recursion equations - définition. Qu'est-ce que recursion equations
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Qu'est-ce (qui) est recursion equations - définition

Alpha recursion; Α-recursion theory

Einstein field equations         
  • EFE on a wall in [[Leiden]], Netherlands
FIELD EQUATIONS IN GENERAL RELATIVITY
Einstein field equation; Einstein's field equations; Einstein's equations; Einstein equation; Einstein Field Equations (EFE); Mass-energy tensor; Vacuum field equations; Einstein's equation; Einstein's field equation; Einstein Field Equations; Einstein equations; Einstein/Maxwell field equations; Einstein-Maxwell equations; Albert Einstein's equation; Einstein's equations of gravity; Einstein Equations; Albert Einstein's field equations; Einstein–Maxwell equations; Einstein gravitational constant
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
Maxwell's equations         
  • '''n'''}} is the [[unit normal]]. (The curl of a vector field doesn't literally look like the "circulations", this is a heuristic depiction.)
  • '''B'''}} field, which has no magnetic charges as shown. The outward [[unit normal]] is '''n'''.
  • 1='''E'''<sub>0</sub> ⋅ '''B'''<sub>0</sub> = 0 = '''E'''<sub>0</sub> ⋅ '''k''' = '''B'''<sub>0</sub> ⋅ '''k'''}}
  • In a [[geomagnetic storm]], a surge in the flux of charged particles temporarily alters [[Earth's magnetic field]], which induces electric fields in Earth's atmosphere, thus causing surges in electrical [[power grid]]s. (Not to scale.)
  • ''Left:'' A schematic view of how an assembly of microscopic dipoles produces opposite surface charges as shown at top and bottom. ''Right:'' How an assembly of microscopic current loops add together to produce a macroscopically circulating current loop. Inside the boundaries, the individual contributions tend to cancel, but at the boundaries no cancelation occurs.
  • Electric field from positive to negative charges
SET OF PARTIAL DIFFERENTIAL EQUATIONS THAT DESCRIBE HOW ELECTRIC AND MAGNETIC FIELDS ARE GENERATED AND ALTERED BY EACH OTHER AND BY CHARGES AND CURRENTS
Maxwell's Equations; Maxwells equations; Maxwell equations; Maxwell equation; Maxwell's equation; Maxwell's laws; Maxwell's theory; Maxwell Laws; Maxwell's field equations; Maxwell Equations; Laws of electromagnetism; Maxwell theory; Maxwell's Equation; Maxwell's Laws; Maxwell field equations; Maxwell electrodynamics; Maxwell’s equations; Maxwell Law; Great papers of James Clerk Maxwell; Great papers of james clerk maxwell; Maxwell's four equations; Maxwell's differential equations; Table of Maxwell equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
System of equations         
FINITE SET OF EQUATIONS TO BE SOLVED SIMULTANEOUSLY (AS A LOGICAL CONJUNCTION), POSSIBLY FOR MULTIPLE UNKNOWNS
Systems of equations; Simultaneous equation; Equation system; X equations y unknowns; Simeltanious equations; Simultaneous linear equation; Simultaneous equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a:

Wikipédia

Alpha recursion theory

In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals α {\displaystyle \alpha } . An admissible set is closed under Σ 1 ( L α ) {\displaystyle \Sigma _{1}(L_{\alpha })} functions, where L ξ {\displaystyle L_{\xi }} denotes a rank of Godel's constructible hierarchy. α {\displaystyle \alpha } is an admissible ordinal if L α {\displaystyle L_{\alpha }} is a model of Kripke–Platek set theory. In what follows α {\displaystyle \alpha } is considered to be fixed.

The objects of study in α {\displaystyle \alpha } recursion are subsets of α {\displaystyle \alpha } . These sets are said to have some properties:

  • A set A α {\displaystyle A\subseteq \alpha } is said to be α {\displaystyle \alpha } -recursively-enumerable if it is Σ 1 {\displaystyle \Sigma _{1}} definable over L α {\displaystyle L_{\alpha }} , possibly with parameters from L α {\displaystyle L_{\alpha }} in the definition.
  • A is α {\displaystyle \alpha } -recursive if both A and α A {\displaystyle \alpha \setminus A} (its relative complement in α {\displaystyle \alpha } ) are α {\displaystyle \alpha } -recursively-enumerable. It's of note that α {\displaystyle \alpha } -recursive sets are members of L α + 1 {\displaystyle L_{\alpha +1}} by definition of L {\displaystyle L} .
  • Members of L α {\displaystyle L_{\alpha }} are called α {\displaystyle \alpha } -finite and play a similar role to the finite numbers in classical recursion theory.
  • Members of L α + 1 {\displaystyle L_{\alpha +1}} are called α {\displaystyle \alpha } -arithmetic.

There are also some similar definitions for functions mapping α {\displaystyle \alpha } to α {\displaystyle \alpha } :

  • A function mapping α {\displaystyle \alpha } to α {\displaystyle \alpha } is α {\displaystyle \alpha } -recursively-enumerable, or α {\displaystyle \alpha } -partial recursive, iff its graph is Σ 1 {\displaystyle \Sigma _{1}} -definable in ( L α , ) {\displaystyle (L_{\alpha },\in )} .
  • A function mapping α {\displaystyle \alpha } to α {\displaystyle \alpha } is α {\displaystyle \alpha } -recursive iff its graph is Δ 1 {\displaystyle \Delta _{1}} -definable in ( L α , ) {\displaystyle (L_{\alpha },\in )} .
  • Additionally, a function mapping α {\displaystyle \alpha } to α {\displaystyle \alpha } is α {\displaystyle \alpha } -arithmetical iff there exists some n ω {\displaystyle n\in \omega } such that the function's graph is Σ n {\displaystyle \Sigma _{n}} -definable in ( L α , ) {\displaystyle (L_{\alpha },\in )} .

Additional connections between recursion theory and α recursion theory can be drawn, although explicit definitions may not have yet been written to formalize them:

  • The functions Δ 0 {\displaystyle \Delta _{0}} -definable in ( L α , ) {\displaystyle (L_{\alpha },\in )} play a role similar to those of the primitive recursive functions.

We say R is a reduction procedure if it is α {\displaystyle \alpha } recursively enumerable and every member of R is of the form H , J , K {\displaystyle \langle H,J,K\rangle } where H, J, K are all α-finite.

A is said to be α-recursive in B if there exist R 0 , R 1 {\displaystyle R_{0},R_{1}} reduction procedures such that:

K A H : J : [ H , J , K R 0 H B J α / B ] , {\displaystyle K\subseteq A\leftrightarrow \exists H:\exists J:[\langle H,J,K\rangle \in R_{0}\wedge H\subseteq B\wedge J\subseteq \alpha /B],}
K α / A H : J : [ H , J , K R 1 H B J α / B ] . {\displaystyle K\subseteq \alpha /A\leftrightarrow \exists H:\exists J:[\langle H,J,K\rangle \in R_{1}\wedge H\subseteq B\wedge J\subseteq \alpha /B].}

If A is recursive in B this is written A α B {\displaystyle \scriptstyle A\leq _{\alpha }B} . By this definition A is recursive in {\displaystyle \scriptstyle \varnothing } (the empty set) if and only if A is recursive. However A being recursive in B is not equivalent to A being Σ 1 ( L α [ B ] ) {\displaystyle \Sigma _{1}(L_{\alpha }[B])} .

We say A is regular if β α : A β L α {\displaystyle \forall \beta \in \alpha :A\cap \beta \in L_{\alpha }} or in other words if every initial portion of A is α-finite.