Tetrahedron maps, Yang–Baxter maps, and partial linearisations

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and Yang–Baxter maps, which are set-theoretical solutions to the quantum Yang–Baxter equation. In particular, we clarify the structure of the nonlinear algebraic relations which define linear (parametric) tetrahedron maps (with nonlinear dependence on parameters), and we present several transformations which allow one to obtain new such maps from known ones. Furthermore, we prove that the differential of a (nonlinear) tetrahedron map on a manifold is a tetrahedron map as well. Similar results on the differentials of Yang–Baxter and entwining Yang–Baxter maps are also presented. Using the obtained general results, we construct new examples of (parametric) Yang–Baxter and tetrahedron maps. The considered examples include maps associated with integrable systems and matrix groups. In particular, we obtain a parametric family of new linear tetrahedron maps, which are linear approximations for the nonlinear tetrahedron map constructed by Dimakis and Müller-Hoissen [9] in a study of soliton solutions of vector Kadomtsev–Petviashvili (KP) equations. Also, we present invariants for this nonlinear tetrahedron map.

Pembentukan Grup Matriks Singular 2×2

The study aims to determine the set of the singular matrix 2×2 that forms the group and describes its properties. The type of research was used exploratory research. Using diagonalization of the singular matrix S, whereas a generator matrix, pseudo-identity, and pseudo-inverse methods, we obtained a group singular matrix 2×2 with standard multiplication operations on the matrix, with conditions namely: (1) closed, (2) associative, (3) there was an element of identity, (4) inverse, there was (A)-1 so A x (A)-1 = (A)-1 x A = Is. The group was the abelian group (commutative group). In addition, in the group, Gs satisfied that if Ɐ A, X, Y element Gs was such that A x X = A x Y then X = Y and X x A = Y x A then X = Y. This show that the group can be applied the cancellation properties like the case in nonsingular matrix group. This research provides further research opportunities on the formation of singular matrix groups 3×3 or higher order.

On (2,3)-generation of matrix groups over the ring of integers, II

Final steps are done in proving that the groups SL ⁡ ( n , Z ) \operatorname {SL}(n,\mathbb {Z}) , GL ⁡ ( n , Z ) \operatorname {GL}(n,\mathbb {Z}) and PGL ⁡ ( n , Z ) \operatorname {PGL}(n,\mathbb {Z}) are ( 2 , 3 ) (2,3) -generated if and only if n ≥ 5 n\ge 5 , and PSL ⁡ ( n , Z ) \operatorname {PSL}(n,\mathbb {Z}) is ( 2 , 3 ) (2,3) -generated if and only if n = 2 n=2 or n ≥ 5 n\ge 5 . In particular, the results cover the remaining cases of n = 8 n=8 , …, 12 12 , and 14 14 .

An Optical Flow Based Left-Invariant Metric for Natural Gradient Descent in Affine Image Registration

Accurate spatial alignment is essential for any population neuroimaging study, and affine (12 parameter linear/translation) or rigid (6 parameter rotation/translation) alignments play a major role. Here we consider intensity based alignment of neuroimages using gradient based optimization, which is a problem that continues to be important in many other areas of medical imaging and computer vision in general. A key challenge is robustness. Optimization often fails when transformations have components with different characteristic scales, such as linear versus translation parameters. Hand tuning or other scaling approaches have been used, but efficient automatic methods are essential for generalizing to new imaging modalities, to specimens of different sizes, and to big datasets where manual approaches are not feasible. To address this we develop a left invariant metric on these two matrix groups, based on the norm squared of optical flow induced on a template image. This metric is used in a natural gradient descent algorithm, where gradients (covectors) are converted to perturbations (vectors) by applying the inverse of the metric to define a search direction in which to update parameters. Using a publicly available magnetic resonance neuroimage database, we show that this approach outperforms several other gradient descent optimization strategies. Due to left invariance, our metric needs to only be computed once during optimization, and can therefore be implemented with negligible computation time.

Tame torsion and the tame inverse Galois problem

Fix a positive integer g and a squarefree integer m. We prove the existence of a genus g curve $$C/{\mathbb {Q}}$$ C / Q such that the mod m representation of its Jacobian is tame. The method is to analyse the period matrices of hyperelliptic Mumford curves, which could be of independent interest. As an application, we study the tame version of the inverse Galois problem for symplectic matrix groups over finite fields.

On Coordinate Expressions of Jet Groups and their Representations

A detailed derivation of the jet composition in local coordinates for jet (differential) groups is presented. A suitable faithful representation in matrix groups is demonstrated. Furthermore, Toupin subgroups which occur in continuum mechanics are demonstrated as an example in which representations can be used effectively.