Search Results (7)

In the Standard Model, ad hoc hypotheses assume the existence of three generations of point-like leptons and quarks, which possess a point-like structure and follow the Dirac equation involving four anti-commutative matrices. In this work, we consider the sedenion hypercomplex algebra as an extension of the Standard Model and show its close link to SU(5), which is the underlying symmetry group for the grand unification theory (GUT). We first consider the direct-product quaternion model and the eight-element octonion algebra model. We show that neither the associative quaternion model nor the non-associative octonion model could generate three fermion generations. Instead, we show that the sedenion model, which contains three octonion sub-algebras, leads naturally to precisely three fermion generations. Moreover, we demonstrate the use of basis sedenion operators to construct twenty-four 5 × 5 generalized lambda matrices representing SU(5) generators, in analogy to the use of octonion basis operators to generate Gell-Mann’s eight 3 × 3 lambda-matrix generators for SU(3). Thus, we provide a link between the sedenion algebra and Georgi and Glashow’s SU(5) GUT model that unifies the electroweak and strong interactions for the Standard Model’s elementary particles, which obey SU(3)$\otimes$SU(2)$\otimes$U(1) symmetry. Full article

In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure. Full article

This article spanned a new, consistent framework for production, archiving, and provision of analysis ready data (ARD) from multi-source and multi-temporal satellite acquisitions and an subsequent image fusion. The core of the image fusion was an orthogonal transform of the reflectance channels from optical sensors on hypercomplex bases delivered in Kennaugh-like elements, which are well-known from polarimetric radar. In this way, SAR and Optics could be fused to one image data set sharing the characteristics of both: the sharpness of Optics and the texture of SAR. The special properties of Kennaugh elements regarding their scaling—linear, logarithmic, normalized—applied likewise to the new elements and guaranteed their robustness towards noise, radiometric sub-sampling, and therewith data compression. This study combined Sentinel-1 and Sentinel-2 on an Octonion basis as well as Sentinel-2 and ALOS-PALSAR-2 on a Sedenion basis. The validation using signatures of typical land cover classes showed that the efficient archiving in 4 bit images still guaranteed an accuracy over 90% in the class assignment. Due to the stability of the resulting class signatures, the fuzziness to be caught by Machine Learning Algorithms was minimized at the same time. Thus, this methodology was predestined to act as new standard for ARD remote sensing data with an subsequent image fusion processed in so-called data cubes. Full article

In the 2-spinor formalism, the gravity can be dealt with curvature spinors with four spinor indices. Here we show a new effective method to express the components of curvature spinors in the rank-2 $4\times 4$ tensor representation for the gravity in a locally inertial frame. In the process we have developed a few manipulating techniques, through which the roles of each component of Riemann curvature tensor are revealed. We define a new algebra ‘sedon’, the structure of which is almost the same as sedenion except for the basis multiplication rule. Finally we also show that curvature spinors can be represented in the sedon form and observe the chiral structure in curvature spinors. A few applications of the sedon representation, which includes the quaternion form of differential Binanchi identity and hand-in-hand couplings of curvature spinors, are also presented. Full article

By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary “i” should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the two-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotation. Full article

Given a 2^N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6 , we first observe that the multiplication table of its imaginary units e_a , 1 ≤ a ≤ 2^N - 1 , is encoded in the properties of the projective space PG(N - 1,2) if these imaginary units are regarded as points and distinguished triads of them {e_a, e_b , e_c} , 1 ≤ a < b < c ≤ 2^N - 1 and e_ae_b = ±e_c , as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b ≠ c . Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG(N - 1,2) , the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration C_N; in particular, C₃ (octonions) is isomorphic to the Pasch (6₂, 4₃) -configuration, C₄ (sedenions) is the famous Desargues (10₃) -configuration, C₅ (32-nions) coincides with the Cayley-Salmon (15₄, 20₃) -configuration found in the well-known Pascal mystic hexagram and C₆ (64-nions) is identical with a particular (21₅, 35₃) -configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic C_N leads to a conjecture that C_N is isomorphic to a combinatorial Grassmannian of type G₂(N + 1). Full article