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The $E_7$ Coxeter graph is the simply-laced seven-vertex tree whose three arms have lengths $1$, $2$, and $3$. Its Coxeter Gram matrix is positive definite and has determinant $2$.

The character group $\hat{G}$ consists of the group homomorphisms $\gamma :G\to {\mathbb{C}}^{\times }$. Pointwise multiplication makes it a finite abelian group isomorphic to $G$.

In the Category of sets, filtered colimits commute with finite limits. Elements in a finite limiting diagram involve only finitely many representatives and finitely many equalities, all of which can be realized at one common stage of a filtered diagram.

A category is weakly filtered when every finite connected diagram in it admits a cocone. Equivalently, each of its connected components is a filtered category.

For a morphism of schemes $f:X\to Y$ and a point $y\in Y$, the scheme-theoretic fiber is the fiber product $X{\times }_Y\mathrm{Spec}\kappa (y)$. It remembers multiplicities and nilpotents that the underlying set-theoretic inverse image cannot detect.

Its underlying fiber is enriched with local-ring and multiplicity information.

The Fermi sea is the zero-temperature many-fermion quantum state with every one-particle state below the Fermi energy occupied and every state above it empty. For a uniform isotropic Fermi gas, the occupied region in momentum space is a solid sphere; its boundary is the Fermi surface. These are momenta, not positions: filling the sphere does not put the particles in a ball in ordinary space. Opposite momentum contributions cancel, so the gas at rest has zero total momentum even though its total kinetic energy is positive.

The filled states are forced upward in energy by the Pauli exclusion principle. This is different from the zero-point energy of one particle confined by a boundary. At fixed number density, the Fermi energy remains positive as the box grows to the thermodynamic limit, while the lowest box confinement energy tends to zero. Low-energy excitations move particles from occupied states near the Fermi surface into nearby empty states; deep states have no nearby available destination. This explains why only a small fraction of electrons contribute to the low-temperature electron heat capacity.^[ref]

For activator diffusion $D$, local linear decay $r$, and coupling $\rho (u-v)$, eliminating the fast inhibitor givesIts pattern-forming threshold is

Choose an orthonormal basis of $T_{p}M$ and put $e(\theta )=(\cos \theta ,\sin \theta )$. Away from the centre and cut locus,defines geodesic polar coordinates centred at $p$.

A normal neighbourhood of $p$ is the diffeomorphic image under ${\exp }_p$ of a star-shaped neighbourhood of $0$ in $T_{p}M$. In a sufficiently small normal ball, radial geodesics from $p$ uniquely minimize length.

The domain of ${\exp }_p$ is the star-shaped open setSmooth dependence for ordinary differential equations makes $(p,v)\mapsto {\exp }_p(v)$ smooth on its open domain in $TM$.

If $N$ is geometric on $\{1,2,\dots \}$ with parameter $p$, independently of unit-rate exponential variables $S_i$, thenis exponential with rate $p$. Its moment-generating function is $p/(p-\theta )$ for $\theta <p$.

For independent exponential variables $S_i$ of rates $q_i$ and $Q={\sum }_i{q}_i$, let $T={\min }_i{S}_i$ and let $K$ be the minimizing index. ThenConsequently $T$ is exponential with rate $Q$, $\mathbb{P}(K=k)=q_k/Q$, and $K$ and $T$ are independent.

If $A$ and $B$ are independent exponential variables with the same rate, then $A/(A+B)$ is uniform on $(0,1)$ and is independent of $A+B$.

If $T\sim \mathrm{Exp}(\mu )$, thenThus an exponential remaining lifetime has the same law regardless of the elapsed lifetime.

For $S_t=S_0{e}^{\mu t+\sigma W_t}$, shifting Brownian drift bymakes $S_t=S_0{e}^{(r-{\sigma }^2/2)t+\sigma W_t^Q}$ under an equivalent martingale measure $Q$. The discounted stock is then a martingale.

Suppose a transient nearest-neighbour jump chain on $\mathbb{N}$ has uniformly bounded expected visits to each state and the holding rate at $i$ grows geometrically. Thenso the continuous-time chain explodes almost surely.

Conditional on a jump path $(Y_n)$, if ${\sum }_{n}1/q_{Y_n}<\infty$, then the sum of exponential holding times has finite expectation and is finite almost surely.

For $p=n^{-2/3}\log n$, the expected number of pairs of $K_4$ copies sharing a vertex isThus, with probability tending to one, all the $K_4$ copies are vertex-disjoint. Together with sparse clique-count concentration, this gives arbitrarily many vertex-disjoint copies with probability tending to one.

For $p=n^{-2/3}\log n$, the number $X$ of copies of $K_4$ satisfiesand $\mathrm{var}(X)=o((\mathbb{E}X{)}^2)$. Two distinct copies have dependent indicators only when they share at least two vertices; pairs sharing two or three vertices contribute respectively $O(n^6{p}^{11})$ and $O(n^5{p}^9)$ to the variance.

The $p$th de Rham cohomology group is the quotient of closed $p$-forms by exact $p$-forms.