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For a continuous local martingale $M$,

For a square-integrable predictable process $H$ and Brownian motion $W$,More generally, for a continuous $L^2$-bounded martingale $M$,

Whenever the integrands are admissible,

For a differential operator $L$, the martingale problem asks for a process $X$ such thatis a local martingale for every test function $f$ in a suitable domain.

A weak solution may choose its probability space and driving Brownian motion as part of the solution. It is weaker than a strong solution, which must be adapted to a prescribed Brownian motion.

If the leading asymptotic is $C{t}^{-1/2}\cos (\Omega t+\gamma )$ with $C>0$, its upward zero crossings satisfyto leading order.

The standard normal distribution function is the cumulative distribution function of a standard normal random variable $Z$:Here $\varphi$ is the standard normal density and $\mathrm{erf}$ is the error function. The symmetric probability distribution gives $\Phi (-x)=1-\Phi (x)$ and $\Phi (0)=1/2$, while differentiation gives ${\Phi }^{\prime }(x)=\varphi (x)$. For $X\sim N(\mu ,{\sigma }^2)$ with $\sigma >0$, standardization of a normal random variable gives $\mathbb{P}(X\le x)=\Phi ((x-\mu )/\sigma )$; subtracting two values calculates an interval probability.

For a point inclusion $i:\{x\}\hookrightarrow X$ and an abelian group $A$, the skyscraper sheaf $i_{\ast }A$ has sections $A$ on opens containing $x$ and zero on other opens. Its restriction maps are identities or maps to zero. If $A\ne 0$, its nonzero stalks occur at every point of the closure $\overline{\{x\}}$: exactly those points whose every neighbourhood contains $x$. Thus the familiar assertion that only the stalk at $x$ is nonzero requires $x$ to be a closed point. It is always flasque, giving the stated skyscraper sheaf cohomology.

Outside a meagre set in $C^{\infty }([0,1])$, every rational $q\in (0,1)$ and every positive integer $M$ admit $m\ge M$ such thatFor fixed $q,M$, the union over $m\ge M$ of these strict-inequality sets is open. It is dense because a perturbation $\delta \cos (K(x-q)-m\pi /2)$ can make the $m$th derivative arbitrarily large while keeping any prescribed finite collection of lower derivatives arbitrarily small. The Baire category theorem applied over the countable pairs $(q,M)$ gives the claim.

If $(f_n)$ is Cauchy in the smooth-function metric, each derivative sequence $(f_n^{(r)})$ converges uniformly to a continuous function $g_r$. Passing to the limit inshows that $g_r^{\prime }=g_{r+1}$. Hence $g_0$ is smooth and $f_n\to g_0$ in the metric.

Over $D=\mathbb{C}\setminus \{z:z^8=1\}$, the two-valued square root is represented byThe maps $\pi (z,w)=z$ and $\mathcal{E}(z,w)=w$ identify this unbranched double cover analytically with the corresponding component of the space of germs.

The evaluation map $\mathcal{E}([f{]}_z)=f(z)$ is holomorphic. On the sheet associated with $(U,f)$, its coordinate expression is exactly the holomorphic function $f$.

For one-hot label $y$ and predicted class probabilities $p$, categorical cross-entropy is $-{\sum }_k{y}_k\log p_k$, the negative categorical log-likelihood.

For $U=(0,\infty )\times {\mathbb{R}}^{n-1}$, the formula $Eu(x_1,x^{\prime })=u(|x_1|,x^{\prime })$ defines a bounded Sobolev extension operator. Its weak normal derivative changes sign across the boundary, while its tangential weak derivatives are reflected unchanged.

For every unit complex number $\omega$ at which the signature form is nonsingular,

A ribbon disk is a slice disk whose radial Morse function has no interior local maxima. It can be built from disks by attaching bands.

Let a real symmetric matrix have an orthonormal eigenbasis $w_1,\dots ,w_n$, withWrite two starting vectors asFor two-column subspace iteration to recover
$\mathrm{span}(w_{n-1},w_n)$, the two projections onto that dominant subspace must be independent. The exact condition isRequiring each of the four coefficients to be nonzero does not imply this determinant condition.

The simplicial mapping space is defined byIf $A\hookrightarrow B$ is a monomorphism and $X$ is a Kan complex, restriction $\underline{\mathrm{Hom}}(B,X)\to \underline{\mathrm{Hom}}(A,X)$ is a Kan fibration.

A Kan complex is a simplicial set in which every simplicial horn, including every outer horn, has a filler.

A quasicategory is a simplicial set in which every inner horn has a filler.