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An associated prime of a module $R/I$ is embedded when it is not minimal over $I$. For example, in $k[x,y]/(x^2,xy)$ the nonzero class of $x$ has annihilator $(x,y)$, while the unique minimal prime is $(x)$. Embedded primes record annihilators which are invisible if one retains only the reduced irreducible components.

These feasible potentials give the lower bound ${\sum }_i{\lambda }_i-{\sum }_j{\mu }_j$ on every assignment cost. An assignment using only equality edges attains this bound and is optimal by weak duality and complementary slackness.

The Hungarian algorithm maintains feasible assignment dual potentials and a matching of equality edges. If the alternating search cannot augment, adjust reachable job and agent potentials by the smallest reduced cost leaving the search set. This preserves feasibility and existing matching equalities while exposing a new equality edge. A resulting perfect matching attains the dual lower bound.

The unit vectors realizing an approximate eigenvalue are approximate eigenvectors: their residual tends to zero, although they need not converge to any genuine eigenvector.

Locally, the ASDYM equations permit a complex gauge with $A_w=A_z=0$. The trace equation then gives $A_{\bar{w}}=K_z$ and $A_{\bar{z}}=-K_w$. The remaining curvature equation is $K_{w\bar{w}}+K_{z\bar{z}}-[K_w,K_z]=0$. A complex gauge generally makes $K$ valued in the complexification of a Lie algebra; reconstructing a real connection requires its inherited reality condition. The gauge/potential construction is local and need not preserve prescribed behaviour at infinity.

Under a length blocking factor $b$, a scalar scaling field transforms as ${\varphi }^{\prime }(x^{\prime })=b^{x_{\varphi }}{\varphi }_{<}(b{x}^{\prime })$. Its critical connected correlation function consequently decays as $r^{-2x_{\varphi }}$. Comparing with $r^{-(D-2+\eta )}$ identifies the anomalous part of the field dimension. The uniform conjugate field has scaling exponent $y_h=D-x_{\varphi }$. Wave-function factors must be defined explicitly before assigning a sign to their logarithmic derivatives.

For equal internal and external pressures, zero axial strain rate and positive surface tension, annular viscous extension gives constant $A=R_e^2-R_i^2$ and thickness $D(t)=R_e(0)-R_i(0)+\gamma t/(2\mu )$. Since $R_i=(A/D-D)/2$, the central hole closes atThe formula holds until closure and requires a genuine initial annulus; the subsequent topology has no inner cylindrical interface.

Over a commutative unital ring, an ideal maximal among annihilators of nonzero elements of a module is a prime ideal. Each such ideal is the annihilator of a module given by the cyclic submodule generated by that element. If $P=\mathrm{Ann}(m)$ and $ab\in P$ with $b\notin P$, then $bm\ne 0$ and $P\subseteq \mathrm{Ann}(bm)$. Maximality makes these annihilators equal, so $a\in P$. A Noetherian ring ensures that a nonzero module has a maximal element annihilator by the ascending chain condition on ideals.

The lower bound makes internal vertical velocity variation negligible; the upper bound makes internal in-plane extensional stress negligible compared with ambient traction. Both are needed for an ambient-controlled interfacial plug current.

Linearity of Stokes flow and Uniqueness of Stokes flow identify the ambient resistance to an expanding thin plug with the reversed perturbation around a stationary disc in an axisymmetric straining flow. The imposed straining flow has zero tangential stress on the disc plane. Each face then resists radial velocity $Er$ with traction $-8\mu Er/[\pi \sqrt{R^2-r^2}]$.

For two mixed points with difference $(p,q)$ and gradient difference $(a,b)$, the input difference is $(p,q+\tau b)$ and the output difference is $(p-\tau a,q)$. Their squared-norm difference is $-2\tau (⟨p,a⟩+⟨q,b⟩)+{\tau }^2(\Vert a{\Vert }^2-\Vert b{\Vert }^2)$. This exact identity is a reusable way to prove averagedness without assuming the two block maps are individually nonexpansive.

For $f(x,y)=h(x,y)+\varphi (y)$, where $h$ is convex with an $L_h$-Lipschitz gradient and $\varphi$ is convex, use a proximal $y$-step for the full section and an explicit $x$-step for $h$. Monotonicity of the two selected subgradients of $\varphi$ combines with cocoercivity of $\nabla h$ to make the full update firmly nonexpansive for $0<\tau L_h\le 1$. Existence of a minimizer then gives convergence, even though the full objective is not smooth.

For an infinite almost disjoint family on omega of size less than the bounding number, select countably many members and remove their intersections with earlier selected members to obtain infinite disjoint sets $C_n$. Bound, eventually and simultaneously, the finite intersections of each member with $C_n$. Choose one point of each $C_n$ above its bound. The resulting infinite set is almost disjoint from every original member, proving the family is not maximal. A selected member has one exceptional infinite intersection, which is ignored in its bounding function.

Let $V_{[1]}\ge \cdots \ge V_{[n]}$ be descending order statistics, with $w_n=0$ and nonnegative decreasing prizes. Decompose the prize vector into awards of $w_k-w_{k+1}$ to each of the best $k$ players. The corresponding truthful multi-unit auction charges each winner the next value $V_{[k+1]}$. Its total payment is $k(w_k-w_{k+1})V_{[k+1]}$. Revenue equivalence transfers the expected payment to the all-pay contest, because the interim allocations and lowest-type utilities coincide. Summing the layers proves the formula.

For effective prizes $A_1\ge A_2>0$ and unit effort costs, equilibrium effort CDFs on $[0,A_2]$ are $G_1(b)=b/A_2$ and $G_2(b)=1-A_2/A_1+b/A_1$. The weaker player has an atom at zero. Incremental utilities are $A_1-A_2$ and zero, and winning probabilities are $1-A_2/(2A_1)$ and $A_2/(2A_1)$. Direct payoff indifference and exclusion of larger bids verify the Nash equilibrium. A further player with effective prize at most $A_2$ cannot profit by entering against these distributions.

A countable algebra of sets generating the intended sigma-algebra. Standard Borel spaces admit such algebras by taking finite Boolean combinations from a countable topological base. Finite linear combinations of its indicators are dense in $L^2$ for every Borel probability measure, by the Monotone class theorem. Countability permits a common full-measure convergence set for all test indicators.

The Alfvén velocity is the signed vector ${\mathbf{v}}_a=\mathbf{B}/\sqrt{{\mu }_0\rho }$, whose magnitude is the Alfvén speed. It follows the orientation of the magnetic field, unlike the nonnegative scalar speed.

The homogeneous Dirichlet boundary condition kernel for $u_t=u_{xx}+\alpha u_x$ on $x>0$. Here $H(r,t)=e^{-r^2/(4t)}/\sqrt{4\pi t}$ is the heat kernel. The exponential reflection weight makes $K_{\alpha }(0,y,t)=0$. Gaussian decay makes its initial-data integral convergent even when the exponentially weighted initial data are not integrable separately.

For a continuously differentiable periodic kernel, integration by parts and the Riemann-Lebesgue lemma give $c_n=o(1/|n|)$. Nonperiodic data $e^{\alpha x}$ with $\alpha \notin i\mathbb{Z}$ have Fourier coefficients asymptotic to a nonzero multiple of $1/n$. Their inverse coefficients $y_n/c_n$ therefore fail square summability, and there is no Hilbert-space inverse or least-squares minimizer. Periodic exponential data $\alpha =im$ instead give the single-mode solution $e^{imx}/c_m$.

If entry is uniform on calendar interval $[{\tau }_a,{\tau }_b]$ and follow-up stops at ${\tau }_c$, the censoring duration is uniform on $[{\tau }_c-{\tau }_b,{\tau }_c-{\tau }_a]$. Its hazard function diverges at the longest possible follow-up. It is independent censoring when entry date is independent of prognosis after the required conditioning.