For $A=P+iQ$ with real matrices $P,Q$, the same linear map viewed over the reals has block matrix $\left(\begin{array}{cc}P& -Q\\ Q& P\end{array}\right)$ in grouped real/imaginary coordinates. Interleaved coordinates merely permute the basis. Its complexification splits into the matrices $A$ and $\overline{A}$, giving the realification determinant identity.

A matrix with the same number of rows and columns represents an endomorphism of a finite-dimensional vector space after choosing a basis. Square matrices support trace, determinant, powers and the matrix exponential.

An intermediate asymptotic region has a scale between two other scales and provides their connecting matched asymptotic expansion. It can be needed because a coefficient changes its dominant form before the narrowest region's highest-derivative balance is reached. For example, a coefficient $x+ϵ$ changes near $x=O(ϵ)$ even when the fast inner expansion has width $O({ϵ}^2)$. The intermediate solution must be matched at both ends rather than assigned an independent boundary value.

An overlap region is a parameter-dependent range in which two neighboring asymptotic expansions have simultaneously ordered common limits. Between scales ${ϵ}^2$ and $ϵ$, it is ${ϵ}^2\ll x\ll ϵ$: the narrow coordinate tends to infinity while the wider coordinate tends to zero. Matching uses these limiting descriptions, not an assertion that either expansion is uniform at all values of the other's scaled variable.

In $\epsilon y^{\prime \prime }+2x{y}^{\prime }-2xy=0$, the drift vanishes at the endpoint $x=0$. Balancing diffusion against drift gives the inner expansion coordinate $z=x/\sqrt{\epsilon }$. The leading layer equation is $Y^{\prime \prime }+2z{Y}^{\prime }=0$, with a bounded error function profile. A regular outer first-order calculation alone cannot impose both endpoint values.

For $\epsilon x{w}^{\prime \prime }+w^{\prime }+2xw=0$ with left endpoint $x=\epsilon$, diffusion there is of size ${\epsilon }^2$. The distinguished coordinate is $z=(x-\epsilon )/{\epsilon }^2$, giving $(1+\epsilon z)W^{\prime \prime }+W^{\prime }+2{\epsilon }^3(1+\epsilon z)W=0$. Matching to one with zero left value gives $W=1-e^{-z}-\epsilon z(z+2)e^{-z}/2+O({\epsilon }^2)$. The layer width is different from the endpoint's distance from zero.

If a leading outer expansion diverges as $\log x$ and a derivative coefficient contains $x-\epsilon z$, balancing that nonlinear correction gives a layer larger than $O(\epsilon )$. Write $x=\epsilon LX$ with $L=\log (1/\epsilon )$ and subtract the large negative constant from $z$. Matching often introduces a $\log L$ correction and yields an endpoint slope of order $1/(\epsilon L)$. The balance must be derived from the actual equation; logarithmic divergence alone does not fix a universal layer width.

Locally in four dimensions, a massless two-form is dual to one real scalar field. In signature $(-+++)$, the first-order density $-H^2/(12g_B^2)+{ϵ}^{\mu \nu \rho \sigma }{H}_{\mu \nu \rho }{\partial }_{\sigma }a/6$ gives $H^{\mu \nu \rho }=g_B^2{ϵ}^{\mu \nu \rho \sigma }{\partial }_{\sigma }a$. Eliminating $H$ in both terms produces $-g_B^2(\partial a{)}^2/2$, so the fixed-normalization kinetic coefficient is inverted. Global flux sectors and scalar periodicities need separate treatment.

A three-form potential in four dimensions has no local massless polarizations. Its four-form field strength can be a constant multiple of the spacetime volume form. The equation $d(g_3^{-2}\ast F_4)=0$ leaves a constant flux parameter, which can affect vacuum energy despite supplying no local wave.

Electron-neutrino scattering has a charged-current contribution absent for muon neutrinos. Fierz rearrangement combined with the external fermionic sign adds a left-electron current: $c_V\to c_V+1$, $c_A\to c_A+1$. The right-electron coupling stays unchanged; the left-electron coupling increases by two.

Crossing a massless neutrino to an antineutrino reverses the antisymmetric part of its spin tensor. The $s^2$ and $u^2$ weights exchange, giving ${\sigma }_{\bar{\nu }}=G_F^{2}s[(c_V-c_A{)}^2+(c_V+c_A{)}^2/3]/(4\pi )$ for the same electron couplings. This is a low-energy contact statement rather than a replacement for a full weak-boson propagator.

For a finite-valued predictable process $K$, this is a stopping time and increases to infinity. The stopped martingale transform uses coefficients $K_s{1}_{\{s\le {\sigma }_j\}}$, which are predictable and bounded by $j$. Thus a transform of a discrete-time martingale is a local martingale without a global boundedness assumption. The stop is before, not after, the first large coefficient is used.

For $Y_t={\sum }_{s\le t}{K}_s(M_s-M_{s-1})$ and fixed finite deterministic $T$, bounded events in ${\mathcal{F}}_{t-1}$ restricting both $Y_{t-1}$ and $K_t$ show that nonnegativity of $Y_t$ forces nonnegativity of $Y_{t-1}$. Induction makes the entire stopped process nonnegative. The nonnegative discrete-time local martingale is a martingale criterion then gives $\mathbb{E}{Y}_T=Y_0=0$. Nonnegative terminal gain is therefore zero almost surely, even without an intermediate wealth bound.

A submartingale with ${\sup }_n\mathbb{E}{Y}_n^{+}<\infty$ converges almost surely to a finite integrable random variable. The positive-part bound also bounds negative parts because $\mathbb{E}{Y}_n\ge \mathbb{E}{Y}_0$. The result specializes to an $L^1$-bounded martingale, and hence to every nonnegative martingale. It does not by itself give convergence in L1; uniform integrability supplies that stronger conclusion.

For order $k$ B-splines, comparison of coefficients in the Marsden identity giveson the basic knot interval. Here $e_m$ is the elementary symmetric polynomial, with $e_0=1$. The $m=0$ case is partition of unity; the $m=1$ coefficients are the Greville abscissae.

The operator that averages a function over the next-state conditional law of a time-homogeneous Markov process. For a random walk with increments of law $\nu$ independent of the past including the initial state, $(Ph)(s)=\int h(s+y)\nu (dy)$. This additive operator preserves convexity when the integrals exist.

Marginalization sums or integrates a joint probability distribution over unwanted coordinates to obtain a marginal distribution.

A Majorana-Weyl spinor satisfies both a Majorana spinor reality condition and a Weyl spinor chirality condition when dimension and signature permit them. In ten-dimensional Lorentzian spacetime it has sixteen real components off shell and eight physical massless spin-one-half polarizations.

If $\kappa$ is Mahlo in $V$, then it is Mahlo in $L$. Inaccessibility is downward absolute, and every club in $L$ remains a club in $V$, where it meets a $V$-inaccessible ordinal.

For an ideal incompressible flow rotating about the axis of the field $\mathbf{B}=(J/2)s\hat{\mathbf{\varphi }}$, use cylindrical Fourier modes $e^{\sigma t+im\varphi }$. The azimuthal derivative of a cylindrical vector Fourier mode reduces induction to $\sigma \mathbf{b}=imJ\mathbf{u}/2$. If the boundary eigenproblem gives $\lambda =i\delta \nu$, with real $|\delta |<1$, eliminating the perturbation gives the displayed dispersion relation. Here $J=\nabla \times \mathbf{B}\cdot \hat{\mathbf{z}}={\mu }_0{J}_{\mathrm{phys}}$ for SI current density; the formula is conditional on that boundary eigenrelation.