Buchholz's psi-functions are a hierarchy of single-argument ordinal functions ψ ν ( α ) {\displaystyle \psi _{\nu }(\alpha )} introduced by German mathematician...

Buchholz (surname) Buchholz hydra, a mathematical game on a labeled tree Buchholz psi functions, a system of ordinal collapsing functions Buchholz's ID...

Psi /ˈ(p)saɪ, ˈ(p)siː/ (P)SY, (P)SEE (uppercase Ψ, lowercase ψ or 𝛙; Greek: ψι psi [ˈpsi]) is the twenty-third and penultimate letter of the Greek alphabet...

Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of the range of Buchholz's psi function and Feferman's theta function....

In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal[citation needed], is a large countable ordinal that is used to measure the proof-theoretic strength...

ordinal, Γ0, up to at least the Bachmann–Howard ordinal. Using Buchholz psi functions one can extend this definition easily to the ordinal of transfinitely...

of infinity (KP). Buchholz's ψ {\displaystyle \psi }  is a hierarchy of single-argument functions ψ ν : O n → O n {\displaystyle \psi _{\nu }:{\mathsf...

proof. Here, ψ 0 {\displaystyle \psi _{0}}  denotes Buchholz's function, and ψ 0 ( ε Ω ω + 1 ) {\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega...

_{v}+1}0=\psi _{0}(\varepsilon _{\Omega _{v}+1})} for v ≤ ω. Yet, while this system is powerful, it does not qualify as an ordinal notation. Buchholz did create...

"Rapidly Growing Ramsey Functions". Annals of Mathematics. 113 (2): 267–314. doi:10.2307/2006985. ISSN 0003-486X. JSTOR 2006985. Buchholz, Wilfried (1984-11-27)...

{\psi }}M\psi +{\bar {\eta }}\psi +{\bar {\psi }}\eta }\,D{\bar {\psi }}\,D\psi =\int e^{\left({\bar {\psi }}+{\bar {\eta }}M^{-1}\right)M\left(\psi +M^{-1}\eta...

1 ) {\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})} . It is the supremum of the range of Buchholz's psi functions. It was first named by...

In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral...

_{1}\psi _{2}+\psi _{1}\psi _{2}\psi _{1})+{\frac {1}{\sqrt {3}}}(\psi _{1}\psi _{2}\psi _{1}+\psi _{2}\psi _{1}\psi _{1}+\psi _{2}\psi _{1}\psi _{1})\right)\\=&{\frac...

ψ ( x → ) {\displaystyle \delta \psi ({\vec {x}})=iq\alpha ({\vec {x}})\psi ({\vec {x}})} where α is some function of position? The Noether charge is...

ψ ( ε Ω + 1 ) {\displaystyle \psi (\varepsilon _{\Omega +1})} in Madore's ψ. 4.^ Uses Madore's ψ rather than Buchholz's ψ. 5.^ Can also be commonly written...

| ψ ⟩ {\displaystyle |\psi \rangle } is no longer a spatial wave function in the traditional sense of a complex-valued function that is defined on a 3-dimensional...

2 ‖ Φ ‖ 2 {\displaystyle P{\big (}[\Psi ],[\Phi ]{\big )}={\frac {|\langle \Psi ,\Phi \rangle |^{2}}{\lVert \Psi \rVert ^{2}\lVert \Phi \rVert ^{2}}}}...

to the equations, universally denoted as ψ or Ψ (Greek psi), are referred to as "wave functions" in the context of RQM, and "fields" in the context of...

gauge functions which satisfy the wave equation ∂ 2 ψ ∂ t 2 = c 2 ∇ 2 ψ {\displaystyle {\frac {\partial ^{2}\psi }{\partial t^{2}}}=c^{2}\nabla ^{2}\psi }...

\end{aligned}}} Since interacting correlation functions can be expressed in terms of free correlation functions, only the latter need to be evaluated in order...

on functions f in classical phase space, then the following properties are usually considered desirable: Q x ψ = x ψ {\displaystyle Q_{x}\psi =x\psi }...

0 ) = x x ( t ) = y e i S D x , {\displaystyle \psi _{t}(y)=\int \psi _{0}(x)K(x-y;t)\,dx=\int \psi _{0}(x)\int _{x(0)=x}^{x(t)=y}e^{iS}\,{\mathcal {D}}x...

often called (causal) Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function). In non-relativistic quantum...

{\displaystyle {\mathcal {S}}=\int {\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }\partial _{\mu }-mc^{2}\right)\psi \,\mathrm {d} ^{4}x} The global symmetry...

condensates must be of the form ⟨ ψ ¯ ψ ⟩ {\displaystyle \langle {\overline {\psi }}\psi \rangle } , where ψ is the fermion field. Similarly a tensor field, Gμν...

_{l}}\langle 0|\mathrm {T} [\Psi _{\beta _{1}}(y_{1})...\Psi _{\beta _{n'}}(y_{n'}){\bar {\Psi }}_{\alpha _{1}}(x_{1})...{\bar {\Psi }}_{\alpha _{n}}(x_{n})]|0\rangle...

}F^{\mu \nu }+i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -e{\bar {\psi }}\gamma ^{\mu }A_{\mu }\psi -m{\bar {\psi }}\psi } = − 1 4 F μ ν F μ ν +...

4 μ 0 F μ ν F μ ν {\displaystyle T={\bar {\psi }}(i\hbar c\gamma ^{\sigma }\partial _{\sigma }-mc^{2})\psi -{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu...

t {\displaystyle \psi ^{*}=\psi +k*d\delta /dt} where ψ ∗ {\displaystyle \psi ^{*}} is the desired yaw rate ψ {\displaystyle \psi } is the actual yaw...