G2quivers

Computational matrix representation modules for linear operators with explicit constructions for a class of lie operators

Scattered data interpolation by radial kernel functions leads to linear equation systems with large, fully populated, ill-conditioned interpolation matrices. A successful iterative solution of such a system requires an efficient matrix-vector multiplication as well as an efficient preconditioner. While multipole approaches provide a fast matrix-vector multiplication, they avoid the explicit setup of the system matrix which hinders the construction of preconditioners, such as approximate inverses or factorizations which typically require the explicit system matrix for their construction. In this paper, we propose an approach that allows both an efficient matrix-vector multiplication as well as an explicit matrix representation which can then be used to construct a preconditioner. In particular, the interpolation matrix will be represented in hierarchical matrix format, and several approaches for the blockwise low-rank approximation are proposed and compared, of both analytical nature (separable expansions)...

NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form. And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves, which can be widely used in computer graphics and industrial design.

Simply subducible groups and ray groups. Explicit matrix representations and the group algebra

In order to do numerical simulations of three dimensional photonic crystals. Because three dimensional photonic crystals is modeled by the Maxwell equations, we used Yee’s finite difference scheme to derive the explicit matrix representation. We get a generalized eigenvalue problem (GEVP). The GEVP corresponding to the photonic crystals with face centered cubic (FCC) lattice and simple cubic (SC) lattice and we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP). Since we want to do numerical simulation, we developed FAME to do this and provided a GUI interface. This thesis is a introduction to FAME. 1

A plasticity-damage theory for large deformation of solids—I. Theoretical formulation

The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a viewpoint of an index theorem by using an explicit matrix representation. For a positive deformation parameter q or q=exp(2πiθ) with an irrational θ, one obtains an index condition dim ker a–dim ker a†=1 which allows only a nonhermitian phase operator with dim ker eiφ–dim ker(eiφ)†=1. For q=exp(2πiθ) with a rational θ, one formally obtains the singular situation dim ker a=∞ and dim ker a†=∞, which allows a hermitian phase operator with dim ker eiΦ–dim ker(eiΦ)†=0 as well as the nonhermitian one with dim ker eiφ– dim ker(eiφ)†=1. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for q=exp(2πiθ).

A plasticity-damage theory for large deformation of solids—II. Applications to finite simple shear

It is shown that polarization formulas have explicit matrix representations. This enables us to prove that polarization formulas of n n -positive maps between C ∗ C^{*} -algebras are coordinatewise positive.

Cubic root of Klein-Gordon equation

We propose a diagonal preconditioning method for automatically selecting the step sizes of a primal-dual splitting method (PDS). The conventional preconditioning method for PDS has several limitations, such as the need to directly access all the entries of the matrices representing the linear operators in the target optimization problem, and the possibility that the proximity operator cannot be solved analytically due to the element-wise preconditioning. In this paper, we establish operator norm-based variable-wise diagonal preconditioning (ON-VW) to resolve these issues. ON- VW has two features that are preferred in real applications. First, the preconditioners constructed by ON-VW are defined using only (an upper bound of) the operator norm of the linear operators, which eliminates the need for their explicit matrix representations. Furthermore, the stepsizes automatically selected by our preconditioners are variable-wise, which allows us to keep the proximity operator computable. We also prove that our preconditioners satisfy the convergence condition of PDS and demonstrate its effectiveness through its application to denoising of hyperspectral images.

In view of increasing collections of available 3D motion capture (mocap) data, the task of automatically annotating large sets of unstructured motion data is gaining in importance. In this paper, we present an efficient approach to label mocap data according to a given set of motion categories or classes, each specified by a suitable set of positive example motions. For each class, we derive a motion template that captures the consistent and variable aspects of a motion class in an explicit matrix representation. We then present a novel annotation procedure, where the unknown motion data is segmented and annotated by locally comparing it with the available motion templates. This procedure is supported by an efficient keyframe-based preprocessing step, which also significantly improves the annotation quality by eliminating false positive matches. As a further contribution, we introduce a genetic learning algorithm to automatically learn the necessary keyframes from the given example motions. For evaluation, we report on various experiments conducted on two freely available sets of motion capture data (CMU and HDM05).

We construct the well-known decomposition of the Lie algebra e8 into representations of e6⊕su(3) using explicit matrix representations over pairs of division algebras. The minimal representation of e6, namely the Albert algebra, is thus realized explicitly within e8, with the action given by the matrix commutator in e8, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in e8, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group E6 ⊂ E8.

We construct an extension of the Poincare group which involves a mixture of internal and space-time supersymmetries. The resulting group is an extension of the superPoincare group with infinitely many generators which carry internal and space-time indices. It is a closed algebra since all Jacobi identities are satisfied and it has therefore explicit matrix representations. We investigate the massless case and construct the irreducible representations of the extended symmetry. They are divided into two sets, longitudinal and transversal representations. The transversal representations involve an infinite series of integer and half-integer helicities. Finally we suggest an extension of the conformal group along the same line.

We present a rigorous mathematical proof of concept demonstrating the theoretical feasibility of quantum linear regression through variational Hamiltonian formulation. Our framework systematically transforms classical linear regression into a quantum optimization problem by encoding regression weights as expectation values of Pauli-Z operators and reformulating the loss function as a quantum Hamiltonian. We provide explicit mathematical constructions, matrix representations, and convergence analysis to establish the theoretical foundation for quantum machine learning applications. Unlike previous quantum linear regression approaches that focus on exponential speedups through quantum linear systems solvers, our variational approach is designed for near-term quantum devices and naturally incorporates regularization through quantum parameter constraints. The mathematical framework demonstrates computational equivalence between classical and quantum approaches while revealing inherent advantages in constrained optimization scenarios. This work provides the theoretical groundwork for future empirical applications to financial time series prediction, including potential implementation on Thai SET Index forecasting using data from Yahoo Finance and quantum computing platforms. Our key contributions include: (1) complete mathematical mapping from classical to quantum regression formulation, (2) explicit Hamiltonian constructions with matrix representations, (3) theoretical convergence analysis for quantum gradient descent, and (4) identification of quantumspecific regularization properties that distinguish this approach from classical methods.