Combinatorics and Graph Theory

2310 Submissions

[2] viXra:2310.0143 [pdf] submitted on 2023-10-29 22:20:07

The Isomorphism of H4 and E8

Authors: J. Gregory Moxness
Comments: 19 Pages.

This paper gives an explicit isomorphic mapping from the 240 real R^8 roots of the E8 Gossett 4_{21} 8-polytope to two golden ratio scaled copies of the 120 root H4 600-cell quaternion 4-polytope using a traceless 8x8 rotation matrix U with palindromic characteristic coefficients and a unitary form e^{iU}}. It also shows the inverse map from a single H4 600-cell to E8 using a 4D<->8D chiral L<->R mapping function, phi scaling, and U^{-1}. This approach shows that there are actually four copies of each 600-cell living within E8 in the form of chiral H4L+phi H4L+H4R+phi H4R roots. In addition, it demonstrates a quaternion Weyl orbit construction of H4-based 4-polytopes that provides an explicit mapping between E8 and four copies of the tri-rectified Coxeter-Dynkin diagram of H4, namely the 120-cell of order 600. Taking advantage of this property promises to open the door to as yet unexplored E8-based Grand Unified Theories or GUTs.
Category: Combinatorics and Graph Theory

[1] viXra:2310.0067 [pdf] submitted on 2023-10-13 20:08:20

Simultaneous Equations of Order Parameter for Ground State Spin Glass Model

Authors: Akira Saito
Comments: 4 Pages. In Japanese (Note by viXra Admin: Please fill in author name in English)

We were able to express the order parameter of the spin glass model in the ground state using simultaneous equations. Although this simultaneous equation is an equation that is valid only in the ground state, it is possible to minimize economic loss from a network system that influences each other to model food chains, organizations, economic social phenomena, etc., and to obtain optimal results from complex conditions. Accurately understanding the ground state provides a wide range of benefits, including combinatorial optimization problems that enable accurate matching. Furthermore, since the spin glass ising model is a basic model, it is also possible to refer to natural world and mathematical problems such as complex systems and P≠NP problems [1]. Here, as a basis for these problems, and as a method for analyzing social phenomena and problems such as combinatorial optimization problems and complex systems, we will introduce simultaneous equation formulation and numerical calculation of simultaneous equation solutions as a solution method for ground state spin glass models. Show the results.
Category: Combinatorics and Graph Theory