Explicit Matrix Representations Research Articles

A new division algebra representation of E6 from E8

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.

Mixed-Type Hypergeometric Bernoulli–Gegenbauer Polynomials

In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We focus our attention on some algebraic and differential properties of this class of polynomials, including its explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relations connecting it with the hypergeometric Bernoulli polynomials. Furthermore, we show that unlike the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the mixed-type hypergeometric Bernoulli–Gegenbauer polynomials do not fulfill either Hanh or Appell conditions.

Implicit Contact Dynamics Modeling With Explicit Inertia Matrix Representation for Real-Time, Model-Based Control in Physical Environment.

Model-based control has great potential for use in real robots due to its high sampling efficiency. Nevertheless, dealing with physical contacts and generating accurate motions are inevitable for practical robot control tasks, such as precise manipulation. For a real-time, model-based approach, the difficulty of contact-rich tasks that requires precise movement lies in the fact that a model needs to accurately predict forthcoming contact events within a limited length of time rather than detect them afterward with sensors. Therefore, in this study, we investigate whether and how neural network models can learn a task-related model useful enough for model-based control, that is, a model predicting future states, including contact events. To this end, we propose a structured neural network model predictive control (SNN-MPC) method, whose neural network architecture is designed with explicit inertia matrix representation. To train the proposed network, we develop a two-stage modeling procedure for contact-rich dynamics from a limited number of samples. As a contact-rich task, we take up a trackball manipulation task using a physical 3-DoF finger robot. The results showed that the SNN-MPC outperformed MPC with a conventional fully connected network model on the manipulation task.

Compact explicit matrix representations of the flexoelectric tensor and a graphic method for identifying all of its rotation and reflection symmetries

Flexoelectricity is an electromechanical phenomenon produced in a dielectric material, with or without centrosymmetric microstructure, undergoing a non-uniform strain. It is characterized by the fourth-order flexoelectric tensor, which links the electric polarization vector with the gradient of the second-order strain tensor. Our previous work [H. Le Quang and Q.-C. He, Proc. R. Soc. A 467, 2369 (2011)] solved the fundamental theoretical problem of determining the number and types of all rotational symmetries that the flexoelectric tensor can exhibit. In the present study, compact explicit matrix representations of the flexoelectric tensor are provided so as to facilitate the use of it with any possible rotational symmetry. The number and types of all reflection symmetries that the flexoelectric tensor can have are also determined. To identify the rotational symmetry and reflection symmetry of a given flexoelectric tensor, a simple and efficient graphic method based on the concept of pole figures is presented and illustrated.

A Note on the Representation of Clifford Algebras

In this note we construct explicit complex and real faithful matrix representations of the Clifford algebras $\Cl_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. In the cases $p+q=4m$, the representation is unique in equivalent sense, and the $1+3$ dimensional space-time corresponds to the simplest and best case. Besides, the relation between the curvilinear coordinate frame and the local orthonormal basis in the curved space-time is discussed in detail, the covariant derivatives of the spinor and tensors are derived, and the connection of the orthogonal basis in tangent space is calculated. These results are helpful for both theoretical analysis and practical calculation. The basis matrices are the faithful representation of Clifford algebras in any $p+q$ dimensional Minkowski space-time or Riemann space, and the Clifford calculus converts the complicated relations in geometry and physics into simple and concise algebraic operations. Clifford numbers over any number field $\mathbb{F}$ expressed by this matrix basis form a well-defined $2^n$ dimensional hypercomplex number system. Therefore, we can expect that Clifford algebras will complete a large synthesis in science.

A computer code for topological quantum spin systems over triangulated surfaces

We derive explicit closed-form matrix representations of Hamiltonians drawn from tensored algebras, such as quantum spin Hamiltonians. These formulas enable us to soft-code generic Hamiltonian systems and to systematize the input data for uniformly structured as well as for un-structured Hamiltonians. The result is an optimal computer code that can be used as a black box that takes in certain input files and returns spectral information about the Hamiltonian. The code is tested on Kitaev’s toric model deployed on triangulated surfaces of genus 0 and 1. The efficiency of our code enables these simulations to be performed on an ordinary laptop. The input file corresponding to the minimal triangulation of genus 2 is also supplied.

Explicit Matrix Representation for the Hamiltonian of the One Dimensional Spin 1/2 Ising Model in Mutually Orthogonal External Magnetic Fields

We derive an explicit matrix representation for the Hamiltonian of the Ising model in mutually orthogonal external magnetic fields, using as basis the eigenstates of a system of non-interacting spin 1/2 particles in external magnetic fields. We subsequently apply our results to obtain an analytical expression for the ground state energy per spin, to the fourth order in the exchange integral, for the Ising model in perpendicular external fields.

GPU-accelerated simulations of isolated black holes

We present a port of the numerical relativity code SpEC which is capable of running on NVIDIA GPUs. Since this code must be maintained in parallel with SpEC itself, a primary design consideration is to perform as few explicit code changes as possible. We therefore rely on a hierarchy of automated porting strategies. At the highest level we use TLoops, a C++ library of our design, to automatically emit CUDA code equivalent to tensorial expressions written into C++ source using a syntax similar to analytic calculation. Next, we trace out and cache explicit matrix representations of the numerous linear transformations in the SpEC code, which allows these to be performed on the GPU using pre-existing matrix-multiplication libraries. We port the few remaining important modules by hand. In this paper we detail the specifics of our port, and present benchmarks of it simulating isolated black hole spacetimes on several generations of NVIDIA GPU.

A New Perspective on the Two-Dimensional Fractional Fourier Transform and Its Relationship with the Wigner Distribution

The fractional Fourier transform \(F_{\theta }(w)\) with an angle \(\theta \) of a function f(t) is a generalization of the standard Fourier transform and reduces to it when \(\theta =\pi /2. \) It has many applications in signal processing and optics because of its close relations with a number of time-frequency representations. It is known that the Wigner distribution of the fractional Fourier transform \(F_{\theta }(w)\) may be obtained from the Wigner distribution of f by a two-dimensional rotation with the angle \(\theta \) in the \(t-w\) plane The fractional Fourier transform has been extended to higher dimensions by taking the tensor product of one-dimensional transforms; hence, resulting in a transform in several but separable variables. It has been shown that the Wigner distribution of the two-dimensional fractional Fourier transform \(F_{\theta ,\phi }(v,w)\) may be obtained from the Wigner distribution of f(x, y) by a simple four-dimensional rotation with the angle \(\theta \) in the \(x-y\) plane and the angle \(\phi \) in the \(v-w\) plane. The aim of this paper is two-fold: (1) To introduce a new definition of the two-dimensional fractional Fourier transform that is not a tensor product of two copies of one-dimensional transforms. The new transform, which is more general than the one that exists in the literature, uses a relatively new family of Hermite functions, known as Hermite functions of two complex variables. (2) To give an explicit matrix representation of a four-dimensional rotation that verifies that the Wigner distribution of the new two-dimensional fractional Fourier transform \(F_{\theta ,\phi }(v,w)\) may be obtained from the Wigner distribution of f(x, y) by a four-dimensional rotation. The matrix representation is more general than the one for the tensor product case and it corresponds to a four-dimensional rotation with two planes of rotations, one with the angle \((\theta +\phi )/2\) and the other with the angle \((\theta -\phi )/2\).

Hierarchical Matrix Approximation for Kernel-Based Scattered Data Interpolation

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)...

Chern–Simons theory encoded on a spin chain

.We construct a one-dimensional (1D) spin chain Hamiltonian with generic interactions and prove that the thermal correlation functions of the model admit an explicit random matrix representation. As an application of the result, we show how the observables of U(N) Chern–Simons theory on S3 can be reproduced with the thermal correlation functions of the 1D spin chain, which is of the XX type, with a suitable choice of exponentially decaying interactions between infinitely many neighbours. We show that for this model, the correlation functions of the spin chain at a finite temperature give the Chern–Simons partition function, quantum dimensions and the full topological S-matrix.

Exact diagonalization of the D-dimensional spatially confined quantum harmonic oscillator

In the existing literature various numerical techniques have been developed to quantize the confined harmonic oscillator in higher dimensions. In obtaining the energy eigenvalues, such methods often involve indirect approaches such as searching for the roots of hypergeometric functions or numerically solving a differential equation. In this paper, however, we derive an explicit matrix representation for the Hamiltonian of a confined quantum harmonic oscillator in higher dimensions, thus facilitating direct diagonalization.

Higher spin extension of cosmological spacetimes in 3D: asymptotically flat behaviour with chemical potentials and thermodynamics

A generalized set of asymptotic conditions for higher spin gravity without cosmological constant in three spacetime dimensions is constructed. They include the most general temporal components of the gauge fields that manifestly preserve the original asymptotic higher spin extension of the BMS$_{3}$ algebra, with the same central charge. By virtue of a suitable permissible gauge choice, it is shown that this set can be directly recovered as a limit of the boundary conditions that have been recently constructed in the case of negative cosmological constant, whose asymptotic symmetries are spanned by two copies of the centrally-extended W$_{3}$ algebra. Since the generalized asymptotic conditions allow to incorporate chemical potentials conjugated to the higher spin charges, a higher spin extension of locally flat cosmological spacetimes becomes naturally included within the set. It is shown that their thermodynamic properties can be successfully obtained exclusively in terms of gauge fields and the topology of the Euclidean manifold, which is shown to be the one of a solid torus, but with reversed orientation as compared with one of the black holes. It is also worth highlighting that regularity of the fields can be ensured through a procedure that does not require an explicit matrix representation of the entire gauge group. In few words, we show that the temporal components of generalized dreibeins can be consistently gauged away, which partially fixes the chemical potentials, so that the remaining conditions can just be obtained by requiring the holonomy of the generalized spin connection along a thermal circle to be trivial. The extension of the generalized asymptotically flat behaviour to the case of spins $s\geq2$ is also discussed.

Zeeman truncation in NMR. I. The role of operator commutation

Hamiltonians are of pivotal importance for describing and analyzing NMR experiments. However, the “exact” spin Hamiltonian operators are in practice not utilized, but merely a simplified form referred to either as the “secular,” “Zeeman‐truncated,” or “high‐field” Hamiltonian. It results after accounting for the dominating role of the Zeeman interaction relative to all other, much smaller NMR interactions, such as chemical shifts, through‐bond, or through‐space spin–spin couplings. In this article and the following one, we introduce the Zeeman truncation process to newcomers to NMR by thoroughly reviewing the options available for reducing the full Hamiltonian of a spin interaction to its Zeeman‐truncated counterpart. The present paper considers time‐independent Hamiltonians, where we discuss the criteria for performing truncation, highlighting the role of operator commutation by a simple formalism that is equivalent to application of lowest‐order static perturbation theory. The validity of the approximations are illustrated by examining the explicit matrix representations of the “exact” and “Zeeman truncated” Hamiltonians, considering the NMR interactions relevant for systems of interacting spin‐1/2 nuclei. © 2015 Wiley Periodicals, Inc. Concepts Magn Reson Part A 43A: 91–108, 2015.

Matrix representations for 3D strain-gradient elasticity

The theory of first strain gradient elasticity (SGE) is widely used to model size and non-local effects observed in materials and structures. For a material whose microstructure is centrosymmetric, SGE is characterized by a sixth-order elastic tensor in addition to the classical fourth-order elastic tensor. Even though the matrix form of the sixth-order elastic tensor is well-known in the isotropic case, its complete matrix representations seem to remain unavailable in the anisotropic cases. In the present paper, the explicit matrix representations of the sixth-order elastic tensor are derived and given for all the 3D anisotropic cases in a compact and well-structured way. These matrix representations are necessary to the development and application of SGE for anisotropic materials.

Matrix representation of the double-curl operator for simulating three dimensional photonic crystals

Three dimensional photonic crystals can be modeled by the Maxwell equations as a generalized eigenvalue problem (GEVP). Simulations based on the numerical solutions of the GEVP are used to reveal physical properties and boost innovative applications of photonic crystals. However, to solve these GEVP remains a computational challenge in both timing and accuracy. The GEVP corresponding to the photonic crystals with face centered cubic (FCC) lattice is one of the challenging eigenvalue problems. From a viewpoint of matrix computation, we demonstrate how such obstacles can be overcome. Our main contribution is an explicit matrix representation of the double-curl operator associated with the photonic crystal with FCC lattice. This particular matrix represents the degenerate coefficient matrix of the discrete GEVP obtained by Yee’s scheme. The explicit matrix leads to an eigendecomposition of the degenerate coefficient matrix and then a fast eigenvalue solver. Promising numerical results in terms of timing and accuracy are reported for solving the discrete GEVP arising in three dimensional photonic crystals with various geometric parameters.

Extensions of the Poincaré group

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.

Matrix Representation of Conflict Resolution in Multiple-Decision-Maker Graph Models with Preference Uncertainty

Explicit matrix representations of solution concepts in a graph model of a multiple-decision-maker conflict with preference uncertainty are developed. In a graph model, the relative preferences of each DM over the available states are crucial in determining which states are stable according to any stability definition (solution concept). Unfortunately, it is often difficult to obtain accurate preference information in practical cases, so models that allow preference uncertainty can be very useful. In this work, stability definitions are extended to apply to graph models with this feature. The extension is easiest to implement using the matrix representation of a conflict model, which was developed to ease the coding of logically-defined stability definitions. Another benefit of matrix representation is that it facilitates modification and extension of the definitions.

Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory

AbstractIn the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel–Darboux formulas are presented for the first time.

Ternary algebras and groups

We construct explicitly groups associated to specific ternary algebras which extend the Lie (super)algebras (called Lie algebras of order three). It turns out that the natural variables which appear in this construction are variables which generate the three-exterior algebra. An explicit matrix representation of a group associated to a peculiar Lie algebra of order three is constructed considering matrices with entry which belong to the three exterior algebra.