner of each matrix within a single set. A diagonal line drawn from a root cell to the opposite corner of the same matrix is a "root diagonal".
The periodicity is defined by "partial square rings" (rings) of cells adjoining a root cell on two sides. All cells within the same ring, (even if they are located in a different matrix) have a similar "period". If a matrix contains (n+1)2 cells then the outermost ring contains "2n+1" cells which are all included in the same period. A ring identification number (n) identifies each period. The root cell is also the smallest ring and is identified as; n = 0. Each subsequent ring (1, 2, 3, etc) has 2n+1 cells (3, 5, 7, etc).
Individual cells contained within a ring are identified by their deviation
from the root diagonal. Each cell within a ring is assigned a deviation number (D). All cells intersected by the root diagonal have; D = 0. All cell locations in a column deviation have positive values of D. All cell locations in a row deviation have negative values of D.
Any cell within a set will require three numbers for the identification of its location;
a is the matrix number
n is the ring number
D is the deviation number
The cell could also have its location identified as;
a is the matrix number
x is the column number (root cell = 0)
y is the row number (root cell = 0)
The two locational systems are analogous to Radial (anD) and Cartesian (axy) systems. Generally this article will use the "anD" locational method.
The contents of any cell must contain data that is periodic in some manner.
Two sets (of compatible construction) may combine so that the root cells on similar sized matrices are adjacent. This is a "set pair" and is identified by a "pair number" (P). The resultant matrices are not square but are 2n x n rectangular.
Four sets may also combine so that all root cells on similar sized matrices are adjacent. This gives a resultant set of square matrices having an even number of cells on each side. All root cells will form a central 2x2 "core" within each resultant matrix. The resultant set is actually two pairs. Each pair forms half of the resultant set. The identifiers (P,S) will tag each quadrant of the resultant set, which is all of the original sets.
P = +½ represents the upper pair
P = -½ represents the lower pair
S = +½ represents the right set of each pair.
S = -½ represents the left set of each pair.
Five identifiers are required to locate any cell in R4 M;
P is the pair number
S is the set number
a is the matrix number
n is the ring number
D is the deviation number
) and particle physics (for example, with sub atomic particles). The resultant set R4 4 is of special interest.
The periodic rings may be associated with quantum harmonic oscillation. A quantum harmonic oscillator has energy (En) defined as; En = (n + ½)ћω. Where; ћ = h/2π and h is Plank's constant, and ω is frequency. The number of cells in each period may be written as; 2En/ћω.
The rings may also be associated with atomic orbitals. If the ring number (n) is equal to the quantum number
for orbital angular momentum (the azimuthal number l ), then the rings (0, 1, 2, 3) correspond to the orbitals (s, p, d, f). The ring number is NOT equal to the principle quantum number (n ). The number of cells per ring is half the number of electrons per orbital due to spin duality of the electrons.
The quantum numbers are;
n is the principal quantum number
l is the quantum number for orbital angular momentum (the azimuthal number)
ml is the orbital magnetic moment
ms is the spin magnetic moment
The spin quantum number (s) is not normally used in chemistry applications as all electrons are; s = ½.
The atomic number (Z) may be expressed as a function of energies which in turn are functions of the quantum numbers.
If a resultant set is R4 4 then the locational numbers correspond to the quantum numbers as follows.
S = ms
n = l
D = ml
The Madelung rule gives the "P" and "a" relationships. This rule may be generalized as follows;
2a - P = n + l + s
(2n + 2l + 2s - 4a)2 = 1
This generalization may also be obtained from J coupling.
If; P = -½
Then; a = ½(n + l)
If; P = +½
Then; a = ½(n + l) + ½
The sub-atomic particles may be grouped as an R4 3 combination.
Z = ZP + ZS + Za + Zn + ZD - ½
The five locator terms are as follows.
ZP = -2a2(P+½)
ZS = -2(n+½)(S+½)
Za = 4a(a+1)(a+½)/3
Zn = -2n(n+½)
ZD = (D+½)
This distribution of atomic numbers in R4 4 is a locationally compliant matrix set of the Periodic Table. The following tables show the resultant matrices populated with the atomic numbers.
R4 4 showing combined matrices 1 to 4 populated with atomic number (Z)
a = 1
a = 2
a = 3
a = 4
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a periodic matrix set is a set of square matrices in which each square matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
is of a different size, and such that each cell within each matrix within a set contains data associated with some type of periodic distribution.
Construction of a set
A set may be specified to contain a fixed number of matrices and is identified by a set number (SM). Where; S is the set identification number and M is the number of matrices included in the set. There is no limit to the number of matrices which may be members of a periodic set.Each matrix within a set has an identification number (a) and must contain a "root cell". A root cell must be located at any corner of a matrix. All root cells must be located at the same corner of each matrix within a single set. A diagonal line drawn from a root cell to the opposite corner of the same matrix is a "root diagonal".
The periodicity is defined by "partial square rings" (rings) of cells adjoining a root cell on two sides. All cells within the same ring, (even if they are located in a different matrix) have a similar "period". If a matrix contains (n+1)2 cells then the outermost ring contains "2n+1" cells which are all included in the same period. A ring identification number (n) identifies each period. The root cell is also the smallest ring and is identified as; n = 0. Each subsequent ring (1, 2, 3, etc) has 2n+1 cells (3, 5, 7, etc).
Individual cells contained within a ring are identified by their deviation
Deviation
Deviation may refer to:* Deviation , the difference between the value of an observation and the mean of the population in mathematics and statistics** Standard deviation, which is based on the square of the difference...
from the root diagonal. Each cell within a ring is assigned a deviation number (D). All cells intersected by the root diagonal have; D = 0. All cell locations in a column deviation have positive values of D. All cell locations in a row deviation have negative values of D.
Any cell within a set will require three numbers for the identification of its location;
a is the matrix number
n is the ring number
D is the deviation number
The cell could also have its location identified as;
a is the matrix number
x is the column number (root cell = 0)
y is the row number (root cell = 0)
The two locational systems are analogous to Radial (anD) and Cartesian (axy) systems. Generally this article will use the "anD" locational method.
The contents of any cell must contain data that is periodic in some manner.
Combined Sets
Combinations of sets are possible, however each set must be conformable for combination. A resultant set (RN M) is the combination of N sets each having M matrices.Two sets (of compatible construction) may combine so that the root cells on similar sized matrices are adjacent. This is a "set pair" and is identified by a "pair number" (P). The resultant matrices are not square but are 2n x n rectangular.
Four sets may also combine so that all root cells on similar sized matrices are adjacent. This gives a resultant set of square matrices having an even number of cells on each side. All root cells will form a central 2x2 "core" within each resultant matrix. The resultant set is actually two pairs. Each pair forms half of the resultant set. The identifiers (P,S) will tag each quadrant of the resultant set, which is all of the original sets.
P = +½ represents the upper pair
P = -½ represents the lower pair
S = +½ represents the right set of each pair.
S = -½ represents the left set of each pair.
Five identifiers are required to locate any cell in R4 M;
P is the pair number
S is the set number
a is the matrix number
n is the ring number
D is the deviation number
Applied sets
Periodic matrix sets have an application to chemistry (for example, in the periodic tablePeriodic table
The periodic table of the chemical elements is a tabular display of the 118 known chemical elements organized by selected properties of their atomic structures. Elements are presented by increasing atomic number, the number of protons in an atom's atomic nucleus...
) and particle physics (for example, with sub atomic particles). The resultant set R4 4 is of special interest.
The periodic rings may be associated with quantum harmonic oscillation. A quantum harmonic oscillator has energy (En) defined as; En = (n + ½)ћω. Where; ћ = h/2π and h is Plank's constant, and ω is frequency. The number of cells in each period may be written as; 2En/ћω.
The rings may also be associated with atomic orbitals. If the ring number (n) is equal to the quantum number
Quantum number
Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
for orbital angular momentum (the azimuthal number l ), then the rings (0, 1, 2, 3) correspond to the orbitals (s, p, d, f). The ring number is NOT equal to the principle quantum number (n ). The number of cells per ring is half the number of electrons per orbital due to spin duality of the electrons.
The quantum numbers are;
n is the principal quantum number
l is the quantum number for orbital angular momentum (the azimuthal number)
ml is the orbital magnetic moment
ms is the spin magnetic moment
The spin quantum number (s) is not normally used in chemistry applications as all electrons are; s = ½.
The atomic number (Z) may be expressed as a function of energies which in turn are functions of the quantum numbers.
If a resultant set is R4 4 then the locational numbers correspond to the quantum numbers as follows.
S = ms
n = l
D = ml
The Madelung rule gives the "P" and "a" relationships. This rule may be generalized as follows;
2a - P = n + l + s
(2n + 2l + 2s - 4a)2 = 1
This generalization may also be obtained from J coupling.
If; P = -½
Then; a = ½(n + l)
If; P = +½
Then; a = ½(n + l) + ½
The sub-atomic particles may be grouped as an R4 3 combination.
Data compliance
A set is considered to be "locationally compliant" if the data contained in each cell is also a function of the location of the cell. Let an R4 4 resultant set be populated with atomic numbers. Each cell contains one atomic number (Z). The number in each cell should be a function of the locators of the cell. If a term is associated with each locator then the atomic number will be the sum of all terms and a constant.Z = ZP + ZS + Za + Zn + ZD - ½
The five locator terms are as follows.
ZP = -2a2(P+½)
ZS = -2(n+½)(S+½)
Za = 4a(a+1)(a+½)/3
Zn = -2n(n+½)
ZD = (D+½)
This distribution of atomic numbers in R4 4 is a locationally compliant matrix set of the Periodic Table. The following tables show the resultant matrices populated with the atomic numbers.
R4 4 showing combined matrices 1 to 4 populated with atomic number (Z)
a = 1
| 2 | 1 |
| 4 | 3 |
a = 2
| 9 | 8 | 5 | 6 |
| 10 | 12 | 11 | 7 |
| 18 | 20 | 19 | 15 |
| 17 | 16 | 13 | 14 |
a = 3
| 28 | 27 | 26 | 21 | 22 | 23 |
| 29 | 35 | 34 | 31 | 32 | 24 |
| 30 | 36 | 38 | 37 | 33 | 25 |
| 48 | 54 | 56 | 55 | 51 | 43 |
| 47 | 53 | 52 | 49 | 50 | 42 |
| 46 | 45 | 44 | 39 | 40 | 41 |
a = 4
| 67 | 66 | 65 | 64 | 57 | 58 | 59 | 60 |
| 68 | 78 | 77 | 76 | 71 | 72 | 73 | 61 |
| 69 | 79 | 85 | 84 | 81 | 82 | 74 | 62 |
| 70 | 80 | 86 | 88 | 87 | 83 | 75 | 63 |
| 102 | 112 | 118 | 120 | 119 | 115 | 107 | 95 |
| 101 | 111 | 117 | 116 | 113 | 114 | 106 | 94 |
| 100 | 110 | 109 | 108 | 103 | 104 | 105 | 93 |
| 99 | 98 | 97 | 96 | 89 | 90 | 91 | 92 |