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An intriguing possibility for going beyond the Standard Model is extending the matter spectrum through the addition of fundamental high spin fields. Here we describe a study of the (j,0)⊕(0,j) family of Lorentz algebra representations where we built a covariant basis for the operators on these spaces. These operators generate an algebraic structure additional to the symmetry algebra, analogous to the Clifford algebra satisfied by the Dirac gamma matrices. This structure, which mathematicians call a Jordan-Lie algebra, is spin-dependent. The construction is based on an analysis of the covariant properties of the parity operator, which for these representations transforms as the completely temporal component of a symmetrical tensor of rank 2j. We make the construction explicit for j=1/2,1 and 3/2, and provide an algorithm for the corresponding calculations for arbitrary j.
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We establish results concerning the existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay and Poisson jumps in the phase space C((-∞,0];Rd) under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition on the coefficients by means of the successive approximation. Compared with the previous results, the results obtained in this paper is based on a other proof and our results can complement the earlier publications in the existing literatures.
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In this paper, we study semi symmetric and pseudo symmetric conditions in S -manifolds, those are RR = 0 , RC = 0 , C R = 0 , C C = 0 , = ( , ) R R L1Q g R , = ( , ) RC L2Q g C , = ( , ) CR L3Q g R , and = ( , ) C C L4Q g C , where C is the Concircular curvature tensor and 1 2 3 4 L ,L ,L ,L are the smooth functions on M , further we discuss about Ricci soliton
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An intriguing possibility for going beyond the Standard Model is extending the matter spectrum through the addition of fundamental high spin fields. Here we describe a study of the (j,0)⊕(0,j) family of Lorentz algebra representations where we built a covariant basis for the operators on these spaces. These operators generate an algebraic structure additional to the symmetry algebra, analogous to the Clifford algebra satisfied by the Dirac gamma matrices. This structure, which mathematicians call a Jordan-Lie algebra, is spin-dependent. The construction is based on an analysis of the covariant properties of the parity operator, which for these representations transforms as the completely temporal component of a symmetrical tensor of rank 2j. We make the construction explicit for j=1/2,1 and 3/2, and provide an algorithm for the corresponding calculations for arbitrary j.
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We establish results concerning the existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay and Poisson jumps in the phase space C((-∞,0];Rd) under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition on the coefficients by means of the successive approximation. Compared with the previous results, the results obtained in this paper is based on a other proof and our results can complement the earlier publications in the existing literatures.
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In this paper, we study semi symmetric and pseudo symmetric conditions in S -manifolds, those are RR = 0 , RC = 0 , C R = 0 , C C = 0 , = ( , ) R R L1Q g R , = ( , ) RC L2Q g C , = ( , ) CR L3Q g R , and = ( , ) C C L4Q g C , where C is the Concircular curvature tensor and 1 2 3 4 L ,L ,L ,L are the smooth functions on M , further we discuss about Ricci soliton
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We make use of the conformal compactification of Minkowski spacetime M# to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime [M#]−1 obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying M# with the projective light cone in (4+2)-dimensional spacetime, we write two independent conformal-invariant functionals of the 6-dimensional Maxwellian field strength tensors - one bilinear, the other trilinear in the field strengths -- which are to enter general nonlinear constitutive equations. We also make some remarks regarding the dimensional reduction procedure as we consider its generalization from linear to general nonlinear theories.
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42. Conic Sections, Circles, Constants in Exponential Functions.notebook
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