A renormalization-group flow is the trajectory of effective couplings as the observation scale changes.

The engineering dimension of a field, parameter or operator is its dimension under the rescaling that leaves the quadratic theory invariant. Interactions can change it into a full scaling dimension.

The momentum-shell renormalization group splits a field into slow modes below $\Lambda /\zeta$ and fast modes in $\Lambda /\zeta <|k|<\Lambda$, integrates out the fast modes, then rescales momenta and fields to restore the cutoff and kinetic normalization.

Wave-function renormalization rescales a field so that its kinetic term or propagator residue has the chosen normalization. Its scale dependence gives the field's anomalous dimension.

Variational regularization balances data fidelity against a lower-semicontinuous penalty, for example by minimizing $\frac{1}{2}\Vert Au-f{\Vert }^2+\alpha J(u)$.

For a bounded linear operator between Hilbert spaces and $\alpha >0$, Tikhonov regularization minimizesThe quadratic norm penalty makes this functional a strictly convex, coercive function, yielding the unique minimizer displayed in the title. Here $A^{\ast }$ is the adjoint operator and $I$ the identity operator. Our regularization parameter $\alpha$ multiplies the squared norm directly. Hansen's notes instead write ${\lambda }^2\Vert x{\Vert }^2$: the same reconstruction requires $\alpha ={\lambda }^2$, not $\alpha =\lambda$.^[ref]

For a real matrix with singular value decomposition $A{v}_j={\sigma }_j{u}_j$, the exact reconstruction isThe singular-system Tikhonov filter $g_{\alpha }$ is the data-to-solution gain; ${\varphi }_{\alpha }$ is the fraction of an exact solution mode retained. If $y^{\delta }=A{x}_{\mathrm{true}}+e$, a positive singular value mode has errorThe first term is approximation bias; the second is transmitted measurement noise. Suppressing a weak mode suppresses its true signal too.

For the two-pixel matrix in the inverse problem example and $\alpha =0.01$ ($\lambda =0.1$), the mean and difference attenuation factors are $100/101$ and $1/2$. Since $y^{\delta }=q_{+}/\sqrt{2}+0.12q_{-}/\sqrt{2}$, the reconstruction isIts noise contribution is $(0.05,-0.05{)}^T$, half the direct inverse's, but its exact-data reconstruction is $(0.74505,0.24505{)}^T$. Consequently its total error Euclidean norm, about $0.28293$, exceeds the direct inverse's $0.14142$: this parameter sacrifices too much contrast. Taking $\alpha =0.001$ instead gives $(1.04496,-0.04595{)}^T$ and error about $0.06429$. Regularization parameter choice matters; attenuation alone does not guarantee accuracy.

A decision stump is a decision tree with one split. A regression stump predicts a constant, usually the sample mean, in each of its two terminal regions.

Phi cubed theory has interaction $-g{\varphi }^3/3!$, giving a cubic vertex $-ig$. In four spacetime dimensions the scalar field has mass dimension one and $g$ has mass dimension one.

Hamming distortion is zero for a correctly reconstructed symbol and one for a mismatch. Its expected distortion is the symbol-error probability:For strings of length $n$, separable distortion divides their Hamming distance by $n$. Thus $001$ reconstructed as $000$ has Hamming distance one and average Hamming distortion $1/3$. Counting only whether the whole string is correct would give a different criterion.

Complete randomization chooses uniformly among all treatment assignments having prespecified treatment-arm sizes.

A Hilbert-space-valued random variable is a measurable map into a Hilbert space. It is square-integrable when $\mathbb{E}\Vert X{\Vert }^2<\infty$, in which case its mean is defined by the Bochner integral.

The diagonal $r$-uniform hypergraph Ramsey number $R^{(r)}(k)$ is the least $N$ such that every red-blue colouring of the $r$-element subsets of an $N$-element set has a monochromatic $k$-element set.

The orthonormal-frame components of Schwarzschild curvature scale as $M/r^3$. The geodesic deviation equation therefore gives radial stretching and transverse compression whose magnitude diverges as $r\to 0$.

R-parity violation permits interactions containing an odd number of superpartners. Renormalizable R-parity-violating interactions can violate lepton number or baryon number, and allowing both generically induces rapid proton decay.

A group is virtually free when it has a free group as a finite-index subgroup. Every finitely generated virtually free group is quasi-isometric to a tree.

If a finitely generated group $G$ maps surjectively to $H$ with finite kernel, the quotient map is a quasi-isometry. After lifting a finite generating set of $H$, the only possible extra length lies in the finite kernel and is therefore uniformly bounded.

If $H$ is a finite-index subgroup of a finitely generated group $G$, then $H$ is finitely generated and its inclusion into $G$ is a quasi-isometry. Finite generation follows from Schreier's lemma, while finitely many coset representatives give coarse surjectivity and uniformly bounded distortion.

The free Dirac propagator is $S_F(p)=i(/p+m)/(p^2-m^2+iϵ)$ and inverts the Dirac kinetic operator with Feynman boundary conditions.

A Wightman function is an unordered vacuum correlation function. For a free scalar mode $f_k(t)$, the positive-frequency two-point function is $G^{>}(t_1,t_2;k)=f_k(t_1)f_k^{\ast }(t_2)$.

The Feynman propagator is the time-ordered Green function selected by the $iϵ$ prescription.