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Abstract: Abstract This paper addresses the study of the complexity of products in geometric algebra. More specifically, this paper focuses on both the number of operations required to compute a product, in a dedicated program for example, and the complexity to enumerate these operations. In practice, studies on time and memory costs of products in geometric algebra have been limited to the complexity in the worst case, where all the components of the multivector are considered. Standard usage of Geometric Algebra is far from this situation since multivectors are likely to be sparse and usually full homogeneous, i.e., having their non-zero terms over a single grade. We provide a complete computational study on the main Geometric Algebra products of two full homogeneous multivectors, that are outer, inner, and geometric products. We show tight bounds on the number of the arithmetic operations required for these products. We also show that some algorithms reach this number of arithmetic operations. PubDate: 2022-11-14
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Abstract: Abstract In this paper, we consider the actions of affine Yangian and \(W_{1+\infty }\) algebra on three cases of symmetric functions. The first one is Schur functions of 2D Young diagrams. It is known that affine Yangian and \(W_{1+\infty }\) algebra can be represented by 1 Boson field with center 1 in this case. The second case is the symmetric functions \(Y_\lambda ({\mathbf{p}})\) of 2D Young diagrams which we defined. They become Jack polynomials when \(h_1=h, h_2=-h^{-1}\) . In this case affine Yangian and \(W_{1+\infty }\) algebra can be represented by 1 Boson field with center \(-h_\epsilon /\sigma _3\) . The third case is 3-Jack polynomials of 3D Young diagrams who have at most N layers in z-axis direction. We show that in this case affine Yangian and \(W_{1+\infty }\) algebra can be represented by N Boson field with center \(-h_\epsilon /\sigma _3\) . At each case, we define the Fermions \(\Gamma _m\) and \(\Gamma _m^*\) and use them to represent the \(W_{1+\infty }\) algebra. PubDate: 2022-11-14
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Abstract: Abstract In this paper, we first define supersymmetric Schur Q-functions and give their vertex operators realization. By means of the vertex operator, we obtain a series of non-linear partial differential equations of infinite order, called the super BKP hierarchy and the super BKP hierarchy governs the supersymmetric Schur Q-functions as the tau functions. Moreover, we prove that supersymmetric Schur Q-functions can be viewed as compound Schur Q-functions. This means that we can study the properties of supersymmetric Schur Q-functions according to Schur Q-functions, such as their applications in representation theory. PubDate: 2022-11-12
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Abstract: Abstract In this paper, we prove the sharp Pitt’s inequality for a generalized Clifford-Fourier transform which is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra. As an application, the Beckner’s logarithmic uncertainty principle for the Clifford-Fourier transform is established. PubDate: 2022-11-12
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Abstract: Abstract Most modern algorithms use convolutional neural networks to classify image data of different kinds. While this approach is a good method to differentiate between natural images of objects, big datasets are needed for the training process. Another drawback is the demand for high computational power. We introduce a new approach which involves classic feature vectors with structural information based on higher order Riesz transform. Following this way we create a framework specialized for texture data like images of rock cross-sections. The key advantages are faster computations and more versatile choices of the underlying machine learning tools while maintaining a comparable accuracy in comparison with state-of-the-art algorithms. PubDate: 2022-10-07
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Abstract: Abstract We generalize the space-time Fourier transform (SFT) (Hitzer in Adv Appl Clifford Algebras 17(3):497–517, 2007) to a special affine Fourier transform (SASFT, also known as offset linear canonical transform) for 16-dimensional space-time multivector Cl(3, 1)-valued signals over the domain of space-time (Minkowski space) \(\mathbb {R}^{3,1}.\) We establish how it can be computed in terms of the SFT, and introduce its properties of multivector coefficient linearity, shift and modulation, inversion, Rayleigh (Parseval) energy theorem, partial derivative identities, a directional uncertainty principle and its specialization to coordinates. All important results are proven in full detail. PubDate: 2022-10-07
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Abstract: Abstract Zeon algebras have proven to be useful for enumerating structures in graphs, such as paths, trails, cycles, matchings, cliques, and independent sets. In contrast to an ordinary graph, in which each edge connects exactly two vertices, an edge (or, “hyperedge”) can join any number of vertices in a hypergraph. Hypergraphs have been used for problems in biology, chemistry, image processing, wireless networks, and more. In the current work, zeon (“nil-Clifford”) and “idem-Clifford” graph-theoretic methods are generalized to hypergraphs. In particular, zeon and idem-Clifford methods are used to enumerate paths, trails, independent sets, cliques, and matchings in hypergraphs. An approach for finding minimum hypergraph transversals is developed, and zeon formulations of some open hypergraph problems are presented. PubDate: 2022-10-07
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Abstract: Abstract In this note, we first derive a Schödinger uncertainty relation for any pair of quaternionic observables and a mixed state. Then, the Wigner–Yanase skew information is introduced in the quaternion setting. Based on the skew information, we establish a new quantum uncertainty inequality for the non-Hermitian quaternionic observables. PubDate: 2022-10-06
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Abstract: Abstract In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras \(\mathcal {G}_{p,q}\) of vector space of dimension \(n=p+q\) . We present basis-free formulas for all characteristic polynomial coefficients in the cases \(n\le 6\) , alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case \(n=4\) , and one of the formulas in the case \(n=5\) . We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases \(n\le 5\) . The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation. PubDate: 2022-09-09 DOI: 10.1007/s00006-022-01232-0
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Abstract: Abstract This paper proposes to go beyond the Einstein General Relativity theory in a noncommutative geometric framework. As a first step, we rewrite the General Relativity theory intrinsically (coordinate-free formulation). As a second step, we rewrite the first and the second Bianchi identities, including torsion, within minimal algebraic hypotheses. Then, in order to extend the General Relativity theory for dealing with noncommutative scalar fields on a manifold \(\mathcal {M}\) , we are forced to adapt the definition of vector fields and connections. It leads to the consideration of one associative product p of vector fields and four derivations ( \(\nabla \) : (vector, vector) to vector, \(\partial \) : (vector, scalar) to scalar, \(\delta \) : (scalar, vector) to scalar, \(\mathcal {C}\) : (scalar, scalar) to scalar). At last, the particular case where scalars are a Clifford numbers motivates future investigations towards a common writing of the Einstein field equations and the Dirac equation. PubDate: 2022-09-05 DOI: 10.1007/s00006-022-01233-z
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Abstract: Abstract A number of new Lévy-Leblond type equations admitting four component spinor solutions have been proposed. The pair of linearized equations thus obtained in each case lead to Hamiltonians with characteristic features like L-S coupling and supersymmetry. The relevant momentum operators have often been understood in terms of Clifford algebraic bases producing Schrödinger Hamiltonians with L-S coupling. As for example, Hamiltonians representing Rashba effect or three dimensional harmonic oscillator have been constructed. Moreover the supersymmetric nature of one dimensional harmonic oscillator emerges naturally in this formulation. PubDate: 2022-09-05 DOI: 10.1007/s00006-022-01239-7
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Abstract: Abstract Let p and q be polynomials with degree 2 over an arbitrary field \(\mathbb {F}\) , and M be a square matrix over \(\mathbb {F}\) . Thanks to the study of an algebra that is deeply connected to quaternion algebras, we give a necessary and sufficient condition for M to split into \(A+B\) for some pair (A, B) of square matrices over \(\mathbb {F}\) such that \(p(A)=0\) and \(q(B)=0\) , provided that no eigenvalue of M splits into the sum of a root of p and a root of q. Provided that \(p(0)q(0) \ne 0\) and no eigenvalue of M is the product of a root of p with a root of q, we also give a necessary and sufficient condition for M to split into AB for some pair (A, B) of square matrices over \(\mathbb {F}\) such that \(p(A)=0\) and \(q(B)=0\) . In further articles, we will complete the study by lifting the assumptions on the eigenvalues of M. PubDate: 2022-09-04 DOI: 10.1007/s00006-022-01241-z
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Abstract: Abstract In the present self-contained paper, we want, first, to construct a fundamental diagram, called (S.C), in homage to Carl Siegel and I. Satake that connects the following groups: \(\mathrm {SU}(m,m)\) , \(\mathrm {SO}^{*}(2m)\) , \(\mathrm {Sp}(2m,\mathbb {R}),\) \(\mathrm {Sp}(4m,\mathbb {R})\) , \(\mathrm {SO}^{*}(4m)\) . Then, we define and study three Clifford algebras related to that diagram. First, we consider the morphism from \(\mathrm {Sp}(2m,\mathbb {R})\) into \(\mathrm {SU}(m,m)\) , shown in the construction of the diagram (S.C.). Then, we define a Clifford algebra \(Cl^{m,m}\) , naturally associated with the group \(\mathrm {U}(m,m).\) Let (E, b) be an m-dimensional skew-hermitian space over \(\mathbb {H}\) . For any \(x,y\in E,\) write \(b(x,y)=h(x,y)+ja(x,y).\) It is well known that h is a skew-hermitian complex form on \(\mathbb {E}_{2m}\) , the complex 2m-dimensional vector space underlying E, and a is a symmetric bilinear complex form on \(\mathbb {E}_{2m}\) . We proved previously in [4] that the special unitary group \(\mathrm {SU}(E,b)\) of a skew-hermitian \(\mathbb {H}\) -right vector space (E, b), m-dimensional over \(\mathbb {H}\) , can be identified with the group \(\mathrm {SO}^{*}(2m)\) defined by E. Cartan. We define a real Clifford algebra, namely \(Cl_{\mathbb {R}}^{*}(2m),\) whose complexified algebra is \(C_{2m}^{+}(\mathbb {E}_{2m},a)\) , the even complex Clifford algebra associated with a. Both algebras are associated with the geometry of the skew-hermitian \(\mathbb {H}\) -space (E, b). Let \(V=(\mathbb {R}^{2m},\mathrm {Sp}(2m,\mathbb {R}))\) be the standard model of a real symplectic space. We present some connections between the geometry of V and the algebras \(Cl^{m,m}\) , \(C_{2m}^{+}(\mathbb {E}_{2m},a)\) , \(Cl_{\mathbb {R}}^{*}(2m)\) . The last section wants ... PubDate: 2022-09-03 DOI: 10.1007/s00006-022-01223-1
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Abstract: Abstract There is increasing demand for multi-level declassification of geographic vector field data in the big data era. Different from traditional encryption, declassification does not aim at making the original data unavailable through perturbation and transformation. During declassification process, the general geospatial features are usually retained but the detailed information is hidden from the perspective of data security. Furthermore, when faced with different levels of confidentiality, different levels of declassification are needed. In this paper, A declassification and reversion method with multi-level schemes is realized under the geometric algebra (GA) framework. In our method, the geographic vector field data is uniformly expressed as a GA object. Then, the declassification methods are proposed for vector field data with the rotor operator and perturbation operator. The declassification methods can progressively hide the detailed information of the vector field by vector rotating and vector perturbating. To make our method more unified and adaptive, a GA declassification operator is also constructed to realize the declassification computing of geographic vector field data. Our method is evaluated quantitatively by comparing the numerical and structure characterization of the declassification results with the original data. Divergence and curl calculating results are also compared to evaluate the reanalysis ability of the declassification results. Experiments have shown that our method can perform effective multi-level controls and has good randomness and a high degree of freedom in numerical and structure characteristics of geophysical vector field data. The method can well capture the application needs of geographic vector field data in data disclosure, secure transmission, encapsulation storage, and other aspects. PubDate: 2022-09-02 DOI: 10.1007/s00006-022-01229-9
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Abstract: Abstract Real geometric algebras distinguish between space and time; complex ones do not. Space-times can be classified in terms of number n of dimensions and metric signature s (number of spatial dimensions minus number of temporal dimensions). Real geometric algebras are periodic in s, but recursive in n. Recursion starts from the basis vectors of either the Euclidean plane or the Minkowskian plane. Although the two planes have different geometries, they have the same real geometric algebra. The direct product of the two planes yields Hestenes’ space-time algebra. Dimensions can be either open (for space-time) or closed (for the electroweak force). Their product yields the eight-fold way of the strong force. After eight dimensions, the pattern of real geometric algebras repeats. This yields a spontaneously expanding space-time lattice with the physics of the Standard Model at each node. Physics being the same at each node implies conservation laws by Noether’s theorem. Conservation laws are not pre-existent; rather, they are consequences of the uniformity of space-time, whose uniformity is a consequence of its recursive generation. PubDate: 2022-08-28 DOI: 10.1007/s00006-022-01235-x
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Abstract: Abstract In the Clifford algebra setting the present study develops three reproducing kernel Hilbert spaces of the Paley–Wiener type, namely the Paley–Wiener spaces, the Hardy spaces on strips, and the Bergman spaces on strips. In particular, we give spectrum characterizations and representation formulas of the functions in those spaces and estimation of their respective reproducing kernels. PubDate: 2022-08-27 DOI: 10.1007/s00006-022-01240-0
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Abstract: Abstract Sylvester-like matrix equations are encountered in many areas of control engineering and applied mathematics. In this paper, we construct some necessary and sufficient conditions for the system of Hermitian mixed type generalized Sylvester matrix equations to have a solution. The closed form formula to compute the general solution is also established when solvability conditions are satisfied. An algorithm and a numerical example are provided to validate our findings. PubDate: 2022-08-26 DOI: 10.1007/s00006-022-01222-2
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Abstract: Abstract The hyperbolic Möbius transformations, which have been defined and proved to be hyperbolic conformal in Golberg and Luna-Elizarrarás (Math Methods Appl Sci 2020, https://doi.org/10.1002/mma.7109), are generalizations of Möbius transformations in complex space \({\mathbb {C}}(i) \) and hyperbolic space \({\mathbb {D}} \) to multidimensional hyperbolic space \({\mathbb {D}}^n \) . In this paper, we study the hyperbolic Möbius transformation in bicomplex space \( {{\mathbb {B}}}{{\mathbb {C}}} \) isomorphic to \({\mathbb {D}}^2 \) in detail, present a conjugacy classification according to the number of fixed points in \(SL(2,{{\mathbb {B}}}{{\mathbb {C}}})\) , and detailedly prove that the cross-ratio is invariant under hyperbolic Möbius transformations. Furthermore, the present paper generalizes the classical results, which have closed relation with fixed points and cross-ratios, to \( {{\mathbb {B}}}{{\mathbb {C}}} \) and may give new energy for the development of hyperbolic Möbius groups. PubDate: 2022-08-25 DOI: 10.1007/s00006-022-01231-1
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Abstract: Abstract This paper presents a three-step program for extension of functions of complex analysis to the biquaternions by means of Cauchy’s integral formula: I. Investigate biquaternion bases consisting of roots of \(-1\) . A complex valued standard function (standardization factor) determines roots of \(-1\) . A root of \(-1\) with a non-zero imaginary part, can uniquely determine a biquaternion ortho-standard basis. II. A single reference basis element determines two subspaces, one the span of scalars and the reference element, the other pure vector biquaternions orthogonal to the reference. The subspaces represent the distinct parts of the generalized Cayley-Dickson form. The Peirce decomposition projects into two subspaces: one is the span of the related idempotents and the other of the nilpotents. III. Using invertible elements in each of these subspaces, biquaternion functional extensions of holomorphic functions follow by Cauchy’s integral formula. Extensions retain analyticity in each biquaternion component. Cauchy integral formula uses separate idempotent and nilpotent representations of biquaternion reciprocals to define holomorphic function extensions. The Peirce projections allow extension to all viable biquaternions. PubDate: 2022-08-19 DOI: 10.1007/s00006-022-01238-8
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