MATRICES AND GRAPHS
OF FUNDAMENTAL MECHANICS

If a man's wit be wandering, let him study the mathematics.

Francis Bacon

Date and time

Matrix and graph of instantaneous condition of the Lattice were considered above in the article "The Fundamental Lattice".

The historical matrix of the Lattice is "archive" of every chronon updated matrices of the instantaneous condition of the Lattice of infinitely remote past till the last moment:

                    or

                    or    ,

where l, m and n – respectively a coordinate x, y and z of tops of the Lattice, t – time coordinate.
Elements of a matrix are equal either plus 1, or minus 1.
The matrix every chronon extends on one column or for one line – how it is written down.
The matrix has border. It means that the future physically does not exist.
The historical graph of the Lattice is isomorphic to a four-dimensional hyper cubic lattice with edge with directional edges. The edges symbolizing fudls have the positive or negative direction, and all edges symbolizing chronons – only positive. Time is unidirectional. And, all "chronon" edges between two next "fudl" layers symbolize the same chronon. Absolute time flows equally everywhere.

Matrix of potential oscillations of elements of the Lattice:

                    or

                 .

0 means lack of attempt of a spontaneous oscillation at the corresponding fudl in this chronon here, and 1 – its existence. Spontaneous oscillations are potential. Actual oscillations are formed of fudls, spontaneous as a result of interactions.

Matrix of actual oscillations of elements of the Lattice:

                    or

                 .

The history of actual oscillations can be traced on a historical matrix.

Matrix of potential interactions of elements of the Lattice:

                 .

uik=0 means that i-th an element of the Lattice immediately does not influence k-th elemt Lattices. For nonadjacent elements of the Lattice always uik=0.
uik=1 means that i-th an element of the Lattice immediately influences k-th the Lattice element. For interfacing elements of the Lattice equiprobable uik=0 and uik=1 options. If uik=uki=1, it is possible to nullify them as in this case interactions are neutralized.
In a graph of potential interactions fudls are symbolized by its tops, and influence of i-th of an element on k-th – the arrow directed from element i-th to k-th.
In a matrix on the main diagonal there could be units. Then loops would be added to all tops of the count. And as the loop is a cycle, the element is nullified.

Matrix of actual interactions of elements of the Lattice:

                 .

This matrix turns out from a matrix of potential interactions by zeroing of its elements meeting a condition of uik=uki=1 and elements, forming cycles. Really, in these cases of interaction are compensated.
vik=1 and vki=0 means that i-th an element of the Lattice made impact on k-th the Lattice element; that is, if the first element oscillated, it compelled the second element to oscillate and if the first element did not oscillate, it compelled the second element not to oscillate.
Graph of actual interactions we receive from graph of potential interactions by removal of cycles from it.

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