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The stationary spectral-density matrix of $dX=-AXdt+bdW$ is

A morphism $i:Z\to X$ is a closed immersion when it identifies $Z$ homeomorphically with a closed subset of $X$ and the morphism ${\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z$ is surjective. Affine-locally it has the form $\mathrm{Spec}(A/I)\to \mathrm{Spec}A$.

A morphism is proper when it is separated, of finite type, and universally closed. Properness is stable under base change and composition.

A morphism $f:X\to Y$ is separated when its diagonal $X\to X{\times }_{Y}X$ is a closed immersion. This is the scheme-theoretic analogue of the Hausdorff property.

A morphism of schemes is separated when its diagonal is a closed immersion. This is a property of the morphism, distinct from the broader morphism or diagonal constructions.

For $C_t=D{C}_{xx}+aC$ with $a>0$, $C(0,t)=0$, and $C_x(L,t)=0$, every mode decays exactly when

If $M\subseteq {\mathbb{R}}^{n+m}$ and $f$ is smooth in the ambient space, then

In a Noetherian ring, the primary component belonging to each isolated prime $\mathfrak{p}$ is uniquely determined byEmbedded primary components need not be unique.

An isolated prime of a minimal primary decomposition is a minimal member of the set of radicals of its primary components.

A rational function is a quotient $p/q$ of polynomials, with $q$ nonzero. After cancelling common factors, it defines a holomorphic map from the Riemann sphere to itself.

A differentiable map on a connected open subset of Euclidean space whose derivative vanishes everywhere is constant. The mean value inequality makes it constant on every ball in the domain, hence locally constant; connectedness then makes the value global.

The same contact does not order Gaussian curvature, because determinant is not monotone under addition of a positive-semidefinite matrix when the original Hessian is indefinite or negative definite. For example,satisfy $g\ge f$ and are tangent at zero, but $K_g(0)=1<4=K_f(0)$.

If $\rho ={\sigma }^{\alpha }{T}_{\alpha }$ and $[T_{\alpha },T_{\beta }]=c^{\gamma }{}_{\alpha \beta }{T}_{\gamma }$, then

The outer product of column vectors $u$ and $v$ is the rank-at-most-one matrix $u{v}^T$, whose $(i,j)$ entry is $u_i{v}_j$.

For the equal point-null mixture,

If $0\notin K$, thendiffers from $K$, while the computable map $n\mapsto 2n+1$ proves $K{\le }_{m}S$.

Magnetic translations along two lattice vectors commute when their cell flux obeys

Every linear character of the upper-triangular subgroup $B\le S{L}_2({\mathbb{F}}_q)$ with $q\ge 4$ is obtained from a multiplicative character $\theta$ of ${\mathbb{F}}_q^{\times }$. Its induction to $S{L}_2({\mathbb{F}}_q)$ is irreducible exactly when ${\theta }^2\ne 1$.

Writing $\rho =\lambda /\mu$, the stationary empty probability is $e^{-\rho }$. Alternating idle periods of mean $1/\lambda$ with busy periods and applying the renewal-reward theorem gives

If $X_0=n$ and $p_t=e^{-\mu t}$, thenwith independent summands. The binomial term counts surviving initial customers; Poisson thinning gives the surviving later arrivals.

In stationarity, the departure process of an $M/M/\infty$ queue is a Poisson process of rate $\lambda$. By time reversibility, the departures before any fixed time are independent of the occupancy at that time.