For a finite-index subgroup of the modular group, take the union of translates of the standard fundamental domain of the modular group over right coset representatives. Boundary identifications are inherited from the subgroup action. The truncated region is compact, and its finitely many cusp ends have exponential parameters; this proves compactness of the compactified modular curve and boundedness of invariant cusp-form norms.

For independent Brownian motions $W,B$, the strong law for Brownian motion makes ${\int }_0^{\infty }{e}^{2W_s-s}ds$ finite almost surely, so bracket localization gives an almost sure limit of the stochastic integral. The arctangent transform of a two-noise affine diffusion and endpoint convergence of a bounded angle diffusion identify its distribution function as $1/2+\arctan (x)/\pi$. Conditional Gaussianity rules out atoms.

At a sharp mush–liquid edge with linear liquidus $T_L=T_m-mC$, marginal equilibrium makes the liquid temperature gradient tangent to the local liquidus: $T_z=-m{C}_z$. Together with $T=T_L$ at the edge and $T\ge T_L$ in the exterior liquid, it supplies the usual absence-of-constitutional supercooling closure. When solute diffusion is neglected inside the mushy layer but retained outside it, interfacial solute conservation can require a small solid fraction jump; imposing unit porosity as an additional finite-diffusivity condition generally overdetermines this sharp-edge reduction.

For $\mathcal{C}=C_0/\Delta C\gg 1$ and inverse Stefan number $\mathcal{S}=O(\mathcal{C})$, a stagnant mushy-layer model has nearly constant effective heat capacity, giving ${\Omega }^2{\theta }_t=\kappa {\theta }_{zz}$ with ${\Omega }^2=1+\mathcal{S}/\mathcal{C}$. With negligible liquid solutal diffusivity, roof value $\theta =-1$, positive liquid superheat ${\theta }_{\infty }$, and edge $h=2\lambda \sqrt{\kappa t}$, the mush profile is $\theta =-1+\mathrm{erf}(\Omega \eta )/\mathrm{erf}(\Omega \lambda )$. Matching heat flux requires $\Omega e^{{\lambda }^2}\mathrm{erfc}\lambda ={\theta }_{\infty }{e}^{{\Omega }^2{\lambda }^2}\mathrm{erf}(\Omega \lambda )$. The square on $\Omega$ in the exponential is essential. The porosity $\mathcal{C}/(\mathcal{C}-\theta )$ records the leading composition balance; the linear thermal profile is asymptotic, not exact at finite $\mathcal{C}$.

Homogenizing the top depth $h$ of a constant-gradient stable stratification over horizontal area $\mathcal{A}$ creates mean density ${\rho }_t+{\rho }_0{N}^{2}h/(2g)$. Comparing gravitational potential energies before and after gives $\Delta P={\rho }_0\mathcal{A}{N}^2{h}^3/12$. The density jump at the base has reduced gravity $N^{2}h/2$. The cubic energy cost leads to a quadratic-in-depth coefficient multiplying the deepening rate.

The Steenrod algebra organizes stable mod-$p$ cohomology operations. At $p=2$ it is generated by the Steenrod squares; at an odd prime it is generated by the Steenrod reduced powers and Bockstein homomorphism. Cohomology is a module over this algebra, and every continuous map induces a homomorphism preserving its action.

Suppose two integral cohomology groups of a space are free of rank one. An integral isomorphism can send each chosen generator only to itself or its negative. After reduction modulo $p$, naturality of a Steenrod reduced power between those groups therefore permits its coefficient to change only by a sign, even though an abstract vector-space change of basis could rescale by any nonzero field element. At $p=5$, coefficients one and two cannot be related by signs, so they obstruct an integral homotopy equivalence. This constraint is stronger than comparing the isolated mod-five Steenrod modules with arbitrary bases.

A spherical squirmer with tangential surface slip velocity $(I-nn)(A+Bn)$, where $B$ is a symmetric second-rank tensor and a traceless second-rank tensor, translates with $U=-2A/3$ and does not rotate in an unbounded quiescent fluid. With $S=x\cdot Bx$ its exterior Stokes flow isThis follows from the Unscaled Papkovich–Neuber representation using harmonic functions as potentials $\Phi =-a^{2}Bx/(2r^3)$ and $\chi =a^{3}A\cdot x/(3r^3)+a^{4}S/(2r^5)$. Matching the surface coefficients proves the formula. The two modes generate a potential dipole and a stresslet, respectively.

For a sphere of radius $a$ with prescribed surface slip $u^{\prime }$ and zero external torque, the Lorentz reciprocal theorem for Stokes flow giveswhere $n$ points from the sphere into the fluid. Use an auxiliary rotating sphere in Stokes flow with arbitrary angular velocity $\hat{\Omega }$ and traction $-3\mu \hat{\Omega }\times n$. The actual torque-free cross-work vanishes. Substituting the boundary velocity $\Omega \times an+u^{\prime }$ into the other cross-work leaves $-8\pi \mu a^3\hat{\Omega }\cdot \Omega -3\mu \hat{\Omega }\cdot \int n\times u^{\prime }=0$. A possible translation contributes zero because the auxiliary net force is zero.

In the joint filtration of Brownian history and an independent volatility history, conditioning on the entire volatility path and then using the tower property gives $\mathbb{E}[\sqrt{S_T}\mid {\mathcal{F}}_t]=\sqrt{S_t}\mathbb{E}[\exp (-\frac{1}{8}{\int }_t^T{\sigma }_u^{2}du)\mid {\mathcal{F}}_t]$. Removing the outer conditional expectation requires the integrated variance to be known at time $t$. Independence alone does not give this measurability; a volatility parameter disclosed later supplies a counterexample.

If $\mathbb{E}[\sqrt{S_T}\mid {\mathcal{F}}_t]=\sqrt{S_t}\exp (-{\int }_t^T{f}_t(u)du)$ and $d{f}_t(T)=A_t(T)dt+B_t(T)d{W}_t$, the stochastic Fubini theorem and Itô formula give $f_t(t)={\sigma }_t^2/8$ and $A_t(T)=B_t(T)({\int }_t^T{B}_t(u)du-{\sigma }_t/2)$. The final term comes from the product cross-variation. Continuity extends the drift equality to the specified continuous versions.

For a homogeneous ideal gas of constant specific-heat ratio $\gamma >1$, the Schwarzschild criterion applied to the square-root exponential atmospheric profile givesFor $T_0>0$ and $\alpha >0$, put $s=\sqrt{T-T_0}$ and $A={\nabla }_{\mathrm{ad}}$. The quadratic inequality $A\alpha s^2-2s+A\alpha T_0<0$ has solutions precisely when $A\alpha \sqrt{T_0}<1$, between $s_{\pm }=[1\pm \sqrt{1-(A\alpha {)}^2{T}_0}]/(A\alpha )$. This is a local instability of the imposed temperature gradient; actual convection adjusts that gradient and the prescribed profile need not survive.

If a group homomorphism section $i:G/N\to G$ satisfies $\pi i=\mathrm{id}$, then each $g\in G$ has a unique form $i(q)n$ with $q\in G/N$ and $n\in N$: take $q=\pi (g)$ and $n=i(q{)}^{-1}g$. Thus $(q,n)\mapsto i(q)n$ is a bijection. It is a homomorphism for the ordinary direct product of groups exactly when the section image centralizes $N$. In general it describes a semidirect product with a nontrivial conjugation action.

For $\Lambda =1+\omega$ with antisymmetric lowered parameters, the spin matrix is $S=1-(i/4){\omega }_{\mu \nu }{\sigma }^{\mu \nu }$, where ${\sigma }^{\mu \nu }=(i/2)[{\gamma }^{\mu },{\gamma }^{\nu }]$. At transformed coordinates this is the full component transformation. At fixed coordinates an additional orbital term $-{\omega }^{\mu }{}_{\nu }{x}^{\nu }{\partial }_{\mu }\Psi$ appears.

The spinor Lorentz matrices preserve the indefinite form defined by ${\gamma }^0$: $S^{†}{\gamma }^{0}S={\gamma }^0$. This follows from the gamma-matrix adjoint identities for infinitesimal generators and extends to finite connected transformations. Consequently the Dirac adjoint transforms with $S^{-1}$, even though a boost matrix is not ordinarily unitary.

The linear spin-wave approximation expands a Holstein–Primakoff transformation about a classical ordered state and retains the quadratic oscillator Hamiltonian. It gives the leading order-$S$ excitation energies above the order-$S^2$ classical energy. In a Heisenberg antiferromagnet, first make a bipartite spin rotation. The approximation also predicts a zero-point reduction of the order parameter; infrared-divergent quantum depletion of Néel order signals that the assumed ordered state is not a valid thermodynamic starting point.

Checkerboard spin decimation retains one parity sublattice of a square-lattice Ising model and sums over the other. The retained primitive vectors are $a(1,1)$ and $a(1,-1)$, so the coarse lattice is again square with spacing $\sqrt{2}a$. A rotation and length rescaling restore the original geometry. Every original nearest-neighbour bond joins the two parity classes, whereas a diagonal next-nearest-neighbour bond stays within one class.

A singular Weyl sequence for a bounded self-adjoint operator $L$ at $\lambda$ is a spectral Weyl sequence that also converges weakly to zero. Such a sequence exists exactly when $\lambda$ is in the essential spectrum of a bounded self-adjoint operator. An infinite-dimensional shifted kernel gives a weakly null orthonormal sequence. If that kernel is finite-dimensional but the shifted range is not closed, choose approximate null vectors in the kernel complement and extract a weakly convergent subsequence; its limit belongs to both kernel and complement, so is zero.

For the Fourier transform ${\hat{E}}_L(\nu )=(2\pi {)}^{-1}{\int }_{-L/2}^{L/2}E(z)e^{-i\nu z}dz$ of a stationary random field, the periodogram measure $(2\pi /L)\mathbb{E}|{\hat{E}}_L(\nu ){|}^{2}d\nu$ converges weakly to the spectral measure of a stationary random field. The expected squared modulus is $(4{\pi }^2{)}^{-1}{\int }_{-L}^L(L-|\zeta |)C(\zeta )e^{-i\nu \zeta }d\zeta$. The factor $2\pi /L$ follows from Parseval identity, and preserves the total power per unit length. It also avoids meaningless squares of Dirac delta distributions for coherent plane-wave components.

For a nonzero nonnegative integrable bandlimited function $f$ with frequency support in $[-\Delta ,\Delta ]$, the displayed construction is even, nonnegative and normalizable. Each factor has half-width frequency support, so their product retains width $\Delta$. Boundedness gives ${\int }_T^{\infty }w(t)dt\le 2C\Vert f{\Vert }_{\infty }{\int }_{T/2}^{\infty }f(u)du$. Thus a positive-time tail estimate becomes a two-sided estimate after evenization, with a rescaled decay constant. A simple average with $f(-t)$ would not establish this from an uncontrolled negative tail.