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A free particle following Langevin dynamics initialized at zero position and velocity has relaxation time $\tau =m/\zeta$ and thermal diffusivity $D=k_{B}T/\zeta$. Its position is the convolution of the random force with $[1-e^{-t/\tau }]/\zeta$. The fluctuation-dissipation relation for a Langevin particle gives force covariance amplitude $2\zeta k_{B}T$, and integrating the squared kernel yields the displayed variance. At short times it grows as $2D{t}^3/(3{\tau }^2)$ because the initial velocity is cold; at long times it grows as $2Dt$. A Maxwellian initial velocity produces a different short-time transient.

For two Bernoulli random variables with joint probabilities $a,b,c,d$ in the order $(0,0),(0,1),(1,0),(1,1)$, their covariance is $d-(c+d)(b+d)=ad-bc$. Zero covariance fixes the $(1,1)$ probability to the product of its marginals; subtracting this from each marginal fixes all three remaining joint probabilities to their products. Thus uncorrelated random variables taking only zero and one are independent, including degenerate marginals. The implication is special to this two-point setting and fails for general distributions.

There are ${\aleph }_1$ pairwise disjoint stationary subsets of the nonzero countable limit ordinals. In a Ulam matrix on omega-one, each row partitions a stationary tail into countably many cells, so some cell is stationary. One column contains stationary cells in uncountably many rows, and the column cells are disjoint. Unions indexed by different subsets of ${\omega }_1$ then differ on a stationary set.

The displayed sum is the first-order projection of a centred U-statistic onto sums of functions of individual observations. Its summands are independent and have mean zero. Subtracting it leaves a remainder with conditional expectation zero in each individual observation, a useful decomposition for variance and normal-limit calculations.

A bulk cell expansion generally leaves an oscillatory Dirichlet mismatch. A boundary corrector can solve the homogeneous oscillatory conductivity equation with boundary data cancelling that mismatch. It restores the prescribed physical boundary trace while leaving the leading bulk cell formula unchanged. Estimates depend on boundary regularity and on the coefficient assumptions.

For conditionally independent paired counts with means ${\lambda }_i$ and ${\lambda }_i{e}^{\beta }$, condition on every pair total $N_i=Y_{i0}+Y_{i1}$. The joint conditional likelihood removes all pair-specific rates, leaving the displayed independent binomial distributions. A zero-total pair contributes no information. With unrestricted fixed baseline rates, this conditional likelihood has the same parameter-dependent part as the ordinary profile likelihood; a specified Gamma–Poisson hierarchical model can retain additional information by imposing structure on those rates.

With $\Theta =T/T_{\min }>1$, the two-dimensional self-gravitating rod equilibrium has the displayed energy per unit length for the potential reference $\psi (R_e)=-GM\log (V/V_0)$. Integration of $-\rho \psi /2$ gives its gravitational energy; its two-dimensional Maxwell-Boltzmann velocity distribution has kinetic energy $NkT$. At fixed area, differentiation gives $C=Nk[2+2\Theta \log (1-{\Theta }^{-1})+\Theta /(\Theta -1)]$. Expanding the logarithm shows $C/(Nk)=1+{\sum }_{j\ge 3}(j-2)/(j{\Theta }^{j-1})>1$. The energy increases continuously from $-\infty$ to $+\infty$, so there is no finite-energy endpoint or turning point of the regular sequence. This excludes the classical gravothermal catastrophe in this model, while leaving a distinct canonical collapse at or below its minimum temperature.

When the birefringence separation of optical propagation constants $\Delta k=2\pi \Delta n/\lambda$ is large compared with the spatial rate of rotation of the optical axes, light with linear polarization in a local eigenpolarization follows the twisting nematic director. Thickness alone is insufficient if birefringence vanishes. Strong field alignment along the propagation direction suppresses the transverse optical anisotropy and hence the quarter-turn polarization rotation.

For isometry families $f_x:U\to U\prime$ and $g_y:U\prime \to U\prime \prime$, the complement of $g_y{f}_x(V)$ decomposes orthogonally as the complement of $g_y(W)$ plus the image under $g_y$ of the complement of $f_x(V)$ in $W$. Their fiberwise one-point compactifications therefore give the same Thom space as the external direct sum. The resulting stage isomorphisms respect structure maps and give $Y{⋉}_g(X{⋉}_{f}E)\cong (Y\times X){⋉}_{g\circ f}E$ in the stable homotopy category.

For an isolated uniform sphere with fixed positive cosmological constant, energy conservation and the virial theorem give $2\eta x^3-(2+\eta )x+1=0$, $x=R_v/R_m$. The physical root starts at $x=1/2$ when $\eta =0$ and decreases for $0<\eta <1$. At the limiting value $\eta =1$, the cubic factors as $(x-1)(2x^2+2x-1)$; the collapsing branch approaches $(\sqrt{3}-1)/2$. Finite turnaround therefore gives a strict lower bound, whose infimum is approached rather than attained. Compared with zero vacuum energy at the same turnaround data, the negative vacuum potential and its virial contribution require stronger final gravitational binding.

For the Brusselator, the stationary spatial threshold is $b_T=(1+a\sqrt{D_u/D_v}{)}^2$ and the homogeneous oscillatory threshold is $b_H=1+a^2$. A genuine Turing instability appears first only if $b_T<b_H$, which rearranges to the displayed diffusivity-ratio bound. Equality gives simultaneous stationary spatial and homogeneous oscillatory neutral modes. Otherwise homogeneous loss of stability occurs first.

On a specified finite domain, only its boundary-compatible wavenumbers can grow. Solve the Brusselator mode determinant for its neutral value of $b$ and minimize over those nonzero modes. Minimizing over all continuous $k$ instead gives the usual threshold and $k_c^2=a/\sqrt{D_u{D}_v}$. The finite-mode minimum must still precede the homogeneous threshold to describe a Turing instability.

For $u_t=D{u}_{xx}-u(u-r)(u-1)-\rho (v-u)$ and $0=v_{xx}-v+u$, a Fourier mode perturbation about zero has $v=u/(1+k^2)$ and growth rate $\sigma (k)=-r-D{k}^2+\rho k^2/(1+k^2)$. If $\rho >D$, its maximum is $-r+(\sqrt{\rho }-\sqrt{D}{)}^2$ at $k^2=\sqrt{\rho /D}-1$; if $\rho \le D$, its maximum is $-r$ at zero. Replacing $(u,v,r)$ by $(1-u,1-v,1-r)$ gives the corresponding criterion at the other homogeneous equilibrium.

For $\overline{w{a}^{\prime }}=-{\eta }_t{B}_{0}f$, $⟨f⟩=1$ and positive diffusivities, the Zeldovich planar flux balance gives $G=({\eta }_t/\eta )B_0^2[1-({\eta }_t/\eta )\mathrm{Var}(f)]$, where $G=⟨|\nabla a^{\prime }{|}^2⟩$. The Cauchy-Schwarz inequality and a Poincaré inequality $⟨a^{\prime 2}⟩\le d^{2}G/{\pi }^2$ give ${\eta }_t^2\mathrm{Var}(f)\le ⟨w^2⟩d^{2}s(1-s)/{\pi }^2$, with $s=({\eta }_t/\eta )\mathrm{Var}(f)\in [0,1]$. Maximizing $s(1-s)$ gives the displayed bound. Large turbulent-to-microscopic diffusivity therefore requires $f$ to be close to one in mean square, without guaranteeing pointwise uniformity.

Let $f_0$ be the triweight kernel and h a twice continuously differentiable density with matching zero mean and variance. Put $e=h-f_0$ and $c=35/32$. Two integrations by parts over its support give $\int e^{\prime \prime }{f}_0^{\prime \prime }=48c[h(1)+h(-1)]+c{\int }_{|x|>1}(360x^2-72)h(x)dx\ge 0$, using the zero mass and second moment of e. Therefore $R(h^{\prime \prime })=R(f_0^{\prime \prime })+R(e^{\prime \prime })+2\int e^{\prime \prime }{f}_0^{\prime \prime }\ge R(f_0^{\prime \prime })$. The endpoint terms matter because the third derivative jumps there.

For real $x>0$, the sine integral approaches $\pi /2$. Exponential damping gives ${\int }_0^{\infty }{e}^{-at}\sin t/tdt=\arctan (1/a)$, so the limiting constant follows by letting $a\downarrow 0$. Integration by parts of the tail gives $\mathrm{Si}(x)=\pi /2-\cos x/x-\sin x/x^2+O(x^{-3})$.

For real $x>0$, its small-argument expansion is $\mathrm{Ci}(x)={\gamma }_E+\log x+O(x^2)$. Integrating ${\mathrm{Ci}}^{\prime }(x)=\cos x/x$ separates the logarithm from the regular correction. Its conventional constant is the Euler--Mascheroni constant; the primary series normalization is dlmf.nist.gov/6.6.E6 .

Consider assemblies of congruent polygonal tiles with same-labelled faces paired by reflection and pure Neumann boundary conditions on unpaired faces. An invertible intertwiner for all face permutation matrices preserves the Euler characteristic. The number of tiles is the dimension of the representation. For each face label the number of edge classes is $(n+\mathrm{tr}{M}_s)/2$. For each vertex label the number of vertex classes is the number of orbits of the subgroup generated by incident-face involutions, hence the dimension of its invariant vectors. Intertwining preserves all these numbers. Summing the vertex and edge counts proves equal $V-E+F$, including boundary corners. This argument uses genuine permutation matrices; signed matrices encoding mixed boundary conditions do not give the same topological count.

Take eight congruent hexagonal tiles with three pairwise nonadjacent distinguished faces. On tile indices modulo eight let $\sigma (j)=1-j$, $t(j)=-j$, and use $u_1(j)=3j$ for one assembly and $u_2(j)=3j+4$ for the other. Pair faces by reflection and leave fixed faces, and the three remaining faces, with Neumann boundary conditions. If $R{e}_j=e_{j+1}$, the displayed matrix intertwines all three face actions: its circulant coefficients satisfy $c_r=c_{-r}=c_{3r+4}$. Its eigenvalues are $1+(-1{)}^k+4\cos (\pi k/4)$, which are nonzero for $0\le k\le 7$. The transplantation theorem gives equal spectra. The second gluing graph is bipartite, with parts $\{0,5,6,7\}$ and $\{1,2,3,4\}$, so it is orientable. The first has the odd cycle $1,3,6,2,7,1$ and is nonorientable. Both graphs are connected with eight vertices and ten edges, and their bordered surfaces retract up to homotopy to the graphs. Both have Euler characteristic $-2$ and fundamental groups free of rank three. Neither is simply connected.

Take four congruent octagonal regular polygons with four alternate sides labelled $a,b,c,d$. The intervening sides have Neumann boundary conditions. Pair equal face parameters in the positive reference boundary direction. The complete gluing data areAll unpaired labelled faces not in a Dirichlet column also have Neumann boundary conditions. Encode these rules by signed permutation matrices, using a positive swap for a paired interface and diagonal $-1$ or $+1$ for an unpaired Dirichlet boundary condition or Neumann boundary condition. The four-by-four Hadamard matrix $C$ above satisfies $C{M}_s=N_{s}C$ for every label. Explicitly, if its columns are $h_0,h_1,h_2,h_3$, applying $N_a,N_b,N_c,N_d$ gives respectively $(h_0,h_1,h_3,h_2)$, $(h_0,h_2,h_1,h_3)$, $(h_1,h_0,-h_2,-h_3)$, and $(h_0,-h_1,h_2,-h_3)$, which are exactly the columns of the corresponding $C{M}_s$. On every intervening face the face matrix is the identity.

Since $C^{T}C=4I$, the orthogonal matrix $C/2$ preserves the $L^2$ inner product and the sum of Dirichlet energies. Its intertwining relations map matching Sobolev traces and zero Dirichlet traces bijectively to those of the other assembly. It therefore preserves the quadratic form domains and the forms themselves. The associated self-adjoint operators are unitarily equivalent, proving equality of all eigenvalues with their multiplicities for the specified mixed boundary conditions.

The topology of disk tiles glued along disjoint boundary arcs shows that $D$, whose gluing graph is $0,1,2,3$ in a path, is a closed disc and is simply connected and orientable. The gluing graph of $B$ has edges $13,12,02,01,23$; it is connected with two independent cycles, so $\chi (B)=-1$ and ${\pi }_1(B)\cong F_2$. Its odd cycle $1,2,3,1$ makes it a nonorientable surface. Each assembled surface has nonempty polygonal boundary. This explicitly realizes isospectrality between a simply-connected orientable bordered surface and a nonorientable, non-simply-connected one. The boundary corners and mixed boundary conditions are essential qualifications; it does not contradict the Euler characteristic from the heat trace of a bordered surface for smooth boundaries and pure boundary conditions.