Unpublished Paper
Movable Band Matrixes in a Circuit Core Model
(2020)
Abstract
A band matrix is a permutation matrix for a band in an odd m adjacency matrix A[m] that has (m-1)/2 bands, B[k,m], where k runs over the odd integers from 1 to m-2. Each B[k,m] has a mXm permutation matrix P[m,k] that represents its non-intersecting simple Hamiltonian circuit components. The latter has the following polynomial of degree m in the variable x:
(1) f[x, k, m] = Det[B[k, m] – x I[m]], where I[m] is the mXm identity matrix.
This polynomial is positive for all x, 0 < x ≤ 1, (0,1]. It shows the net gain to a business enterprise in (2), with U and V positive m-vectors, U for inputs and V for outputs. This interpretation requires nonnegative x in (0,1].
(2) Net Gain = U’ B[k,m] V – Cost,
Revenue is U’ B[k, m] V and Cost is x U’ V. Net gains of the (m-1)/2 enterprises move with the variable x. Net gains can differ among these enterprises; some with losses, some break even and some with profits. These results depend on x as if it were a random variable drawn from a pdf on the interval (0,1]. This random x is a common force that affects all bands as business enterprises.
My essay also discusses even m-polygons that have different properties than those for odd m-polygons. Even m-polygons have Eulerian circuits whose characteristic polynomials are degree n < m and are negative on (0,1]. An Eulerian circuit can use an arrow at most once but can use vertexes many times. Since they do not have partitions, they do not include all the arrows. Excluded arrows measure structural unemployment. This unemployment rate has a cyclical pattern. It behavior over economic growth poses a new problem to the theory of circuit core models. Including even m-polygons with properties markedly different than odd m-polygons adds to circuit-core models for all fair m-polygons.
Keywords
- Circuit Core,
- Band Characteristic Function,
- Eulerian Cycle
Disciplines
Publication Date
September 6, 2020
Citation Information
Circuit Core Models, Business Fluctuations, Structural Unemployment