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Register a new userA Generalization of the Idea of Disjunction
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Authors
Kerry M. Soileau
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We generalize the concept of disjunction.
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One of the main goals of these notes is to explain how rotations in reals^n are induced by the action of a certain group, Spin(n), on reals^n, in a way that generalizes the action of the unit complex numbers, U(1), on reals^2, and the action of the unit quaternions, SU(2), on reals^3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and reals^n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra, Cl_n, the Clifford algebra associated with reals^n. Since the spinor groups are certain well chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Cl_{p, q} associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful 8-periodicity theorem of Elie Cartan and Raoul Bott (with proofs).
We present a common ground for infinite sums, unordered sums, Riemann integrals, arc length and some generalized means. It is based on extending functions on finite sets using Hausdorff metric in a natural way.
In the paper a new generalization of the Golden mean, as a further generalization in relation to Stakhov (1989) and to Spinadel (1999), is presented. Also it is first observed that the Euler divine equation represents a possible generalization of Golden mean. In this fourth version Figure A1 and Tables A3-A10 are added.
In this paper we prove that Neutrosophic Set (NS) is an extension of Intuitionistic Fuzzy Set (IFS) no matter if the sum of single-valued neutrosophic components is < 1, or > 1, or = 1. For the case when the sum of components is 1 (as in IFS), after applying the neutrosophic aggregation operators one gets a different result from that of applying the intuitionistic fuzzy operators, since the intuitionistic fuzzy operators ignore the indeterminacy, while the neutrosophic aggregation operators take into consideration the indeterminacy at the same level as truth-membership and falsehood-nonmembership are taken. NS is also more flexible and effective because it handles, besides independent components, also partially independent and partially dependent components, while IFS cannot deal with these. Since there are many types of indeterminacies in our world, we can construct different approaches to various neutrosophic concepts. Also, Regret Theory, Grey System Theory, and Three-Ways Decision are particular cases of Neutrosophication and of Neutrosophic Probability. We extended for the first time the Three-Ways Decision to n-Ways Decision, and the Spherical Fuzzy Set to n-HyperSpherical Fuzzy Set and to n-HyperSpherical Neutrosophic Set.
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Xiao-Jun Yang (School of Mathematics
, China University of Mining andn Technology
, Xuzhou
2021
In this article an alternative infinite product for a special class of the entire functions are studied by using some results of the Laguerre-P{o}lya entire functions. The zeros for a class of the special even entire functions are discussed in detail. It is proved that the infinite product and series representations for the hyperbolic and trigonometric cosine functions, which are coming from Euler, are our special cases.
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