Graph symmetry detection, similarity, and identity measures have been extensively studied in graph automorphism and isomorphism problems. Nevertheless, graph isomorphism and automorphism detection remain an open (unsolved) problem for many decades. In this paper, a new optinalytic coefficient termed as an optical moment coefficient was introduced for optinalysis. Its characteristic efficiency was tested for bijective property, invariance, deterministic polynomiality and non-polynomiality. The test results show that the new optical moment coefficient is very efficient for symmetry detections, similarity and identity measures between two isometric isomorphs and automorphs; and deterministic on polynomial and non-polynomial graph models.

Measures of graph symmetry, similarity, and identity have been extensively studied in graph automorphism and isomorphism detection problems. Nevertheless, graph isomorphism detection remains an open (unsolved) problem for many decades. In this paper, a new and efficient methodological paradigm, called optinalysis, is proposed for symmetry detections, similarity, and identity measures between isometric isomorphs or automorphs. Optinalysis is explained and expressed in clearly stated definitions and prove theorems, which conform to the definitions and theorems of isometry, isomorphism, and automorphism. Analogous to the polynomiality formalization for an efficient algorithm for graph isomorphism detection, optinalysis is however deterministic on polynomial and non-polynomial graph models.

This paper describes a polynomial time algorithm for solving graph isomorphism and automorphism. We introduce a new tree data structure called Walk Length Tree. We show that such tree can be both constructed and compared with another in polynomial time. We prove that graph isomorphism and automorphism can be solved in polynomial time using Walk Length Trees.

This paper will explore analogs of groups formed by the transformations of two ropes as presented by Minh-Tam Quang Trinh.1 The two operations acting on these ropes are T and S, defined as twists and clockwise turns, respectively. These actions intertwine the ropes, creating tangles. T and S also have the ability to untangle the rope when used in certain combinations. In the two-rope case, these actions expressed as a group presentation were shown to be isomorphic to SL₂(Z). We explored the analog of these opera- tions, T and S, and conjectured the corresponding group to which they are isomorphic. Partial proofs for this conjecture are included and future work would include proving our conjecture as well as generalizing results to k ropes.

Zero-Knowledge Proofs ZKP provide a reliable option to verify that a claim is true without giving detailed information other than the answer. A classical example is provided by the ZKP based in the Graph Isomorphism problem (GI), where a prover must convince the verifier that he knows an isomorphism between two isomorphic graphs without publishing the bijection. We design a novel ZKP exploiting the NP-hard problem of finding the algebraic ideal of a multivariate polynomial set, and consequently resistant to quantum computer attacks. Since this polynomial set is obtained considering instances of GI, we guarantee that the protocol is at least as secure as the GI based protocol.

Collatz conjecture (3x+1 problem) is a natural phenomenon in set theory that may be reconstructed using known combinatorics and order theory. Construction begins by selecting a specific order isomorphism with a bijectional order-embedding. Mapping by 3x+1 induces a unique property to only members of the mapped embedding. Under selective congruence from recursive division by two, that complex controls sequence behavior. This demonstration uses an order isomorphism consisting of two linear orders: the (1) odd positive integers and the always odd (2) Rule 50 Jacobsthal numbers, as the embedding. The argument proceeds by cardinality. When the order isomorphism is mapped under 3x+1, all Rule 50 Jacobsthal numbers are mapped to all the powers of four. This one-to-one correspondence guarantees that every odd integer is assigned to a Rule 50 Jacobsthal number, and subsequently, to a power of four. To align this construction with the conjecture, expand the set to all positive integers by showing that: (1) the powers of four embed the powers of two, in a natural way, and (2) the unique factorization of any positive even integer, not congruent to a power of two, is simply an odd integer, once the factorized power of two is divided out. In other words, if the initial choice for a positive integer is not congruent to a power of two, then recursion continues until a Rule 50 Jacobsthal number (guaranteed by cardinality) is attained. Since this value mapped by 3x+1 is always a power of four, repeated division by two will always send the sequence to one. Because this same process occurs for any choice of positive integer, Collatz conjecture is true.

The PLM concept implies the use of heterogeneous information resources at different stages of the product life cycle, the joint work of which allows you to effectively solve the problems of product quality and various costs. According to the principle of isomorphism of regularities of complex systems, an effective PLM system must have these regularities. Unfortunately, this principle is not often fundamental when designing PLM systems. The purpose of the work is to show, using a simple example, the principles of development, operation and use of an educational multi-agent PLM system, the main purpose of which is to study and research these regularities in the life cycle of a special threaded connection. The multi-agent approach to the development of a PLM system provides the necessary prerequisites for the emergence of system-wide regularities in it. The parallel work of agents is implemented using the actor model and the Ray Python-package. Agents for the logical inference of knowledge base facts, CAD/FEA/CAM/SCADA agents, agents for optimization by various methods, and other agents have been developed. Open source software was used to develop the system. Each agent has relatively simple behavior, implemented by its rule function, and can interact with other agents. The system can work in interactive mode with the user or in automatic mode according to a simple algorithm: the rule functions of all agents are executed until at least one of them returns a value other than None. Examples of the operation of the system are given and such system-wide regularities as emergence, historicity and self-organization are demonstrated in it.

In this article, we investigate limitations of importing methods based on algorithmic information theory from monoplex networks into multidimensional networks (such as multilayer networks) that have a large number of extra dimensions (i.e., aspects). In the worst-case scenario, it has been previously shown that node-aligned multidimensional networks with non-uniform multidimensional spaces can display exponentially larger algorithmic information (or lossless compressibility) distortions with respect to their isomorphic monoplex networks, so that these distortions grow at least linearly with the number of extra dimensions. In the present article, we demonstrate that node-unaligned multidimensional networks, either with uniform or non-uniform multidimensional spaces, can also display exponentially larger algorithmic information distortions with respect to their isomorphic monoplex networks. However, unlike the node-aligned non-uniform case studied in previous work, these distortions in the node-unaligned case grow at least exponentially with the number of extra dimensions. On the other hand, for node-aligned multidimensional networks with uniform multidimensional spaces, we demonstrate that any distortion can only grow up to a logarithmic order of the number of extra dimensions. Thus, these results establish that isomorphisms between finite multidimensional networks and finite monoplex networks do not preserve algorithmic information in general and highlight that the algorithmic information of the multidimensional space itself needs to be taken into account in multidimensional network complexity analysis.

We define degree of dependence of two events A and B in a probability space by using Boltzmann-Shannon entropy function of an appropriate probability distribution produced by these events and depending on one parameter (the probability of intersection of A and B) varying within a closed interval I. The entropy function attains its global maximum when the events A and B are independent. The important particular case of discrete uniform probability space motivates this definition in the following way. The entropy function has a minimum at the left endpoint of I exactly when one of the events and the complement of the other are connected with the relation of inclusion (maximal negative dependence). It has a minimum at the right endpoint of I exactly when one of these events is included in the other (maximal positive dependence). Moreover, the deviation of the entropy from its maximum is equal to average information that carries one of the binary trials defined by A and B with respect to the other. As a consequence, the degree of dependence of A and B can be expressed in terms of information theory and is invariant with respect to the choice of unit of information. Using this formalism, we describe completely the screening tests and their reliability, measure efficacy of a vaccination, the impact of some events from the financial markets to other events, etc. A link is available for downloading an Excel program which calculates the degree of dependence of two events in a sample space with equally likely outcomes.