3 papers + 1 book related to Clifford algebras

Note: Communicated by Nek Valous, National Center for Tumor Diseases (NCT), Heidelberg.


Title: On the hardness of the noncommutative determinant
Journal: Computational Complexity, March 2018, Volume 27, Issue 1, pp 1–29
Authors: V. Arvind, Srikanth Srinivasan
Link: https://link.springer.com/article/10.1007/s00037-016-0148-5
Abstract: In this paper, we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below:
○ We show that if the noncommutative determinant polynomial has small noncommutative arithmetic circuits then so does the noncommutative permanent. Consequently, the commutative permanent polynomial has small commutative arithmetic circuits.
○ For any field F we show that computing the n×n  permanent over F is polynomial-time reducible to computing the 2n×2n (noncommutative) determinant whose entries are O(n2)×O(n2) matrices over the field F.
○ We also derive as a consequence that computing the   n×n  permanent over nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant over Clifford algebras of nO(1) dimension.
Our techniques are elementary and use primarily the notion of the Hadamard Product of noncommutative polynomials.

Title: Modified Rodrigues Parameters: An Efficient Representation of Orientation in 3D Vision and Graphics
Journal of Mathematical Imaging and Vision, March 2018, Volume 60, Issue 3, pp 422–442
George Terzakis, Manolis Lourakis, Djamel Ait-Boudaoud
Modified Rodrigues parameters (MRPs) are triplets in R^3  bijectively and rationally mapped to quaternions through stereographic projection. We present here a compelling case for MRPs as a minimal degree-of-freedom parameterization of orientation through novel solutions to prominent problems in the fields of 3D vision and computer graphics. In our primary contribution, we show that the derivatives of a unit quaternion in terms of its MRPs are simple polynomial expressions of its scalar and vector part. Furthermore, we show that updates to unit quaternions from perturbations in parameter space can be computed without explicitly invoking the parameters in the computations. Based on the former, we introduce a novel approach for designing orientation splines by configuring their back-projections in 3D space. Finally, in the general topic of nonlinear optimization for geometric vision, we run performance analyses and provide comparisons on the convergence behavior of MRP parameterizations on the tasks of absolute orientation, exterior orientation and large-scale bundle adjustment of public datasets.

Title: Multistability Analysis of Quaternion-Valued Neural Networks With Time Delays
Journal: IEEE Transactions on Neural Networks and Learning Systems, 02 March 2018, doi: 10.1109/TNNLS.2018.2801297
Authors: Qiankun Song, Xiaofeng Chen
Link: http://ieeexplore.ieee.org/document/8306303/
This paper addresses the multistability issue for quaternion-valued neural networks (QVNNs) with time delays. By using the inequality technique, sufficient conditions are proposed for the boundedness and the global attractivity of delayed QVNNs. Based on the geometrical properties of the activation functions, several criteria are obtained to ensure the existence of 81n equilibrium points, 16n of which are locally stable. Two numerical examples are provided to illustrate the effectiveness of the obtained results.


Title: Advances in Geometry and Lie Algebras from Supergravity
Publisher: Springer
Authors: Pietro Giuseppe Frè
Link: http://www.springer.com/us/book/9783319744902
About the book:

This book aims to provide an overview of several topics in advanced differential geometry and Lie group theory, all of them stemming from mathematical problems in supersymmetric physical theories. It presents a mathematical illustration of the main development in geometry and symmetry theory that occurred under the fertilizing influence of supersymmetry/supergravity. The contents are mainly of mathematical nature, but each topic is introduced by historical information and enriched with motivations from high energy physics, which help the reader in getting a deeper comprehension of the subject.
– Finite Groups and Lie Algebras: The ADE Classification and Beyond
– Isometries and the Geometry of Coset Manifolds
– Complex and Quaternionic Geometry
– Special Geometries
– Solvable Algebras and the Tits Satake Projection
– Black Holes and Nilpotent Orbits
– E7, F4 and Supergravity Scalar Potentials
– (Hyper)Kähler Quotients, ALE-Manifolds and Cn/Γ Singularities
– Epilogue

Source: Email from N. Velous, nek.valous_AT_nct-heidelberg.de, 10 Mar. 2018.


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