Attach weights $mg$ at the joint and $\lambda mg$ at each outer endpoint of the right-angle two-rod resistance matrix geometry. With the first rod initially horizontal and the second vertically upwards, quasistatic Stokes flow gives $\dot{\theta }=(1-\lambda )mg(\cos \theta -\sin \theta )/(4C{L}^2)$. For $\lambda <1$ the orientation tends to $\pi /4$; for $\lambda >1$ it tends to $-3\pi /4$. Eliminating time gives $x-x_0=(2L/3)(1+\sin \theta -\cos \theta )$ and the same rightward net displacement $2L/3$ in both cases. For $\lambda >1$ the initial drift is leftwards before reversing. At $\lambda =1$ there is no rotation or lateral drift, illustrating noncommuting infinite-time and parameter limits.

Along a unit-speed minimizing geodesic of positive length $L$, take perpendicular parallel orthonormal fields $E_i$ and set $V_i=\sin (\pi t/L)E_i$. The second variation of geodesic energy makes each index form nonnegative, while $\mathrm{Ric}\ge (n-1)\kappa g$ bounds their sum as displayed. For $n\ge 2$ and $\kappa >0$, lengths greater than $\pi /\sqrt{\kappa }$ are impossible. Completeness is what supplies a minimizing geodesic between arbitrary points in the Bonnet-Myers theorem.

An approximate Riemannian heat kernel with the correct delta initial limit and a residual extending with enough regularity to time zero. A cutoff Gaussian times transport coefficients supplies a local construction. Increasing the expansion order, or summing the full asymptotic series smoothly, makes the residual regular enough for a Volterra parametrix correction.

On a closed manifold with a Riemannian metric, an orthonormal eigenbasis for the positive Laplace-Beltrami operator gives this formula, with eigenvalues counted with multiplicity. Polynomial elliptic bounds and exponential time decay give convergence of every derivative away from time zero. The conjugate is necessary for a complex basis.

For a finite free action by Riemannian isometries, the Riemannian heat kernel on the quotient is the image sum shown above. Integrating over a fundamental domain combines all images into the integral on the covering manifold, proving the initial condition and showing that there is no averaging factor in the kernel. The heat trace does have an averaging factor $1/|U|$, because the integral of a quotient function over the cover is $|U|$ times its quotient integral. Mere freeness of a general nondiscrete group action is not enough for this covering formula.

In a high-orbital eccentricity interior $k:1$ mean-motion resonance, the resonant argument $\varphi =k{\lambda }_p-\lambda -(k-1)\varpi$ places successive rotating-frame apocentres at $\pi -[\varphi +(2j+1)\pi ]/k$. Avoiding astronomical conjunction at these slow outer excursions favors $\varphi =0$ for $k=2$ and $\varphi =\pi$ for $k=3$. This geometrical phase protection suggests stable resonant-argument libration centres; actual dynamical stability requires the resonant perturbation problem.

For a Hermitian matrix $X$ and nonreal $z$, this is the negative of the usual resolvent of an operator convention $(zI-X{)}^{-1}$. Its normalized trace is the Stieltjes transform of a measure of the empirical spectral measure with kernel $(x-z{)}^{-1}$. Naming the convention prevents sign errors in self-consistency identities.

On a compact oriented Riemannian manifold without boundary, a smooth $(n-p)$-form $\chi$ annihilates all harmonic $p$-forms under wedge integration exactly when its harmonic projection vanishes. Indeed ${\int }_M\phi \wedge \chi =⟨\ast \phi ,\chi {⟩}_{L^2}$, and the Hodge star maps ${\mathcal{H}}^p$ isomorphically onto ${\mathcal{H}}^{n-p}$. By the Hodge decomposition theorem, this annihilator isThus both exact and coexact additions are allowed; a harmonic representative is unique. The potentials of these additions need not be unique, and out-of-range degree spaces are zero. For a concrete coexact ambiguity, take the torus $(\mathbb{R}/2\pi \mathbb{Z}{)}^2$ with metric $d{x}^2+d{y}^2$. The form $\chi =\delta (\sin ydx\wedge dy)=\cos ydx$ annihilates every harmonic one-form by orthogonality, yet $d\chi =\sin ydx\wedge dy$ is nonzero. Thus an annihilating representative need not even be closed, let alone exact.

On a complete category that is locally small, a set-valued functor is representable exactly when it preserves small categorical limits and its category of elements has a weakly initial set. Categorical limit preservation makes that comma category complete. The initial-object lemma for complete categories with a weakly initial set then supplies a universal element, hence a representation of a functor.

A regression outlier has a response unusually far from its fitted conditional mean. A standardized regression residual adjusts for regression leverage; an externally studentized residual also estimates noise variance with the observation deleted. Selecting the largest residual among many observations changes its reference distribution. An outlier need not be an influential observation, and unusual valid data should not be discarded solely for being unusual.

An influential observation substantially changes a fitted model or its predictions when removed or perturbed. In a linear regression, influence depends jointly on residual size and regression leverage. Large predictor leverage with a small residual and a large residual at ordinary leverage can have different consequences. Regression diagnostics examine this sensitivity; influence is not synonymous with an invalid observation.

With $t=\mu Y/(2\lambda )$, the lower root is $\mu /(1+t+\sqrt{t(t+2)})$ and the upper root is ${\mu }^2/x_{-}$. Rationalizing the lower-root formula avoids subtracting nearly equal large numbers when $Y$ is large.

The Jensen zero-count bound gives $O(T\log T)$ zeros up to ordinate $T$. Since their real parts lie in $(0,1)$, a fixed-point real logarithmic derivative summand is $O(|\Im \rho {|}^{-2})$. Dyadic ordinate bands then contribute $O(j{2}^{-j})$. This proves absolute convergence of the real sum, while the unpaired complex sum of reciprocals need not converge absolutely.

For a modulus with no infinite component, suppose the ordinary ideal class group is trivial. Then the ray class group is trivial exactly when the global unit group maps surjectively onto the product of local unit residue groups at the finite modulus. Negative real units are allowed: positivity restrictions occur only when the modulus includes the corresponding real places.

In a polynomial representation, every matrix coefficient is a polynomial in the matrix entries. The action extends to the monoid of all matrices. The scalar action splits it into homogeneous polynomial degrees; each homogeneous degree-$n$ category is the module category of the Schur algebra $S(m,n)$. Its irreducibles are the Schur modules $D_{\lambda }({\mathbb{C}}^m)$ with $\lambda ⊢n$ and at most $m$ rows.

Tensoring a rational representation of the general linear group with a determinant power shifts every highest-weight coordinate by the same integer $r$. Positive twists can clear determinant denominators; negative twists supply rational representations that do not extend to singular matrices. Twisting preserves irreducibility and dimension.

Split a rational $S{L}_m$ representation by its finite scalar center: on $W_k$, $\rho (\zeta I)={\zeta }^{k}I$ for ${\zeta }^m=1$. The displayed extension is independent of the scalar-root choice and is a group homomorphism. A matrix coefficient of $\rho {|}_{W_k}$ can be represented on $S{L}_m$ by a sum of homogeneous polynomials $P_d$ with $d\equiv k(\mathrm{mod}m)$, by averaging over the scalar center. Its extension is ${\sum }_d(\det g{)}^{(k-d)/m}{P}_d(g)$, a regular function on $G{L}_m$. Every invariant subspace remains invariant under this extension, so irreducibility is preserved in both directions.

The irreducible rational $G{L}_m$ modules are $D_{\lambda }(V)=(\det V{)}^{{\lambda }_m}\otimes D_{\lambda -{\lambda }_m(1,\dots ,1)}(V)$. The second factor is a Schur module. Clearing determinant denominators reduces completeness to the Schur algebra classification of polynomial modules. Distinct dominant integer tuples have distinct highest torus weights. Their characters are symmetric Laurent polynomials; only polynomial modules have characters defined at every singular endomorphism.

For a dominant integer tuple $\lambda$, choose $k=\max (0,-{\lambda }_m)$ and define $D_{\lambda }={\det }^{-k}\otimes D_{\lambda +k(1,\dots ,1)}$, using the polynomial Schur module on the right. The alternant character formula for the general linear group shows that a larger admissible shift yields the same irreducible representation.

A quadratic homogeneous polynomial in the fibre variables of the cotangent bundle is a first integral of the geodesic Hamiltonian exactly when its symmetric lowered coefficients form a rank-two Killing tensor. For $v=g^{-1}p$, the Poisson bracket is $\{K,H\}={\nabla }_{(a}{K}_{bc)}{v}^a{v}^b{v}^c$, so the equivalence follows by comparing cubic coefficients at every point.