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An algebraic variety whose irreducible components all have the same dimension of an algebraic set. The components of a generic fibre in the fiber dimension theorem have this property even when that fibre is reducible.

For a diffusion process with diffusion generator $L$, the Itô formula makes $\phi (X_t)-\phi (X_0)-{\int }_0^{t}L\phi (X_s)ds$ a local martingale. Under stopping and integrability conditions making it a true martingale, taking expectations gives the displayed identity. For Brownian motion, $L=\frac{1}{2}\Delta$, recovering the usual Brownian formula.

After diffusion arrest of Gaussian packet compression, define the logarithmic area increase $A_t=\frac{1}{2}\log [\det J(t)/\det J(0)]$. It is nonnegative. Assume $A_t/t$ has a large deviation principle with convex rate function $\mathcal{I}$, minimized at $\lambda >0$. The exact amplitude relation $f(t)=f(0)e^{-A_t}$ and the Laplace principle for probability measures give the displayed moment decay rate. The rate function refers to area, so rare width fluctuations are included rather than discarded without justification. For differentiable strictly convex $\mathcal{I}$, the interior minimizer satisfies ${\mathcal{I}}^{\prime }(s_{\mu })=-\mu$. If ${\mu }_{\ast }=-{\mathcal{I}}^{\prime }(0)$ is finite, then ${\gamma }_{\mu }=\mathcal{I}(0)$ for $\mu \ge {\mu }_{\ast }$: rare nearly unstretched packets dominate high moments. Below the threshold, $d({\gamma }_{\mu }/\mu )/d\mu =-\mathcal{I}(s_{\mu })/{\mu }^2<0$. These statements concern a positive Gaussian packet with flow-only averaging; a homogeneous signed random initial scalar requires a different averaging kernel. Arbitrary random $C(t)$ alone does not imply this statistical model.

The Rayleigh criterion calls two equal, mutually incoherent optical point sources just resolved when one circular-pupil Airy disk's centre coincides with the other's first zero. It is a specified discrimination convention, distinct from full width at half maximum and from precision in estimating one source's position.

For this circular optical pupil, the Rayleigh criterion uses angular separation $s_{\mathrm{R}}={\theta }_1\simeq 1.22\lambda /D$, or $s_{\mathrm{R}}=r_1\simeq 1.22\lambda f/D$ in the paraxial approximation to the focal plane. The point spread functions still overlap.^[ref]

For the Airy disk design example with $D=5.00\mathrm{mm}$, $f=100\mathrm{mm}$ and $\lambda =550\mathrm{nm}$, the angular convention gives $s_{\mathrm{R}}=1.34164\times 10^{-4}\mathrm{rad}$, or about $27.67$ arcseconds, and the image-plane separation is $13.4164\mu \mathrm{m}$. Two equally bright, mutually incoherent optical point sources at separations $0.7$, $1$ and $1.5$ times this scale would have image separations $9.391$, $13.416$ and $20.125\mu \mathrm{m}$ respectively. These calculated cases are compared in incoherent two-source Airy profile.

The Rayleigh criterion is useful because it ties an easily recognized two-peak convention to a predictable optical scale. It does not assert that all information vanishes below that separation. Inferring two unknown positions also depends on photon shot noise, background, unequal brightness and uncertainty in the point spread function; locating a single isolated source is a different task. A theoretical analysis shows that alternative optical measurements can estimate the separation of two incoherent optical point sources below this scale.^[ref]

The mutual optical coherence condition is essential. In incoherent imaging, time-averaged light intensities add. In coherent imaging, wave amplitudes add before squaring, leaving a wave interference term controlled by relative wave phase. Two holes illuminated by the same coherent laser therefore do not implement the equal-incoherent-source convention merely because their individual patterns resemble Airy disks. The coherent single-aperture laboratory test establishes the response of each aperture; it does not establish the pair's mutual incoherence.

Perturb the Couette flow between walls moving at +-U by a small pressure gradient -G. The mean shear rate remains U/h, and the stress variation across the gap is -Gy. Linearizing the monotone generalized Newtonian fluid law divides this variation by differential viscosity, giving the displayed velocity and flux per unit width. Require $Gh\ll (U/h){\eta }_d(U/h)$ and a smooth constitutive law on the perturbed rates. For a power-law fluid, the denominator is $nk(U/h{)}^{n-1}$.

Linearizing the differential pom-pom model about its relaxed state leaves no first-order stretch perturbation in incompressible flow. Its anisotropic orientation obeys $\dot{a}+a/{\tau }_1=A_1$, producing the displayed relaxation modulus. In a slow-flow expansion, $A=I+{\tau }_1{A}_1+{\tau }_1^2(2A_1^2-A_2)+O(3)$. Trace normalization and the first stretch correction affect only isotropic stress at this order, giving the displayed second-order fluid coefficients. Linear small-amplitude response allows arbitrary frequency, whereas this local second-order expansion also requires slow time dependence.

For extension rate $s>0$ and $x=s{\tau }_1<1/2$, the steady conformation tensor is $\mathrm{diag}((1-2x{)}^{-1},(1+x{)}^{-1},(1+x{)}^{-1})$. Its normalized orientation has axial component $(1+x)/[3(1-x)]$ and transverse components $(1-2x)/[3(1-x)]$. Stretch is finite only if $s^2{\tau }_1{\tau }_2<1-x$, when ${\lambda }_{\infty }=(1-x)/(1-x-s^2{\tau }_1{\tau }_2)$ and the displayed extensional viscosity follows. Above $x=1/2$, the unnormalized axial tensor grows exponentially but $B\to \mathrm{diag}(1,0,0)$; then stretch remains bounded only if $s{\tau }_2<1$. Orientation normalization and stretch relaxation therefore impose different thresholds.

For a real-valued smooth function on a smooth manifold, its differential of a smooth map is a differential one-form: $d{f}_p$ maps a tangent vector to the directional derivative of the function. In local coordinates, $df={\sum }_i({\partial }_{i}f)d{x}^i$. The exterior derivative extends this operation from functions to higher-degree differential forms.

Take $X$ with continuous uniform distribution on $(0,1)$ and set $Y=X^2$. This is an increasing invertible transformation on that interval. Its cumulative distribution function is $F_Y(y)=\sqrt{y}$, givingAlthough the probability density diverges near zero, it is integrable. The logarithmic stretch is integrable too:The inverse function $X=\sqrt{Y}$ recovers every input in the interval. The changed differential entropy records a changed coordinate volume, rather than a failure of recovery.

// Target: probability-and-statistics.bigb

The Fresnel equations give reflection and transmission amplitudes for both transverse-electric and transverse-magnetic polarization of an electromagnetic wave at a planar interface. They follow by applying interface conditions for electromagnetic fields to incident, reflected and refracted plane electromagnetic waves.

For a computably enumerable first-order theory $T$ extending Robinson arithmetic, choose the standard arithmetic arithmetical provability predicate ${\mathrm{Prov}}_T$. The diagonal lemma, carried out in the weak base theory, gives a sentence $G$ withIf $T$ is consistent, $T$ cannot prove $G$: an actual finite proof would be verifiable in the weak base, so $T$ would prove both its provability and its unprovability. There is therefore no actual proof of $G$, and the displayed equivalence, true in the standard natural numbers, shows that this canonical $G$ is true there.

If $T$ is omega-consistent, it cannot prove $\neg G$ either. Such a proof would imply the existence of a proof code for $G$, while each standard numeral can be individually verified not to be such a code. Ordinary consistency suffices for unprovability of $G$; the original argument for unprovability of its negation uses the stronger hypothesis. Requiring the fixed-point equivalence in a sound weak base is important: an arbitrary equivalence provable only in an unsound extension need not yield a true sentence.

A logistic regression fitted to ungrouped Bernoulli observations has residual deviance $D=-2\ell (\hat{\beta })$. Comparing it automatically with ${\chi }_{n-p}^2$ is generally invalid: the saturated dimension grows with $n$, with one observation per fitted probability. If the true model has only an intercept and success probability $1/2$, then $D/n\to 2\log 2$, whereas ${\chi }_{n-1}^2/n\to 1$. Fixed-dimensional nested-model likelihood-ratio tests can still use Wilks theorem under regularity. For absolute fit, use meaningful grouping, residual diagnostics, or a fitted-model parametric bootstrap.

With the axiom of choice, enumerate both players' binary-move strategies in an infinite game in length $\mathfrak{c}$. Each strategy admits $\mathfrak{c}$ compatible plays. At stage $\alpha$, reserve a fresh play following I's strategy as a loss for I, and a fresh play following II's strategy as a win for I. Avoid all previously reserved plays, which number less than $\mathfrak{c}$. The collection of the second reservations is a payoff set against which neither player has a winning strategy in an infinite game. This uses no regularity assumption on $\mathfrak{c}$.

Every infinite game of perfect information on a set-sized move alphabet with a Borel payoff in its discrete product topology has a winning strategy in an infinite game for one player. A proof strengthens the induction hypothesis to existence of arbitrary-depth-preserving game coverings unravelling the payoff. The closed-set announcement construction starts the induction. For countable unions, successive coverings stabilize each finite level and their inverse limit makes every constituent payoff clopen; their union is open and can be unravelled once more.

If a Sylow p-subgroup $P$ is abelian, a hypothetical central subgroup $Q\le G^{\prime }$ of order $p$ has a nontrivial linear character extending to $P$. Inducing this extension to $G$ gives a character of degree prime to $p$, hence an irreducible constituent of such degree. On $Q$ its determinant character is a nontrivial power of the chosen linear character, but every determinant character is trivial on $G^{\prime }$. This contradiction proves the displayed restriction.

The derivative of the determinant at an invertible matrix is $D{\det }_A(K)=\det A\mathrm{tr}(A^{-1}K)$. At $A=I+H$, use $\det (I+H)=1+\mathrm{tr}H+O(\Vert H{\Vert }^2)$ and $(I+H{)}^{-1}=I-H+O(\Vert H{\Vert }^2)$. Differentiating the first derivative gives the displayed symmetric bilinear map. More generally $D^2{\det }_A(H,K)=\det A[\mathrm{tr}(A^{-1}H)\mathrm{tr}(A^{-1}K)-\mathrm{tr}(A^{-1}H{A}^{-1}K)]$.

For an arbitrary square matrix $A$, the derivative of the determinant in direction $H$ is the displayed expression. Differentiating the determinant's multilinear expansion gives the sum of entries of $H$ weighted by their cofactors, equal to the displayed trace. If $A$ is invertible, $\mathrm{adj}(A)=(\det A)A^{-1}$ gives $(\det A)\mathrm{tr}(A^{-1}H)$, and division by $\det A$ gives the Jacobi determinant derivative formula for the logarithmic derivative. The adjugate expression remains valid at singular matrices.

Use the unnormalized Fourier transform and $|f|\le 1$. Normalize each derivative correlation by $N$ and align its phase. Their absolute sum is at least $\alpha N$. One Cauchy-Schwarz inequality, expansion by shift/frequency differences, and a second Cauchy-Schwarz inequality bound its fourth power by $N$ times the additive energy of a frequency graph. Parseval identity on a finite group supplies the factor $N$; exact equality of both shift sums and frequency sums is retained.

An inner derivation of a Lie algebra is ${\mathrm{ad}}_x:y\mapsto [x,y]$. The Jacobi identity makes this a derivation of a Lie algebra. For every derivation of a Lie algebra $D$, $[D,{\mathrm{ad}}_x]={\mathrm{ad}}_{Dx}$, so the inner derivations form an ideal of a Lie algebra in the derivation Lie algebra. The kernel of $x\mapsto {\mathrm{ad}}_x$ is the center of a Lie algebra.

Over a field of characteristic zero, a nilpotent operator $D$ which is a derivation of a Lie algebra exponentiates to a Lie algebra automorphism. The iterated rulefollows by induction using Pascal's identity. Regrouping the finite expansion of $[\exp (D)x,\exp (D)y]$ by total degree gives $\exp (D)[x,y]$. The inverse is $\exp (-D)$. This factorial construction must not be reduced naively to small positive characteristic.