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Ifthen truncating the sum after $N$ terms gives finite-rank Hermitian $T_N$ with

A Matsubara correlator is obtained from the retarded spectral density by a Cauchy transform evaluated at an imaginary frequency.

Let $L(\lambda )=L_0+(\lambda -{\lambda }_0)B$ be a real self-adjoint family on a common domain, and let $y_0$ be a simple null eigenfunction of $L_0$, normalized in the chosen inner product. For the nonlinear problem $L(\lambda )y=y^{m+1}$ with integer $m\ge 1$, put $y=\epsilon y_0+O({\epsilon }^2)$ and $\lambda -{\lambda }_0={\epsilon }^m{\mu }_0+o({\epsilon }^m)$. Projecting the leading equation onto $y_0$ givesIf the denominator is nonzero, this determines the leading shift. For the ordinary $L^2$ pairing and $B=I$, it reduces to ${\mu }_0=\int y_0^{m+2}dx$, the familiar special case. For the convention $L(\lambda )y=-(p{y}^{\prime }{)}^{\prime }+qy-\lambda wy$, one instead has $B=-w$ and the sign and denominator change. If $⟨y_0,B{y}_0⟩=0$, the scaling or higher-order solvability analysis must be reconsidered; the numerator alone is not a universal shift formula.

For $y^{\prime \prime }+\lambda y=y^3$ on $(0,1)$ with zero endpoint values, $y_0=\sqrt{2}\sin (\pi x)$ and ${\lambda }_0={\pi }^2$. Here $B=I$, $m=2$ and ${\int }_0^1{y}_0^{4}dx=3/2$, so the branch has leading shift $\lambda ={\pi }^2+(3/2){\epsilon }^2+o({\epsilon }^2)$ under the stated amplitude normalization.

For $1\le p<\infty$, no bounded map from $L^p({\mathbb{R}}^n)$ to $L^p({\mathbb{R}}^{n-1})$ can agree with restriction on continuous functions. Indeed,has a fixed nonzero trace while $\Vert u_k{\Vert }_p=k^{-1/p}\Vert \varphi {\Vert }_p\Vert \psi {\Vert }_p\to 0$.

For $1\le p<\infty$, boundary restriction on smooth functions extends uniquely to a bounded linear map $T:W^{1,p}({\mathbb{R}}_{+}^n)\to L^p({\mathbb{R}}^{n-1})$. The estimate follows by applying the fundamental theorem of calculus on normal line segments and then integrating over the boundary variables.

A Hölder space consists of functions satisfying a Hölder condition. A bounded function belongs to $C^{0,\alpha }({\mathbb{R}}^n)$ when

The translation mechanism obstructs compactness when a domain contains arbitrarily distant translates of one fixed compact support, as ${\mathbb{R}}^n$ does. Choose a nonzero smooth bump $\varphi$ and mutually disjoint translates $u_j=\varphi (\cdot -a_j)$. Their Sobolev norms are equal, whileThus no subsequence is Cauchy in L2 space. If $|a_j|\to \infty$, the same sequence vanishes on every fixed bounded region eventually, exhibiting loss of compactness at infinity. An arbitrary unbounded domain need not admit translates of a fixed bump; the geometric room for them is part of the example.

The Gagliardo-Nirenberg interpolation inequalities bound an intermediate derivative norm by powers of a higher derivative norm and a lower norm. In two-dimensional whole space, one useful case for $f\in H^2({\mathbb{R}}^2)$ isApply Ladyzhenskaya's inequality to the components of $\nabla f$, then combine their estimates by the Cauchy-Schwarz inequality. Density of smooth functions in a Sobolev space extends the result from smooth compactly supported functions to the stated Sobolev space.

The domain and boundary assumptions matter. On a bounded domain, the affine function $f(x)=x_1$ has nonzero gradient but $D^{2}f=0$, so this homogeneous inequality cannot hold for every $H^2$ function there. A bounded-domain version needs appropriate boundary restrictions or an additional lower-order term.

If a periodic function $u$ has zero mean, its nonzero Fourier frequencies giveThus inversion of the Laplacian on mean-zero periodic functions gains two Sobolev derivatives.

Let $V$ be the Snell envelope of integrable adapted rewards $Z_0,\dots ,Z_N$, and use the Doob decomposition in discrete time $V=M-A$, with $A_0=0$ and $A$ nondecreasing. Definewith $\inf \varnothing =\infty$. Since $A_{n+1}$ is ${\mathcal{F}}_n$-measurable, $\tau$ is a stopping time.

Before the first increase, $A_{n\wedge \tau }=0$, so $V_{n\wedge \tau }=M_{n\wedge \tau }$ is a martingale. If $\tau <N$, its next compensator increment is positive and the Snell envelope recursion forces $V_{\tau }=Z_{\tau }$; if $\tau =N$, this is the terminal condition. The optional stopping theorem gives $\mathbb{E}{Z}_{\tau }=\mathbb{E}{V}_0$, attaining the supermartingale majorant bound. This rule may wait through earlier ties between reward and continuation; it need not be the first time $V_n=Z_n$.

For the Doob compensator of a Snell envelope, $(V_n-Z_n)(A_{n+1}-A_n)=0$: compensation grows only on the stopping region $V_n=Z_n$.

Every reduced variety over a perfect field has a dense open smooth locus. At the generic point of each irreducible component, the local ring is a field and is regular. Over a perfect field, regularity of a finite-type local algebra is equivalent to smoothness there. The smooth locus is open and contains all these generic points, so its closure is the whole variety.^[ref]

For an irreducible affine variety of dimension $d$ embedded in ${\mathbb{A}}^N$, this can also be expressed through the Jacobian criterion. Separability over the perfect ground field gives generic Jacobian rank $N-d$. A nonzero minor of that size stays nonzero on a nonempty Zariski-open set, producing explicit smooth points. This argument does not assume in advance that minimum tangent dimension equals dimension. Perfection matters: a purely inseparable field extension can define a regular scheme with no smooth points over the original field.

At a closed point $p$ of a variety over an algebraically closed field, singularity means failure of the regular local ring condition:For an irreducible variety of dimension $d$, the right side is $d$. Thus the usual test is $\dim T_{p}X>d$. The Zariski tangent space measures first-order directions, while the local dimension measures the actual local components; excess tangent directions detect singularity.^[ref]

For the ordinary cusp $y^2=x^3$, both first derivatives vanish at the origin, giving tangent dimension two although the curve has dimension one. On a variety with components of different dimensions, compare with $\dim {\mathcal{O}}_{X,p}$, not the largest dimension elsewhere. Over a perfect field, the nonregular and nonsmooth point notions agree; over an imperfect field they need not, as explained in singular locus.

For a canonical inflaton with $V(\varphi )=\lambda {\varphi }^n/n$, $\lambda >0$, $n>0$ and $\varphi >0$, use natural units and reduced Planck mass $M=(8\pi G{)}^{-1/2}$. The slow-roll equations giveWith unreduced Planck mass $m_P=\sqrt{8\pi }M$, this is $N\simeq 4\pi ({\varphi }_i^2-{\varphi }_f^2)/(n{m}_P^2)$. The number of e-folds is dimensionless and measures logarithmic expansion.

The conventional potential endpoint estimate ${ϵ}_V=n^2{M}^2/(2{\varphi }_f^2)=1$ gives ${\varphi }_f^2\simeq n^2{M}^2/2$ and $N\simeq (n/4)[({\varphi }_i/{\varphi }_f{)}^2-1]$. A different formal extrapolation, equating slow-roll kinetic energy ${\dot{\varphi }}^2/2$ to $V$, gives ${\varphi }_f^2\simeq n^2{M}^2/6$ and the coefficient $n/12$ instead. These endpoint estimates use slow roll where it is already failing.

The exact scalar-dominated acceleration boundary is ${\dot{\varphi }}^2=V$, equivalently ${ϵ}_H=1$, rather than equality of kinetic and potential energy. Precise endpoints require the full inflaton equation of motion. Keeping ${\varphi }_f$ explicit separates the reliable large-field integral from its approximate endpoint convention.

The slice genus is the smallest genus of a smooth compact connected oriented surface properly embedded in $B^4$ with boundary $K$.

A slice disk is a smooth proper embedding $D^2\hookrightarrow B^4$ whose boundary is the knot in $S^3=\partial B^4$.

Near a finite-time viscous-capillary pinch-off, balancing time dependence, viscous normal stress, capillary pressure, and axial lubrication flow can select self-similar radius and axial scales.

For $a=(a_1,a_2,a_3{)}^T$, the map $b\mapsto a\times b$ has matrixIt has trace zero and annihilates $a$, so its determinant is zero.

Dependent percolation allows the open states of different vertices or edges to be statistically dependent.

The method of multiple scales describes motion with widely separated time or length scales by treating them as independent variables during an asymptotic expansion. With fast time $t$ and slow time $T=ϵt$, write $u(t;ϵ)=U(t,T;ϵ)$ and useAt each order, a solvability condition in the method of multiple scales removes resonant forcing forcing that would generate a secular term. The resulting amplitude-phase equations describe accumulated changes over many fast periods.^[ref]

For $u^{\prime \prime }+2ϵu^{\prime }+u=0$, put $U_0=A(T)\cos t+B(T)\sin t$. The first correction obeysIts cosine and sine coefficients vanish precisely when $A_T=-A$ and $B_T=-B$. For $u(0)=1$, $u^{\prime }(0)=0$, the leading result is $e^{-ϵt}\cos t$ with bounded corrections. A fixed-time expansion instead contains $-ϵt\cos t$, which becomes order one when $t=O({ϵ}^{-1})$. This explains the slow variable's purpose; eliminating resonances is a construction step, while an error estimate is needed to establish uniform accuracy on a stated time interval.