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Let $S_0=0$ be a simple symmetric random walk, let $T_a$ be its first hitting time of the positive integer $a$, and reverse every increment after $T_a$. The resulting path is again a simple symmetric random walk. Consequently,

The continuous analogue is the existing Reflection principle (Wiener process); its tail formula has no discrete one-unit lattice correction.

For a stopping time $T$,The indicator is predictable, so a stopped integrable martingale is a martingale.

For a square-integrable martingale transform and positive conditional increment variance,

A right-continuous continuous-time martingale with ${\sup }_{t\ge 0}\mathbb{E}|M_t|<\infty$ has an almost-sure finite limit as $t\to \infty$.

A Majorana zero mode is a spatially localized Majorana operator that commutes with a gapped quadratic Hamiltonian, exactly or up to exponentially small finite-size corrections.

For a vertical magnetic field in a Keplerian disk, axisymmetric vertical MRI modes are unstable when $k^2{v}_A^2<3{\Omega }^2$. The shortest unstable wavelength is ${\lambda }_{\mathrm{crit}}=2\pi v_A/(\sqrt{3}\Omega )$.

A rotational discontinuity changes the tangential direction of the magnetic field and velocity while leaving density, pressure, and normal components continuous. The normal flow is Alfvénic.

The de Hoffmann-Teller frame is a tangentially boosted shock frame in which the electric field vanishes. Ideal MHD then makes the fluid velocity parallel to the magnetic field on both sides.

For constant $B$,

A Lie algebra is nilpotent when its lower central series eventually becomes zero.

Combining $d(\gamma m\mathbf{v})/dt=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$ with $m{c}^{2}d\gamma /dt=q\mathbf{E}\cdot \mathbf{v}$ gives

For $\mathcal{E}=\gamma m{c}^2$, the temporal component of the covariant Lorentz-force equation givesThe magnetic field does no work because $\mathbf{v}\cdot (\mathbf{v}\times \mathbf{B})=0$.

In uniform perpendicular electric fields and magnetic fields, a charged particle has the transverse guiding centre driftFor nonrelativistic motion, write $\mathbf{v}={\mathbf{v}}_D+\mathbf{w}$. The vector triple product gives ${\mathbf{v}}_D\times \mathbf{B}=-\mathbf{E}$, so the Lorentz force equation becomesThe residual perpendicular motion is circular, while the mean drift is independent of charge and mass. Parallel motion along $\mathbf{B}$ can be added. ^[ref]

Relativistically, the drift frame exists for magnetically dominated crossed fields,A Lorentz boost at ${\mathbf{v}}_D$ then eliminates the electric field and leaves $B^{\prime }=\sqrt{B^2-E^2/c^2}$. The particle can execute cyclotron motion in that frame. At or above $E=cB$, the same formal ratio does not supply a subluminal zero-electric-field frame, so the orbit statement needs this restriction.

For example, $\mathbf{E}=E{\mathbf{e}}_x$ and $\mathbf{B}=B{\mathbf{e}}_z$ give ${\mathbf{v}}_D=-(E/B){\mathbf{e}}_y$ for either sign of the particle's electric charge.

For a positive semidefinite trace ball ${\mathcal{S}}_s$, the quadratic-form hypothesis class isIts empirical logistic risk is convex as a function of $M$.

For positive-definite Hermitian matrices, the matrix logarithm preserves the Loewner order:No commutativity is required. With the natural logarithm, the spectral theorem gives the convergent integral representationIf $0<C\le E$, then $E^{-1/2}C{E}^{-1/2}\le I$; taking its inverse and applying the reverse congruence gives $C^{-1}\ge E^{-1}$. HenceThis proves operator monotonicity by a resolvent comparison.

For singular positive semidefinite matrices, the ordinary finite matrix logarithm is not defined at a zero eigenvalue. If $A$ and $B$ have the same kernel of a linear map, compress to its orthogonal complement and apply the theorem there. More generally $\log (A+\epsilon I)\le \log (B+\epsilon I)$ holds for every $\epsilon >0$, but divergent logarithms cannot simply be replaced by finite operators on different supports. The spectral definition and this support qualification are essential.^[ref]

For $x>0$, concavity of the natural logarithm givesEquivalently, ${\log }_{b}x\le ({\log }_{b}e)(x-1)$ for $b>1$.

The natural logarithm $\ln x$ is the logarithm with base Euler's number, and the inverse function of the real exponential function. A particularly geometric definition isFor $x>1$, it is the area under the hyperbola $y=1/t$ from $1$ to $x$. For $0<x<1$, the reversed integration limits give a negative value. The fundamental theorem of calculus gives $(\ln x{)}^{\prime }=1/x$, so the natural logarithm is increasing, while $(\ln x{)}^{\prime \prime }=-1/x^2$ gives concavity.

The change of variables $t=au$ shows ${\int }_a^{ab}dt/t={\int }_1^{b}du/u=\ln b$ for $a,b>0$. Thus the intervals from $1$ to $2$ and from $2$ to $4$ each contribute area $\ln 2$, explaining $\ln 4=2\ln 2$. Equal ratios give equal increments on a logarithmic scale, even though the ordinary differences grow. This makes the natural logarithm useful for comparing multiplicative changes and solving for time in exponential growth or exponential decay. The integral construction and inverse function properties are discussed in OpenStax, the natural logarithm as an integral.

A locally uniform limit of continuous real-valued functions is continuous. Moreover, local uniform convergence is uniform on every compact subset: choose finitely many neighbourhoods from the local uniformity cover and take the largest of their convergence indices.

A line bundle on a scheme is a locally free sheaf of rank one. Its global sections can define a morphism to projective space when they have no common zero.

Local polynomial weights of Degree $p$ reproduce every multivariate polynomial of total degree at most $p$: applying the estimator to its design values returns its value at the target point.