Hamiltonian mechanics with two times: part 1

To start with, we start with Lagrangian mechanics. Lagrangian mechanics states that the variation of the “action” with respect to every dynamical variable is zero. Suppose the action can be written in the form$$S=\int Ld\tau d\sigma$$ Then standard variational methods give the Euler-Lagrange equation$$\frac{\partial L}{\partial x}=\frac{\partial}{\partial\tau}\frac{\partial L}{\partial \frac{\partial x}{\partial \tau}}+\frac{\partial}{\partial\sigma}\frac{\partial L}{\partial \frac{\partial x}{\partial \sigma}}$$ For […]

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Cyclotomic fields and lattices

Adam P. Goucher discusses lattices constructed from cyclotomic fields. In particular, one possible lattice that can be constructed is the Leech lattice. As the Leech lattice has a notorious lack of simple constructions, it’s worth considering this in more detail. If $\zeta_n$ is a primitive $n$th root of unity, the $n$th cyclotomic ring $\mathbb Z[\zeta_n]$

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Ideal OCF attempt

Ideal OCF \(\kappa\) is hereditarily \(\Pi_n\)-reflecting if and only if it is \(\Pi_n\)-reflecting or \(\Pi_m\)-reflecting on the hereditarily \(\Pi_n\)-reflecting ordinals. \(C_\kappa(\alpha)\) is the closure of \(\kappa\cup{\text{least }\Pi_n\text{-reflecting}\mid n>1}\) under \(\beta,\gamma\mapsto\beta+\gamma\) and \(\beta,\gamma,\delta\mapsto\chi_\beta(\gamma,\delta)\) provided \(\gamma<\alpha\). Suppose \(\kappa\) is \(\Pi_{n+1}\)-reflecting (\(n>0\)) but not \(\Pi_{n+2}\)-reflecting. Then set \(\mathcal A_\kappa^\alpha\) to be the set of \(\pi<\kappa\) such that: If

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Simplified Reflection OCF

I’m back. Surprise! I’m making a simplified collapsing function reaching $\Pi_\omega$-ref. $K$ denotes the least $+1$-stable, First, let’s set up our reflection configurations. We define them by stating that for ordinal $\kappa$, integer $n>0$, ordinal $\alpha$, and reflection configurations $\mathbb{X},\ldots$, $(\kappa,n,\alpha,\mathbb{X},\ldots)$ is a reflection configuration, provided $\kappa$ is $\Pi_{n+1}$-reflecting on $\mathcal A\mathbb X\cap\ldots$. The variables

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Beyond the stability OCFs

The post is meant to extend https://googology.fandom.com/wiki/User_blog:Hyp_cos/Attempt_of_OCF_up_to_Stability. Firstly, I’ll verify that it’s worth extending. I compared it with the other OCFs (Stegert, Duchhardt). It is virtually identical to Stegert, if we associate the second reflection instance element be the position in the hierarchy. The only worrying difference is that Stegert’s hierarchies require their elements to

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