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Replacing a discrete gradient by a second-difference operator $H:X\to X^4$ gives the global-norm penalty $\alpha \Vert Hu{\Vert }_2$. Its reconstruction is $g-P_{C}g$ for $C=H^{\ast }{B}_{\alpha }$. Components in $\ker H$ are preserved; affine-image membership of this kernel depends on the boundary stencil. Pixelwise Hessian penalties instead use a product of four-component balls.

Mean-field variational inference restricts the trial probability density function to $q(x)={\prod }_i{q}_i(x_i)$. With the other factors fixed, $q_j^{\ast }(x_j)\propto \exp ({\mathbb{E}}_{q_{-j}}\log \pi (x_j,X_{-j}))$, provided this expression has a finite positive normalization constant. Subtracting the objective at $q_j^{\ast }$ leaves $\mathrm{KL}(q_j\Vert q_j^{\ast })\ge 0$, proving the update is optimal when the terms are well defined.

Markov chain variational inference optimizes an initial parametric probability distribution ${\nu }_{\eta }$ by minimizing $\mathrm{KL}(K^t{\nu }_{\eta }\Vert \pi )$ after a fixed number of Markov kernel updates. The best initial probability distribution can depend on $t$; Kullback-Leibler divergence contraction does not imply that optimization commutes with applying the Markov kernel.

A reparameterization gradient differentiates an expected value by representing the simulated random variable as a differentiable transformation of parameter-independent noise. For $X_{\eta }=g_{\eta }(\epsilon )$, the identity ${\nabla }_{\eta }\mathbb{E}[F_{\eta }(X_{\eta })]=\mathbb{E}[{\nabla }_{\eta }{F}_{\eta }(g_{\eta }(\epsilon ))]$ requires justified differentiation under the integral sign. For a normal distribution, use $X=m+e^{\rho }\epsilon$ with $\epsilon$ drawn from the standard normal distribution.

Suppose an integrable orthonormal scaling function has a low-pass filter of a multiresolution analysis that is $C^{p-1}$ near $\pi$, and its Fourier transform is $C^{p-1}$ near zero. If the associated wavelet has $p$ integrable vanishing moments, then $m^{(k)}(\pi )=0$ for $k<p$. Indeed, moment differentiation of the Fourier transform makes ${\hat{\psi }}^{(k)}(0)=0$, while $|\hat{\phi }(0)|=1$. In $\hat{\psi }(2t)=e^{-it}\overline{m(t+\pi )}\hat{\phi }(t)$, division by the nonzero smooth factor proves the conclusion. With only continuity of that factor one still obtains a Peano zero, but not automatically higher ordinary derivatives.

A Vandermonde matrix has entries $V_{ij}={\lambda }_i^{j-1}$. Distinct nodes give determinant ${\prod }_{i<j}({\lambda }_j-{\lambda }_i)\ne 0$, explaining independence of polynomial evaluations and distinct-eigenvalue Krylov subspaces.

If an element $x$ of the fraction field is absent from a valuation ring $V$, its inverse $y$ belongs to the maximal ideal. A monic equation for $x$, multiplied by a power of $y$, would put $1$ in that maximal ideal. Thus $V$ is an integrally closed domain, without a discrete or Noetherian hypothesis.

Every local subring $A$ of a field $K$ is dominated by a valuation ring with fraction field $K$. Order local overrings by inclusion and contraction of their maximal ideals; chain unions are local, so Zorn's lemma gives a maximal one. For each $z\in K^{\times }$, at least one of the extensions of its maximal ideal to $V[z]$ and $V[z^{-1}]$ is proper: otherwise two least-degree relations for $1$ with maximal-ideal coefficients reduce one another to a smaller-degree relation. Localizing the proper extension gives a dominating overring, so maximality forces either $z$ or $z^{-1}$ into $V$.

A perturbative correlator factors into an exponential of connected Vacuum Feynman diagrams times a sum of Feynman diagrams attached to external insertions. Dividing its path integral by the vacuum generating functional at zero source cancels that exponential, including all disconnected products of vacuum components. The corresponding operator formalism uses the same denominator in the vacuum expectation of the Dyson series. This leaves externally disconnected correlators intact; the connected generating functional removes their products.

A square matrix is strictly upper triangular if every entry on or below its diagonal is zero. A product of $n$ such $n$-by-$n$ matrices vanishes, so they form a nilpotent Lie algebra under the commutator. Multiplication by an upper triangular matrix preserves strict upper triangularity and zero trace.

For $x>q$ and $(a,q)=1$, the prime-counting function in an arithmetic progression satisfiesThe denominator involves the Euler totient function and the length-to-modulus ratio. In particular, for $q\le x^{1-\eta }$ with fixed $\eta >0$, this is $O_{\eta }(x/(\varphi (q)\log x))$.

Near the stable Adler phase equation point $\alpha =\arcsin (\omega /ϵ)$, linearized noise obeys an Ornstein-Uhlenbeck process with relaxation rate $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. Its harmonic-model raw correlation is ${\alpha }^2+(T_{\mathrm{eff}}/\kappa )e^{-\kappa |\tau |}$. Nonlinear drift also shifts the local mean at order $T_{meff}$. This describes times after relaxation but before appreciable phase slips; it is not a stationary infinite-time correlation of the unwrapped phase at fixed nonzero noise.

For a locally path-connected subspace $A\subseteq X$, a selected component of the inverse image in a universal cover covers the corresponding component $A_0$ of $A$. Its covering subgroup is $\ker ({\pi }_1(A,a_0)\to {\pi }_1(X,a_0))$; surjectivity onto all of $A$ additionally requires $A$ to be path connected.

The subgroup $\{U\in U(2n):UJ{U}^T=J\}$, with $J=\left(\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right)$, is the compact symplectic group, often denoted $Sp(n)$ or $USp(2n)$. It preserves both a positive Hermitian metric and a nondegenerate complex alternating form.

On the Hilbert space ${\ell }^2(\mathbb{N})$, the left shift deletes the first coordinate. It is the adjoint operator of the unilateral shift operator $S$. The identities $LS=I$ and $SLf=f-f_1{e}_1$ show that it is also the Moore–Penrose inverse of an operator for $S$; both maps have operator norm one.

Put $Q=2|\Omega |c/(\pi G{\Sigma }_0)$. The uniformly rotating gas-sheet dispersion relation has growing waves precisely for $Q<1$, at $q_{-}<|k|<q_{+}$ with $q_{\pm }=(\pi G{\Sigma }_0/c^2)(1\pm \sqrt{1-Q^2})$. Pressure stabilizes short waves and rotation stabilizes long waves. Equality $Q=1$ is marginal; $Q\ge 1$ is the gas Toomre stability criterion.

At fixed ${\Sigma }_0,\Omega \ne 0$, slowly decreasing $c$ first reaches a zero of the wave-frequency minimum at $c=\pi G{\Sigma }_0/(2|\Omega |)$. The corresponding wavelength is $\ell ={\pi }^{2}G{\Sigma }_0/(2{\Omega }^2)$. It estimates the spacing and scale of growing density concentrations just below marginality; linear theory does not determine the final nonlinear clump radius.

A nondivergence-form elliptic operator applies its principal coefficients directly to the Hessian matrix: $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u+cu$. Only the symmetric part of $a$ contributes to the second-derivative term. Uniform ellipticity bounds its quadratic form below by $\lambda |\xi {|}^2$ with a constant $\lambda >0$ independent of position.

On a connected locally finite recurrent graph, root Wilson's algorithm at a fixed graph vertex $r$ and process a countable enumeration of the graph vertices. Every walk hits the existing tree almost surely, since it contains $r$. Each attached graph path is finite, and the union is a connected acyclic spanning subgraph. Its law is the infinite uniform spanning tree.

It is also the common free and wired finite-volume limit of a sequence along every graph exhaustion. For a finite edge set, run a finite initial segment of the chosen enumeration containing all its endpoints. Its membership is then permanently decided, because later attachments have new internal graph vertices. Only finitely many finite random-walk trajectories were used, and their graph vertices and graph neighbours fit inside all sufficiently large exhaustion sets. Coupling then gives convergence of those finite-dimensional edge laws for both boundary conventions. Finite Wilson's algorithm is root/order independent, so the limiting law is too. There is no uniform counting measure on all infinite spanning trees; the definition is this limiting probability law.

The complement of an $\ell$-uniform hypergraph on $V$ has all $\ell$-subsets of $V$ that are not hyperedges of the original hypergraph. An edge in every $r$-set of one hypergraph is equivalent to the absence of a complete $\ell$-uniform $r$-vertex hypergraph in its complement.