The right angle to look at orthogonal sets

If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in (X,Y) has a normal hyperdefinable X-internal subgroup N such that G/N is Y-internal; N is unique up to commensurability. In order to make sense of this statement, local simplicity theory for hyperdefinable sets is developped. Moreover, a version of Schlichting's Theorem for hyperdefinable families of commensurable subgroups is shown

Model Order Reduction for Rotating Electrical Machines

The simulation of electric rotating machines is both computationally expensive and memory intensive. To overcome these costs, model order reduction techniques can be applied. The focus of this contribution is especially on machines that contain non-symmetric components. These are usually introduced during the mass production process and are modeled by small perturbations in the geometry (e.g., eccentricity) or the material parameters. While model order reduction for symmetric machines is clear and does not need special treatment, the non-symmetric setting adds additional challenges. An adaptive strategy based on proper orthogonal decomposition is developed to overcome these difficulties. Equipped with an a posteriori error estimator the obtained solution is certified. Numerical examples are presented to demonstrate the effectiveness of the proposed method

P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and approaches for this question. The first one is equivalence of satisfaction of 3-SAT problem with the question of reaching zero of a nonnegative degree 4 multivariate polynomial (sum of squares), what could be tested from the perspective of algebra by using discriminant. It could be also approached as a continuous global optimization problem inside $[0,1]^n$, for example in physical realizations like adiabatic quantum computers. However, the number of local minima usually grows exponentially. Reducing to degree 2 polynomial plus constraints of being in $\{0,1\}^n$, we get geometric formulations as the question if plane or sphere intersects with $\{0,1\}^n$. There will be also presented some non-standard perspectives for the Subset-Sum, like through convergence of a series, or zeroing of $\int_0^{2\pi} \prod_i \cos(\varphi k_i) d\varphi$ fourier-type integral for some natural $k_i$. The last discussed approach is using anti-commuting Grassmann numbers $\theta_i$, making $(A \cdot \textrm{diag}(\theta_i))^n$ nonzero only if $A$ has a Hamilton cycle. Hence, the P$\ne$NP assumption implies exponential growth of matrix representation of Grassmann numbers. There will be also discussed a looking promising algebraic/geometric approach to the graph isomorphism problem -- tested to successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure

Geometric aspects of the symmetric inverse M-matrix problem

We investigate the symmetric inverse M-matrix problem from a geometric perspective. The central question in this geometric context is, which conditions on the k-dimensional facets of an n-simplex S guarantee that S has no obtuse dihedral angles. First we study the properties of an n-simplex S whose k-facets are all nonobtuse, and generalize some classical results by Fiedler. We prove that if all (n-1)-facets of an n-simplex S are nonobtuse, each makes at most one obtuse dihedral angle with another facet. This helps to identify a special type of tetrahedron, which we will call sub-orthocentric, with the property that if all tetrahedral facets of S are sub-orthocentric, then S is nonobtuse. Rephrased in the language of linear algebra, this constitutes a purely geometric proof of the fact that each symmetric ultrametric matrix is the inverse of a weakly diagonally dominant M-matrix. Review papers support our belief that the linear algebraic perspective on the inverse M-matrix problem dominates the literature. The geometric perspective however connects sign properties of entries of inverses of a symmetric positive definite matrix to the dihedral angle properties of an underlying simplex, and enables an explicit visualization of how these angles and signs can be manipulated. This will serve to formulate purely geometric conditions on the k-facets of an n-simplex S that may render S nonobtuse also for k>3. For this, we generalize the class of sub-orthocentric tetrahedra that gives rise to the class of ultrametric matrices, to sub-orthocentric simplices that define symmetric positive definite matrices A with special types of k x k principal submatrices for k>3. Each sub-orthocentric simplices is nonobtuse, and we conjecture that any simplex with sub-orthocentric facets only, is sub-orthocentric itself.Comment: 42 pages, 20 figure

Visualizing Magnitude and Direction in Flow Fields

In weather visualizations, it is common to see vector data represented by glyphs placed on grids. The glyphs either do not encode magnitude in readable steps, or have designs that interfere with the data. The grids form strong but irrelevant patterns. Directional, quantitative glyphs bent along streamlines are more effective for visualizing flow patterns. With the goal of improving the perception of flow patterns in weather forecasts, we designed and evaluated two variations on a glyph commonly used to encode wind speed and direction in weather visualizations. We tested the ability of subjects to determine wind direction and speed: the results show the new designs are superior to the traditional. In a second study we designed and evaluated new methods for representing modeled wave data using similar streamline-based designs. We asked subjects to rate the marine weather visualizations: the results revealed a preference for some of the new designs

The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems

We review the general problem of finding a global rotation that transforms a given set of points and/or coordinate frames (the "test" data) into the best possible alignment with a corresponding set (the "reference" data). For 3D point data, this "orthogonal Procrustes problem" is often phrased in terms of minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean distance measure relating the two sets of matched coordinates. We focus on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; we focus on these exact solutions to expose the structure of the entire eigensystem for the traditional 3D spatial alignment problem. We then explore the structure of the less-studied orientation data context, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. We conclude with a brief discussion of the combined 3D translation-orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices, and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. Supplementary Material covers extensions of quaternion methods to the 4D problem.Comment: This replaces an early draft that lacked a number of important references to previous work. There are also additional graphics elements. The extensions to 4D data and additional details are worked out in the Supplementary Material appended to the main tex