Permutation Theory Research Articles

New structure of algebras using permutations in symmetric groups

The permutation BG-algebras were first introduced as a novel kind of algebra. In this work, their basic qualities were investigated to better understand how they relate to one another and how they might be combined to construct various sorts of superstructures. We investigated permutation BG-algebras, permutation group-derived, permutation BGsubalgebra, and permutation BG-normal. Furthermore, we explored certain novel concepts in permutation theory for the initial time. We additionally looked into BG-algebra isomorphism theorems, BG-algebra homomorphism, quotient permutation BG-algebras, and equivalence relations.

Identifying the a priori time-to-failure distribution of unique highly vital elements using the expert method

Abstract. Aim. Systematic failures, unlike random hardware failures, cannot be described using the mathematics of the probability theory and the dependability theory. However, such failures are the biggest problem due to their unpredictability. In the case of describing systematic failures of unique highly vital systems, a solution is presented by an approach that involves taking into account the quantitative criteria of functional performance of a facility in time that are defined, for example, by prescribing a set of parameters for each function that characterize its ability to perform, as well as acceptable limits for such parameters’ variation. The paper aims to develop an approach to the use of expert evaluation for the purpose of identifying the type and parameters of the distribution of the time to failure of unique highly vital elements. The author examined an approach to determining the a priori distribution of the time to failure of unique highly vital elements by pairwise comparison that would be useful for improving the accuracy of their dependability indicators.Methods. Hierarchy analysis, fuzzy logic and permutation theory were used. Fuzzy variables were introduced, the degrees of belonging to which are interpreted as subjective probabilities of the time to failure and its characteristics within different time intervals. Methods were proposed for accounting for the accuracy of expert evaluation and for solving the cluster sampling problem.

A Partially-Sorted Concentric Layout for Efficient Label Localization in Augmented Reality.

A common approach for Augmented Reality labeling is to display the label text on a flag planted into the real world element at a 3D anchor point. When there are more than just a few labels, the efficiency of the interface decreases as the user has to search for a given label sequentially. The search can be accelerated by sorting the labels alphabetically, but sorting all labels results in long and intersecting leader lines from the anchor points to the labels. This paper proposes a partially-sorted concentric label layout that leverages the search efficiency of sorting while avoiding the label display problems of long or intersecting leader lines. The labels are partitioned into a small number of sorted sequences displayed on circles of increasing radii. Since the labels on a circle are sorted, the user can quickly search each circle. A tight upper bound derived from circular permutation theory limits the number of circles and thereby the complexity of the label layout. For example, 12 labels require at most three circles. When the application allows it, the labels are presorted to further reduce the number of circles in the layout. The layout was tested in a user study where it significantly reduced the label searching time compared to a conventional single-circle layout.

Development of framework for performance prediction of flexible road pavement in Nigeria using Fuzzy logic theory

ABSTRACT Though several attempts have been made in the past to develop performance prediction models for flexible road pavement in Nigeria, insufficient dataset remained the major problem of using such models. This study used Fuzzy logic theory which is capable of handling the challenge of imprecise dataset to develop a framework for performance prediction of flexible road pavement in Nigeria. A Fuzzy Inference System was developed using MATLAB software. The attributes used included; Initial Pavement Surface Condition, Age of pavement, Resilient Modulus of subgrade soil, Average Truck load per day, Average Annual Air Temperature and Rainfall to predict the Future Pavement Surface Condition (FPSC). The model was calibrated using observed logical behaviour of pavement to fit engineering experience and judgement. A goodness-of-fit test showed high level of accuracy at 81.63% which was validated at 74.13% using extrapolated dataset of the same source. The study proposed 5120 mutually exclusive decision rules for performance prediction of flexible road pavement based on permutation theory. Though there was no current and well-spread dataset that described the present pavement condition to calibrate the decision rules, a framework for performance prediction of flexible road pavement using Fuzzy logic theory was developed for pavement engineers in Nigeria.

BARNUMISM OF PUZZLES, or HOW TO FOOL WISELY?

The study examines the contradictory nature of the culture of intellectual leisure, attention is paid to the conflict between the rational nature of cognitive procedures for solving tasks and the intuitive-emotional side of the creative nature of a person. The subject of attention is those methods of creating the context of laughing, which are themselves based on an intellectual foundation. The effect of the organic conjunction of imaginary antagonists - culture and intellectuality - is examined using the example of puzzles, which exemplify one of the most obvious specimens of the culture of intellectual leisure. The principal barnumism (funny cheating, humbugs) of puzzles considered on the example of puzzles associated with the name of Samuel Loyd, the American author of numerous entertaining tasks and one of the coryphaei of intellectual leisure. These puzzles not only clearly demonstrate the arsenal of the barnumism – gimmicks, trickery, they retain an intellectual foundation, remaining, for example, good illustrations of aspects of covering the plane with circles (“Barnum”s Magic Donkeys”), mathematical analysis (“Teddy and Lions”), or permutation theory and combinatorics in general (“The 14–15 Puzzle”). The term “barnumism” (like the “Barnum effect”) originated from the name of Phineas Taylor Barnum – the showman, the entrepreneur, the founder of the American circus “The Greatest Show on Earth”. The potential of the barnumism of puzzles of to overcome of the conflict between the rational and entertaining sides of intellectual leisure is substantiated.

Type synthesis of 90°/180° dual-function upenders for large shell units of nuclear reactor

In this paper, unit vectors collinear with the axes of large shell units of nuclear reactors were used to represent the poses of the workpieces. Based on the motion operator interpretation of the rotation matrix, 60 kinds of rotation matrix permutations with 90° turns as the step length to achieve 180° flipping of the workpiece were obtained. After eliminating invalid permutations and merging homogenous permutations, class I, II, and III flipping motion models were obtained. By calculating the Lie algebra of each step of rigid body motion in the flipping motion model, the twist and the motion pair property for each step were obtained. By means of the permutation and combination theory, the sequence of the motion pairs for different configurations was determined, and the rules between sequential transformations and axis changes of the motion pairs under the initial configuration were stipulated. Three classes of 19 initial configurations of 90°/180° dual-function upenders were constructed. For three of them, taking into account the characteristics of the axial dimension variation of workpieces, prismatic pairs were added along the workpiece’s axis (vector), and 15 overall configurations of 90°/180° dual-function upenders were synthesized. This provides the basic theoretical support for the innovative design of upenders with independent intellectual property rights.

Detecting nonlinear stochastic systems using two independent hypothesis tests.

Various systems in the real world can be nonlinear and stochastic, but because nonlinear time series analysis has been developed to distinguish nonlinear deterministic systems from linear stochastic systems, there have been no appropriate methods developed so far for testing the nonlinear stochasticity for a given system. Thus, here we propose a set of two hypothesis tests, one for the nonlinearity and one for the stochasticity, independent of each other. The test for the linearity is based on Fourier-transform-based surrogate data with a nonlinear test statistic, while the test for determinism depends on the theory of ordinal patterns or permutations recently developed intensively. We demonstrate the proposed set of tests with time series generated from toy models. In addition, we show that both a foreign exchange market and a temperature series in Tokyo could be nonlinear and stochastic, as well as sometimes with determinism beyond pseudoperiodicity.

Settings and misunderstandings in mathematics

This paper pursues two goals. Its first goal is to clear up the “identity problem” faced by the structuralist interpretation of mathematics. Its second goal, through the consideration of examples coming in particular from the theory of permutations, is to examine cases of misunderstandings in mathematics fit to cast some light on mathematical understanding in general. The common thread shared by these two goals is the notion of setting. The study of a mathematical object almost always goes together with the choice of a particular setting, and the understanding of the workings of mathematical settings is an essential component of mathematical knowledge. It is claimed that the recognition of mathematical settings, as features distinct from both mathematical structures and the systems which instantiate those structures, allows one to classify most of understandable misunderstandings in mathematics, and also to solve the identity problem.

Carathéodory, Helly, and Radon Numbers for Sublattice and Related Convexities

The Carathéodory, Helly, and Radon numbers are three main invariants in convexity theory. These invariants have been determined, exactly or approximately, for a number of different convexity structures. We consider convexity structures defined by the sublattices and by the convex sublattices of finite-dimensional Euclidian, integer, and Boolean spaces. Such sublattices arise in submodular optimization (lattice programming) and in monotone comparative statics of optimization and fixed point problems. We also consider integral L-natural convexities, induced by dual network flow constraint systems. We determine the exact Carathéodory, Helly, and Radon numbers of most of these convexities, and very close upper and lower bounds for the other Carathéodory numbers. Our results imply, for example, that if a set can be obtained with unions and intersections from a given family of subsets of a finite set then it can be obtained with unions and intersections from a small subfamily. We also show that finding the Carathéodory number of integral L-natural convexities reduces to an extremal problem in the theory of permutations, solved in a companion paper. We leave as open problems the determination of the Helly and Radon numbers of the integer convex sublattice convexity.

Permutations, power operations, and the center of the category of racks

ABSTRACTRacks and quandles are rich algebraic structures that are strong enough to classify knots. Here we develop several fundamental categorical aspects of the theories of racks and quandles and their relation to the theory of permutations. In particular, we compute the centers of the categories and describe power operations on them, thereby revealing free extra structure that is not apparent from the definitions. This also leads to precise characterizations of these theories in the form of universal properties.

Largest Minimal Inversion-Complete and Pair-Complete Sets of Permutations

We solve two related extremal problems in the theory of permutations. A set 𝑄 of permutations of the integers 1 to 𝑛 is inversion-complete (resp., pair-complete) if for every inversion (𝑗, 𝑖), where 1 ≤ 𝑖 < 𝑗 ≤ 𝑛, (resp., for every pair (𝑖, 𝑗), where 𝑖 ≠ 𝑗) there exists a permutation in 𝑄 where 𝑗 is before 𝑖. It is minimally inversion-complete if in addition no proper subset of 𝑄 is inversion-complete; and similarly for pair completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Caratheodory numbers for certain abstract convexity structures on the (𝑛 − 1)-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide whenever 𝑛 ≥ 4.

Flavour singlets in gauge theory as permutations

Gauge-invariant operators can be specified by equivalence classes of permutations. We develop this idea concretely for the singlets of the flavour group $SO(N_f)$ in $U(N_c)$ gauge theory by using Gelfand pairs and Schur-Weyl duality. The singlet operators, when specialised at $N_f =6$, belong to the scalar sector of ${\cal N}=4$ SYM. A simple formula is given for the two-point functions in the free field limit of $g_{YM}^2 =0$. The free two-point functions are shown to be equal to the partition function on a 2-complex with boundaries and a defect, in a topological field theory of permutations. The permutation equivalence classes are Fourier transformed to a representation basis which is orthogonal for the two-point functions at finite $N_c , N_f$. Counting formulae for the gauge-invariant operators are described. The one-loop mixing matrix is derived as a linear operator on the permutation equivalence classes.

A novel Morse code-inspired method for multiclass motor imagery brain–computer interface (BCI) design

Motor imagery (MI)-based brain–computer interfaces (BCIs) allow disabled individuals to control external devices voluntarily, helping us to restore lost motor functions. However, the number of control commands available in MI-based BCIs remains limited, limiting the usability of BCI systems in control applications involving multiple degrees of freedom (DOF), such as control of a robot arm. To address this problem, we developed a novel Morse code-inspired method for MI-based BCI design to increase the number of output commands. Using this method, brain activities are modulated by sequences of MI (sMI) tasks, which are constructed by alternately imagining movements of the left or right hand or no motion. The codes of the sMI task was detected from EEG signals and mapped to special commands. According to permutation theory, an sMI task with N-length allows 2×(2N−1) possible commands with the left and right MI tasks under self-paced conditions. To verify its feasibility, the new method was used to construct a six-class BCI system to control the arm of a humanoid robot. Four subjects participated in our experiment and the averaged accuracy of the six-class sMI tasks was 89.4%. The Cohen's kappa coefficient and the throughput of our BCI paradigm are 0.88±0.060 and 23.5bits per minute (bpm), respectively. Furthermore, all of the subjects could operate an actual three-joint robot arm to grasp an object in around 49.1s using our approach. These promising results suggest that the Morse code-inspired method could be used in the design of BCIs for multi-DOF control.

Combining p-values in replicated single-case experiments with multivariate outcome

Interest in combining probabilities has a long history in the global statistical community. The first steps in this direction were taken by Ronald Fisher, who introduced the idea of combining p-values of independent tests to provide a global decision rule when multiple aspects of a given problem were of interest. An interesting approach to this idea of combining p-values is the one based on permutation theory. The methods belonging to this particular approach exploit the permutation distributions of the tests to be combined, and use a simple function to combine probabilities. Combining p-values finds a very interesting application in the analysis of replicated single-case experiments. In this field the focus, while comparing different treatments effects, is more articulated than when just looking at the means of the different populations. Moreover, it is often of interest to combine the results obtained on the single patients in order to get more global information about the phenomenon under study. This paper gives an overview of how the concept of combining p-values was conceived, and how it can be easily handled via permutation techniques. Finally, the method of combining p-values is applied to a simulated replicated single-case experiment, and a numerical illustration is presented.

Maximum values of total differentials and linear hulls of block symmetric ciphers

The approach to obtaining the estimates of provable block symmetric ciphers security against differential and linear cryptanalysis attacks, represented by the maximum values of differential and linear probabilities of multi-cycle enciphering transformations is proposed in the paper. The main objective of the research is to obtain the design ratios to calculate these indicators. As it is known, today these indicators are represented as estimated values, significantly differing from true that was determined using the new methodology for estimating the indicators of provable block symmetric ciphers security against differential and linear cryptanalysis attacks, developed recently. The basic mathematical apparatus of the random permutation theory is described, and the calculation results of provable security indicators, obtained using this mathematical apparatus for ciphers with 128-bit input, and in particular for the cipher Rijndael and ciphers, presented at the recent Ukrainian competition are given in the paper. The obtained results allow to obtain more objective data on ciphers security indicators, which can be used for improving the quality of expert solutions and conclusions on the improvement degree of the projects, submitted to the contest, as well as for selecting new promising solutions and developments on building ciphers. It is concluded that all these ciphers have the same parameters both for the differential and linear properties, almost equal to 2-121.

Asymptotically Optimal Algorithms for One-to-One Pickup and Delivery Problems With Applications to Transportation Systems

The Stacker Crane Problem is NP-Hard and the best known approximation algorithm only provides a 9/5 approximation ratio. The objective of this paper is threefold. First, by embedding the problem within a stochastic framework, we present a novel algorithm for the SCP that: (i) is asymptotically optimal, i.e., it produces, almost surely, a solution approaching the optimal one as the number of pickups/deliveries goes to infinity; and (ii) has computational complexity $O(n^{2+\eps})$, where $n$ is the number of pickup/delivery pairs and $\eps$ is an arbitrarily small positive constant. Second, we asymptotically characterize the length of the optimal SCP tour. Finally, we study a dynamic version of the SCP, whereby pickup and delivery requests arrive according to a Poisson process, and which serves as a model for large-scale demand-responsive transport (DRT) systems. For such a dynamic counterpart of the SCP, we derive a necessary and sufficient condition for the existence of stable vehicle routing policies, which depends only on the workspace geometry, the stochastic distributions of pickup and delivery points, the arrival rate of requests, and the number of vehicles. Our results leverage a novel connection between the Euclidean Bipartite Matching Problem and the theory of random permutations, and, for the dynamic setting, exhibit novel features that are absent in traditional spatially-distributed queueing systems.

Stability and periodicity of discrete Hopfield neural networks with column arbitrary-magnitude-dominant weight matrix

Stability and periodicity of neural networks is important behavior in biological and cognitive activities. In order to better simulate a biological genuine model, a special kind of discrete Hopfield neural networks (SDHNNs) in which every neuron has only one input is considered. By applying permutation theory and mathematical induction, we prove that the SDHNN always converges to a stable state or a limit cycle. The SDHNN is extended to the discrete Hopfield neural networks with column arbitrary-magnitude-dominant weight matrix (DHNNCAMDWM) in which there only exits a magnitude-dominant element in every column. Some important results, especially the periodic stability of the DHNNCAMDWM, are obtained. And the XOR problem is successfully solved by the results.

Order-Restricted Inference with Linear Rank Statistics in Microarray Data

The classification of subjects with unknown distribution in a small sample size often involves order-restricted constraints in multivariate parameter setups. Those problems make the optimality of a conventional likelihood ratio based statistical inferences not feasible. Fortunately, Roy (1953) introduced union-intersection principle(UIP) which provides an alternative avenue. Multivariate linear rank statistics along with that principle, yield a considerably appropriate robust testing procedure. Furthermore, conditionally distribution-free test based upon exact permutation theory is used to generate p-values, even in a small sample. Applications of this method are illustrated in a real microarray data example (Lobenhofer et al., 2002).

Quantum algorithm for the Laughlin wave function

We construct a quantum algorithm that creates the Laughlin state for an arbitrary number of particles $n$ in the case of filling fraction one. This quantum circuit is efficient since it only uses $n(n-1)/2$ local qudit gates and its depth scales as $2n-3$. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. Finally, we discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary filling fraction.

Robust Inference for Testing Order-Restricted Inference

Classification of subjects with unknown distribution in small sample size setup may involve order-restricted constraints in multivariate parameter setups. Those problems makes optimality of conventional likelihood ratio based statistical inferences not feasible. Fortunately, Roy (1953) introduced union-intersection principle(UIP) which provides an alternative avenue. Redescending M-estimator along with that principle yields a considerably appropriate robust testing procedure. Furthermore, conditionally distribution-free test based upon exact permutation theory is used to generate p-values, even in small sample. Applications of this method are illustrated in simulated data and read data example (Lobenhofer et al., 2002)