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At a nonzero Euclidean momentum $k$, the longitudinal projector of a vector field is $P_L^{ab}=k^a{k}^b/k^2$. Its Lorentzian mixed-index form is $(P_L{)}^a{}_b=p^a{p}_b/p^2$ for non-null four-momentum. Together with the transverse projector of a vector field, it obeys $P_T+P_L=I$, $P_T{P}_L=0$, $P_T^2=P_T$, and $P_L^2=P_L$. This formula is not a pointwise decomposition at null four-momentum.

At a nonzero Euclidean momentum $k$, the transverse projector of a vector field is $P_T^{ab}={\delta }^{ab}-k^a{k}^b/k^2$. In a Lorentzian Minkowski metric, its mixed-index form is $(P_T{)}^a{}_b={\delta }_b^a-p^a{p}_b/p^2$, defined for $p^2\ne 0$. It obeys $P_{T}p=0$ and $P_T^2=P_T$. A nonzero null four-momentum is insufficient for this particular decomposition because the denominator vanishes.

At $\xi =1$, the Euclidean photon propagator is $D_{\mu \nu }(k)={\delta }_{\mu \nu }/k^2$. Its Lorentz-index contraction simplifies loop numerators in quantum electrodynamics.

Let $\dot{x},{\dot{a}}_1,\dots ,{\dot{a}}_n$ be forcing names and let $\phi$ be a formula. The namebelongs to the ground model by axiom schema of separation. The forcing theorem shows that its value in a generic extension is exactlyConsequently every generic extension satisfies the separation schema.

The syntactic forcing relation is defined recursively inside the ground model from the ranks of forcing names and the logical complexity of $\phi$. The forcing theorem proves that it agrees with the semantic forcing relation.

For a countable transitive model $M$, semantic forcing declares $p⊩\phi$ when every generic filter $G$ over $M$ containing $p$ gives $M[G]\models \phi$.

For two identical spin-one-half fermions with no additional antisymmetric internal factor, the antisymmetric spin-one-half singlet state combines with an even-$\ell$ symmetric orbital state, while the symmetric spin-one-half triplet state combines with an odd-$\ell$ antisymmetric orbital state.

Backpropagation applies the chain rule from a neural network's output toward its input to compute the gradient of a loss with respect to every parameter.

If $B(y,R)$ is an exterior ball tangent to a planar domain at $x_0$, thenis a harmonic function in the domain, vanishes at $x_0$, and is positive at every other point of its closure. It is therefore a barrier for the Dirichlet problem and proves that $x_0$ is a regular boundary point.

A $T_0$ space $E$ is exponentiable in ${\mathbf{Top}}_0$ if and only if the functor ${\mathbf{Top}}_0(-\times E,S)$ to sets is representable, where $S$ is the Sierpiński space. Every $T_0$ space is an equalizer of maps between powers of $S$, so a representing object for maps into $S$ constructs exponentials for every target by products and equalizers.

If the terminal object is also initial and $E$ is exponentiable, the left adjoint $-\times E$ preserves the initial object. But $0\times E\cong E$ because $0$ is terminal, while preservation of initiality gives $0\times E\cong 0$. Hence $E\cong 0$.

If $A$ and $B$ are exponentiable objects, thenhas the composite of their exponential functors as a right adjoint. The terminal object supplies the empty product, so exponentiable objects are closed under finite products.

A probability measure-preserving transformation $T$ is weakly mixing when, for every pair of measurable sets $A,B$, the average absolute correlation discrepancy tends to zero:Equivalently, the correlations converge to their products in convergence in density of a sequence, its Cartesian product with itself is an ergodic transformation, or its product with every ergodic transformation is ergodic. The absolute values are essential: signed Cesaro convergence of a sequence of the correlations to their products characterizes ergodic transformations and does not imply weak mixing. For example, the transformation interchanging two equally weighted points has discrepancy $(-1{)}^n/4$ for a singleton $A=B$: its signed average tends to zero while its average absolute discrepancy stays $1/4$.

Let $T$ be an ergodic transformation preserving a probability measure $\mu$ on a countable set. Some state $x$ has $\mu (\{x\})>0$. Along its forward orbit,An infinite sequence of distinct states with masses bounded below by $\mu (\{x\})$ is impossible, so the orbit eventually reaches a finite cycle $Y$. Since $Y\subseteq T^{-1}Y$ and invariance gives $\mu (T^{-1}Y)=\mu (Y)>0$, the difference is null. The cycle is invariant modulo null sets, and ergodicity therefore gives $\mu (Y)=1$.

For a measure-preserving action of $\mathbb{Z}$ and a finite partition $\xi$,The reverse inequality to the interval formula follows from Shearer's inequality applied to translates of $F$ along long intervals; the uncovered boundary has sublinear size.

The Kolmogorov-Sinai entropy is $h_{\mu }(T)={\sup }_{\xi }{h}_{\mu }(T,\xi )$ over finite measurable partitions.

For $E:y^2=x(x+1)(x+4)$,The points $(-2,2)$ and $(-1,0)$ generate the two factors, and a two-descent on an elliptic curve proves that the Mordell-Weil group has rank zero.

Let $E$ be a genus one curve over an algebraically closed field and choose $O\in E$. The mapis an isomorphism. For a degree-zero divisor $D$, Riemann--Roch gives a nonzero section of $D+O$, so $D\sim P-O$ for some point $P$. If $P-O\sim P^{\prime }-O$ with $P\ne P^{\prime }$, a rational function with divisor $P-P^{\prime }$ would give a degree-one map to the projective line, contradicting that $E$ has genus one. The Abel--Jacobi map transports addition in the divisor class group to the elliptic-curve group law.

For p-regular elements $g,h$,where $S$ ranges over the simple modules and $P_S$ is the projective cover of $S$.

For $\mathbf{u}=u_r{\mathbf{e}}_r+u_{\theta }{\mathbf{e}}_{\theta }$,