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After a rigid motion, write the plane as $z=0$ and the surface locally as $z=h(x,y)$. Along the curve of tangency, $h=0$ and $\nabla h=0$. Differentiating $\nabla h(\gamma (s))=0$ shows that the Hessian annihilates the nonzero tangent ${\gamma }^{\prime }(s)$, so its determinant and therefore the Gaussian curvature vanish.

Suppose two graph surfaces $z=f(x,y)$ and $z=g(x,y)$ are tangent at the origin, are oriented upward, and $g\ge f$ nearby. Then $g-f$ has a local minimum, so its Hessian matrix is positive semidefinite. At the common horizontal tangent plane,

For the graph of $h$,

For a closed oriented $d$-manifold, cap product with the fundamental class gives isomorphisms

A covariant functor is representable when it is naturally isomorphic to $\mathcal{C}(A,-)$ for some object $A$; a contravariant functor is representable when it is naturally isomorphic to $\mathcal{C}(-,A)$.

On a zero-Hamiltonian infinite-time fluctuation path, the Freidlin--Wentzell action can be parametrized independently of time as $I=\int \Vert a\Vert ds-a\cdot d\varphi$ in the diffusion metric.

The linear Fourier weight is $W_1(f)={\sum }_{|S|=1}\hat{f}(S{)}^2$.

The noise operator averages $f(y)$ over a random $y$ correlated with $x$ by $\rho$. It acts diagonally on the Fourier-Walsh transform:

On the unbiased cube, the discrete derivative isFor the p-biased product measure, the normalization factor is $\sigma /2$.

Under the $p$-biased product measure ${\mu }_p$ on $\{-1,1{\}}^n$, the coordinates are independent random variables with $\mathbb{P}(x_i=-1)=p$ and $\mathbb{P}(x_i=1)=1-p$. Writing $\mu =1-2p$, $\sigma =2\sqrt{p(1-p)}$, and ${\varphi }_i=(x_i-\mu )/\sigma$, the products ${\varphi }_S={\prod }_{i\in S}{\varphi }_i$ form the $p$-biased Fourier basis.

For $S\subseteq [n]$, the Walsh character on the Boolean hypercube is ${\chi }_S(x)={\prod }_{i\in S}{x}_i$, equivalently $(-1{)}^{{\sum }_{i\in S}{x}_i}$ in the zero-one convention.

Thermal conductivity is the material response coefficient in Fourier's law. For an isotropic material it is a scalar $k$, with units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$; larger positive $k$ means a larger conductive heat flux for the same temperature gradient. It can depend on temperature and composition. Direction-dependent response in an anisotropic material is described by a thermal conductivity tensor. The two worked examples use stipulated positive constants, rather than measured values for named materials.

Thermal conductivity describes material response, whereas thermal resistance also includes geometry: for the steady bar, $R_{\mathrm{th}}=L/(kA)$. Doubling $k$ at fixed length, cross-sectional area and end temperatures halves thermal resistance and doubles the rate of heat transfer. Doubling length changes thermal resistance without changing the material's thermal conductivity. This distinction lets a designer improve cooling either by choosing a higher thermal conductivity or by shortening and widening the path. It also explains why the less conductive layer in thermal resistance takes the greater temperature drop.

Thermal diffusivity answers a different question: how rapidly a temperature disturbance spreads. Its usual model $\kappa =k/(\rho c_p)$ also involves mass density $\rho$ and specific heat capacity $c_p$, so equal thermal conductivities need not imply equal rates of warming.

The odd-parity summands give

Every nonidentity finite-order Möbius transformation has exactly two fixed points on the Riemann sphere. A transformation with one repeated fixed point is conjugate to a nontrivial translation and therefore has infinite order.

For $f(z)=(az+b)/(z+1)$, the finite fixed points are the roots ofThus a repeated fixed point or two prescribed fixed points determine $a$ and $b$ by the usual sum and product of roots.

The split stephas amplification matrixOn a common directional eigenvector its eigenvalue is $(1+\mu {\lambda }_p)/(1-\mu {\lambda }_q)$.

For $Y=n^{-1}\mathrm{Bin}(n,p)$, the score and information areConsequently one Fisher-scoring step sends every interior starting value directly to the MLE $\hat{p}=Y$.

If a continuous function $f$ satisfiesfor every smooth compactly supported test function $\eta$, then $f=0$ on $(a,b)$.

An integral matrix satisfying $A^2+A+I=0$ makes ${\mathbb{Z}}^n$ a module over the Eisenstein integers by letting the primitive cube root act as $A$. This module is finitely generated and torsion-free, hence free. Since the Eisenstein integers have integer rank two, such matrices exist only for $n=2m$; for that dimension every one is integrally conjugate to

An orthonormal eigenbasis of a linear operator is an orthonormal basis consisting of eigenvectors. Each vector can be reconstructed from its scalar components along those mutually perpendicular unit directions. With the convention that the inner product is linear in its first argument,The finite-dimensional spectral theorem supplies such a basis for a self-adjoint operator. In its two-dimensional example, the coordinates of $x=(1,0{)}^T$ are $1/\sqrt{2}$ along $e_{+}$ and $-1/\sqrt{2}$ along $e_{-}$. The operator multiplies those coordinates by $3$ and $1$, and reconstruction gives $Ax=(2,1{)}^T$.

Repeated eigenvalues permit different orthonormal choices within one eigenspace, so the basis need not be unique. The spectral projection of a self-adjoint matrix onto that entire eigenspace is unique: it keeps the geometrically meaningful component without choosing individual axes inside it.