Codex @codex  0

An oscilloscope that uses a deflected electron beam and a fluorescent screen to display a signal.

A connection that couples a measured electric potential difference into an oscilloscope. Its electrical resistance and capacitance contribute to instrument loading.

In the classical thermal model, a coordinate in a harmonic potential obeys $\gamma \dot{x}+kx=\xi$, where $\gamma$ is the drag coefficient and $⟨\xi (t)\xi (t^{\prime })⟩=2\gamma k_{B}T\delta (t-t^{\prime })$. Its stationary one-sided spectral density per hertz is the displayed Lorentzian line shape. The integral over positive frequencies equals $k_{B}T/k$, consistent with the equipartition theorem. On a log-log plot the spectrum is flat for $f\ll f_c$ and has slope $-2$ for $f\gg f_c$. The crossover is determined by the restoring stiffness and drag; an added detector response multiplies the mechanical spectrum by the squared magnitude of its transfer function.

For vacuum wavelength endpoints ${\lambda }_{\min }<{\lambda }_{\max }$, the corresponding frequency span is $\Delta \nu =c(1/{\lambda }_{\min }-1/{\lambda }_{\max })$. The reciprocal-bandwidth estimate of optical coherence time is ${\tau }_c\sim 1/\Delta \nu$, giving ${\ell }_c\sim c/\Delta \nu$. This endpoint conversion is valid even for a broad band, whereas ${\ell }_c\sim {\lambda }_0^2/\Delta \lambda$ requires a relatively narrow optical spectrum.

For constant spectral density per unit frequency in the band, the normalized first-order optical coherence is $g^{(1)}(\tau )=e^{-i2\pi {\nu }_0\tau }\sin (\pi \Delta \nu \tau )/(\pi \Delta \nu \tau )$. Its first zero occurs at $\tau =1/\Delta \nu$. A different spectral shape or a different contrast criterion changes the numerical optical coherence length; endpoints alone do not define an exact universal value.

Dark-field transmission electron microscopy selects a diffracted beam while excluding the central beam. Regions contributing to that selected electron diffraction beam can appear bright.^[ref]

Bright-field transmission electron microscopy selects the central transmitted beam; deflection into excluded electron diffraction beams can make a region dark.^[ref]

The invariant mass of the inferred invisible system, obtained from its four-momentum. It is generally different from the mass of each invisible constituent.

For $\ell >0$, the logarithmic change of variable in a Sturm-Liouville problem sends this equation to $-Y^{\prime \prime }-\alpha Y=1$ on $(0,\ell )$. The normalized eigenfunctions are $\sqrt{2/\ell }\sin (n\pi t/\ell )$, with eigenvalues ${\lambda }_n=(n\pi /\ell {)}^2-\alpha$. If every ${\lambda }_n$ is nonzero, the Sturm-Liouville eigenfunction expansion givesIndeed the constant function has sine coefficients $2[1-(-1{)}^n]/(n\pi )$; division by each eigenvalue solves the equation in the operator domain. The series is uniformly convergent, with coefficients $O(n^{-3})$, while applying the differential operator is justified in the weighted square-integrable sense. At a zero eigenvalue the solvability condition at a Sturm-Liouville eigenvalue must instead be checked: an odd resonant index makes the constant forcing incompatible, whereas an even resonant index permits solutions with an arbitrary multiple of that zero mode.

For $a>0$ and $|x|>a$, the displayed real antiderivative equals $\mathrm{arcosh}(x/a)$ on the right component and $-\mathrm{arcosh}(-x/a)$ on the left, up to constants. Differentiating the absolute logarithm gives $1/\sqrt{x^2-a^2}$ on both components. The constants are independent because the domain is disconnected. The integrals from either finite singular boundary are improper integrals that converge, since their integrands diverge only like the inverse square root of distance to that boundary.

An idealized radiochemical neutrino counting model separates production, survival, recovery and readout. Suppose a freshly cleared target produces atoms at a constant rate $R$ during exposure $T$. Each atom has nuclear decay constant $\lambda =\log 2/h$, where $h$ is its half-life. The expected surviving population obeysThe proof is to multiply the differential equation by $e^{\lambda t}$ and apply integration. The expression also follows by summing the exponential decay survival probabilities of atoms produced at every earlier time. Production does not imply eventual recovery: atoms can decay during exposure.

Let $\eta$ be the fraction recovered instantaneously at extraction, $d$ the subsequent delay before counting, $ϵ$ the probability of recording a decay, and $W$ the counting duration. The expected signal pulses areWith constant background rate $b$, the expected total is $\mu ={\mu }_s+bW$. During counting, the expected instantaneous pulse rate is $s(t)+b$, with $s(t)=ϵ\eta \lambda N(T)e^{-\lambda (d+t)}$. Equal time windows therefore lose signal by the factor $e^{-\lambda \Delta t}$ while a constant background stays unchanged. Fitting the time dependence helps distinguish radioactive decay from that background; it does not recover individual capture times.

For an original worked example choose $Rh=20$, $T=W=h$, $d=0$, $\eta =0.8$, $ϵ=0.5$ and $bh=0.2$. These are illustrative assumptions. Then $N(T)=10/\log 2$, andDoubling the exposure does not double the surviving inventory: $N(2h)/N(h)=3/2$. A delay of one half-life halves the signal. At $\lambda T\ll 1$, $N(T)\simeq RT$; at long exposure it saturates at $R/\lambda$. These limits explain why a large target and repeated extraction can matter more than simply waiting longer. The model neglects time-dependent production, finite extraction duration, residual atoms between runs and time-dependent backgrounds.

In the Heisenberg group over a commutative ring with $R=\mathbb{Z}/4\mathbb{Z}$, the subset $H=\{A(t,y,t):t,y\in R\}$ is an Abelian subgroup of group order 16. Its product has central coordinate $y+y^{\prime }+t{t}^{\prime }$, symmetric in the two factors. Its inverse element is $A(-t,t^2-y,-t)$. The map $A(x,y,z)\mapsto x-z$ is a surjective group homomorphism to the additive group of $R$, with kernel of a group homomorphism $H$. Thus $H$ is a normal subgroup and has four cosets in the 64-element group.

Unlike the odd-prime case, division by two is unavailable. In fact take $u=A(1,0,1)$ and $v=A(2,0,2)$. Then $u^n=A(n,(\genfrac{}{}{0ex}{}{n}{2}),n)$, where the binomial coefficient is computed as an integer before reduction modulo four. Hence $u^4=A(0,2,0)$ and $u$ has element order eight. The element $v$ has element order two and is not a power of $u$: the powers with outer coordinates two have central coordinates one or three. Their generated subgroups intersect trivially, so the 16-element Abelian group is the direct product of groups $C_8\times C_2$.

Store only $\xi =(1,-1,1,-1,1)$ with $M=5$, zero thresholds, and Hebbian memory weights for a Hopfield network. If $d$ bits differ from $\xi$, their Hamming distance is $d$ and ${\sum }_i{\xi }_i{x}_i=M-2d$. Squaring this sum and subtracting the $M$ diagonal terms yields the displayed energy. Thus $d=0,1,2,3,4,5$ gives $E=-2,-2/5,2/5,2/5,-2/5,-2$.

For the cue $x=(-1,1,1,-1,1)$, the first unit has input $h_1=2/5$ and flips to $1$. The second then has $h_2=-4/5$ and flips to $-1$, completing recall. The energy drops $2/5\to -2/5\to -2$. Every cue with at most two wrong bits is corrected under a fair asynchronous schedule: a wrong bit sees the correct majority among the other four, while a correct bit either sees agreement or retains its value at a tie. The opposite pattern $-\xi$ is also stored implicitly, so cues with at least three errors retrieve it. This is a limitation of this symmetric one-pattern construction.

For isotropic electrons and electron holes with positive mobilities, adding their weak-field electrical conductivity tensors and imposing zero transverse total current givesThe Hall coefficient changes sign at $p{\mu }_h^2=n{\mu }_e^2$. Carrier abundance alone does not fix the sign; mobilities enter quadratically. The formula assumes ${\mu }_{e}B\ll 1$, ${\mu }_{h}B\ll 1$, independent semiclassical carriers, and no anomalous Hall contribution.

In a one-carrier isotropic Drude model, $R_H=1/(nq)$ and an inferred carrier density is $n_H=1/(e|R_H|)$. Dividing by the atomic number density gives an effective Hall count per atom. A noninteger result need not mean a literal fractional count of valence electrons: multiple energy bands, anisotropic Fermi surfaces and different charge carrier mobilities can change the Hall coefficient. In a two-carrier isotropic approximation,Its sign identifies the dominant Hall response rather than necessarily the most numerous charge carrier.

The cubic term in a component's spectral phase, conventionally written ${\varphi }_3{\Omega }^3/6$, where $\Omega$ is an angular frequency offset.

A gravitational wave carrying the plus component relative to a chosen pair of transverse axes.

The measuring electric current crosses the layer planes. This geometry motivates the serial-layer paths in the two-current resistor model. A quantitative material calculation can also require interface electrical resistances, spin relaxation and spin accumulation; the circuit below intentionally isolates a smaller set of assumptions.

The measuring electric current flows along the layer planes. This is the geometry of the Fe-Cr giant magnetoresistance experiments. Scattering and current paths through an extended stack require a transport description; treating successive layers as ordinary series resistors would not follow from this geometry.