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The Gagliardo-Nirenberg interpolation inequalities bound an intermediate derivative norm by powers of a higher derivative norm and a lower norm. In two dimensions, one useful case is

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.

The first time the Snell-envelope compensator is about to become positive is an optimal stopping time; before it the martingale part equals both value and reward at stopping.

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 $V_n=Z_n$.

If $X\subseteq {\mathbb{A}}^n$ is irreducible of dimension $d$, choose a nonzero $(n-d)$-rowed minor of a Jacobian matrix at one point of minimum tangent dimension. Its nonvanishing locus is a nonempty Zariski-open set on which the Jacobian has rank $n-d$, so every point there is smooth. Irreducibility makes every nonempty Zariski-open set dense.

A point $p$ of an irreducible variety $X$ is singular when $\dim T_{p}X>\dim X$. Equivalently, its local ring is not regular. The singular locus is the complement of the smooth locus.

For $V(\varphi )=\lambda {\varphi }^n/n$ under the stated Cambridge Tripos normalization,If slow roll ends by kinetic-potential equality, then

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 treats fast and slow variables such as $t$ and $T=ϵt$ as independent, then removes resonant forcing to obtain slow amplitude and phase equations.

A matched asymptotic expansion constructs approximations in regions with different distinguished limits and fixes their free constants by requiring agreement in an overlap region. The inner expansion resolves rapid variation; the outer expansion describes the surrounding region. Matching compares their common limiting behavior, rather than imposing distant boundary conditions directly on a local approximation.^[ref]

For the singular perturbation problem $ϵy^{\prime \prime }+y^{\prime }=0$, $y(0)=0$, $y(1)=1$, the leading outer solution is $y_0=1$. The inner variable $X=x/ϵ$ gives $Y_0^{\prime \prime }+Y_0^{\prime }=0$ and $Y_0(0)=0$, hence $Y_0=A(1-e^{-X})$. In the overlap region $ϵ\ll x\ll 1$, the inner variable tends to infinity while the outer coordinate tends to zero. The asymptotic matching condition isso $A=1$. One concrete path through the overlap region is $x=\sqrt{ϵ}$: then $x\to 0$ and $X={ϵ}^{-1/2}\to \infty$. Matching selects the free constant that the left boundary condition alone could not determine.

Why choose the right boundary condition for the outer solution? If instead one selects $y_0=0$ from the left, a proposed right boundary layer with $Z=(1-x)/ϵ$ satisfies $U_0^{\prime \prime }-U_0^{\prime }=0$. Its general solution is $A+B{e}^Z$. Matching to zero as $Z\to \infty$ forces $A=B=0$, contradicting $U_0(0)=1$. The direction in which the exponential function decays determines which endpoint can carry the boundary layer; endpoint selection is part of the calculation.

Compatible orientations of all top-dimensional simplices make their signed sum a cycle. For a connected pseudomanifold this cycle generates top homology over the integers; if no compatible orientation exists, top integral homology is zero.

A simplicial abelian group is a simplicial object in abelian groups. Its underlying simplicial set is always a Kan complex.

A simplicial set is a functor from the opposite of the simplex category to sets. Its face and degeneracy maps satisfy the simplicial identities.

The simplex category has objects the finite nonempty ordered sets $[n]=\{0<\cdots <n\}$ and morphisms the order-preserving maps.

The vertices of the barycentric subdivision of a simplicial complex are its nonempty faces, and its simplices are strict chains of faces.

The Euler characteristic compresses a finite CW complex's homology into one integer: count its topological cells with alternating signs, or count its Betti numbers with the same signs. A filled triangle has Euler characteristic one; its cycle graph boundary has Euler characteristic zero. The missing face changes the answer even though the vertices and edges are identical.

For a finite CW complex $X$, including the geometric realization of a finite simplicial complex,Here $b_i={\dim }_{\mathbb{Q}}{H}_i(X;\mathbb{Q})$ is the rational Betti number. More generally this ordinary homology definition makes sense whenever every rational homology group is finite-dimensional and only finitely many are nonzero. An arbitrary infinite CW complex need not satisfy these assumptions; its formal infinite alternating cell count is not a definition of this invariant.

The usefulness is that a finite description can reveal something about a space without computing every homology group. By the cell-count formula for Euler characteristic, count pieces rather than solve all the boundary operators. If two spaces give different answers, topological invariance of Euler characteristic rules out a homeomorphism, and even a homotopy equivalence. Equal answers are less decisive: the Euler characteristic of an annulus and the Euler characteristic of a torus are both zero, while their homology groups differ. This is a compact diagnostic, with a clearly defined limit on what it can tell us.