For $L>0$ and $t>0$, the Sturm-Liouville eigenfunction expansion for $u_t=u_{xx}$ on $(0,L)$ with homogeneous Neumann boundary conditions givesThe constant eigenfunction mode preserves the spatial integral. This is the $\beta =0$ limit of the weighted Neumann heat kernel with constant drift, and its boundary inputs have the signs in the Neumann boundary-forcing formula.

For standard Brownian motion in ${\mathbb{R}}^d$, $p_t(y-x)$ is the transition density from $x$ to $y$. It is the heat kernel for generator $\frac{1}{2}\Delta$. For every unit vector $e$, $\int |{\partial }_e{p}_t(x)|dx=\sqrt{2/\pi }/\sqrt{t}$, which bounds the change in this density under a spatial translation and proves the Gaussian heat-kernel proof of the harmonic Liouville theorem.

For the heat equation with zero Dirichlet boundary conditions on $(0,L)$,It is also obtained by the method of images, taking the difference between Gaussian sources at $y+2jL$ and $-y+2jL$. The eigenfunction expansion is useful at long times; the image expansion is useful at short times.

For $x>0$ and the heat equation on a half-line, the zero-initial-data solution with prescribed Dirichlet boundary condition $g$ is ${\int }_0^{t}P(x,t-s)g(s)ds$, whereIts mass concentrates at $t=0$ as $x\downarrow 0$, recovering the boundary trace.

The normalized Gaussian mass of an interval isIt is the convolution of an interval indicator with the heat kernel, and has an error function representation. As the variance tends to zero it converges to the interval indicator away from its endpoints, with value one half at an endpoint. This is a concrete approximate identity.

The zero-phase velocity ansatz $w=(\mathrm{sech}z{)}^k(\tanh z{)}^{1-k}$ solves the Taylor–Goldstein equation away from its critical level of an internal gravity wave exactly when $J=k(1-k)$. Let $y=\tanh z$ and $a=1-k$. Then $w^{\prime }/w=a/y-y$ and $w^{\prime \prime }/w=a(a-1)/y^2-a-1+2y^2$, so the differential-equation residual divided by $w$ is $[J-k(1-k)](y^{-2}-1)$. The curve has maximum $1/4$ at $k=1/2$. The endpoint $k=0$ does not decay at infinity; fractional powers at the critical level require a branch and regularity qualification.

For a Schwarzschild black hole, a greybody factor is a mode's transmission probability through the exterior scattering potential. The spectrum of Hawking radiation at future null infinity is the thermal horizon occupation multiplied by this factor, with a sum over angular modes.

The late outgoing rays from a nonextremal collapsing black hole have an exponential relation between a regular early affine null coordinate $U$ and late retarded time $u$. Logarithmic divergence of the tortoise coordinate at the horizon gives $U_H-U=A{e}^{-\kappa u}$ up to subleading terms. Backward propagation turns $e^{-i\omega u}$ into $(U_H-U{)}^{i\omega /\kappa }$; its positive- and negative-frequency Fourier parts give a thermal Bogoliubov transformation.

Black-hole evaporation is mass loss through Hawking radiation. For a large isolated Schwarzschild black hole, the leading mass dependence is $dM/dt\propto -M^{-2}$ and the evaporation timescale is proportional to $M^3$. This semiclassical scaling cannot determine the endpoint when the curvature reaches quantum-gravitational scales.

The first integral singular homology of the Hawaiian earring has a direct summand isomorphic to $\mathbb{Q}$. This consequence of its structural homology theorem suffices, through the universal coefficient theorem for cohomology and nonzero Ext of the rationals with integer coefficients, to show that its integral $H^2$ is nonzero. The precise structural theorem is Theorem 3.1 of www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/eda-kawamura2.pdf.

For a full-rank ordinary least squares design matrix, the hat matrix $H$ and residual projection $G=I-H$ satisfy $HG=GH=0$. Hence fitted values and regression residuals are perpendicular as observed vectors. Under isotropic errors their cross-covariance matrix is zero, and under a normal linear model the two vectors are independent.

The Hartmann number measures the strength of magnetic coupling relative to viscous and resistive effects in a channel:Here $L$ is the channel half-width, $\nu$ the kinematic viscosity, $\eta$ the magnetic diffusivity and ${\sigma }_c$ the electrical conductivity. In Hartmann flow, large $H$ produces thin magnetic-viscous wall layers and a flatter, slower core for fixed pressure gradient.

A Hartmann layer is the thin viscous-magnetic boundary layer next to a wall crossed by the applied magnetic field. For channel half-width $L$ and large Hartmann number $H$, its thickness is $\delta \sim L/H=\sqrt{{\mu }_0\rho \nu \eta }/|B_0|$. It restores the no-slip boundary condition and the specified magnetic wall condition outside the nearly uniform Hartmann flow core.

For a streamwise pressure gradient $-\rho G$ and walls at $z=\pm L$ with zero tangential magnetic field, the volumetric flow rate per unit span isIts zero-field value is $2G{L}^3/(3\nu )$; it decreases strictly for $H>0$ and is asymptotic to $2G{L}^3/(\nu H)$. The partial-fraction expansion of the hyperbolic cotangent proves monotonicity directly.

The attaching sphere of an index-$h$ handle $D^h\times D^{n-h}$ is $S^{h-1}\times \{0\}$ in the previous boundary. Its framing of an embedded sphere identifies a neighbourhood with $S^{h-1}\times D^{n-h}$ and specifies the handle attachment.

The belt sphere of an index-$h$ handle is $\{0\}\times S^{n-h-1}$ in the boundary after attachment. It is the attaching sphere of the dual index-$(n-h)$ handle. Its intersections with the attaching spheres of later handles detect passages through this handle.

A handle slide replaces the framed attaching sphere of one handle by a band sum with a parallel copy of another handle of the same index. It changes the attaching data while preserving the diffeomorphism type of the resulting manifold. On the handle incidence matrix it performs an elementary row operation, or a column operation when applied to the dual decomposition.

Suppose $v_i=\pm 1/\sqrt{n}$ encodes the true partition and $\hat{v}$ is a unit estimated eigenvector. Every coordinate of the wrong sign contributes at least $1/n$ to ${\min }_{\sigma =\pm 1}\Vert \hat{v}-\sigma v{\Vert }_2^2=2(1-|{\hat{v}}^{\top }v|)$. Since this is at most $2(1-|{\hat{v}}^{\top }v{|}^2)$, which is the squared Frobenius distance between the rank-one orthogonal projection matrices, the bound follows. Coordinates estimated as zero may be assigned consistently to either group.

An inclusion-maximal separated set of separation $an$ covers the Hamming cube by strict-radius $an$ balls. The Hoeffding lower-tail bound for Hamming balls then gives at least $e^{2n(1/2-a{)}^2}$ selected vertices. At $a=1/8$ this yields $e^{9n/32}$ vertices, all separated by at least $n/8$.

For $0<\rho \le 1/2$, the number of binary words within Hamming distance $\rho n$ of a fixed word is at most $e^{nh(\rho )}$, with the binary entropy function in natural logarithms. In $1={\sum }_j\left(\genfrac{}{}{0ex}{}{n}{j}\right){\rho }^j(1-\rho {)}^{n-j}$, each summand weight for $j\le \rho n$ is at least ${\rho }^{\rho n}(1-\rho {)}^{n(1-\rho )}=e^{-nh(\rho )}$. Also $h^{\prime \prime }(\rho )=-1/(\rho (1-\rho ))\le -4$, so $\log 2-h(1/2-a)\ge 2a^2$. These two bounds give the approximate-recovery denominator in the list-decoding Fano inequality.