Orthonormal Basis Sets in Clifford Algebras

LoRa, a low-power long-range physical-layer communication method, is one of the emerging technologies in low-power wide-area networks (LPWANs). In LoRa, a linearly varying-frequency up chirp and its cyclic shifts act as an orthonormal basis set for message representation. The number of orthonormal basis functions dictates the achievable bit rate of a LoRa system. This article aims to increase the achievable data rate of a conventional LoRa system by adding a set of new orthonormal basis functions to the existing set. Specifically, we use a linearly varying-frequency down chirp and its cyclic shifts to generate the second set of orthonormal basis functions in the proposed scheme, named slope-shift-keying LoRa (SSK-LoRa) modulation. We also develop both low-complexity optimum coherent and noncoherent detection algorithms for the proposed SSK-LoRa modulation. We then obtain closed-form analytical and tight approximations for the bit and symbol error probabilities of noncoherent detection in a Rayleigh fading environment. Our proposed SSK-LoRa scheme is shown to outperform the state-of-the-art LoRa-based modulation schemes.

Clifford algebra is introduced as a theoretical foundation for network topology expression and algorithm construction. Network nodes are coded with basis vectors in a vector space , and the edges and k‐walk routes can be expressed by 2‐blades and k‐blades, respectively, in the Clifford algebra Cl(n,0). The topologies among nodes, edges, and routes of networks can be directly calculated, and the network routes can be extended and traversed with oriented join products. The network algorithm construction processes based on Clifford algebra are instantiated by the single source shortest path algorithm. The experimental results on different scale random networks suggest that Clifford algebra is suited for network expression and relation computation. The Clifford algebra‐based shortest path algorithm is vivid and clear in geometric meaning and has great advantage on temporal and spatial complexity. Copyright © 2013 John Wiley & Sons, Ltd.

Unital quadratic Jordan algebras J(q, I) determined by nondegenerate quadratic forms with basepoints over a field axe called full Jordan Clifford algebras. In characteristic 2 they have ample outer ideals which are also simple; they come in 3 sizes, tiny, small, and large, where the large are full Clifford algebras but the tiny and small algebras are lacking some of their parts. The simple algebras played a role in Zelmanov's solution of the Burnside Problem. In this paper we will analyze these in more detail, determining their centroids and their local algebras; this is important in the classification of prime Jordan triples of Clifford type in arbitrary char-acterstics. In addition we make a careful charcterization of the tiny, small, and large Clifford algebras. We use this to straighten one or two missteps in a proof from the classification of simple algebras. An important role in our characterization is played by commutators, and we describe the Jor­dan commutator products and Bergmann formulas for Clifford algebras in general.

On the application of complete orthonormal atomic basis sets to x-rays diffracted by a thermally excited molecule

Plane waves have unparalleled simplicity and systematic convergence by a single monotonic parameter, the energy cutoff, but they are limited to speriodic systems and require Fourier transforms that scale as N(2)logN, where N is the number of atoms. Real-space methods for order-N scaling are computationally complex and convergence depends on several variables. Here we introduce and demonstrate "Lagrange functions" as a family of analytical, complete, and orthonormal basis sets that are suitable for efficient, accurate, real-space, order-N electronic structure calculations. Convergence is controlled by a single monotonic parameter, the dimension of the basis set, and computational complexity is lower than that of plane waves.

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras.

Measurement of the angular power spectrum of the cosmic microwave background is most often based on a spherical harmonic analysis of the observed temperature anisotropies. Even if all-sky maps are obtained, however, it is likely that the region around the Galactic plane will have to be removed as a result of its strong microwave emissions. The spherical harmonics are not orthogonal on the cut sky, but an orthonormal basis set can be constructed from a linear combination of the original functions. Previous implementations of this technique, based on Gram—Schmidt orthogonalization, were limited to maximum Legendre multipoles of lmax≲50, as they required all the modes have appreciable support on the cut-sky, whereas for large lmax the fraction of modes supported is equal to the fractional area of the region retained. This problem is solved by using a singular value decomposition to remove the poorly supported basis functions, although the treatment of the non-cosmological monopole and dipole modes necessarily becomes more complicated. A further difficulty is posed by computational limitations — orthogonalization for a general cut requires operations and storage and so is impractical for lmax≳200 at present. These problems are circumvented for the special case of constant (Galactic) latitude cuts, for which the storage requirements scale as and the operations count scales as . Less clear, however, is the stage of the data analysis at which the cut is best applied. As convolution is ill-defined on the incomplete sphere, beam-deconvolution should not be performed after the cut and, if all-sky component separation is as successful as simulations indicate, the Galactic plane should probably be removed immediately prior to power spectrum estimation.

The quantumness of Dirac particles for quantized fields in a dilatonic black hole is estimated by means of quantum channel. A general Bloch vector representation of quantum channel in black hole spacetimes beyond single mode approximation is developed. The nonclassicality of Dirac particles can be measured by the minimization of quantum coherence over all orthonormal basis sets. The quantumness of the channel decreases as the dilaton field increases. The interplay between the external reservoir noise and dilaton black hole on the dynamical behavior of quantum coherence and steerability is investigated in the Pauli basis. The external environment is modeled by a random telegraph noise channel. The monotonous decay of quantum nonlocality occurs in the weak coupling case. The degradation and revival of quantum nonlocality are observed in the strong coupling condition. It is found that quantum fluctuation effects of the external reservoir can protect quantum coherence and steerability from the information loss of the black hole.

In this chapter, the classification and representation of Clifford algebras are introduced and discussed. Some important theorems regarding the structure of Clifford algebras are presented; later on, these are used for the classification and construction of the representations of the Clifford algebras as well. The periodicity theorems and their generalisation are demonstrated and exemplified. The Clifford algebras up to dimension 8 in all metric signatures are explicitly derived; hence, with the aid of the periodicity theorem, all Clifford algebras of finite dimension are classified. In addition, procedures to explicitly build Clifford algebra matrix representations are presented, through the idempotent structure.

It is proved that every finite-dimensional algebra is embeddable in a simple finite-dimensional algebra (a suitable isotope of a matrix algebra). An isotope of the 2nd order matrix algebra over an infinite extension of the ground field may contain a trivial ideal. Every one-sided isotope of a simple unital alternative or Jordan algebra is a simple algebra. Besides, any isotope of a central simple non-Lie Maltsev algebra of characteristic other than 2 and 3 is a simple algebra. But an isotope of a simple Jordan algebra of the symmetric bilinear form on the infinite dimensional space may contain a trivial ideal.

The complete reduction of simple Clifford algebras is spanned by matrix bases. Considered as generators of an algebra the basis elements pij satisfy the product relations pijpkl= δjkpil. It follows that subsets of pij with subscript or superscript fixed span minimal left or right ideals. The expression of the basis elements pij in the usual multivector basis is derived from isomorphic matrix algebras. Spinors and Hermitian conjugate spinors are defined as members of minimal left and right ideals. Spinor scalar products are defined, these scalar products are in fact scalar multiples of an idempotent.

The hidden universal Clifford algebra structure of M4(R) is ambiguous in the sense that this matrix algebra is the universal geometric algebra belonging to each of the real four-dimensional quadratic vector spaces R1,3 and R2,2. As a non-universal Clifford algebra, however, M4(R) is a geometric algebra of the unique five-dimensional quadratic vector space R2,3.

Multi-level basis selection of wavelet packet decomposition tree for heart sound classification

The paper develops a new heteroskedasticity and autocorrelation robust test in a time series setting. The test is based on a series long‐run variance matrix estimator that involves projecting the time series of interest onto a set of orthonormal bases and using the sample variance of the projection coefficients as the long‐run variance estimator. When the number of orthonormal bases K is fixed, a finite‐sample‐corrected Wald statistic converges to a standard F distribution. When K grows with the sample size, the usual uncorrected Wald statistic converges to a chi‐square distribution. We show that critical values from the F distribution are second‐order correct under the conventional increasing smoothing asymptotics. Simulations show that the F approximation is more accurate than the chi‐square approximation in finite samples.

Our purpose in this chapter is to introduce three algebras of importance in the theory of quadratic forms, the Clifford algebra, the quaternion algebra, and the Hasse algebra. The Clifford algebra will be developed from first principles and its main use for us will be in the definition of an invariant called the spinor norm. The quaternion algebra and the Hasse algebra play an important role in the arithmetic theory of quadratic forms. The definition of the Hasse algebra depends on some of the structure theory of central simple algebras, in particular it needs Wedderburn’s theorem and the theory of similarity of algebras that is normally used in defining the Brauer group. We have therefore included a proof of Wedderburn’s theorem and some of its consequences. Also included as a convenience to the reader is a brief discussion of the tensor product of finite dimensional vector spaces1.