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A dominant invariant subspace is spanned by eigenvectors whose eigenvalue moduli exceed those outside the subspace.

A proper submodule $N⊊M$ is primary when $rm\in N$ and $m\notin N$ imply $r^{k}M\subseteq N$ for some positive integer $k$.

For simple symmetric random walk stopped on hitting $-a$ or $b$ and $0<z<1$, choose $w>1$ byThen $z^n{w}^{\pm X_n}$ are martingales. Solving the two boundary equations gives

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.