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For a centered shell Gaussian measure of a real scalar field with positive shell kernel $K(p)=\kappa p^2+r$, its Gaussian functional integral gives $⟨\tilde{\varphi }(p)\tilde{\varphi }(q)⟩=(2\pi {)}^D{\delta }^{(D)}(p+q)/K(p)$ on the shell. Inverse Fourier transforms therefore yield $G_{>}(x-y)={\int }_{\Lambda /b<|p|\le \Lambda }{d}^{D}p(2\pi {)}^{-D}{e}^{ip\cdot (x-y)}/K(p)$. The kernel must be positive throughout the shell. Translation invariance makes the separation essential; a numerator involving $x$ alone is valid only with $y=0$ or if $x$ is redefined as the separation.

For interaction $V=u\int {\varphi }^4/4!$, split $\varphi =\phi +\eta$ into slow and fast Fourier modes, with a centered fast Gaussian measure. The first cumulant expansion term is $⟨V⟩=(u/24)\int [{\phi }^4+6G_{>}(0){\phi }^2+3G_{>}(0{)}^2]$ by Wick theorem. The constant shifts the free energy, the quadratic term gives $\delta r=u{G}_{>}(0)/2$, and there is no gradient correction at this order. Rescaling gives $r^{\prime }=b^2[r+u{G}_{>}(0)/2]+O(u^2)$ and $u^{\prime }=b^{4-D}u+O(u^2)$. The mass shift mixes thermal and quartic perturbations; the interacting critical tuning need not have zero bare mass.

For centered fast Fourier modes and $V=u\int {\varphi }^4/4!$, the two-fast-field vertex is $(u/4)\int {\phi }^2{\eta }^2$. Its connected second cumulant uses $⟨{\eta }^2(x){\eta }^2(y){⟩}_c=2G_{>}(x-y{)}^2$, producing $-u^2\int {\phi }^2(x){\phi }^2(y)G_{>}(x-y{)}^2/16$. Extracting the local zero-external-momentum quartic coefficient gives $u^{\prime }=b^{4-D}(u-3u^2{I}_2/2)+\cdots$, where $I_2={\int }_{\mathrm{shell}}{d}^{D}p(2\pi {)}^{-D}K(p{)}^{-2}>0$. At $D=4$ its sign makes weak positive quartic coupling marginally irrelevant for increasing length scale, after tuning the critical mass. This local truncation omits generated derivative and higher-order operators; it is not the exact complete finite-shell effective action.

For a Markov jump-process generator $L$, differentiating an integrable observable gives $d\mathbb{E}[f(X_t)]/dt=\mathbb{E}[Lf(X_t)]$, when the generator identity and interchange are justified. Choosing powers gives equations for moments. These need not form a closed finite system.

The second moment of a real random variable is $\mathbb{E}[X^2]$. Its variance is $\mathbb{E}[X^2]-(\mathbb{E}X{)}^2$ when these quantities are finite.

Quantizing a moduli-space approximation gives wavefunctions with measure $\sqrt{\det g}{d}^{d}q$ and, under the minimal scalar ordering, the Hamiltonian operator $-{\hslash }^2{\Delta }_g/2+V$, where ${\Delta }_g$ is the Laplace-Beltrami operator. Global identifications, statistics and regularity supply additional restrictions on wavefunctions. Curvature-dependent ordering terms or quantum corrections are extra choices, not determined by the classical kinetic energy alone.

A module isomorphism is a bijection of modules over the same ring preserving addition and scalar multiplication. Its inverse preserves these operations too. When a linear operator $T$ gives a vector space an $F[t]$-module structure by $t\cdot v=T(v)$, a module isomorphism is exactly an invertible linear map intertwining the two operators. This translates classification of modules into matrix similarity and rational canonical form.

For any finite-index subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ and rational positive-determinant $\gamma$, the intersection $S{L}_2(\mathbb{Z})\cap {\gamma }^{-1}\Gamma \gamma$ has finite index. Multiply $\gamma$ by a positive rational scalar to obtain an integral matrix $A$ with positive integer determinant $D$. For $h=I+DB\in \Gamma (D)$, $Ah{A}^{-1}=I+AB\mathrm{adj}(A)$ is integral, so $\Gamma (D)$ lies in $S{L}_2(\mathbb{Z})\cap {\gamma }^{-1}S{L}_2(\mathbb{Z})\gamma$. This intersection has finite index. Its further intersection with ${\gamma }^{-1}\Gamma \gamma$ has relative index at most $[S{L}_2(\mathbb{Z}):\Gamma ]$. No assumption that $\Gamma$ itself is a congruence subgroup is needed.

A continuous modification of a stochastic process is a modification of a stochastic process whose paths are continuous outside a single null event. For a process originally given only on a countable dense set, the analogous construction gives a continuous extension agreeing simultaneously at all original indices. The Kolmogorov continuity theorem and dyadic increment chaining are standard ways to obtain it.

Put $u_j={\dot{p}}_j+{\partial }_i{W}_{ij}$. The additive independent noises give the forward actionFor a time-even order parameter and time-odd flux, the backward action replaces $u,W$ by $-u,-W$ while keeping $\mu$ fixed on the corresponding configurations. Common normalization and time-reversal invariance of a path Jacobian then giveAt the two thermal noise strengths, periodic boundary conditions cancel the two spatial terms by integration by parts, leaving $-(F_2-F_1)/(k_{B}T)$. Thus the joint dynamics satisfies microscopic reversibility.

The Taylor swimming sheet is an infinite two-dimensional microswimmer whose prescribed traveling deformation drives Stokes flow. A transverse wave $y_s=ϵ\sin (x-t)$ of small amplitude swims opposite to its direction of propagation, with speed $U={ϵ}^2/2+O({ϵ}^4)$ when lengths are scaled by inverse wavenumber and velocities by wave speed. Its biharmonic stream function for planar Stokes flow has first-order part ${\psi }_1=(1+y)e^{-y}\sin (x-t)$.

For a flagellum with axial resistance matrix of a slender helix and a head with resistances $A_0=6\pi \mu a$, $D_0=8\pi \mu a^3$, neglect interactions between the parts. A motor with relative angular velocity $\omega$ givesThe head rotates at $\Omega -\omega$. A large head creates excessive translational resistance; a small head supplies too little rotational resistance, allowing the head to counterrotate while the flagellum barely moves relative to the fluid.

This metric identifies $\mathbb{R}$ isometrically with the open interval $(-\pi /2,\pi /2)$ in the usual real distance. It induces the usual topology but is incomplete: the integer sequence has transformed limit $\pi /2$ outside that interval. This illustrates that completeness depends on the metric, not solely on its topology.

A metric is compatible with a given topology when its open balls generate exactly that topology. Compatible metrics on a compact topological space are uniformly equivalent: the identity maps between the resulting compact metric spaces are uniformly continuous. Consequently they define the same proximality and the same asymptotic-pair relation for a fixed continuous map.

For two equal spheres of radius $a$ separated by $\mathbf{R}$, let the first have applied force $\mathbf{F}$ and let the second be force-free and torque-free. With ${\mathbf{U}}_0=\mathbf{F}/(6\pi \mu a)$, the leading reflected stresslet changes the first velocity byThe distant sphere's freely translating and rotating rigid motions remove its monopole force and torque fields; its incident rate-of-strain tensor produces the first self-mobility correction.

In the preceding two-sphere geometry, holding the second sphere's angular velocity at zero needs a torque $\mathbf{G}=-4\pi \mu a^3{\mathbf{\omega }}_{\infty }$. Its rotlet addsto the first sphere's velocity, in addition to the self-mobility correction from a distant force-free sphere. This leading constraint correction is transverse to the line of centres.

Strict increase alone does not suffice in the preceding theorem. Let ${\sigma }_j=1-1/j$, $c_j=2^{-j}$ and $f(y)=\eta (y){\sum }_{j\ge 1}{2}^{-j}{y}^{1-1/j}$, where $\eta$ is continuous, equals one on $(0,1]$ and vanishes on $[2,\infty )$. After division by the next power, every finite remainder extends continuously to zero. Nevertheless the Mellin transform has genuine poles at $-{\sigma }_j$ accumulating at $-1$, so it cannot be a meromorphic function on the whole plane.

Two-sample Mendelian randomization combines genetic variant associations with an exposure estimated in one sample and their associations with an outcome estimated in another. It requires the instrumental variable assumptions and compatible associations across the populations; sample overlap can change the behaviour of weak instruments.

This weighted median of genetic variant ratio estimates consistently estimates a common causal effect when more than half of its asymptotic weight comes from valid instrumental variables, subject to the usual sampling regularity conditions. It is a sensitivity analysis for horizontal pleiotropy, not an unconditional guarantee of validity.

MR-Egger regression fits genetic outcome associations to genetic exposure associations with a free intercept. Its slope estimates a causal effect under the InSIDE assumption and appropriate instrument-strength assumptions; its intercept assesses average directional horizontal pleiotropy. A nonsignificant intercept, especially with few genetic variants, does not prove the exclusion restriction for every instrument.