New Definition Of Complexity Research Articles

The measure of complexity in charged celestial bodies in f(R,T,RμνTμν) gravity

In this paper, we investigate irregularities in a cylindrical self-gravitating system which contains the properties of an imperfect matter and electromagnetic field. For $f(R,T,Q)$ theory, in which $R$ represents the Ricci scalar and $T$ shows the trace of matter stress-energy tensor while $Q\equiv R_{\gamma\delta}T^{\gamma\delta}$, the field equations containing electric charge, mass functions and Darmois junction conditions at the hypersurface are examined. We have adopted new definition of complexity introduced by Herrera \cite{herrera2018new}, generalized it for the static charged cylindrically symmetric case in $f(R,T,Q)$ theory by performing a detailed analysis on the orthogonal splitting of the Riemann curvature tensor. One of the effective scalars, $Y_{TF}$, has been recognized as a complexity factor. This factor is comprised of certain physical components of the fluid such as irregularity in energy density, locally pressure anisotropy and electric charge (arranged in a specific way). In addition, the effects of extra curvature terms of modified gravity are examined by making the relations among the complexity factor, Weyl scalar and Tolman mass.

Complexity factor for static anisotropic self-gravitating source in f(R) gravity

In a recent paper, Herrera (Phys Rev D 97:044010, 2018) have proposed a new definition of complexity for static self-gravitating fluid in general relativity. In the present article, we implement this definition of complexity for static self-gravitating fluid to case of f(R) gravity. Here, we found that in the frame of f(R) gravity the definition of complexity proposed by Herrera, entirely based on the quantity known as complexity factor which appears in the orthogonal splitting of the curvature tensor. It has been observed that fluid spheres possessing homogenous energy density profile and isotropic pressure are capable to diminish their the complexity factor. We are interested to see the effects of f(R) term on complexity factor of the self-gravitating object. The gravitating source with inhomogeneous energy density and anisotropic pressure have maximum value of complexity. Further, such fluids may have zero complexity factor if the effects of inhomogeneity in energy density and anisotropic pressure cancel the effects of each other in the presence of f(R) dark source term. Also, we have found some interior exact solutions of modified f(R) field equations satisfying complexity criterium and some applications of this newly concept to the study of structure of compact objects are discussed in detail. It is interesting to note that previous results about the complexity for static self-gravitating fluid in general relativity can be recovered from our analysis if f(R)=R, which general relativistic limit of f(R) gravity.

New definition of complexity for self-gravitating fluid distributions: The spherically symmetric, static case

We put forward a new definition of complexity, for static and spherically symmetric self--gravitating systems, based on a quantity, hereafter referred to as complexity factor, that appears in the orthogonal splitting of the Riemann tensor, in the context of general relativity. We start by assuming that the homogeneous (in the energy density) fluid, with isotropic pressure is endowed with minimal complexity. For this kind of fluid distribution, the value of complexity factor is zero. So, the rationale behind our proposal for the definition of complexity factor stems from the fact that it measures the departure, in the value of the active gravitational mass (Tolman mass), with respect to its value for a zero complexity system. Such departure is produced by a specific combination of energy density inhomogeneity and pressure anisotropy. Thus, zero complexity factor may also be found in self--gravitating systems with inhomogeneous energy density and anisotropic pressure, provided the effects of these two factors, on the complexity factor, cancel each other. Some exact interior solutions to the Einstein equations satisfying the zero complexity criterium are found, and prospective applications of this newly defined concept, to the study of the structure and evolution of compact objects, are discussed.

A new definition of complexity in a risk analysis setting

In this paper, we discuss the concept of complexity in a risk analysis context. Inspired by the work of Johansen and Rausand, a new perspective on complexity is presented which includes several common definitions of complexity as special cases. The idea is to link complexity to activities, to the knowledge about the activity and its consequences (including scenarios, events, outcomes), and to the knowledge about the sub-activities: an activity is considered complex if we have poor knowledge about the consequences of the activity, even if we have strong knowledge about the consequences of its sub-activities. In the paper, we show how this perspective provides new insights about the complexity concept and related risk understanding, risk assessment and risk management. The perspective allows in particular for novel ways of characterizing the risk of complex systems and activities. Some examples are used to illustrate the perspective and the discussion.

A NEW CLASSIFICATION OF NON-OVERLAPPING SYMMETRIC BINARY FRACTAL TREES USING EPSILON HULLS

This paper presents an analysis of non-overlapping symmetric binary fractal trees using methods of computational topology. In particular, we study the topological aspects of the epsilon-hulls of the trees as epsilon ranges over the non-negative real numbers. The complexity of a tree is defined in terms of the maximum range of the number of levels of holes that can be present for a given ∊. All self-contacting trees have infinite complexity, with the important exceptions of the self-contacting trees with branching angles 90° and 135° (which are space-filling). These two angles have been identified by Mandelbrot and Frame as being topologically critical, and our analysis provides further support to that claim. We will show that for trees with branching angles 90° or 135°, there is a finite upper bound to complexity. For all other branching angles there is no upper bound to complexity. Self-avoiding trees are topologically equivalent because they are all contractible. Our new definition of complexity provides a new way to compare self-avoiding trees.

On meta complexity of propositional formulas and propositional proofs

A new approach to defining complexity of propositional formulas and proofs is suggested. Instead of measuring the size of these syntactical structures in the propositional language, the article suggests to define the complexity by the size of external descriptions of such constructions. The main result is a lower bound on proof complexity with respect to this new definition of complexity.

Response to Disturbance and Abundance of Final State: aMeasure for Complexity?

We propose a new definition of complexity. The definition shows thatwhen a system evolves to a final state via a transient state, itscomplexity depends on the abundance of both the final state andtransient state. The abundance of the transient state may be describedby the diversity of the response to disturbance. We hope that thisdefinition can describe a clear boundary between simple systems andcomplex systems by showing that all the simple systems have zerocomplexity, and all the complex systems have positive complexity. Someexamples of the complexity calculations are presented, which supportsour hope.

Structural thinking: Organizing complexity through disciplined activity

In today's world, large-scale systems are frequently involved in levels of complexity that have a serious and global impact on productivity. A context model is set forth that facilitates a new definition of complexity, providing a background against which a science of complexity can be developed, and describing an empirical process from which the complexity of any particular situation can be quantified through a situation complexity index. In moving toward a science of complexity, the desirability of incorporating semiotics as a component of the science is indicated, because of the contributions semiotics makes to understanding the foundations and ubiquity of modelling. In the development of models, as semiotics indicates, the connection between what is being modelled and the language of description in the model is critical. Structural analysis clarifies the inappropriateness of models comprised only of prose to convey a description of a complex situation. Illustrative examples from the practice of interactive management (a system of management that supports the development and interpretation of structural models of complex situations, and design of improved systems) show the significance of structural thinking as the primary intellectual mode required to manage or cope with complexity. Group activity that would otherwise be invalidated by spreadthink is converted into a powerful approach to in-depth learning about a complex situation, which then provides a well-supported foundation for an organized attack to bring a complex situation under control.

Structural Thinking: Organizing Complexity Through Disciplined Activity

In today's world, large-scale systems are frequently involved in levels of complexity that have a serious and global impact on productivity. A context model is set forth that facilitates a new definition of complexity, providing a background against which a science of complexity can be developed, and describing an empirical process from which the complexity of any particular situation can be quantified through a situation complexity index. In moving toward a science of complexity, the desirability of incorporating semiotics as a component of the science is indicated, because of the contributions semiotics makes to understanding the foundations and ubiquity of modelling. In the development of models, as semiotics indicates, the connection between what is being modelled and the language of description in the model is critical. Structural analysis clarifies the inappropriateness of models comprised only of prose to convey a description of a complex situation. Illustrative examples from the practice of interactive management (a system of management that supports the development and interpretation of structural models of complex situations, and design of improved systems) show the significance of structural thinking as the primary intellectual mode required to manage or cope with complexity. Group activity that would otherwise be invalidated by spreadthink is converted into a powerful approach to in-depth learning about a complex situation, which then provides a well-supported foundation for an organized attack to bring a complex situation under control.

MEASURING COMPLEXITY: A RELATIVE ASPECT OF PATTERN RECOGNITION

We propose a new definition of complexity. The degree of complexity depends on the observer’s knowledge and is expressed by templates and their interactions. Complexity of a pattern is measured in a procedure of tiling the pattern with the templates. A new relation of complexity vs. entropy is obtained wherein entropy becomes a function of complexity.