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The sum of $n$ independent exponential variables of rate $r$ has the Erlang distribution with densityConvolving with one more exponential proves the formula inductively.

If $X_0$ embeds compactly into $X$ and $X$ embeds continuously into $X_1$, then boundedness in $L^p(0,T;X_0)$ together with a suitable time-derivative bound in $L^q(0,T;X_1)$ makes a family relatively compact in $L^p(0,T;X)$. A standard case is

Attach Möbius bands to a torus along the classes $ab$ and $a^2{b}^3$. If their core generators are $x,y$, the resulting presentation isBecause the exponent vectors $(1,1)$ and $(2,3)$ form a unimodular basis of ${\mathbb{Z}}^2$, eliminating $a,b$ givesThis group maps onto $S_3$ by $x\mapsto (12)$ and $y\mapsto (23)$, so it is nonabelian.

A group presentation gives a two-dimensional cell complex with one vertex, one oriented loop for each generator, and one 2-cell attached along each relator. Its fundamental group is the presented group.

This equation characterizes a minimal surface presented as a graph.

The graph of $h$ is minimal exactly whenEquivalently, it is the Euler--Lagrange equationfor the area functional $\int \sqrt{1+|\nabla h{|}^2}$.

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.