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The Fermi energy is the highest occupied one-particle energy in the ground state of a noninteracting Fermi gas at zero temperature, measured here relative to the lowest one-particle energy. The Pauli exclusion principle allows only one fermion in each complete one-particle quantum state, including its spin, so adding more particles fills progressively higher energy eigenstates. A cold gas can therefore have substantial kinetic energy even when its temperature is zero. In the thermodynamic limit, with the same energy reference, $E_F=\mu (T=0)$ where $\mu$ is the chemical potential.^[ref]

For a uniform, three-dimensional gas of noninteracting electrons, with spin one-half, equal populations of the two spin states, no applied magnetic field, and nonrelativistic particles of mass $m=m_e$,Here $n$ is the total number density, $V$ the volume, $p_F$ the Fermi momentum, $\hslash =h/(2\pi )$ the reduced Planck constant, and $k_B$ the Boltzmann constant. The electron Fermi temperature $T_F$ expresses the same energy scale in kelvin; it is not the actual gas temperature. Strong degeneracy of a degenerate electron gas means $T\ll T_F$. The momentum formula counts states and survives changing the energy–momentum relation, whereas $E_F=p_F^2/(2m)$ requires $p_F\ll mc$.^[ref]

The reference matters: the displayed $E_F$ excludes rest energy and sets the zero at $p=0$. Including rest energy shifts both the one-particle energy and the zero-temperature chemical potential by $m{c}^2$, leaving occupations unchanged. More generally, adding a constant $C$ to every one-particle energy also adds $C$ to $\mu$; the Fermi-Dirac distribution depends only on their difference. In a finite quantum system the chemical potential at zero temperature can lie between the last occupied and first empty energy eigenstates; its equality with the filling edge is a continuum statement.

At positive temperature, $E_F$ remains the zero-temperature scale associated with the specified number density. The Fermi level $\mu (T)$ must instead satisfy the particle-number conditionwhere $g(E)$ is the extensive density of states, including both spin states. Smearing the Fermi-Dirac distribution while arbitrarily holding $\mu =E_F$ generally changes $N$. This distinction matters when interpreting electron heat capacity and any experiment that infers temperature from an occupation profile.^[ref]

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.

In the toric fan construction, a toric cone is a rational polyhedral convex conewith $v_i$ in a toric lattice $N$, satisfying $\sigma \cap (-\sigma )=\{0\}$. The last condition is strong convexity: the toric cone contains no nonzero linear subspace. The zero toric cone is allowed and corresponds to the dense algebraic torus chart.

A simplicial polyhedral cone has linearly independent primitive toric ray generators. For its affine toric variety to be smooth, those generators must extend to a lattice basis. The toric cone generated by $(1,0),(1,2)$ in ${\mathbb{Z}}^2$ is simplicial but not smooth, because the determinant is two rather than $\pm 1$. Thus real linear independence alone does not decide smoothness.^[ref]

Let $G$ be the real Heisenberg group, let $Z$ be its center of a group, and let $Z_0$ be the integer subgroup of $Z$. Every finite-dimensional continuous representation of a topological group $\rho$ of $G/Z_0$ kills the central circle group $Z/Z_0$, hence is not a faithful representation.

To prove this, lift $\rho$ to $G$ and differentiate. The Heisenberg Lie algebra has $[a,b]=c$ with $c$ central. Applying the Lie theorem to the complexified Lie algebra representation makes all representing matrices upper triangular. Its derived algebra is consequently represented by strictly upper triangular matrices, so $N=d\rho (c)$ is nilpotent. Choose the normalization for which $\exp (c)$ generates $Z_0$. Descent to $G/Z_0$ gives $e^N=I$. Since the finite matrix logarithm of $e^N$ is $N$, this forces $N=0$. Hence $\rho (\exp (tc))=e^{tN}=I$ for all real $t$, proving the claim. This distinguishes the faithful Lie algebra representation supplied by Ado theorem from a faithful group representation of a prescribed global Lie group.

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

The hyperbolic cotangent is $\coth x=\cosh x/\sinh x$ where $\sinh x\ne 0$.

The hyperbolic sine is the odd functionIts derivative is $\cosh x\ge 1$, so it increases strictly from $-\infty$ to $\infty$ and is a bijection of $\mathbb{R}$ onto itself. For example, $\sinh 0=0$, $\sinh 1\approx 1.175201$, and $\sinh (-1)\approx -1.175201$. It is the vertical coordinate on the unit hyperbola parametrized by $(\cosh x,\sinh x)$.

The Taylor seriesshows that $\sinh x\approx x$ near zero. Its tangent line there is $y=x$; unlike the ordinary sine, the hyperbolic sine grows without bound. Its antiderivative is $\cosh x+C$. In the catenary $y=a\cosh (x/a)$, the hyperbolic sine gives both the slope, through the derivative $y^{\prime }=\sinh (x/a)$, and the arc length $s(x)=a\sinh (x/a)$ from the lowest point for $x\ge 0$.