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Degeneracy pressure is the pressure of a fermionic system that remains at zero temperature because the Pauli exclusion principle forces particles to occupy states with nonzero momentum.

For spin-one-half particles in three dimensions with dispersion $\epsilon =pc$, the density of states isAt zero temperature,

The Fermi energy is the energy of the highest occupied one-particle state of a noninteracting Fermi gas at zero temperature. Equivalently, it is the zero-temperature chemical potential.

On $([0,1],\lambda )$, the functionsconverge pointwise to zero but satisfy $\int f_{n}d\lambda =1$. Hence the two sides of Fatou lemma can be zero and one.

SetThen $0\le g_n\uparrow {\mathrm{liminf}}_k{f}_k$. The monotone convergence theorem and $g_n\le f_k$ for every $k\ge n$ give

A fan is complete when the union of all its cones is the whole ambient real vector space. A toric variety is proper exactly when its fan is complete.

A ray of a fan is a one-dimensional cone in a fan in toric geometry. It has the form ${\mathbb{R}}_{\ge 0}v$, where $v$ is a primitive lattice vector, and occurs as a face of a polyhedral cone.

A ray of a fan is a one-dimensional cone. It has a unique primitive lattice generator $v_{\rho }$ pointing along it.

A cone in toric geometry is a convex cone generated by finitely many lattice vectors and containing no nonzero linear subspace. Such cones are the pieces of a fan in toric geometry.

A cone in toric geometry is the nonnegative real span of finitely many lattice vectors. It is rational when those generators lie in a lattice and strictly convex when it contains no nonzero linear subspace.

Let $G$ be the real upper unitriangular three-by-three group, let $Z$ be its centre, and let $Z_0$ be the integer subgroup of $Z$. Every finite-dimensional continuous complex representation of $G/Z_0$ kills the central circle $Z/Z_0$, and hence is not faithful.

Commuting complex involutions are simultaneously diagonalizable, so $(C_2{)}^n$ can act faithfully in dimension $d$ only if $n\le d$. A group containing $(C_2{)}^n$ for arbitrarily large $n$, such as the full permutation group of $\mathbb{N}$, has no faithful finite-dimensional complex representation.

For a $p$-cycle $g\in S_p$, each $g^k$ with $1\le k<p$ is conjugate to $g$. In a faithful complex representation, $\rho (g)$ has a nontrivial $p$th-root eigenvalue $\lambda$, and similarity of $\rho (g)$ and $\rho (g{)}^k$ puts all $p-1$ values ${\lambda }^k$ in its spectrum. Hence the representation has dimension at least $p-1$.

The left regular representation acts on the basis $(e_h{)}_{h\in G}$ by $\rho (g)e_h=e_{gh}$. It is faithful.

If a group acts transitively on a set of prime size and $N$ is normal, then the $N$-orbits form a $G$-invariant block system of equal size. They are therefore either singletons or the whole set. In a faithful action the singleton case forces $N=1$, so every nontrivial normal subgroup is transitive.

A cell-free layer is a near-wall region of plasma depleted of suspended blood cells. Because plasma is less viscous than the cell-rich core, the layer can substantially reduce hydraulic resistance.

The complete bipartite graph $K_{3,3}$ has vertex sets $A=\{a_1,a_2,a_3\}$ and $B=\{b_1,b_2,b_3\}$, with every edge between $A$ and $B$ and none inside either part. Its six vertices support $3\cdot 3=9$ edges and no triangles in a graph: among any three vertices, two belong to the same independent set and have no joining edge.

Add the single edge $a_1{a}_2$. Each $b_j$ is a common neighbour of its endpoints, producing the triangle in a graph $a_1{a}_2{b}_j$. These are exactly the three new triangles in a graph, because every triangle in a graph must use the only edge inside a part. The new count is ten edges; nine was the last possible count without a triangle in a graph on six vertices.

In general, every simple graph with a finite set of $n\ge 0$ vertices that is triangle-free has at mostedges. This is the $r=2$ case of Turán's theorem, with forbidden clique $K_3$. Equality holds exactly for the balanced complete bipartite graph $K_{\lfloor n/2\rfloor ,\lceil n/2\rceil }$, up to graph isomorphism; empty parts are allowed for $n<2$.^[ref]

The floor function gives the exact integer bound. Splitting $n$ vertices into two independent sets of sizes $a$ and $n-a$, and putting in every cross edge, givesThe count is largest when $a$ is an integer nearest $n/2$. For $n=2k$ it is $k^2$; for $n=2k+1$ it is $k(k+1)$. Both equal $\lfloor n^2/4\rfloor$. This proves attainability for every $n$, including the empty graph.

A short proof of the upper bound uses degrees of vertices. In a triangle-free graph, the endpoints $u,v$ of an edge have disjoint vertex neighbourhoods: a common neighbour would complete a triangle in a graph. Thus $d(u)+d(v)\le n$. Summing over edges counts $d(v)$ once for each of the $d(v)$ edges incident to $v$, givingFor $n>0$, the Cauchy-Schwarz inequality and handshaking lemma yieldIf $m>0$, division gives $m\le n^2/4$; $m=0$ is immediate. Since $m$ is an integer, apply the floor function. The case $n=0$ is immediate too. This proves the bound; the full equality characterization follows from Turán's theorem.

The useful reversal is that more than $\lfloor n^2/4\rfloor$ edges force a triangle in a graph, whatever their placement. A global count certifies a local pattern. The extremizer is bipartite, but a triangle-free graph need not be bipartite: a five-vertex graph cycle has five edges, below the bound of six. Historically, Willem Mantel posed the triangle-exclusion question in 1907.^[ref]

The Trouton ratio is extensional viscosity divided by shear viscosity. It approaches three for an incompressible Newtonian fluid in uniaxial extension.

If $L=K(\alpha )$ is a finite separable extension and $v$ is an absolute value on $K$, the extensions of $v$ to $L$ correspond to the irreducible factors of the minimal polynomial of $\alpha$ over the completion of $K$ at $v$.

For $w=a_0\cdots a_{n-1}$, a witnessing sequence from $p_0$ to $p_n$ satisfies $p_{i+1}\in \Delta (p_i,a_i)$. Induction on word length shows that $q^{\prime }\in \hat{\Delta }(q,w)$ exactly when such a sequence runs from $q$ to $q^{\prime }$.

The maximum disk hypothesis assigns the largest stellar mass-to-light ratio compatible with a measured rotation curve, conventionally making the disk supply about $85\%$ of the circular speed near $2.2R_d$.

The hyperbolic secant is