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For a sufficiently smooth unit-speed curve with $\kappa >0$, the Frenet frame is the positively oriented orthonormal frame

Assume nondegenerate hydrogen and helium in chemical equilibrium with a black-body radiation field at temperature $T$. Neglect doubly ionized helium, excited-state populations and the small ion/atom mass difference in translational factors. DefineThe Saha ionization equation and ground-state statistical weights givewhere $I_H$ and $I_{He}$ are the first ionization energies. The factor four comes from the ground-state degeneracy of neutral helium: the electronic spin singlet state has weight one, whereas the singly ionized helium ground state and free electron each have weight two. For hydrogen, the corresponding degeneracy ratio is one when nuclear spin is counted consistently.^[ref]

Both equations share the same free-electron abundance. Combining them with the free-electron fraction during hydrogen--helium recombination closes the algebraic system. As cosmological recombination falls out of chemical equilibrium, a kinetic evolution replaces these instantaneous Saha relations.

For a frame $L$, an $L$-valued matrix $f:A⇝B$ is a function $A\times B\to L$. Composition is matrix multiplication with join as addition and meet as multiplication.

An integral ideal is a fractional ideal of an integral domain $A$ that is contained in $A$. For a number field, take $A$ to be its ring of integers ${\mathcal{O}}_K$.

The cutoffs ${\theta }_n(\xi )=(1-|\xi |/n{)}_{+}$ increase pointwise to one, while their Fourier transforms are nonnegative and have integral $2\pi$ under the angular-frequency convention.

The squared sinc function is nonnegative and integrable; it occurs as the Fourier transform of a triangular function.

It is the pointwise square of the sinc function, with its normalization inherited from that function.

The triangular functionhas angular-frequency Fourier transform $4{\sin }^2(\xi /2)/{\xi }^2$.

Equivalently, this is the convolution of two unit-width rectangular functions: the overlap length is $1-|x|$ when $|x|<1$, and zero otherwise. Time rescaling and amplitude changes give triangular pulses with different widths and heights. The convolution theorem also obtains the displayed transform by squaring $\sin (\xi /2)/(\xi /2)$, with continuous value one at zero.

For $y^{\prime }=i\omega y$ with real nonzero $\omega$, explicit Euler has amplification factor $1+i\omega h$, whose modulus $\sqrt{1+{\omega }^2{h}^2}$ exceeds one for every $h>0$.

If $C$ is Hermitian positive definite and $D$ is Hermitian, every eigenvalue of $CD$ is real. Equivalently, $CD$ is similar to the Hermitian matrix $C^{1/2}D{C}^{1/2}$.

If a real Fourier polynomial $c(x)$ is strictly positive, then $C_{nm}={\hat{c}}_{n-m}$ is Hermitian positive definite becausefor every nonzero $z$.

The degree-$d$ moment curve is $\gamma (t)=(t,t^2,\dots ,t^d)$ in ${\mathbb{R}}^d$. Its tangent vectors at distinct parameters are linearly independent:Factor $1,2,\dots ,d$ from the rows and apply the Vandermonde determinant formula. Parameter separation bounded below therefore gives quantitative transversality of the tangent directions.

For radial position vectors, the determinant instead equals $({\prod }_j{t}_j){\prod }_{i<j}(t_j-t_i)$. They are independent only when all parameters are nonzero as well as distinct. In particular $\gamma (0)=0$ supplies no radial direction. For $d=2$, the curve is the parabola, with tangent determinant $2(t_2-t_1)$. Its nondegeneracy is a basic source of curvature in Fourier restriction theory and decoupling inequalities.

Kakeya inequalities bound overlaps of long thin tubes with controlled directions.

These tube estimates are used to study Kakeya sets and their dimension.

Constructive interference occurs when many oscillatory phases nearly agree, making their exponential terms add with comparable arguments and producing a large value of the sum.

Fourier localization controls the variation of a demodulated envelope at reciprocal scales. Suppose the Fourier support lies in a rectangular box centered at ${\xi }_0$, with widths of order $N$ in spatial-frequency coordinates and $N^2$ in a time-frequency coordinate. Set $g(x,t)=e^{-i{\xi }_0\cdot (x,t)}f(x,t)$. Choose a Schwartz function whose transform is one on the centered box and rescale it to obtain a kernel $K_N$. Thenfor $d$ spatial coordinates. The convolution averages on spatial scale $N^{-1}$ and temporal scale $N^{-2}$. For bounded $g$, differentiating the kernel gives $\Vert {\partial }_{x_j}g{\Vert }_{\infty }\le CN\Vert g{\Vert }_{\infty }$ and $\Vert {\partial }_{t}g{\Vert }_{\infty }\le C{N}^2\Vert g{\Vert }_{\infty }$.

These are quantitative variation bounds, not literal constancy on boxes. Removing the center frequency is essential: multiplying a slowly varying envelope by $e^{iMx}$ can make its phase rotate rapidly while leaving its Fourier-support width unchanged. The principle controls a localized envelope or a smoothed magnitude, rather than forbidding carrier oscillations.

A decoupling inequality controls an $L^p$ norm of a function with Fourier support near a curved set by an ${\ell }^2$ sum of norms of pieces supported near smaller caps.

For an integrable function and the angular-frequency convention,so a translation multiplies each frequency component by a factor of modulus one. The amplitude of every component stays the same, while its wave phase changes by $-ka$. For example, delaying $\cos (2\pi \nu t)$ by $a$ produces $\cos (2\pi \nu t-2\pi \nu a)$: faster oscillations accumulate a larger wave phase shift for the same delay. Substituting $y=x-a$ in the defining Fourier transform is the entire proof. Conversely, this multiplier in the Fourier transform produces a translation after Fourier inversion.

For the angular-frequency convention,Consequently cosine transforms to the half-sum of the two shifted deltas, while sine transforms to their signed difference divided by $2i$.

For a Schwartz function, with the angular-frequency convention,More generally, the identity holds when integration by parts is justified and the boundary term vanishes. Repeated differentiation gives $\widehat{f^{(m)}}(k)=(ik{)}^m\hat{f}(k)$. The short proof transfers the derivative from $f$ to $e^{-ikx}$ by integration by parts. High frequencies receive larger multipliers, so rapid ripples contribute more strongly to a derivative. For the heat equation, the second derivative becomes multiplication by $-k^2$, directly explaining the faster decay of fine spatial structure.

When $\hat{f}\in L^1$, its inverse Fourier integral is continuous by dominated convergence. If $f$ is also continuous and Fourier inversion identifies the two functions almost everywhere, then they agree everywhere.

A variable-based grammar has disjoint finite terminal and variable alphabets $\Sigma ,V$, a start variable $S\in V$, and productions $\alpha \to \beta$ in which $\alpha ,\beta \in (\Sigma \cup V{)}^{\ast }$ and $\alpha$ contains at least one variable.