Permutation Representations Research Articles

Communication: A benchmark-quality, full-dimensional ab initio potential energy surface for Ar-HOCO

A full-dimensional, global ab initio potential energy surface (PES) for the Ar-HOCO system is presented. The PES consists of a previous intramolecular ab initio PES for HOCO [J. Li, C. Xie, J. Ma, Y. Wang, R. Dawes, D. Xie, J. M. Bowman, and H. Guo, J. Phys. Chem. A 116, 5057 (2012)], plus a new permutationally invariant interaction potential based on fitting 12 432 UCCSD(T)-F12a/aVDZ counterpoise-corrected energies. The latter has a total rms fitting error of about 25 cm−1 for fitted interaction energies up to roughly 12 000 cm−1. Two additional fits are presented. One is a novel very compact permutational invariant representation, which contains terms only involving the Ar-atom distances. The rms fitting error for this fit is 193 cm−1. The other fit is the widely used pairwise one. The pairwise fit to the entire data set has an rms fitting error of 427 cm−1. All of these potentials are used in preliminary classical trajectory calculations of energy transfer with a focus on comparisons with the results using the benchmark potential.

Lorenz, Gödel and Penrose: new perspectives on determinism and causality in fundamental physics

Despite being known for his pioneering work on chaotic unpredictability, the key discovery at the core of meteorologist Ed Lorenz’s work is the link between space-time calculus and state-space fractal geometry. Indeed, properties of Lorenz’s fractal invariant set relate space-time calculus to deep areas of mathematics such as Gödel’s Incompleteness Theorem. Could such properties also provide new perspectives on deep unsolved issues in fundamental physics? Recent developments in cosmology motivate what is referred to as the ‘cosmological invariant set postulate’: that the universe can be considered a deterministic dynamical system evolving on a causal measure-zero fractal invariant set in its state space. Symbolic representations of are constructed explicitly based on permutation representations of quaternions. The resulting ‘invariant set theory’ provides some new perspectives on determinism and causality in fundamental physics. For example, while the cosmological invariant set appears to have a rich enough structure to allow a description of (quantum) probability, its measure-zero character ensures it is sparse enough to prevent invariant set theory being constrained by the Bell inequality (consistent with a partial violation of the so-called measurement independence postulate). The primacy of geometry as embodied in the proposed theory extends the principles underpinning general relativity. As a result, the physical basis for contemporary programmes which apply standard field quantisation to some putative gravitational lagrangian is questioned. Consistent with Penrose’s suggestion of a deterministic but non-computable theory of fundamental physics, an alternative ‘gravitational theory of the quantum’ is proposed based on the geometry of , with new perspectives on the problem of black-hole information loss and potential observational consequences for the dark universe.

Memetic algorithms for optimal task allocation in multi-robot systems for inspection problems with cooperative tasks

Multi-robot task allocation means to distribute and schedule a set of tasks to be accomplished by a group of robots to minimize cost while satisfying operational constraints. It can be challenging to solve a large number of tasks and becomes even more challenging when tightly coupled multi-robot tasks are also taken into account. For example, it is more complex to solve problems that include tasks that have to be carried out jointly by two robots due to the resulting temporal and spatial constraints. Additionally, the complexity of task allocation increases exponentially with rising task variety. This paper focuses on multi-robot task allocation in inspection problems with both single- and two-robot tasks. A novel memetic algorithm is proposed combining a genetic algorithm with two local search schemes. Using permutation representation, eight approaches based on four basic coding strategies are designed for multi-robot task allocation of inspection problems with two-robot tasks. The performance of the memetic algorithm is evaluated in case studies on inspecting a large storage tank area of a petroleum refinery.

On the automaton complexity of wreath powers of non-abelian finite simple groups

We show that for every n⩾5 the infinite permutational wreath power of the alternating group of degree n with its natural permutation representation is topologically generated by a 2-state automaton, answering the question on the existence of a minimal automaton realization for an infinite wreath power of a non-trivial group. We also extend this result to some 2-generated perfect groups. Finally, we show that every non-abelian finite simple group admits a faithful and transitive action on a finite set such that the corresponding wreath power has an almost minimal automaton realization, extending the result from [15].

A note on barely transitive groups

We study some properties for barely transitive group.

The Pauli Exclusion Principle. Can It Be Proved?

The modern state of the Pauli exclusion principle studies is discussed. The Pauli exclusion principle can be considered from two viewpoints. On the one hand, it asserts that particles with half-integer spin (fermions) are described by antisymmetric wave functions, and particles with integer spin (bosons) are described by symmetric wave functions. This is a so-called spin-statistics connection. The reasons why the spin-statistics connection exists are still unknown, see discussion in text. On the other hand, according to the Pauli exclusion principle, the permutation symmetry of the total wave functions can be only of two types: symmetric or antisymmetric, all other types of permutation symmetry are forbidden; although the solutions of the Schrodinger equation may belong to any representation of the permutation group, including the multi-dimensional ones. It is demonstrated that the proofs of the Pauli exclusion principle in some textbooks on quantum mechanics are incorrect and, in general, the indistinguishability principle is insensitive to the permutation symmetry of the wave function and cannot be used as a criterion for the verification of the Pauli exclusion principle. Heuristic arguments are given in favor that the existence in nature only the one-dimensional permutation representations (symmetric and antisymmetric) are not accidental. As follows from the analysis of possible scenarios, the permission of multi-dimensional representations of the permutation group leads to contradictions with the concept of particle identity and their independence. Thus, the prohibition of the degenerate permutation states by the Pauli exclusion principle follows from the general physical assumptions underlying quantum theory.

Quantum mechanics and permutation invariants of finite groups

We study quantum behavior from a constructive "finite" point of view, since the introduction of continuum or other actual infinities into physics poses serious conceptual and technical difficulties without any need for these concepts in physics as an empirical science. Taking this approach, we can show that the quantum-mechanical problems can be formulated in the invariant subspaces of permutation representations of finite groups, while the quantum interferences occur as phenomena that are observable in these subspaces. The scalar products in the invariant subspaces (which are needed for formulating the Born rule – the main postulate of quantum mechanics that links mathematical description with experiment) are linear combinations of independent bilinear invariant forms of the permutation representation. A complete set of such forms for any permutation group can be easily calculated by a simple algorithm. Slightly more sophisticated algorithms are required for expressing quantum observables in terms of these forms.

A simple optimal binary representation of mosaic floorplans and Baxter permutations

Mosaic floorplans are rectangular structures subdivided into smaller rectangular sections and are widely used in VLSI circuit design. Baxter permutations are a set of permutations that have been shown to have a one-to-one correspondence to objects in the Baxter combinatorial family, which includes mosaic floorplans. An important problem in this area is to find short binary string representations of the set of n-block mosaic floorplans and Baxter permutations of length n. The best known representation is the Quarter-State Sequence which uses 4n bits. This paper introduces a simple binary representation of n-block mosaic floorplan using 3n−3 bits. It has been shown that any binary representation of n-block mosaic floorplans must use at least (3n−o(n)) bits. Therefore, the representation presented in this paper is optimal (up to an additive lower order term).

On the group [formula omitted] as the multiplication group of a loop

We prove that the group PSL(n,q) in its natural permutation representation is the multiplication group of a loop if and only if n≥3, (n,q)≠(3,2), and gcd(n,q−1)=1.

The absolute order of a permutation representation of a Coxeter group

A permutation representation of a Coxeter group W naturally defines an absolute order. This family of partial orders (which includes the absolute order on W) is introduced and studied in this paper. Conditions under which the associated rank generating polynomial divides the rank generating polynomial of the absolute order on W are investigated when W is finite. Several examples, including a symmetric group action on perfect matchings, are discussed. As an application, a well-behaved absolute order on the alternating subgroup of W is defined.

Hybrid evolutionary computation methods for quay crane scheduling problems

Quay crane scheduling is one of the most important operations in seaport terminals. The effectiveness of this operation can directly influence the overall performance as well as the competitive advantages of the terminal. This paper develops a new priority-based schedule construction procedure to generate quay crane schedules. From this procedure, two new hybrid evolutionary computation methods based on genetic algorithm (GA) and genetic programming (GP) are developed. The key difference between the two methods is their representations which decide how priorities of tasks are determined. While GA employs a permutation representation to decide the priorities of tasks, GP represents its individuals as a priority function which is used to calculate the priorities of tasks. A local search heuristic is also proposed to improve the quality of solutions obtained by GA and GP. The proposed hybrid evolutionary computation methods are tested on a large set of benchmark instances and the computational results show that they are competitive and efficient as compared to the existing methods. Many new best known solutions for the benchmark instances are discovered by using these methods. In addition, the proposed methods also show their flexibility when applied to generate robust solutions for quay crane scheduling problems under uncertainty. The results show that the obtained robust solutions are better than those obtained from the deterministic inputs.

Solving resource-constrained project scheduling problems: Conceptual validation of FLP formulation and efficient permutation-based ABC computation

In this paper, we propose two alternative approaches, applying the facility layout problem (FLP) concept and integrating the permutation-based artificial bee colony (PABC) algorithm, to effectively tackle the resource-constrained project scheduling problem (RCPSP). In the FLP formulation, the constraints are expressed to design the activities in the space constructed by resource and temporal restrictions, without violating the precedence relationships and overlaps between the activities. For dodging the difficulty of the FLP-based model to treat large-sized instances of NP-hard RCPSP, the permutation representation scheme of the PABC algorithm is in turn introduced utilizing the artificial bee colony (ABC) process to search the best solution for RCPSP. In the procedure, a crossover operator and an insert operator following the update equation of the ABC algorithm are devised to augment the effectiveness of computation, whereas a shift operator subject to the resource utilization ratio value is suggested to diversify the solutions. The makespan is then obtained and improved with the assistance of a serial scheduling scheme and a double justification skill. Subsequently, the computational experiments conducted substantiate the conceptual validity of the proposed facility layout formulation for RCPSP and the comprehensive simulation shows the effectiveness of the PABC algorithm for RCPSP.

On the refined counting of graphs on surfaces

Ribbon graphs embedded on a Riemann surface provide a useful way to describe the double-line Feynman diagrams of large N computations and a variety of other QFT correlator and scattering amplitude calculations, e.g. in MHV rules for scattering amplitudes, as well as in ordinary QED. Their counting is a special case of the counting of bi-partite embedded graphs. We review and extend relevant mathematical literature and present results on the counting of some infinite classes of bi-partite graphs. Permutation groups and representations as well as double cosets and quotients of graphs are useful mathematical tools. The counting results are refined according to data of physical relevance, such as the structure of the vertices, faces and genus of the embedded graph. These counting problems can be expressed in terms of observables in three-dimensional topological field theory with Sd gauge group which gives them a topological membrane interpretation.

Sharply 2-Transitive Linear Groups

A group Γ is sharply 2-transitive if it admits a faithful permutation representation that is transitive and free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. a skew field that is distributive only from the left, see Definition 2) such that ⁠. This is well known in the finite case. We prove this conjecture when Γ<GLn(F) is a linear group, where F is any field with char(F)≠2 and that p−char(Γ)≠2 (see Definition 2.2).

Finite groups and quantum physics

Concepts of quantum theory are considered from the constructive “finite” point of view. The introduction of a continuum or other actual infinities in physics destroys constructiveness without any need for them in describing empirical observations. It is shown that quantum behavior is a natural consequence of symmetries of dynamical systems. The underlying reason is that it is impossible in principle to trace the identity of indistinguishable objects in their evolution—only information about invariant statements and values concerning such objects is available. General mathematical arguments indicate that any quantum dynamics is reducible to a sequence of permutations. Quantum phenomena, such as interference, arise in invariant subspaces of permutation representations of the symmetry group of a dynamical system. Observable quantities can be expressed in terms of permutation invariants. It is shown that nonconstructive number systems, such as complex numbers, are not needed for describing quantum phenomena. It is sufficient to employ cyclotomic numbers—a minimal extension of natural numbers that is appropriate for quantum mechanics. The use of finite groups in physics, which underlies the present approach, has an additional motivation. Numerous experiments and observations in the particle physics suggest the importance of finite groups of relatively small orders in some fundamental processes. The origin of these groups is unclear within the currently accepted theories—in particular, within the Standard Model.

An Aschbacher–OʼNan–Scott theorem for countable linear groups

The purpose of this note is to extend the classical Aschbacher–OʼNan–Scott theorem on finite groups to the class of countable linear groups. This relies on the analysis of primitive actions carried out in Gelander and Glasner (2008) [GG08]. Unlike the situation for finite groups, we show here that the number of primitive actions depends on the type: linear groups of almost simple type admit infinitely (and in fact unaccountably) many primitive actions, while affine and diagonal groups admit only one. The abundance of primitive permutation representations is particularly interesting for rigid groups such as simple and arithmetic ones.

A new encoding scheme-based hybrid algorithm for minimising two-machine flow-shop group scheduling problem

This article investigates the two-machine flow-shop group scheduling problem (GSP) with sequence-dependent setup and removal times, and job transportation times between machines. The objective is to minimise the total completion time. As known, this problem is an NP-hard problem and generalises the typical two-machine GSPs. In this article, a new encoding scheme based on permutation representation is proposed to transform a random job permutation to a feasible permutation for GSPs. The proposed encoding scheme simultaneously determines both the sequence of jobs in each group and the sequence of groups. By reasonably combining particle swarm optimisation (PSO) and genetic algorithm (GA), we develop a fast and easily implemented hybrid algorithm (HA) for solving the considered problems. The effectiveness and efficiency of the proposed HA are demonstrated and compared with those of standard PSO and GA by numerical results of various tested instances with group numbers up to 20. In addition, three different lower bounds are developed to evaluate the solution quality of the HA. Limited numerical results indicate that the proposed HA is a viable and effective approach for the studied two-machine flow-shop group scheduling problem.

Watermarking Images in the Frequency Domain by Exploiting Self-Inverting Permutations

In this work we propose efficient codec algorithms for watermarking images that are intended for uploading on the web under intellectual property protection. Headed to this direction, we recently suggested a way in which an integer number w which being transformed into a self-inverting permutation, can be represented in a two dimensional (2D) object and thus, since images are 2D structures, we have proposed a watermarking algorithm that embeds marks on them using the 2D representation of w in the spatial domain. Based on the idea behind this technique, we now expand the usage of this concept by marking the image in the frequency domain. In particular, we propose a watermarking technique that also uses the 2D representation of self-inverting permutations and utilizes marking at specific areas thanks to partial modifications of the image’s Discrete Fourier Transform (DFT). Those modifications are made on the magnitude of specific frequency bands and they are the least possible additive information ensuring robustness and imperceptiveness. We have experimentally evaluated our algorithms using various images of different characteristics under JPEG compression. The experimental results show an improvement in comparison to the previously obtained results and they also depict the validity of our proposed codec algorithms.

Classical and quantum discrete dynamical systems

We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for these concepts to physics as an empirical science. For a consistent description of the symmetries of dynamical systems at different times and the symmetries of the various parts of such systems, we introduce discrete analogs of the gauge connections. Gauge structures are particularly important to describe the quantum behavior. We show that quantum behavior is the result of a fundamental inability to trace the identity of indistinguishable objects in the process of evolution. Information is available only on invariant statements and values, relating to such objects. Using mathematical arguments of a general nature we can show that any quantum dynamics can be reduced to a sequence of permutations. Quantum interferences occur in the invariant subspaces of permutation representations of symmetry groups of dynamical systems. The observable values can be expressed in terms of permutation invariants. We also show that for the description of quantum phenomena, instead of a nonconstructive number system --- the field of complex numbers, it is enough to use cyclotomic fields --- the minimal extentions of natural numbers suitable for quantum mechanics. Finite groups of symmetries play a central role in this article. The interest in such groups has an additional motivation in physics. Numerous experiments and observations in particle physics point to an important role of finite groups of relatively low orders in a number of fundamental processes.

Genetic algorithm for balancing reconfigurable machining lines

We consider the problem of designing a reconfigurable machining line. Such a line is composed of a sequence of workstations performing specific sets of operations. Each workstation is comprised of several identical CNC machines (machining centers). The line is required to satisfy the given precedence order, inclusion, exclusion and accessibility constraints on the given set of operations. Inclusion and exclusion are zoning constraints which oblige or forbid certain operations to be performed on the same workstation. The accessibility constraints imply that each operation has a set of possible part positions under which it can be performed. All the operations performed on the same workstation must have a common part position. Workstation times are computed taking into account processing and setup times for operations and must not exceed a given bound. The number of CNC machines at one workstation is limited, and the total number of machines must be minimized. A genetic algorithm is proposed. This algorithm is based on the permutation representation of solutions. A heuristic decoder is suggested to construct a solution from a permutation, so that the output solution is feasible w.r.t. precedence, accessibility, cycle time, and exclusion constraints. The other constraints are treated with a penalty approach. For a local improvement of solutions, a mixed integer programming model is suggested for an optimal design of workstations if the order of operations is fixed. An experimental evaluation of the proposed GA on large scale test instances is performed.