6,689 research outputs found
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
A remarkable number of different numerical algorithms can be understood and
analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
Scattering equations and virtuous kinematic numerators and dual-trace functions
Inspired by recent developments on scattering equations, we present a
constructive procedure for computing symmetric, amplitude-encoded, BCJ
numerators for n-point gauge-theory amplitudes, thus satisfying the three
virtues identified by Broedel and Carrasco. We also develop a constructive
procedure for computing symmetric, amplitude-encoded dual-trace functions (tau)
for n-point amplitudes. These can be used to obtain symmetric kinematic
numerators that automatically satisfy color-kinematic duality. The S_n symmetry
of n-point gravity amplitudes formed from these symmetric dual-trace functions
is completely manifest. Explicit expressions for four- and five-point
amplitudes are presented.Comment: 24 pages; v2: minor sign corrections, added references; v3: minor
corrections, published versio
Tensor and Matrix Inversions with Applications
Higher order tensor inversion is possible for even order. We have shown that
a tensor group endowed with the Einstein (contracted) product is isomorphic to
the general linear group of degree . With the isomorphic group structures,
we derived new tensor decompositions which we have shown to be related to the
well-known canonical polyadic decomposition and multilinear SVD. Moreover,
within this group structure framework, multilinear systems are derived,
specifically, for solving high dimensional PDEs and large discrete quantum
models. We also address multilinear systems which do not fit the framework in
the least-squares sense, that is, when the tensor has an odd number of modes or
when the tensor has distinct dimensions in each modes. With the notion of
tensor inversion, multilinear systems are solvable. Numerically we solve
multilinear systems using iterative techniques, namely biconjugate gradient and
Jacobi methods in tensor format
Mechanics of Systems of Affine Bodies. Geometric Foundations and Applications in Dynamics of Structured Media
In the present paper we investigate the mechanics of systems of
affinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry.
Certain physical applications are possible in modelling of molecular crystals,
granular media, and other physical objects. Particularly interesting are
dynamical models invariant under the group underlying geometry of degrees of
freedom. In contrary to the single body case there exist nontrivial potentials
invariant under this group (left and right acting). The concept of relative
(mutual) deformation tensors of pairs of affine bodies is discussed. Scalar
invariants built of such tensors are constructed. There is an essential novelty
in comparison to deformation scalars of single affine bodies, i.e., there exist
affinely-invariant scalars of mutual deformations. Hence, the hierarchy of
interaction models according to their invariance group, from Euclidean to
affine ones, can be considered.Comment: 50 pages, 4 figure
- …