In terms of the trace-reversed metric perturbation and in the Lorenz gauge in linearized gravity, the linearized Einstein equations on Minkowski spacetime are

Neutrino free streaming transports neutrinos out of small-scale overdensities. Massive neutrinos therefore contribute to the homogeneous matter density while clustering inefficiently below their free-streaming scale, slowing the growth of cold-matter perturbations.

The debiased Lasso adds an approximate inverse-Gram correction to a Lasso estimate so that each corrected coordinate has an asymptotically linear expansion.

Comparing the Lasso objective at an estimate $\hat{\beta }$ and a target ${\beta }^0$, then expanding the squared loss, bounds the prediction error by a noise-score term and the decrease in the ${\ell }^1$ penalty.

The effective field theory of large-scale structure represents unresolved short-scale dynamics by an effective stress tensor. Its leading deterministic correction is proportional to $k^2{\delta }^{(1)}$ and renormalizes ultraviolet-sensitive loop contributions.

The transverse-field Ising model is a quantum spin model with competing Ising exchange and a perpendicular field, for example $H=-J{\sum }_i{X}_i{X}_{i+1}-h{\sum }_i{Z}_i$. In one dimension its ordered and paramagnetic phases meet at $|h/J|=1$.

The normal equation for $Au=f$ is $A^{\ast }Au=A^{\ast }f$. Its solutions minimize $\Vert Au-f\Vert$; when they exist, the unique solution orthogonal to $\ker A$ is the Moore--Penrose solution $A^{†}f$.

Regularization replaces sensitive inversion by a controlled approximation. For a bounded linear operator between Hilbert spaces, a linear regularization is a family of bounded linear operators $R_{\alpha }$ satisfying $R_{\alpha }y\to A^{†}y$ as $\alpha \downarrow 0$ for each datum in the domain of the Moore–Penrose inverse of an operator. Noisy-data convergence additionally requires a suitable regularization parameter choice.^[ref] Nonlinear variational regularization can instead use nonquadratic penalties.

The noise-bias decomposition for linear regularization separates two costs: amplifying measurement noise and changing the exact-data reconstruction. Truncated singular value decomposition discards weak modes; Tikhonov regularization attenuates them smoothly. A quadratic norm penalty expresses a preference for smaller reconstructions. Neither method manufactures a missing null space component: with complete averaging, the minimum-norm least-squares solution for $(0.5,0.5{)}^T$ is $(0.5,0.5{)}^T$, even if the original image was $(1,0{)}^T$. The preference selects a representative; extra observations or justified constraints are needed to identify the original.

For Tikhonov regularization, the Tikhonov stability bound limits transmitted measurement noise of norm at most $\delta$ to $\delta /(2\sqrt{\alpha })$. Taking $\alpha \to 0$ while $\delta /\sqrt{\alpha }\to 0$ gives a sufficient rule for convergent regularization of an inverse problem. Holding $\alpha$ fixed leaves approximation bias; letting it vanish too quickly can admit overwhelming measurement noise.

An inverse scattering problem reconstructs a medium or obstacle from measured scattered waves. Under the Born approximation, fixed-frequency far-field data sample the Fourier transform of the scattering potential on a restricted frequency surface.

At a stationary slope, an internal wave preserves frequency and tangential wavenumber. Unlike specular reflection, the reflected energy ray preserves its angle to the vertical; a supercritical slope can reverse its horizontal propagation direction.

For a uniformly stratified fluid, internal-wave phase velocity is parallel to the wavevector while group velocity is perpendicular to it. Energy from a localized monochromatic source propagates along four beams forming a St Andrew's cross.

The Mellin transform is an integral transform suited to changes of scale: replacing $f(x)$ by $f(ax)$ multiplies its transform by $a^{-s}$ for $a>0$. For a locally integrable function $f:(0,\infty )\to \mathbb{C}$, definewhere the improper integral converges. The complex number $x^{s-1}=\exp ((s-1)\log x)$ uses the real natural logarithm because $x>0$. Its absolute value is $x^{\sigma -1}$, so absolute convergence depends on the real part $\sigma$ rather than the imaginary part $\tau$. The Mellin fundamental strip records the open range of such real parts.^[ref]

A familiar example is the exponential function $f(x)=e^{-x}$:This is exactly the Gamma integral. Near zero $e^{-x}$ tends to one, so absolute convergence requires $\sigma >0$; at infinity $x^r{e}^{-x}\to 0$ for every real $r$, so there is no upper boundary. The Gamma function has a meromorphic continuation with simple poles at $0,-1,-2,\dots$, but a finite continued value elsewhere in $\sigma \le 0$ does not make the defining improper integral converge.^[ref]

The Mellin transform of an interval indicator gives an elementary example with absolute convergence for every complex number $s$. The Mellin transform in logarithmic coordinates explains why changes of scale become translations, connecting the scaling property of the Mellin transform to the Fourier transform.

Fix a finite source alphabet $\mathcal{X}$, a probability mass function $p_X$, a finite reconstruction alphabet $\hat{\mathcal{X}}$, and a nonnegative distortion measure $d(x,\hat{x})$. The information rate-distortion function isThe source probability distribution stays fixed; the conditional distribution of the reconstruction is what we optimize. Each feasible test channel must keep expected distortion at most $D$. The objective is mutual information, measured in bits when the logarithms have base two. If no test channel satisfies the constraint, the infimum is $+\infty$.^[ref]

Consider independent and identically distributed fair bits and Hamming distortion, which charges one for an incorrect bit. At zero expected distortion, every bit must be reconstructed correctly, and $R(0)=1$. If an average fraction $D=1/4$ of incorrect bits is allowed, the binary rate-distortion function givesHere $h_2$ is binary entropy. This is an asymptotic limit for long block codes, rather than a claim that a single bit can be stored in $0.18872$ physical bits. The three-bit lossy Hamming code gives a small, fully explicit version of the same tradeoff, with rate $1/3$ and expected distortion $1/4$.

Rate-distortion theory separates the question of what is possible from the design of a practical encoder and decoder. It also forces the fidelity criterion to be explicit. Hamming distortion treats all symbol mismatches alike; mean squared error instead penalizes larger numerical errors more strongly. Those objectives can favor different reconstructions of the same data. An average error criterion does not by itself control the error on every individual input.

Claude Shannon's Coding Theorems for a Discrete Source With a Fidelity Criterion by Claude Shannon was published in 1959. Its formulation fixes a cost for replacing one source letter by another and studies the communication rate compatible with the allowed average cost. The important shift is to quantify which distinctions may be discarded, rather than requiring exact recovery of every letter.^[ref]

For independent $U,V,V^{\prime }$ with $V$ and $V^{\prime }$ identically distributed,This additive form of entropy submodularity follows from data processing and invariance of conditional entropy under translations.

The closed time contour of the in-in formalism gives four bulk propagators: time-ordered $G_{RR}$, anti-time-ordered $G_{LL}$, and the two Wightman functions $G_{LR}$ and $G_{RL}$. Vertices on the two contour branches carry opposite factors of $i$.

A projective resolution of an $R$-module $M$ is an exact sequencein which every $P_i$ is a projective module.

The Hom functor sends two $R$-modules to the abelian group of $R$-linear maps between them. It is contravariant in its first argument and covariant in its second.

A cochain complex is a sequence of abelian groups or modules and maps $d^n:C^n\to C^{n+1}$ satisfying $d^{n+1}{d}^n=0$. Its cohomology is $H^n(C)=\ker d^n/\mathrm{im}{d}^{n-1}$.

A group $G$ is residually finite when every nonidentity $g\in G$ survives in some finite quotient: there are a finite group $Q$ and a homomorphism $\varphi :G\to Q$ with $\varphi (g)\ne 1$.

Group homology is obtained by tensoring a projective resolution of the trivial $\mathbb{Z}G$-module with a coefficient module and taking homology.