Steady plasma flows in a periodic non-symmetric domain

Steady plasma flows have been studied almost exclusively in systems with continuous symmetry or in open domains. In the absence of continuous symmetry, the lack of a conserved quantity makes the study of flows intrinsically challenging. In a toroidal domain, the requirement of double periodicity for physical quantities adds to the complications. In particular, the magnetohydrodynamics (MHD) model of plasma steady state with the flow in a non-symmetric toroidal domain allows the development of singularities when the rotational transform of the magnetic field is rational, much like the equilibrium MHD model. In this work, we show that steady flows can still be maintained provided the rotational transform is close to rational and the magnetic shear is weak. We extend the techniques developed in carrying out perturbation methods to all orders for static MHD equilibrium by Weitzner (Phys. Plasmas, vol. 21, 2014, p. 022515) to MHD equilibrium with flows. We construct perturbative MHD equilibrium in a doubly periodic domain with nearly parallel flows by systematically eliminating magnetic resonances order by order. We then utilize an additional symmetry of the flow problem, first discussed by Hameiri (J. Math. Phys., vol. 22, 1981, pp. 2080–2088, § III), to obtain a generalized Grad–Shafranov equation for a class of non-symmetric three-dimensional MHD equilibrium with flows both parallel and perpendicular to the magnetic field. For this class of flows, we can obtain non-symmetric generalizations of integrals of motion, such as Bernoulli's function and angular momentum. Finally, we obtain the generalized Hamada conditions, which are necessary to suppress singular currents in such a system when the magnetic field lines are closed. We do not attempt to address the question of neoclassical damping of flows.

Research and Application of Improved Clustering Algorithm in Retail Customer Classification

Clustering is a major field in data mining, which is also an important method of data partition or grouping. Clustering has now been applied in various ways to commerce, market analysis, biology, web classification, and so on. Clustering algorithms include the partitioning method, hierarchical clustering as well as density-based, grid-based, model-based, and fuzzy clustering. The K-means algorithm is one of the essential clustering algorithms. It is a kind of clustering algorithm based on the partitioning method. This study’s aim was to improve the algorithm based on research, while with regard to its application, the aim was to use the algorithm for customer segmentation. Customer segmentation is an essential element in the enterprise’s utilization of CRM. The first part of the paper presents an elaboration of the object of study, its background as well as the goal this article would like to achieve; it also discusses the research the mentality and the overall content. The second part mainly introduces the basic knowledge on clustering and methods for clustering analysis based on the assessment of different algorithms, while identifying its advantages and disadvantages through the comparison of those algorithms. The third part introduces the application of the algorithm, as the study applies clustering technology to customer segmentation. First, the customer value system is built through AHP; customer value is then quantified, and customers are divided into different classifications using clustering technology. The efficient CRM can thus be used according to the different customer classifications. Currently, there are some systems used to evaluate customer value, but none of them can be put into practice efficiently. In order to solve this problem, the concept of continuous symmetry is introduced. It is very important to detect the continuous symmetry of a given problem. It allows for the detection of an observable state whose components are nonlinear functions of the original unobservable state. Thus, we built an evaluating system for customer value, which is in line with the development of the enterprise, using the method of data mining, based on the practical situation of the enterprise and through a series of practical evaluating indexes for customer value. The evaluating system can be used to quantify customer value, to segment the customers, and to build a decision-supporting system for customer value management. The fourth part presents the cure, mainly an analysis of the typical k-means algorithm; this paper proposes two algorithms to improve the k-means algorithm. Improved algorithm A can get the K automatically and can ensure the achievement of the global optimum value to some degree. Improved Algorithm B, which combines the sample technology and the arrangement agglomeration algorithm, is much more efficient than the k-means algorithm. In conclusion, the main findings of the study and further research directions are presented.

Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams

A continuous measure of symmetry and the Voronoi entropy of 2D patterns representing Voronoi diagrams emerging from the Penrose tiling were calculated. A given Penrose tiling gives rise to a diversity of the Voronoi diagrams when the centers, vertices, and the centers of the edges of the Penrose rhombs are taken as the seed points (or nuclei). Voronoi diagrams keep the initial symmetry group of the Penrose tiling. We demonstrate that the continuous symmetry measure and the Voronoi entropy of the studied sets of points, generated by the Penrose tiling, do not necessarily correlate. Voronoi diagrams emerging from the centers of the edges of the Penrose rhombs, considered nuclei, deny the hypothesis that the continuous measure of symmetry and the Voronoi entropy are always correlated. The Voronoi entropy of this kind of tiling built of asymmetric convex quadrangles equals zero, whereas the continuous measure of symmetry of this pattern is high. Voronoi diagrams generate new types of Penrose tiling, which are different from the classical Penrose tessellation.

Supersymmetric spectroscopy on AdS4 × S7 and AdS4 × S6

New techniques based on Exceptional Field Theory have recently allowed for the calculation of the Kaluza-Klein spectra of certain AdS4 solutions of D = 11 and massive IIA supergravity. These are the solutions that consistently uplift on S7 and S6 from vacua of maximal four-dimensional supergravity with SO(8) and ISO(7) gaugings. In this paper, we provide an algorithmic procedure to compute the complete Kaluza-Klein spectrum of five such AdS4 solutions, all of them $$ \mathcal{N} $$ N = 1, and give the first few Kaluza-Klein levels. These solutions preserve SO(3) and U(1) × U(1) internal symmetry in D = 11, and U(1) (two of them) and no continuous symmetry in type IIA. Together with previously discussed cases, our results exhaust the Kaluza-Klein spectra of known supersymmetric AdS4 solutions in D = 11 and type IIA in the relevant class.

A new family of AdS4 S-folds in type IIB string theory

We construct infinite new classes of AdS4× S1× S5 solutions of type IIB string theory which have non-trivial SL(2, ℤ) monodromy along the S1 direction. The solutions are supersymmetric and holographically dual, generically, to $$ \mathcal{N} $$ N = 1 SCFTs in d = 3. The solutions are first constructed as AdS4× ℝ solutions in D = 5 SO(6) gauged supergravity and then uplifted to D = 10. Unlike the known AdS4× ℝ S-fold solutions, there is no continuous symmetry associated with the ℝ direction. The solutions all arise as limiting cases of Janus solutions of d = 4, $$ \mathcal{N} $$ N = 4 SYM theory which are supported both by a different value of the coupling constant on either side of the interface, as well as by fermion and boson mass deformations. As special cases, the construction recovers three known S-fold constructions, preserving $$ \mathcal{N} $$ N = 1, 2 and 4 supersymmetry, as well as a recently constructed $$ \mathcal{N} $$ N = 1 AdS4× S1× S5 solution (not S-folded). We also present some novel “one-sided Janus” solutions that are non-singular.

Exponential decay of transverse correlations for O(N) spin systems and related models

We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary non-zero values of the external magnetic field and arbitrary spin dimension $$N > 1$$ N > 1 . Our result is new when $$N > 3$$ N > 3 , in which case no Lee–Yang theorem is available, it is an alternative to Lee–Yang when $$N = 2, 3$$ N = 2 , 3 , and also holds for a wide class of multi-component spin systems with continuous symmetry. The key ingredients are a representation of the model as a system of coloured random paths, a ‘colour-switch’ lemma, and a sampling procedure which allows us to bound from above the ‘typical’ length of the open paths.

Symmetry aspects in the macroscopic dynamics of magnetorheological gels and general liquid crystalline magnetic elastomers

We investigate theoretically the macroscopic dynamics of various types of ordered magnetic fluid, gel, and elastomeric phases. We take a symmetry point of view and emphasize its importance for a macroscopic description. The interactions and couplings among the relevant variables are based on their individual symmetry behavior, irrespective of the detailed nature of the microscopic interactions involved. Concerning the variables we discriminate between conserved variables related to a local conservation law, symmetry variables describing a (spontaneously) broken continuous symmetry (e.g., due to a preferred direction) and slowly relaxing ones that arise from special conditions of the system are considered. Among the relevant symmetries, we consider the behavior under spatial rotations (e.g., discriminating scalars, vectors or tensors), under spatial inversion (discriminating e.g., polar and axial vectors), and under time reversal symmetry (discriminating e.g., velocities from polarizations, or electric fields from magnetic ones). Those symmetries are crucial not only to find the possible cross-couplings correctly but also to get a description of the macroscopic dynamics that is compatible with thermodynamics. In particular, time reversal symmetry is decisive to get the second law of thermodynamics right. We discuss (conventional quadrupolar) nematic order, polar order, active polar order, as well as ferromagnetic order and tetrahedral (octupolar) order. In a second step, we show some of the consequences of the symmetry properties for the various systems that we have worked on within the SPP1681, including magnetic nematic (and cholesteric) elastomers, ferromagnetic nematics (also with tetrahedral order), ferromagnetic elastomers with tetrahedral order, gels and elastomers with polar or active polar order, and finally magnetorheological fluids and gels in a one- and two-fluid description.

Quantum Metrics for Continuous Shape Measures of Molecules

The field of continuous molecular shape and symmetry (dis)similarity quantifiers habitually called measures (specifically continuous shape measures - CShM or continuous symmetry measures - CSM) is obfuscated by the combinatorial numerical algorithms used in the field which restricts the applicability to the molecules containing up to twenty equivalent atoms. In the present paper we analyze this problem using various tools of classical probability theory as well as of one-particle and many-particle quantum mechanics. Applying these allows us to lift the combinatorial restriction and to identify the adequate renumbering of atoms (vertices) without considering all N! permutations of an N-vertex set so that in the end purely geometric molecular shape (dis)similarity quantifier can be defined. Developed methods can be easily implemented in the relevant computer code.<br>

Quantum Metrics for Continuous Shape Measures of Molecules

The field of continuous molecular shape and symmetry (dis)similarity quantifiers habitually called measures (specifically continuous shape measures - CShM or continuous symmetry measures - CSM) is obfuscated by the combinatorial numerical algorithms used in the field which restricts the applicability to the molecules containing up to twenty equivalent atoms. In the present paper we analyze this problem using various tools of classical probability theory as well as of one-particle and many-particle quantum mechanics. Applying these allows us to lift the combinatorial restriction and to identify the adequate renumbering of atoms (vertices) without considering all N! permutations of an N-vertex set so that in the end purely geometric molecular shape (dis)similarity quantifier can be defined. Developed methods can be easily implemented in the relevant computer code.<br>