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If $s>n/2+\alpha$ with $0<\alpha \le 1$, then Fourier inversion, $|e^{it}-1|\le C_{\alpha }|t{|}^{\alpha }$, and Cauchy-Schwarz giveThe weighted frequency integral converges precisely because $s-\alpha >n/2$.

On a bounded regular domain, a bounded subset of $C^{k,\alpha }$ is relatively compact in $C^{l,\beta }$ whenever $k+\alpha >l+\beta$. This is an Arzelà-Ascoli theorem argument applied successively to the derivatives.

On a ball of radius $R$, lower derivative norms of $u\in C^{l,\alpha }$ are bounded by an arbitrarily small multiple of the top Hölder seminorm plus a multiple of $\Vert u{\Vert }_{C^0}$. One endpoint form is

The fiber class $F$ is the divisor class of a fiber of ${\mathbb{F}}_k\to {\mathbb{P}}^1$. Its complete linear system gives the ruling.

The negative section $S\subset {\mathbb{F}}_k$ has self-intersection $S^2=-k$. Together with a fiber $F$ of the ruling, it generates the Picard group and satisfies $F^2=0$ and $S\cdot F=1$.

For a Hilbertian basis $(e_i)$,

For a chosen additive function such as module length, the Poincare series of a graded module is the generating function of that function on its homogeneous components. In this usage it is a Hilbert series.

Unitary gauge uses a gauge transformation to remove the Goldstone fields from the Higgs multiplet, leaving the physical radial Higgs excitation and massive vector fields.

A critical point of a real function with continuous second partial derivatives is nondegenerate when its Hessian matrix is an invertible matrix there, equivalently when none of its eigenvalues is zero. A positive-definite matrix gives a strict local minimum, a negative definite Hessian gives a strict local maximum, and opposite-sign eigenvalues give a saddle point of a scalar function. A nondegenerate critical point need not be an extremum: $x^2-y^2$ has Hessian matrix $\mathrm{diag}(2,-2)$, which is an invertible matrix.

Such critical points are the local building blocks of Morse theory.

A Toeplitz matrix is constant along each diagonal, so its entries have the form $T_{nm}=t_{n-m}$.

A set $x$ is reasonable when every member of $\mathrm{TC}(\{x\})$ is countable. In ZFC this is equivalent to $x$ being hereditarily countable.

On $\mathbb{C}[x]$, the operators $d/dx$, multiplication by $x$, and the identity satisfy $[d/dx,x]=1$. Sending $a,b,c$ to these operators gives a faithful irreducible infinite-dimensional representation of the Heisenberg Lie algebra.

This is a finite-group realization of an Iwahori-Hecke algebra. For a finite group $G$ with a BN-pair, the Iwahori-Hecke algebra over $k$ is the endomorphism algebra of the permutation module $k[G/B]$, up to the conventional opposite algebra. Its standard basis is indexed by the Bruhat double cosets.

Fixing the cyclic momenta $p_{\psi }=\lambda$ and $p_{\varphi }=\mu$ leaves the canonical pair $(\theta ,p_{\theta })$ and a one-degree-of-freedom Hamiltonian with effective potential

For $A=\mathrm{diag}(1,\gamma )$, $\alpha =1/\gamma$, and $\beta =(1-{\gamma }^{-1/2}{)}^2$, every error-propagation block has spectral radius at most $1-{\gamma }^{-1/2}$, and equality occurs.

For initial data $u_0$ on the real line, Fourier transformation or convolution with the fundamental solution gives

For a bounded function $u_0$, the convolution is well defined at every positive time because the heat kernel has integral one. The error-function diffusion front uses a Heaviside step function initial condition and gives one smoothed interface. The heat equation with interval-indicator initial data instead has two interfaces; subtracting their two error functions produces the finite-interval profile.

The centered normal densitiesform an approximate identity: for every $f\in L^1(\mathbb{R})$, $f\ast g_t\to f$ at almost every point where the Lebesgue differentiation theorem applies. Their Fourier transforms are $e^{-t{\xi }^2/2}$ in the angular-frequency convention.

In the classical high-temperature limit, each independent harmonic oscillator contributes $k_B$ to the heat capacity at constant volume, where $k_B$ is the Boltzmann constant. A three-dimensional crystalline solid with $N$ chemical atoms has $3N$ vibrational normal modes in the bulk harmonic approximation. Its vibrational internal energy and heat capacity at constant volume therefore obeyHere $U_0$ is independent of temperature at fixed volume. Thus the Dulong-Petit law predicts $C_V\simeq 3N{k}_B$ for a real solid in this regime. Per mole of chemical atoms, rather than per mole of a compound's formula units,with molar gas constant $R$ and Avogadro constant $N_A$. A formula unit containing $r$ chemical atoms consequently has a classical vibrational prediction of $3rR$ per mole of formula units.

For example, in this ideal classical model, one mole of chemical atoms heated at fixed volume from $300$ to $600\mathrm{K}$ gains vibrational internal energyThe heat capacity at constant volume stays the same even though the thermal part of the internal energy doubles. These temperatures illustrate the stipulated classical model; they do not assert that every real solid is classical at $300\mathrm{K}$.

The proof outline is the equipartition theorem: one classical harmonic oscillator has a quadratic kinetic energy and a quadratic potential energy, each with mean $k_{B}T/2$. Their sum is $k_{B}T$, whose derivative with respect to temperature is $k_B$. Summing over $3N$ normal modes gives $3N{k}_B$. The classical result does not require equal angular frequencies: each harmonic normal mode contributes the same amount once it is classical. Overall rigid motions give only a finite-size correction to the bulk count. This independence from the individual angular frequencies makes the Dulong-Petit law useful as a quick estimate of a solid's heating requirement.

Experiments on freely expanding solids often measure the heat capacity at constant pressure $C_P$, whereas the harmonic prediction is for the heat capacity at constant volume $C_V$. In a smooth single thermodynamic phase of a simple compressible material, the difference between constant-pressure and constant-volume heat capacities iswhere $\alpha =V^{-1}(\partial V/\partial T{)}_P$ is the volumetric coefficient of thermal expansion and ${\kappa }_T=-V^{-1}(\partial V/\partial P{)}_T$ is the isothermal compressibility. Using measured $C_P$ as an approximation to $C_V$ requires this correction to be small compared with $C_V$. ^[ref]

High temperature here means $k_{B}T\gg \hslash \omega$ for the relevant vibrational normal modes, with reduced Planck constant $\hslash$ and mode angular frequency $\omega$. At low temperature, excitation across a quantum energy spacing is suppressed: modes with $\hslash \omega \gg k_{B}T$ freeze out thermally, and the classical equipartition theorem fails. The Einstein model of a solid makes this loss of vibrational heat capacity explicit. The Dulong-Petit law is therefore an ideal high-temperature vibrational limit, not a statement that all solids have the same heat capacity at every temperature. Other contributions and departures from the harmonic approximation can matter even at high temperature. ^[ref]

Historically, Petit and Dulong's 1819 experiments compared the specific heat capacities of several elemental solids and found an approximately constant product with their relative atomic weights. Their original empirical regularity came before the classical harmonic explanation; it suggested that counting chemical atoms reveals a simpler pattern than comparing equal masses. ^[ref]

Along a ray $\mathbf{x}(t)$, the derivative of a field $f(\mathbf{x},t)$ is

The WKB method seeks a rapidly oscillating field in the formand determines the phase and slowly varying amplitude order by order in $\epsilon$.