Using an elementary method, we give a new proof of the all-associativity of octonions. As some applications, the known Taylor theorem is improved, and a new definition and new properties of octonionic determinant are ...Using an elementary method, we give a new proof of the all-associativity of octonions. As some applications, the known Taylor theorem is improved, and a new definition and new properties of octonionic determinant are also obtained.展开更多
The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “...The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial Octonionic geometry embedded in a larger, nonlinear “pilot model” (semi classical) embedding structure. The Penrose suggestion of recycled space time avoiding a “big crunch” is picked as the embedding structure, so as to avoid the “instants” of time issue. Getting Octionic gravity as embedded in a larger, Pilot theory embedding structure may restore Quantum Gravity to its rightful place in early cosmology without the complication of then afterwards “Schrodinger equation” states of the universe, and the transformation of Octonionic gravity to existing space-time is explored via its possible linkage to a new version of the HUP involving metric tensors. We conclude with how specific properties of Octonion numbers algebra influence the structure and behavior of the early-cosmology model. This last point is raised in Section 14, and is akin to a phase transition from Pre-Octonionic geometry, in pre-Planckian space-time, to Octonionic geometry in Planckian space-time. A simple phase transition is alluded to;making this clear is as simple as realizing that Pre-Octonionic is for Pre-Planckian Space-time and Octonionic is for Planckian Space-time. We state that the Standard Model of physics occurs during Planckian Space-time. We also argue that the Standard Model does not apply to Pre Planckian Space-time. This is commensurate with the Octonion number system NOT applying in pre-Planckian space-time, but applying in Plankian space-time. And the last line of Equation (54) gives a minimum time step in pre-Planckian space-time when we do NOT have the Standard Model of physics, or Octonionic Geometry.展开更多
We look at early universe space-time which is characterized by a transition from Pre-Planckian to Planckian space-time. In doing so we also invoke the geometry of Octonionic non-commutative structure and when it break...We look at early universe space-time which is characterized by a transition from Pre-Planckian to Planckian space-time. In doing so we also invoke the geometry of Octonionic non-commutative structure and when it breaks down. Doing so is also equivalent to a speculation given earlier by the author as to the kinetic energy of Pre-Planckian space-time being significantly larger than the Potential energy, which is the opposite of what happens after the onset of Inflation, with the assumption as to how this is justified given in a (Pre- Planckian) Hubble Parameter set as of Equation (16), and we close with a comparison of this proposal with string cosmology, as represented in the 2nd reference in this paper.展开更多
In an explicit, unified, and covariant formulation of an octonion algebra, we study and generalize the electromagnetic chiral fields equations of massive dyons with the split octonionic representation. Starting with 2...In an explicit, unified, and covariant formulation of an octonion algebra, we study and generalize the electromagnetic chiral fields equations of massive dyons with the split octonionic representation. Starting with 2 × 2 Zorn's vector matrix realization of split-octonion and its dual Euclidean spaces, we represent the unified structure of split octonionic electric and magnetic induction vectors for chiral media. As such, in present paper, we describe the chiral parameter and pairing constants in terms of split octonionic matrix representation of Drude-Born-Fedorov constitutive relations. We have expressed a split octonionic electromagnetic field vector for chiral media, which exhibits the unified field structure of electric and magnetic chiral fields of dyons. The beauty of split octonionic representation of Zorn vector matrix realization is that, the every scalar and vector components have its own meaning in the generalized chiral electromagnetism of dyons. Correspondingly, we obtained the alternative form of generalized Proca–Maxwell's equations of massive dyons in chiral media. Furthermore, the continuity equations, Poynting theorem and wave propagation for generalized electromagnetic fields of chiral media of massive dyons are established by split octonionic form of Zorn vector matrix algebra.展开更多
The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the ...The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffier problem is discussed.展开更多
Given two left Oc-analytic functions f, g in some open set Ω of R8, we obtain some sufficient conditions for fg is also left Oc-analytic in Ω. Moreover, we prove?fλ that is a left Oc-analytic function for any const...Given two left Oc-analytic functions f, g in some open set Ω of R8, we obtain some sufficient conditions for fg is also left Oc-analytic in Ω. Moreover, we prove?fλ that is a left Oc-analytic function for any constants λ∈Oc if and only if is a complex Stein-Weiss conjugate harmonic system. Some applications and connections with Cauchy-Kowalewski product are also considered.展开更多
Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. A...Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.展开更多
As it is known, Binomial expansion, De Moivre’s formula, and Euler’s formula are suitable methods for computing the powers of a complex number, but to compute the powers of an octonion number in easy way, we need to...As it is known, Binomial expansion, De Moivre’s formula, and Euler’s formula are suitable methods for computing the powers of a complex number, but to compute the powers of an octonion number in easy way, we need to derive suitable formulas from these methods. In this paper, we present a novel way to compute the powers of an octonion number using formulas derived from the binomial expansion.展开更多
The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “...The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial octonionic geometry embedded in a larger, nonlinear (semi-classical) embedding structure. We detail some of what the quantum HUP is, in terms of deterministic 5-dimensional geometry and show that the projection of 5 dimensions into four is when the octonionic structure kicks in as an emergent gravity phenomenon. The example of such is to consider what would happen if there was an aftermath to a presumed initial causal discontinuous structure, after math being the generation of millions of Planck mass black holes, which would in themselves generate emergent gravity.展开更多
The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4 + 4)-space. Split octonionic representation of SO(4, 4) and Spin(4, 4) groups and the trilinear inva...The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4 + 4)-space. Split octonionic representation of SO(4, 4) and Spin(4, 4) groups and the trilinear invariant form are explicitly written and compared with Clifford algebraic matrix representation. It is noted that the complete algebra of split octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements. Lagrangians on split octonionic fields that generalize Dirac and Maxwell systems are constructed using group invariant forms. It is shown that corresponding equations are related to split octonionic analyticity conditions.展开更多
When the task is to find the roots of octonions or quaternions numbers,we immediately think of Euler’s and De Moivre’s formulas.In this work,we show that the task can be accomplished numerically using Newton-Raphson...When the task is to find the roots of octonions or quaternions numbers,we immediately think of Euler’s and De Moivre’s formulas.In this work,we show that the task can be accomplished numerically using Newton-Raphson method.展开更多
Category is put to work in the non-associative realm in the article.We focus on atypical example of non-associative category.Its objects are octonionic bimodules,morphisms are octonionic para-linear maps,and compositi...Category is put to work in the non-associative realm in the article.We focus on atypical example of non-associative category.Its objects are octonionic bimodules,morphisms are octonionic para-linear maps,and compositions are non-associative in general.The octonionic para-linear map is the main object of octonionic Hilbert theory because of the octonionic Riesz representation theorem.An octonionic para-linear map f is in general not octonionic linear since it subjects to the rule Re(f(px)-pf(x))=0.The composition should be modified as f◎g(x):=f(g(x))-7∑j=1ejRe(f(g(e_(i)x))-f(e_(i)g(x)))j=1 so that it preserves the octonionic para-linearity.In this non-associative category,we introduce the Hom and Tensor functors which constitute an adjoint pair.We establish the Yoneda lemma in terms of the new notion of weak functor.To define the exactness in a non-associative category,we introduce the notion of the enveloping category via a universal property.This allows us to establish the exactness of the Hom functor and Tensor functor.展开更多
The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G2 manifold of very small diameter with G2 being the automorphism group of the octonions.The multidimensional o...The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G2 manifold of very small diameter with G2 being the automorphism group of the octonions.The multidimensional octonion analysis is initiated in this article,which extends the theory of several complex variables,such as the Bochner–Martinelli formula,the theory of non-homogeneous Cauchy–Riemann equations,and the Hartogs principle,to the non-commutative and non-associative realm.展开更多
The three-line theorem on the octonions is obtained, which generalizes the result of J. Peetre and P. Sj?lin from the associative Clifford algebra to non-associative octonion algebra.
In this paper, for rings R, we introduce complex rings C（R）, quaternion rings H（R）, and octonion rings O（R）, which are extension rings of R; R C（R） H（R） O（R）. Our main purpose of this paper is to sho...In this paper, for rings R, we introduce complex rings C（R）, quaternion rings H（R）, and octonion rings O（R）, which are extension rings of R; R C（R） H（R） O（R）. Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius Mgebras and if R is a quasi-Frobenius ring, then C（R） and H（R） are quasi-Frobenius rings and, when Char（R）=2, O（R） is also a quasi-Frobenius ring.展开更多
There are two schools of“measurement-only quantum computation”.The first(Phys.Rev.Lett.86(22),5188-5191(2001))using prepared entanglement(cluster states)and the second(Phys.Rev.Lett.101(1),010501(2008))using collect...There are two schools of“measurement-only quantum computation”.The first(Phys.Rev.Lett.86(22),5188-5191(2001))using prepared entanglement(cluster states)and the second(Phys.Rev.Lett.101(1),010501(2008))using collections of anyons which,according to how they were produced,also have an entanglement pat-tern.We abstract the common principle behind both approaches and find the notion of a graph or even continuous family of equiangular projections.This notion is the leading character in the paper.The largest continuous family,in a sense made pre-cise in Corollary 4.2,is associated with the octonions and this example leads to a universal computational scheme.Adiabatic quantum computation also fits into this rubric as a limiting case:nearby projections are nearly equiangular,so as a gapped ground state space is slowly varied,the corrections to unitarity are small.展开更多
八元数偏移线性正则变换(octonion offset linear canonical transform,OOLCT)作为八元数线性正则变换(octonion linear canonical transform,OLCT)的推广形式,在高维非平稳信号时频调控中具有优势,但八元数非结合性使传统概率统计量定...八元数偏移线性正则变换(octonion offset linear canonical transform,OOLCT)作为八元数线性正则变换(octonion linear canonical transform,OLCT)的推广形式,在高维非平稳信号时频调控中具有优势,但八元数非结合性使传统概率统计量定义失效,且现有研究多聚焦四元数域,缺乏三维OOLCT(3D-OOLCT)域的严谨概率框架。本文将基础概率理论引入3D-OOLCT领域,构建兼容八元数特性的概率体系:首先,定义3D-OOLCT域中八元数值概率密度函数、分布函数、均值及特征函数;其次,证明特征函数定理;最后,通过算例推导验证特定概率密度函数在3D-OOLCT域的特征函数表达式。该研究填补OOLCT域与概率理论结合的空白,完善八元数变换理论,为三维高维随机信号统计分析提供新工具,也为后续相关工程应用奠定数理基础。展开更多
基金Supported in part by the Doctoral Station Grant of Chinese Education Committee (20050574002), P. R. China
文摘Using an elementary method, we give a new proof of the all-associativity of octonions. As some applications, the known Taylor theorem is improved, and a new definition and new properties of octonionic determinant are also obtained.
文摘The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial Octonionic geometry embedded in a larger, nonlinear “pilot model” (semi classical) embedding structure. The Penrose suggestion of recycled space time avoiding a “big crunch” is picked as the embedding structure, so as to avoid the “instants” of time issue. Getting Octionic gravity as embedded in a larger, Pilot theory embedding structure may restore Quantum Gravity to its rightful place in early cosmology without the complication of then afterwards “Schrodinger equation” states of the universe, and the transformation of Octonionic gravity to existing space-time is explored via its possible linkage to a new version of the HUP involving metric tensors. We conclude with how specific properties of Octonion numbers algebra influence the structure and behavior of the early-cosmology model. This last point is raised in Section 14, and is akin to a phase transition from Pre-Octonionic geometry, in pre-Planckian space-time, to Octonionic geometry in Planckian space-time. A simple phase transition is alluded to;making this clear is as simple as realizing that Pre-Octonionic is for Pre-Planckian Space-time and Octonionic is for Planckian Space-time. We state that the Standard Model of physics occurs during Planckian Space-time. We also argue that the Standard Model does not apply to Pre Planckian Space-time. This is commensurate with the Octonion number system NOT applying in pre-Planckian space-time, but applying in Plankian space-time. And the last line of Equation (54) gives a minimum time step in pre-Planckian space-time when we do NOT have the Standard Model of physics, or Octonionic Geometry.
文摘We look at early universe space-time which is characterized by a transition from Pre-Planckian to Planckian space-time. In doing so we also invoke the geometry of Octonionic non-commutative structure and when it breaks down. Doing so is also equivalent to a speculation given earlier by the author as to the kinetic energy of Pre-Planckian space-time being significantly larger than the Potential energy, which is the opposite of what happens after the onset of Inflation, with the assumption as to how this is justified given in a (Pre- Planckian) Hubble Parameter set as of Equation (16), and we close with a comparison of this proposal with string cosmology, as represented in the 2nd reference in this paper.
文摘In an explicit, unified, and covariant formulation of an octonion algebra, we study and generalize the electromagnetic chiral fields equations of massive dyons with the split octonionic representation. Starting with 2 × 2 Zorn's vector matrix realization of split-octonion and its dual Euclidean spaces, we represent the unified structure of split octonionic electric and magnetic induction vectors for chiral media. As such, in present paper, we describe the chiral parameter and pairing constants in terms of split octonionic matrix representation of Drude-Born-Fedorov constitutive relations. We have expressed a split octonionic electromagnetic field vector for chiral media, which exhibits the unified field structure of electric and magnetic chiral fields of dyons. The beauty of split octonionic representation of Zorn vector matrix realization is that, the every scalar and vector components have its own meaning in the generalized chiral electromagnetism of dyons. Correspondingly, we obtained the alternative form of generalized Proca–Maxwell's equations of massive dyons in chiral media. Furthermore, the continuity equations, Poynting theorem and wave propagation for generalized electromagnetic fields of chiral media of massive dyons are established by split octonionic form of Zorn vector matrix algebra.
基金Supported by the National Natural Science Foundation of China(11171298)the Zhejiang Natural Science Foundation of China(Y6110425)
文摘The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffier problem is discussed.
文摘Given two left Oc-analytic functions f, g in some open set Ω of R8, we obtain some sufficient conditions for fg is also left Oc-analytic in Ω. Moreover, we prove?fλ that is a left Oc-analytic function for any constants λ∈Oc if and only if is a complex Stein-Weiss conjugate harmonic system. Some applications and connections with Cauchy-Kowalewski product are also considered.
文摘Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.
文摘As it is known, Binomial expansion, De Moivre’s formula, and Euler’s formula are suitable methods for computing the powers of a complex number, but to compute the powers of an octonion number in easy way, we need to derive suitable formulas from these methods. In this paper, we present a novel way to compute the powers of an octonion number using formulas derived from the binomial expansion.
文摘The quantum gravity problem that the notion of a quantum state, representing the structure of space-time at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having initial octonionic geometry embedded in a larger, nonlinear (semi-classical) embedding structure. We detail some of what the quantum HUP is, in terms of deterministic 5-dimensional geometry and show that the projection of 5 dimensions into four is when the octonionic structure kicks in as an emergent gravity phenomenon. The example of such is to consider what would happen if there was an aftermath to a presumed initial causal discontinuous structure, after math being the generation of millions of Planck mass black holes, which would in themselves generate emergent gravity.
文摘The known equivalence of 8-dimensional chiral spinors and vectors, also referred to as triality, is discussed for (4 + 4)-space. Split octonionic representation of SO(4, 4) and Spin(4, 4) groups and the trilinear invariant form are explicitly written and compared with Clifford algebraic matrix representation. It is noted that the complete algebra of split octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements. Lagrangians on split octonionic fields that generalize Dirac and Maxwell systems are constructed using group invariant forms. It is shown that corresponding equations are related to split octonionic analyticity conditions.
文摘When the task is to find the roots of octonions or quaternions numbers,we immediately think of Euler’s and De Moivre’s formulas.In this work,we show that the task can be accomplished numerically using Newton-Raphson method.
文摘Category is put to work in the non-associative realm in the article.We focus on atypical example of non-associative category.Its objects are octonionic bimodules,morphisms are octonionic para-linear maps,and compositions are non-associative in general.The octonionic para-linear map is the main object of octonionic Hilbert theory because of the octonionic Riesz representation theorem.An octonionic para-linear map f is in general not octonionic linear since it subjects to the rule Re(f(px)-pf(x))=0.The composition should be modified as f◎g(x):=f(g(x))-7∑j=1ejRe(f(g(e_(i)x))-f(e_(i)g(x)))j=1 so that it preserves the octonionic para-linearity.In this non-associative category,we introduce the Hom and Tensor functors which constitute an adjoint pair.We establish the Yoneda lemma in terms of the new notion of weak functor.To define the exactness in a non-associative category,we introduce the notion of the enveloping category via a universal property.This allows us to establish the exactness of the Hom functor and Tensor functor.
基金supported by the National Basic Research Program of China (Grant No. 1999075105)the National Natural Science Foundation of China (Grant No. 10471002)Research Foundation for Doctoral Programm (Grant No. 20050574002)
文摘The Paley-Wiener theorem in the non-commutative and non-associative octonion analytic function space is proved.
基金This work was supported by the NNSF of China(11071230),RFDP(20123402110068).
文摘The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G2 manifold of very small diameter with G2 being the automorphism group of the octonions.The multidimensional octonion analysis is initiated in this article,which extends the theory of several complex variables,such as the Bochner–Martinelli formula,the theory of non-homogeneous Cauchy–Riemann equations,and the Hartogs principle,to the non-commutative and non-associative realm.
基金Research supported by the NNSF (19631080)973 Project of China (1999075105)+1 种基金NSF of Guangdong (030497,020586)NSF of Guangzhou (232)
文摘The three-line theorem on the octonions is obtained, which generalizes the result of J. Peetre and P. Sj?lin from the associative Clifford algebra to non-associative octonion algebra.
文摘In this paper, for rings R, we introduce complex rings C（R）, quaternion rings H（R）, and octonion rings O（R）, which are extension rings of R; R C（R） H（R） O（R）. Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius Mgebras and if R is a quasi-Frobenius ring, then C（R） and H（R） are quasi-Frobenius rings and, when Char（R）=2, O（R） is also a quasi-Frobenius ring.
文摘There are two schools of“measurement-only quantum computation”.The first(Phys.Rev.Lett.86(22),5188-5191(2001))using prepared entanglement(cluster states)and the second(Phys.Rev.Lett.101(1),010501(2008))using collections of anyons which,according to how they were produced,also have an entanglement pat-tern.We abstract the common principle behind both approaches and find the notion of a graph or even continuous family of equiangular projections.This notion is the leading character in the paper.The largest continuous family,in a sense made pre-cise in Corollary 4.2,is associated with the octonions and this example leads to a universal computational scheme.Adiabatic quantum computation also fits into this rubric as a limiting case:nearby projections are nearly equiangular,so as a gapped ground state space is slowly varied,the corrections to unitarity are small.
文摘八元数偏移线性正则变换(octonion offset linear canonical transform,OOLCT)作为八元数线性正则变换(octonion linear canonical transform,OLCT)的推广形式,在高维非平稳信号时频调控中具有优势,但八元数非结合性使传统概率统计量定义失效,且现有研究多聚焦四元数域,缺乏三维OOLCT(3D-OOLCT)域的严谨概率框架。本文将基础概率理论引入3D-OOLCT领域,构建兼容八元数特性的概率体系:首先,定义3D-OOLCT域中八元数值概率密度函数、分布函数、均值及特征函数;其次,证明特征函数定理;最后,通过算例推导验证特定概率密度函数在3D-OOLCT域的特征函数表达式。该研究填补OOLCT域与概率理论结合的空白,完善八元数变换理论,为三维高维随机信号统计分析提供新工具,也为后续相关工程应用奠定数理基础。
