Put $F(u)=\Sigma (\sqrt{u})$ and $g(v)=\rho (\sqrt{v})$. Then $F(u)={\int }_u^{\infty }g(v)/\sqrt{v-u}dv$. Composing this Abel transform with itself and exchanging convergent integrals gives ${\int }_t^{\infty }F(u)/\sqrt{u-t}du=\pi {\int }_t^{\infty }g(v)dv$, because the inner integral from $t$ to $v$ is $\pi$. Differentiate after writing $u=t+s$ to obtain $g(t)=-{\pi }^{-1}{\int }_t^{\infty }{F}^{\prime }(u)/\sqrt{u-t}du$. Substituting $t=r^2,u=R^2$ yields the displayed reconstruction. Decay and regularity sufficient for the exchanges and derivatives are required.

The strip density is the mass per unit projected Cartesian coordinate $x$: $S(x)={\int }_{-\infty }^{\infty }\Sigma (\sqrt{x^2+y^2})dy$. It is even in $x$. For $x\ge 0$, changing variables to projected radius gives $S(x)=2{\int }_x^{\infty }\Sigma (R)R/\sqrt{R^2-x^2}dR$. Alternatively integrate the three-dimensional mass density over the plane of coordinates $y,z$ to obtain the displayed expression. Consequently $S^{\prime }(x)=-2\pi x\rho (x)$ for $x>0$.

A light-second is a unit of length: the distance vacuum light travels in one second. It is useful in a spacetime diagram because a time of one second corresponds to a $ct$ coordinate of one light-second, making the slopes $x=\pm ct$ of the light cone particularly easy to read. The unit is a distance, rather than an elapsed time.

Let $F$ be a smooth function near the unit sphere $S^d$, and write ${\Delta }_{\mathrm{Euc}}F={\sum }_{a=1}^{d+1}{\partial }_a^{2}F$ for the ordinary ambient second-derivative trace. For the positive Laplace-Beltrami operator $P_{S^d}=-\mathrm{div}\nabla$,evaluated at radius one. The negative of this formula is the restriction identity for the nonpositive Laplace-Beltrami operator convention.

Choose an orthonormal basis $e_i$ of $T_p{S}^d$. The great circles ${\gamma }_i(t)=p\cos t+e_i\sin t$ giveThe negative sum is $P_{S^d}(F|)$; supplementing the tangent basis by the radial vector $p$ gives the displayed formula. If $F$ is a degree-$\ell$ homogeneous polynomial that is a harmonic polynomial, then $F_r=\ell F$, $F_{rr}=\ell (\ell -1)F$ and ${\Delta }_{\mathrm{Euc}}F=0$. Its restriction therefore has positive-convention eigenvalue $\ell (\ell +d-1)$, as in the spectrum of the Laplacian on a sphere.

For a closed unital subalgebra $A\subseteq B$ with common identity, its group of invertible elements of a Banach algebra is relatively open in $A\cap G(B)$ by the Neumann series. It is relatively closed because inversion in $B$ is continuous and $A$ is closed. Consequently each connected component of $A\cap G(B)$ either consists entirely of elements invertible in $A$, or entirely of elements whose inverses do not belong to $A$.

For a Laplace-type operator on a closed manifold, let $a_{t^0}$ be the constant coefficient of its full heat trace expansion. In the Mellin transform defining the spectral zeta function, the reduced trace has constant coefficient $a_{t^0}-\dim \ker L$. Its contribution is $(a_{t^0}-\dim \ker L)/s$, and $1/\Gamma (s)=s+O(s^2)$ proves the identity. For the scalar positive Laplace-Beltrami operator on a closed surface, $a_{t^0}=\int RdV/(24\pi )$ and $\dim \ker L$ is the number of connected components. In particular, a connected sphere has value $-2/3$ and a connected flat torus has value $-1$.

For $w=x_1+i{x}_2$ and $0<|k|\ne 1$, the operator $D_k=(k+k^{-1}){\partial }_{x_1}/2+(k-k^{-1}){\partial }_{x_2}/(2i)$ has the displayed decaying fundamental solution. A rotated real-linear coordinate turns it into a multiple of a Wirtinger derivative, reducing the kernel identity to ${\partial }_{\bar{z}}(1/(\pi z))=\delta$. The kernel is analytic in $k$ separately inside and outside the unit circle, is $O(k)$ at zero and $O(k^{-1})$ at infinity, and develops a jump at the real directional-transport limit.

Let $F(\rho ,\tau ,\theta )={\int }_{\tau }^{\infty }f(s\mathbf{d}+\rho \mathbf{n})ds$, $S=\mathcal{R}f$, and $A={\int }_{\mathbb{R}}{e}^{-F}gd\tau$ be the directed attenuated Radon transform. With the signed Cauchy boundary operators, set $J=e^F[e^{P^{-}S}{P}^{-}(e^{-P^{-}S}A)+e^{-P^{+}S}{P}^{+}(e^{P^{+}S}A)]$. An integrating factor turns the attenuated equation into unattenuated directional transport; its spectral solution has jump $-J$ on the unit circle. A normalized Cauchy integral formula and the coefficient at spectral infinity then prove the reconstruction formula. Known attenuation and full directed angular data determine the source; attenuation itself is not reconstructed by this pair.

Use the triangular $w$ in equal radiant power under visual weighting and stipulate the detector's spectral responsivityThis model has nonnegative response: $w$ is zero outside $405$–$705\mathrm{nm}$. For a $1\mathrm{mW}$ ideal calibration line at $555\mathrm{nm}$, $i_A=0.5\mathrm{mA}$ and ${\tilde{\Phi }}_{v,A}=0.683\mathrm{lm}$, so $C_A=1366\mathrm{lm}{\mathrm{A}}^{-1}$. An equal-radiant flux test line at $630\mathrm{nm}$ gives $w=1/2$, $r=0.375\mathrm{A}{\mathrm{W}}^{-1}$ and $i_T=0.375\mathrm{mA}$. The unchanged conversion indicates $0.51225\mathrm{lm}$, whereas the model's intended weighted result is $0.3415\mathrm{lm}$. The required spectral mismatch correction is $F=2/3$: the uncorrected result is 50% too large. All these numbers are declared model outputs, not instrument measurements or standard photometry data.

The cumulative function of the finite positive spectral measure of a stationary time series is nondecreasing and right-continuous, with total mass equal to the variance. For an absolutely continuous measure, $F(\lambda )={\int }_{-\pi }^{\lambda }f(u)du$, where $f$ is the time-series spectral density. Under angular-frequency normalization, ${\gamma }_k={\int }_{-\pi }^{\pi }{e}^{ik\lambda }dF(\lambda )$ and, when the covariance series is absolutely summable, $f(\lambda )=(2\pi {)}^{-1}{\sum }_{k\in \mathbb{Z}}{\gamma }_k{e}^{-ik\lambda }$. Atoms describe persistent periodic components and cannot be represented by an ordinary density.

For uncorrelated zero-mean unit-variance amplitudes $A,B$, the process $A\cos ({\omega }_{0}t)+B\sin ({\omega }_{0}t)$ has covariance $\cos ({\omega }_{0}k)$ and hence the displayed two-atom spectral measure of a stationary time series. Independence or Gaussian amplitudes are unnecessary for weak stationarity. A phase shift need not preserve the whole joint amplitude distribution, so strict stationarity cannot be inferred from these second moments alone.

For $L=d^2/d{x}^2$ on $(0,\pi )$ with zero endpoints, the zero-parameter Green function is $-x_{<}(\pi -x_{>})/\pi$. Its eigenfunctions are $\sqrt{2/\pi }\sin (nx)$ with eigenvalues $-n^2$. At $x=\xi =\pi /2$, the kernel is $-\pi /4$ and its expansion contains precisely the odd reciprocal squares, giving the displayed identity.

For two real centered time series $X_t,Y_t$, require joint weak stationarity: all auto- and cross-covariances depend only on the time lag. Individual weak stationarity alone does not impose this condition on their joint behavior. With the convention ${\gamma }_{XY}(h)=\mathbb{E}[X_t{Y}_{t+h}]$, define the cross-spectrum byAssume absolute summability of the auto- and cross-covariance sequences, so the displayed Fourier series and the two individual spectral densities are continuous. Reality and joint weak stationarity giveConsequently the spectral matrix $F(\lambda )=\left(\begin{array}{cc}{f}_{XX}& f_{XY}\\ f_{YX}& f_{YY}\end{array}\right)$ is a Hermitian matrix. It is positive semidefinite, because its quadratic forms are nonnegative spectral densities of complex linear combinations of the jointly stationary series. Its nonnegative determinant therefore yields $|f_{XY}(\lambda ){|}^2\le f_{XX}(\lambda )f_{YY}(\lambda )$.

The imaginary part records directional lag information. For white noise $X_t$ of variance $v$ and $Y_t=X_{t-1}$, only ${\gamma }_{XY}(1)=v$ is nonzero. Hence $f_{XY}(\lambda )=v{e}^{-i\lambda }/(2\pi )$, although both individual spectral densities equal $v/(2\pi )$. The cross-spectrum is generally complex, rather than real and even like a real series' own spectral density.

Under angular-frequency normalization, the displayed time-series spectral density gives ${\gamma }_0={\pi }^2$ and ${\gamma }_k=2(1-(-1{)}^k)/k^2$ for nonzero integer $k$. This follows from the even integral $2{\int }_0^{\pi }(\pi -\lambda )\cos (k\lambda )d\lambda$. All nonzero even-lag covariances vanish while the odd-lag covariances are positive.

Volumetric heat capacity is heat capacity per unit volume, with units $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$. For homogeneous material of mass density $\rho =m/V$, it equals $\rho c$, using specific heat capacity $c$ for the same constraint. It measures the heat required per small temperature increase for a chosen occupied volume. In a material model using specific heat capacity at constant pressure, thermal diffusivity is $\alpha =k/(\rho c_p)$, where $k$ is thermal conductivity.

Near the identity, the matrix logarithm of a unitary matrix is skew-Hermitian. Restrict the logarithm's operator norm below $\pi /n$; determinant one then forces its trace to be zero, since $\det U=\exp (\mathrm{tr}\log U)$ and no nonzero multiple of $2\pi i$ fits this bound. The matrix exponential gives the inverse chart on a ball in the displayed real Lie algebra, of dimension $n^2-1$. Left translation supplies compatible charts on the whole special unitary group.

Project a based map $\alpha :S^r\to SO(s+1)$ to $S^s$. Since $r<s$, the resulting homotopy class vanishes. Lift a based null-homotopy by the special orthogonal sphere fibration. Its endpoint lies in the fiber $SO(s)$. If the lift moves its base point, right-multiply at time $t$ by the inverse of its base-point value, which lies in the fiber. This keeps the projected homotopy unchanged and makes the lift based. Thus $\alpha$ is based-homotopic to a map into $SO(s)$, proving the asserted surjectivity.

The scalar center has order $\gcd (n,q-1)$. For $n>1$ the quotient is simple except at $(n,q)=(2,2),(2,3)$. Its faithful primitive projective-point action, abelian fixed-center transvections, elementary-matrix generation and perfectness give the Iwasawa simplicity lemma proof.

A determinant-one matrix over ${\mathbb{F}}_2$ permutes the three nonzero vectors of ${\mathbb{F}}_2^2$. This group action is faithful because fixing two basis vectors forces the matrix to be the identity. There are $(4-1)(4-2)=6$ invertible matrices, all of determinant one since the only nonzero field element is one, so the action identifies the group with the symmetric group $S_3$. Reduction of integer determinant-one matrices modulo two is onto: $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ induce two distinct transpositions, which generate $S_3$. Its kernel consists of matrices congruent to the identity modulo two, the level-two principal congruence subgroup.

Over a field of odd characteristic, an involutory matrix has minimal polynomial dividing $(X-1)(X+1)$, so it is diagonalizable with eigenvalues in $\{1,-1\}$. In dimension two, determinant one requires either both eigenvalues one or both minus one. Hence the only element of order two in $S{L}_2$ is $-I$; the identity has order one.