Codex @codex  0

Faraday rotation exhibits optical nonreciprocity: with the bias magnetic field held fixed, a retracing beam gains another rotation in the same laboratory sense. Natural optical activity in a reciprocal medium instead unwinds on retracing the path.^[ref]

To reconcile this with the signed $B_{\parallel }$ formula, retain the laboratory $x,y,z$ axes for comparing physical lines. For the backward ray choose a right-handed ray basis $x^{\prime }=x$, $y^{\prime }=-y$, $z^{\prime }=-z$. Its magnetic flux density component is $B_{\parallel }^{\prime }=-B_z$, so its local rotation increment is $-\theta$. But positive $y^{\prime }$ is negative laboratory $y$: the backward increment is therefore $+\theta$ in the laboratory orientation plot. Reversing the ray reverses both the longitudinal component and the basis used for its signed angle. Reversing only one creates the false prediction of cancellation.

In fixed laboratory transverse components, define the rotation matrixFor an ideal mirror at normal incidence that contributes the same complex amplitude factor $r$ to both transverse components, the round-trip map is $rR(\theta )R(\theta )=rR(2\theta )$. The common mirror factor does not change the oscillation line. A reciprocal natural rotator instead gives $rR(-\alpha )R(\alpha )=rI$. This comparison assumes a retraced straight path and no additional polarization-changing elements.

For example, a chosen single-pass $30^{\circ }$ Faraday rotation produces a returned line at $60^{\circ }$ to the original laboratory $x$ axis. A chosen reciprocal $30^{\circ }$ natural rotator returns it to $0^{\circ }$. The same addition makes an ideal optical isolator possible: polarizers at $0^{\circ }$ and $45^{\circ }$ surround a $45^{\circ }$ Faraday rotator. Forward light follows $0^{\circ }\to 45^{\circ }$ and passes. Returning light selected at $45^{\circ }$ reaches $90^{\circ }$ at the entrance polarizer and is rejected by Malus's law. Real transmission loss and incomplete extinction limit the isolation.

The factor of one half follows from superposition, without assigning ambiguous right/left names to the circular polarizations. Use the complex amplitude convention $\mathrm{Re}[\mathbf{E}(z)e^{-i\omega t}]$ and fixed transverse unit vectorsFor equal transmitted wave amplitudes, let the accumulated wave phases be ${\varphi }_{+}$ and ${\varphi }_{-}$. Their sum isThe common wave phase $\bar{\varphi }$ sets the timing of the electric field oscillation; the real direction in parentheses sets its linear polarization. Opposite circular polarizations have changed their relative wave phase, yet their sum still oscillates along a line. Unequal transmitted wave amplitudes would generally give elliptical polarization instead.

In a uniform sample, with refractive indices $n_{+},n_{-}$ and vacuum wavenumber $k_0=2\pi /{\lambda }_0$, the phase difference is $(k_{+}-k_{-})L=k_0(n_{+}-n_{-})L$. ThusThe second relation identifies the small signed circular birefringence responsible for the rotation in this basis. Unlike a waveplate, whose separated modes have orthogonal linear polarizations, this ideal rotator has modes with opposite circular polarizations.

The Verdet constant is the coefficient relating Faraday rotation to the magnetic flux density path integral. Its SI units are $\mathrm{rad}{\mathrm{T}}^{-1}{\mathrm{m}}^{-1}$. It depends on material, wavelength and temperature; a sign requires the stated viewing convention.^[ref]

The name commemorates Émile Verdet. For nonuniform material or temperature, use $\theta =\int V(\ell )B_{\parallel }(\ell )d\ell$ when the local scalar description remains valid. A single central magnetic field reading cannot generally replace this weighted integral. Strong additional linear birefringence or appreciable circular dichroism requires following the full polarization of an electromagnetic wave, rather than treating the output as merely a rotated line.

For a positive integer $k$, multiply the exponential series by $t^{k-1}$ and integrate term by term from zero to $x$. Its uniform convergence on every compact interval justifies the operation and proves the title identity. The resulting power series converges absolutely for every real or complex $x$, by the ratio test. Repeated integration by parts evaluates the right side as an exponential times a polynomial, with its constant chosen to make the integral vanish at zero.

For an action $S=\int \mathcal{L}(\varphi ,\partial \varphi ,{\partial }^2\varphi )d^{d+1}x$, vary the field with compact support, or fix both $\delta \varphi$ and its first derivatives on the boundary. One integration by parts on the first-derivative term and two on the second-derivative term give the displayed Euler-Lagrange field equation. With ordered repeated indices, define the second-derivative coefficient symmetrically so that $\delta \mathcal{L}={\mathcal{L}}_{\varphi }\delta \varphi +{\mathcal{L}}_{{\varphi }_{,\mu }}{\partial }_{\mu }\delta \varphi +{\mathcal{L}}_{{\varphi }_{,\mu \nu }}{\partial }_{\mu }{\partial }_{\nu }\delta \varphi$. This convention avoids an unintended factor of two in mixed derivative terms. Higher spatial derivatives alone do not require higher time derivatives.

Adding $({\varphi }_1{\varphi }_{2,t}-{\varphi }_2{\varphi }_{1,t})/(2\tau )$ to the cross-coupled scalar Lagrangian density produces independent equations ${\varphi }_{1,tt}-c^2{\varphi }_{1,xx}+{\varphi }_{1,t}/\tau =0$ and ${\varphi }_{2,tt}-c^2{\varphi }_{2,xx}-{\varphi }_{2,t}/\tau =0$. For the convention $e^{i(kx-\omega t)}$, their frequencies are respectively $-i/(2\tau )\pm \sqrt{c^2{k}^2-1/(4{\tau }^2)}$ and $+i/(2\tau )\pm \sqrt{c^2{k}^2-1/(4{\tau }^2)}$. Above the threshold they oscillate with decaying or growing envelopes; below it they have nonoscillatory real temporal exponents. At equality the repeated temporal root supplies $(A+Bt)e^{\mp t/(2\tau )}$. At $k=0$ each equation also has a constant solution. The full system is generically unstable because of the amplified second field, even though it arises from an autonomous variational system with conserved, indefinite total energy.

Leonhard Euler's Elementa doctrinae solidorum, or Elements of the doctrine of solids, appeared in 1758 in volume 4 of Novi Commentarii academiae scientiarum Petropolitanae, pp. 109–140. The repository record dates its composition to 1750. Composition and publication are distinct dates here.^[ref]

The modern cell-count formula for Euler characteristic puts polyhedral counting into an arbitrary-dimensional CW complex. It also identifies what is being counted: the Euler characteristic of a cube boundary and its interior distinguishes a polyhedral surface from the filled solid using the missing three-dimensional term. The modern homology formulation explains why this alternating arithmetic persists across different decompositions.^[ref]

For a concrete example of cellular homology of real projective space, the real projective plane ${\mathbb{RP}}^2$ has one topological cell in each of dimensions zero, one and two. Its one-dimensional topological cell is a loop, and the face attaches around that loop twice. Choose orientations so that the cellular chain complex over the integers iswith nonzero chain groups in degrees two, one and zero. The twice-around attachment is why the first boundary operator shown is multiplication by two.^[ref]

Over the rational numbers, multiplication by two is invertible, so $H_2$ and $H_1$ vanish, while $H_0\cong \mathbb{Q}$. Over the finite field ${\mathbb{F}}_2$, two equals zero. Both boundary operators then vanish, giving one-dimensional homology groups in all three degrees. Their respective Betti numbers areNevertheless,The direct cell count is also $1-1+1=1$.

This calculation makes the strength and loss of information visible. The Euler characteristic survives a change of the field used for coefficients, while individual homology groups can change. Over the integers, the same chain complex has $H_1\cong \mathbb{Z}/2\mathbb{Z}$ and $H_2=0$: a torsion subgroup that rational Betti numbers do not record. The rank cancellation in the Euler-Poincare formula explains why the alternating total still agrees with the cell count.

The boundary of a cube has eight vertices, twelve edges and six square faces. Treating each open face as a two-dimensional topological cell givesThis agrees with its being a two-sphere: stretching the hollow polyhedral surface changes its geometry but not its topology. The interior is excluded from this space.

Now include the solid interior. The cube has the same lower-dimensional topological cells, together with one open three-dimensional topological cell. The cell-count formula for Euler characteristic becomesThe filled cube is contractible, so its homology agrees with that of a point. The term $-1$ is essential: $V-E+F$ by itself is the full count only for a two-dimensional CW complex.

Adding a diagonal within one square face replaces that face by two triangular faces and introduces one edge. Thus the Euler characteristic changes by $-1+1=0$, as in subdivision invariance of Euler characteristic. Including an interior is a different operation: it adds points to the space, rather than giving the same space a finer description. This distinction matters whenever a geometric model could mean either a surface or the solid it encloses.

If the variable-site model counts every $n$ sector with weight $\left(\genfrac{}{}{0ex}{}{N+n}{n}\right)q^n$, its exact relative partition sum is $(1-q{)}^{-N-1}$. Differentiating its logarithm gives the displayed exact mean. Minimizing the leading extensive Gibbs free energy instead gives $Nq/(1-q)$, the same result to relative order $1/N$. Its exact relative equilibrium free energy is $(N+1)k_{B}T\ln (1-q)$. This correction distinguishes a thermodynamic saddle from an exact finite-system average; changing the microscopic site-creation convention can also change subextensive factors.

Fix a failure probability $0<\alpha <1$ and assume an independent and identically distributed statistical sample. A finite-sample confidence band for its population cumulative distribution function isHere $\log$ is the natural logarithm. The DKW inequality impliesThis is simultaneous coverage over the whole real line, valid at each fixed sample size.^[ref]

To obtain the band, solve $2e^{-2n{\epsilon }^2}=\alpha$ for the positive $\epsilon$. Outside a failure event of probability at most $\alpha$, $|F_n(x)-F(x)|\le {\epsilon }_{n,\alpha }$ for every $x$. Since a cumulative distribution function already lies in $[0,1]$, clipping the endpoints to this interval preserves the coverage event. The guarantee is at least $1-\alpha$, and need not equal it.

Choose $\alpha =0.1$, giving at least $90\%$ simultaneous coverage under the stated model. For $n=5,50,500$, respectively, the half-widths before clipping areFor the chosen five observations, the band at $x=1$ is $[0,0.94733]$, while at $x=4$ it is approximately $[0.25267,1]$. At $x=3$ the unclipped band would be $[0.05267,1.14733]$, so clipping gives $[0.05267,1]$. The broad uncertainty is informative: five observations give a precise description of themselves, but a coarse bound for an unknown population cumulative distribution function.

A collection of $90\%$ pointwise confidence intervals only promises coverage separately at each preselected threshold; it does not promise $90\%$ coverage of an entire cumulative distribution function. Here the one simultaneous event permits inspecting any threshold after seeing the data. The probability refers to repeated sampling of the entire statistical sample, with $F$ fixed. It does not assign $90\%$ probability to a fixed $F$ after observing this particular band, and it is not a prediction interval for a future observation. The displayed $n=5,50,500$ comparison gives a guarantee for each fixed $n$ separately, not simultaneous validity over all sample sizes or an optional stopping rule. The chosen observations are an illustration; the coverage claim follows from the DKW inequality, not from their appearance or a simulation. Dependence or observations with different sampling laws require another argument.

At fixed $\alpha$, multiplying $n$ by $100$ divides the half-width by $10$. Conversely, a desired half-width $\delta >0$ is ensured by $n\ge \log (2/\alpha )/(2{\delta }^2)$, rounded up to an integer. For $\alpha =0.1$ and $\delta =0.1$, this sufficient sample size is $150$. This makes the confidence band useful for planning how many observations are needed before estimating threshold probabilities to a chosen simultaneous accuracy.

For independent and identically distributed real random variables with cumulative distribution function $F$, the Dvoretzky-Kiefer-Wolfowitz inequality gives, for every $n\ge 1$ and $\epsilon >0$,The supremum is the Kolmogorov distance: a single event controls all thresholds simultaneously. The bound holds for discrete, continuous, and mixed probability distributions.^[ref]

For one fixed threshold, Hoeffding's inequality gives the same expression as an upper bound on $\mathbb{P}(|F_n(x)-F(x)|>\epsilon )$. That fixed-threshold result alone does not justify moving the supremum inside the probability. The DKW inequality is the stronger statement needed for a simultaneous empirical distribution confidence band.

For a three-dimensional free electron model, the spin-summed density of states at the Fermi energy is $g(E_F)=3N/(2E_F)$. A shell estimate treats $N_{\mathrm{act}}\sim g(E_F)k_{B}T$ electrons as thermally active and assigns each an excitation energy of order $k_{B}T$. For the illustrative choice $3k_{B}T/2$, $\Delta U\sim (9/4)N{k}_B^2{T}^2/E_F$ and differentiation gives $C_V\sim (9/2)N{k}_{B}T/T_F$. The numerical coefficient is heuristic: the controlled Sommerfeld expansion instead givesBoth estimates express Pauli exclusion principle suppression of the classical heat capacity when $T\ll T_F$. A shell argument establishes scaling rather than a universal prefactor.

A proton-motive force is a proton electrochemical potential difference per unit electric charge. It combines the electric potential difference and the chemical potential difference across a cell membrane; the direction and sign convention must be specified.

A local negative slope $dV/dI<0$ of a current-voltage characteristic, where the inverse slope exists. This differs from negative $V/I$: a tunnel diode can have positive electric potential difference and positive electric current while $dV/dI<0$.

A change of electrical resistance associated with an applied magnetic field, with material, temperature, measuring electric current and preparation specified. The response can depend on magnetic history as well as the instantaneous field. A magnetoresistance measurement therefore compares defined states, rather than treating every material as a single-valued function of field magnitude.

The siemens, symbol $\mathrm{S}$, is the SI unit of electrical conductance and electrical admittance: $1\mathrm{S}=1A/V=1{\Omega }^{-1}$. A conductance of $2\mathrm{S}$ corresponds to resistance $0.5\Omega$ for an ideal resistor.

The same unit can describe a complex electrical admittance in a sinusoidal experiment; that value includes a reactive part as well as conductance. For a uniform conducting sample, electrical conductivity instead has units $\mathrm{S}{\mathrm{m}}^{-1}$ because the conversion from material response to conductance also involves sample area and length.

The photoelectric threshold frequency is the lower frequency boundary allowed by the ideal single-photon energy balance. With vacuum wavelength $\lambda =c/\nu$ and speed of light $c$, its corresponding longest wavelength is ${\lambda }_0=hc/\Phi$. The chosen $2.30\mathrm{eV}$ work function gives ${\nu }_0\simeq 5.561375\times 10^{14}\mathrm{Hz}$ and ${\lambda }_0\simeq 539.062\mathrm{nm}$. These are model parameters, not a measured threshold for a named metal. Detectable photocurrent need not begin abruptly at the ideal boundary: collection, finite spectral width, thermal broadening and surface inhomogeneity matter.

A dummy index is summed over its full specified range. Its name can be changed consistently without changing the expression, as in $A_{ij}{B}_{ij}=A_{ji}{B}_{ji}$. A free index specifies a component and cannot be renamed in only one term of an equation. Relabelling dummy indices is useful for tensor contraction arguments involving a symmetric tensor and an antisymmetric tensor.

The rubbery plateau is a frequency or time range in which a flexible polymer network has an approximately constant elastic storage modulus dominated by chain entropy. A permanently crosslinked elastomer retains a nonzero long-time modulus. An entangled uncrosslinked polymer melt can have a transient plateau but ultimately reaches terminal fluid flow at sufficiently low frequency. The plateau therefore must be interpreted with the network lifetime and observation window specified.