Atomistic Mathematical Theory for Metaheuristic Structures of Global Optimization Algorithms in Evolutionary Machine Learning for Power Systems

Global Optimization in the 4D nonlinear landscape generates kinds and types of particles, waves and extremals of power sets and singletons. In this chapter these are demonstrated for ranges of optimal problem-solving solution algorithms. Here, onts, particles, or atoms, of the ontological blueprint are generated inherently from the fractional optimization algorithms in Metaheuristic structures of computational evolutionary development. These stigmergetics are applicable to incremental machine learning regimes for computational power generation and relay, and information management systems.


Introduction
The evolution of Algorithms from a simple route, to complexified paths requires maps from zones of optimal utilization, to be solved sufficiently, in a given amount of time.
These algorithms are constructed for the purpose of building and advancing a continuity for the next location of optimal utilization, in order to realize the importance to form workable nodes and circuits, that are discrete and exact algorithm criteria in a time-basis. Therefore, a complete network on a nonlinear surface and related machine learning epochs is built.
These criteria are based on Fermat's Theorem proving global extrema locations either at stationary or bounding points, based ultimately upon the Pythagorean Theorem, where: Let N be the set of natural numbers 1, 2, 3, … , let Z be the set of integers 0, AE1, AE2, … , and let Q be the set of rational numbers a/b, where a and b are in Z with b 6 ¼ 0. In what follows we will call a solution to x n + y n = z n where one or more of x, y,orz is zero a trivial solution. A solution where all three are non-zero will be called a non-trivial solution [1].

Algorithm definition
The Metaheuristic Algorithm is defined by the Author [Jonah Lissner] as. An ontological mechanism to generate or activate decision paths [algorithms] and make decision potential to solve [essentially two-state] paradoxes, a computational physical network and topos, for practical effort or application. Therefore can be constructed a theoretical or hypothetical guideway from objects and particles to advance ontological gradations of relevance and value, through a logical progression.
A relevant algorithm to solve for discrete stigmergetics in nonlinear optimization challenges for graphing algorithms of power systems has been demonstrated in Ant Colony Optimization [ACO]: Here in general formula where It is a basis for optimization schedules that there is an asymmetrical velocity, mass and gravity of said scope of systems. At various times in the computational history, particle optimization on the manifolds evolve at a faster rate [or slower rate] than before. Hence the given incremental and discrete rate of increase, in valleys and peaks accelerates and stabilizes at a higher positive, null or negative value and result in extremal mechanics and nonlinear dynamics. An example can be demonstrated utilizing power faults and extremals on the electrical circuits [3].
These problems of prediction for probability of choice of one object or particle of a set, for pariwise sets and in algorithms, have been demonstrated in Arrow's Impossibility Theorem and for Algorithmic Information Theory [AIT] whence we can replace voter for global optimization particle and replace group with set: 1. If every voter prefers alternative X over alternative Y, then the group prefers X over Y.

Building the algorithm
In a praexological theory [5] this is proposed because of the inherent general inaccuracy of specific problems, learning rubrics, and Macrodynamic properties of a given performance landscape, and ultimately inefficient of any algorithmic system, given isomorphic [atomistic or non-atomistic] qualities of rulebase, algorithmic structure, weights, and variables [6]. These in turn can be represented as information sets, materiel, work, and symbolic representation and/or power in specific qualia of Historical Rule of Perpetuation of Information Inequalities set to various scales and models.
Clerc has demonstrated a general Metaheuristic algorithm where for f:  n !  essentially f(a) ≤ f(b). S includes the number of particles in the swarm having specific position and velocity in the search-space: for each particle i = 1, … ,Sdo Initialize the particle's position with a uniformly distributed random vector: xi $ U(blo, bup) Initialize the particle's best known position to its initial position: pi xi if f(pi) < f(g) then update the swarm's best known position: g pi Initialize the particle's velocity: vi $ U(À|bup-blo|, |bup-blo|) while a termination criterion is not met do: for each particle i = for system conditions, system boundaries, number and density of particles in the total Information Natural Dynamics [IND] of the Global Optimization Algorithm [GOA]. These are applied to algorithmic manifold for the candidate solution on the given search spaces. It can be argued that given the extremes of information disequilibrium applied to macrodynamic disequilibrium models, there are inevitably generated extremals of various degrees of power, in the incremental Information Dynamics.

Complex adaptive evolutionary system: weighting
These differentiable functions can be further defined c.f. Dense heterarchy in Complex Systems Algorithms of a coupled oscillators, where in general formula.
Here in a differential equation we can demonstrate.
These can be demonstrated in Particle Swarm Optimization [PSO], and Macrodynamic models of Meta-optimization of Particle Swarm Optimization [PSO] [7], c.f.
þ n 2Á r 2Á p best À x i t ðÞ ÀÁ (12) for each set of given epoch or evolutionary landscape scenario prediction in analytical and expectation weighting parameter formula algorithm optimization [Meissner, et al., ibid].

Complex adaptive evolutionary system: thermodynamics
Regarding bounding definitions, Chaitin demonstrated in Algorithmic Information Theory [AIT] algorithmic decomposition given Boltzmann-Shannon entropy, where in general formula to set the integral.
indicate the possibilites and types of information physical mechanics for possible variables of the landscape extremals as particles within the min-max parameters [8]. A particle-discrete control function of the node degrees on the evolutionary landscape can therefore be defined where essentially ] which can be fractional off the prime polynomial root modulos, from the initial power conditions and therefore generate the discrete information inequalities. These can be demonstrated in Hensellian numbers, and secondly, derivable fractional functions, inherent in any given complex topos of an complex adaptive evolutionary system [9]. Nagata defined thusly: A local ring R with maximal ideal m is called Henselian if Hensel's lemma holds. This means that if P is a monic polynomial in R[x], then any factorization of its image P in (R/m)[x] into a product of coprime monic polynomials can be lifted to a factorization in R[x] [10].

Complex adaptive evolutionary system: networks
The network of circuits then form the basis for Complex Network Systems [CNS] from Simple networks L ∝ log N and adaptive complex or dynamic systems and increasingly complex or quantum probability mechanics. where.

Conclusion
These Metaheuristics for Global Optimization Algorithms [GOA] are for purpose of achievement of the theoretical completion between two and more nodes on the network landscape, and ultimately the given requirements for the applied electrical grid. This theory can be utilized to derive, add, multiply, subtract, or divide units designated as necessary to accurately define the parameters for control of the electrical grid, and for control of network extremals. Some theoretical requirements for Power System applications and machine learning algorithm libraries for solving heuristic challenge for power requirements and control on manifolds have been demonstrated: