The determinant is a smooth Lie group homomorphism from the real general linear group to ${\mathbb{R}}^{\times }$. Its derivative at the identity is the trace, and the flow-defined exponential on ${\mathbb{R}}^{\times }$ is the ordinary scalar exponential. Naturality of the Lie group exponential therefore proves the identity without diagonalizability assumptions. The series ${\sum }_{k\ge 0}{t}^k{A}^k/k!$ is the flow-defined matrix exponential because it solves $E^{\prime }=EA$, $E(0)=I$, and has inverse $E(-t)$.

At rest the two complex supercharges have anticommutators $2m{\delta }_{ab}$. A normalized scalar Clifford vacuum generates four states at occupation levels $0,1,2$: two spin-zero bosonic states and one spin-half pair. Products of three creation charges vanish. The one-creation normalization is $(2m{)}^{-1/2}$ and the two-creation normalization is $(2m{)}^{-1}$. The resulting on-shell content is a complex scalar and a massive fermion with equal masses.

A mass eigenstate is a quantum state with a definite mass. For a self-adjoint operator $M$ representing this quantum observable, it satisfies $M|n⟩=m_n|n⟩$; a measurement of that observable in the normalized state gives $m_n$ with probability one.

In a flavor eigenstate basis, the matrix representing the mass observable can have off-diagonal entries. Diagonalizing that matrix identifies the mass eigenstates as linear combinations of the flavor eigenstates. For a finite-dimensional self-adjoint operator, these states can be chosen to form an orthonormal eigenbasis. Propagation of unstable particles can instead be modeled by an effective complex matrix; its eigenvectors need not have the orthogonality of a self-adjoint observable. The numerical finite-element mass matrix is a different construction: an L2 inner product matrix used in discretization.

On a graph, a binary configuration has probability proportional to $\exp (\alpha {\sum }_v{x}_v+\beta {\sum }_{\{v,w\}}{x}_v{x}_w)$, with each edge counted once. The full conditional probability of a one is logistic with linear predictor $\alpha +\beta {\sum }_{w\sim v}{x}_w$. Local neighbor sums eliminate the need to evaluate the normalizing constant.

Reversible-jump Markov chain Monte Carlo samples a posterior on a disjoint union of model-specific parameter spaces. A dimension-matching bijection between old parameters plus auxiliary variables and new parameters plus reverse auxiliary variables supplies a reversible proposal. Its acceptance ratio includes posterior-density, model-selection, proposal-density, and Jacobian determinant factors. The equal-dimension reversible-jump acceptance probability is the ordinary density-based special case.

A nonnegative row vector $x$, not necessarily of total mass one, is an invariant measure for a Markov chain if $x_j={\sum }_i{x}_i{p}_{ij}$. On an irreducible Markov chain, every nonzero invariant measure has positive coordinates: iterate stationarity and use a positive Transition probability along a path.

An ice-edge band is an elongated concentration of ice floes separated from neighbouring ice by open water. In one mechanism, an off-ice wind opens polynyas; short wind waves then exert wave radiation stress on their downwind floes. Wave forcing compacts floes into bands, opposed by incoming longer-period swell. Collisions and mergers can produce fewer, larger bands.

Compute observed-minus-expected event contributions within each stratum's risk set, then add them across independent strata. Under the null, the corresponding conditional variances add. Standardizing gives an approximate normal distribution test, or its square gives a one-degree-of-freedom chi-squared distribution test. The comparison requires the usual exchangeability and independent censoring assumptions within strata.

Two local frames of a vector bundle differ on their overlap by an invertible matrix-valued function. These functions satisfy the Čech cocycle condition. For a holomorphic vector bundle they are holomorphic, and for a holomorphic line bundle they are scalar holomorphic units.

A setting-independent hidden variable fixes responses for every measurement choice, and each response depends only on the local setting. Stochastic local hidden-variable theories can be represented as mixtures of these assignments by including local random seeds in the hidden variable. Deterministic counterfactual assignments are the starting point of the chained modular Bell inequality.

Let a nontrivially valued local field have compact valuation ring $\mathcal{O}$, and choose any $t$ with $0<|t|<1$. The quotient $\mathcal{O}/t\mathcal{O}$ is finite. A fixed set of representatives containing zero gives every field element a unique convergent digit expansion, with finitely many nonzero negative-index terms. Recursive reduction modulo $t\mathcal{O}$ gives existence; comparing the first differing digit modulo this ideal gives uniqueness. The element $t$ need not be a uniformizer.

Scalar advection preserves every finite Lp norm if its complete differentiable characteristic flow preserves volume and the initial datum belongs to the corresponding Lp space. In general the $p$th power of the norm is the initial $|f_0{|}^p$ weighted by the phase-space flow Jacobian. Completeness alone preserves the essential supremum, but does not preserve finite-$p$ norms for compressible transport.

If a velocity operator satisfies $\mathrm{Re}⟨Lh,h⟩\ge \delta \Vert h{\Vert }_2^2$, then sufficiently regular solutions of ${\partial }_{t}f+v\cdot {\nabla }_{x}f+Lf=0$ obey the displayed estimate when spatial boundary fluxes vanish. The free-transport part contributes zero to the energy derivative. Integrating the resulting differential inequality avoids dividing by a possibly zero norm.

For the constant-velocity linear transport equation $u_t+c\cdot {\nabla }_{x}u=0$, a Dirac delta distribution initially at zero moves along $x=ct$. Its spacetime distribution pairs by $⟨u,\phi ⟩=\int \phi (ct,t)dt$, and its initial trace is obtained by weak convergence of distributions. The Fourier representation of the Dirac delta function writes it as an oscillatory integral with phase function $(x-ct)\cdot \theta$. Its singular support is precisely the trajectory; it has order-zero distribution regularity as a measure, without becoming a smooth function.

Choose a finite trial family of polynomial-exponential or trigonometric functions for a particular solution, substitute it into a constant-coefficient linear equation, and solve for its coefficients. If the forcing exponent is a characteristic root of multiplicity $m$, multiply the ordinary trial by $t^m$ to leave the homogeneous kernel. The method is efficient for forcing families preserved by differentiation.

A uniform dilation about the origin multiplies every vector by one positive real scale $s$. The linear map $\mathbf{x}\mapsto s\mathbf{x}$ multiplies lengths by $s$ and $d$-dimensional volumes by $s^d$. This differs from stretching different axes by different factors. The case $s=1$ is the identity map.

A horizontal shear fixes the horizontal axis pointwise and sends $(x,y)$ to $(x+ky,y)$. Its determinant is one and both eigenvalues are one. A nonzero shear preserves area but generally changes lengths and angles. Composing a shear with a planar rotation can produce a symmetric stretch; having determinant one alone does not make a linear map a shear.

A linear isometry from a subspace of a finite-dimensional Hilbert space into the same ambient space extends to a unitary operator. Complete an orthonormal basis of the input subspace and its isometric image to full orthonormal bases, then map the first to the second. The equal complement dimensions make this possible. Orthogonality of the specified image columns, not normalization alone, is essential. In an infinite-dimensional space arbitrary isometries need not be surjective; the finite-dimensional same-space hypothesis cannot be dropped.

For pressureless sub-Hubble perturbations, combine the linearized cosmological continuity equation, cosmological Euler equation and gravitational Poisson equation with matter as its source. The expansion contributes a $2H$ friction term and matter self-gravity contributes the negative coefficient of $\delta$. The background expansion can include radiation or dark energy; treating their perturbations as additional sources requires a more complete system.

With physical matter and radiation densities, primordial amplitudes and spatial flatness fixed, extra smooth late dark energy increases $H_0$ but keeps ${\Omega }_m{H}_0^2$ fixed. Provided the dark energy is negligible at equality, the early cosmological transfer function and equality wavenumber in physical units are unchanged. Earlier accelerated expansion suppresses the linear growth factor, lowering present dimensional matter power by its squared growth ratio. Units involving $h$ can additionally relabel the horizontal and vertical axes.