For fixed positive ridge regression penalty, choose a unit eigenvector $v$ of $X^{\mathsf{T}}X$ with eigenvalue $d>0$, and signal ${\beta }^0=Bv$. The directional ridge risk is $({\lambda }^2{B}^2+{\sigma }^{2}d)/(d+\lambda {)}^2$, whereas ordinary least squares has risk ${\sigma }^2/d$. Letting $|B|$ grow makes the ridge squared bias arbitrarily large.

The total price seen by a route is the sum of resource congestion prices on its links. With weighted logarithmic utility $w_r\log x_r$, maximizing utility minus linear expenditure gives the demand $x_r=w_r/p_r$. This relation links user decisions to multiplicative resource-price dynamics.

With constant mean motions and $P_c/P_b=(1+\delta )(p+q)/p$, mean conjunctions advance by $\Delta \Lambda =2\pi /(P_c/P_b-1)$ each synodic period. Tracking one branch every $q$ conjunctions and subtracting $2\pi p$ givesThe usual formula $-2\pi p\delta /(q{P}_b)$ keeps only first order in $\delta$. A labelled branch takes $q{P}_b(1+\delta )/(p|\delta |)$ to turn once, whereas the full set of $q$ interchangeable directions repeats after $P_b(1+\delta )/(p|\delta |)$. Finite orbital eccentricity introduces true-conjunction corrections.

In the weak, non-crossing, leading-eccentricity planar model, exterior orders one to three have positive resonant disturbing function coefficients $B_q$. Interior eccentricity-type coefficients of these orders have sign $(-1{)}^q$. A local resonant Hamiltonian $H=-A{J}^2/2-B{e}^q\cos \varphi$, with $A>0$, gives $\ddot{\varphi }=AB{e}^q\sin \varphi$. Linear stability analysis therefore selects $\varphi =\pi$ for $B>0$ and $\varphi =0$ for $B<0$. Exterior symmetric centres are $\pi$; interior centres are $0,\pi ,0$ for orders one, two and three. These are conditional leading-harmonic predictions, not universal centres at arbitrary orbital eccentricity; asymmetric resonant-argument libration changes the picture.

Libration about a resonant phase distinct from the symmetric values zero and $\pi$. For example, in a negative-Kepler-curvature reduction with disturbing potential $R=A\cos \varphi +B\cos 2\varphi$, $A>0$, $B>A/4$, stationary points obey $\sin \varphi (A+4B\cos \varphi )=0$. At $\pi$, $R^{\prime \prime }=A-4B<0$, so it is unstable, whereas at $\cos \varphi =-A/(4B)$ one has $R^{\prime \prime }=4B-A^2/(4B)>0$, giving two stable asymmetric centres. Actual branch thresholds depend on the resonance coefficients and evolving orbital elements.

Keeping one resonant harmonic and treating orbital eccentricity as fixed reduces mean-motion resonance dynamics to a physical pendulum equation. For the inner first-order term $\mathcal{R}=(G{M}_2/a_2)f(\alpha )e\cos \varphi$, leading Lagrange planetary equations give $\ddot{\varphi }\simeq 3j^2{\mu }_2{n}^2\alpha fe\sin \varphi$. Around its stable phase, the small-amplitude libration frequency is $jn\sqrt{3{\mu }_2\alpha |f|e}$. Evolution of eccentricity, other harmonics, and approach to a separatrix change this approximation.

A scale-location plot compares the square root of the absolute standardized regression residual with the fitted values. Systematic changes in level or spread suggest heteroscedasticity. It assesses the error scale, while a residual-versus-fitted plot primarily reveals mean patterns as well as changing spread.

In a full-rank normal linear model with $n$ observations and $p$ mean coefficients, the residual standard error estimates the common error standard deviation. Its square is the unbiased estimator $\mathrm{RSS}/(n-p)$; the square root itself is not generally unbiased. It measures unexplained response variation, whereas a regression coefficient's standard error measures uncertainty in that coefficient.

Every functor $F:\mathcal{C}\to \mathbf{Set}$ on a small category is the colimit of the covariant representable functors $\mathcal{C}(A,-)$ indexed by the opposite of its category of elements. The cocone indexed by $(A,x)$ sends $f:A\to B$ to $F(f)(x)$. By the Yoneda lemma, a compatible cocone into $G$ is exactly a natural transformation $F\to G$. The opposite indexing direction is essential because precomposition reverses the representing-object arrow.

A categorical presheaf is representable if its elements projection has a colimit and the presheaf preserves the opposite-indexed categorical limit of that projection. Compatibility of all distinguished elements then supplies a universal element at the colimit object. The preservation requirement must cover this diagram even when its indexing category is large.

A representable functor $\mathcal{C}(R,-)$ preserves every small categorical limit that exists. A map from $R$ into a limit is exactly a compatible family of maps from $R$ to the diagram objects, and such a family is exactly an element of the corresponding limit of sets. The comparison bijection is the one induced by the limit projections.

If $\mathcal{C}$ has small coproducts in a category and $U\cong \mathcal{C}(R,-)$, then $F(S)={\coprod }_{s\in S}R$ is a left adjoint to $U$. Maps from this coproduct to $X$ are families of maps $R\to X$, naturally equivalent to functions $S\to UX$. Conversely, a set-valued right adjoint $U$ is represented by $F(1)$, because maps from a singleton evaluate to elements.

For $a>0$, the heat equation on a half-line with $u_t+2a{u}_x+a^{2}u=0$ at the left endpoint admits $u=e^{-ax+a^{2}t}$. The solution is spatially decaying and temporally growing. The boundary resolvent has a double pole at $p=a^2$, allowing generalized contributions proportional to $t{e}^{a^{2}t}$. For $a<0$ the root $\sqrt{p}=a$ is absent on the principal square root branch.

A discrete renormalization group with length factor $b>1$ and positive relevant thermal multiplier ${\lambda }_t>1$ sends $t$ to ${\lambda }_{t}t$ and the correlation length to $\xi /b$. Combining this with $\xi \sim |t{|}^{-\nu }$ gives the displayed correlation-length critical exponent. The corresponding scaling dimension is $y_t=\log {\lambda }_t/\log b=1/\nu$. Relevance of discrete multipliers is determined by magnitude relative to one, unlike the sign criterion for continuous-flow eigenvalues.

The relative discriminant of a finite extension of number fields is the relative norm of a fractional ideal applied to its different ideal. Consequently, at $\mathfrak{p}$,where $f_{\mathfrak{P}/\mathfrak{p}}$ is the residue-field degree and $d_{\mathfrak{P}}$ is the different exponent. Over $\mathbb{Q}$, $N({\mathfrak{D}}_{E/\mathbb{Q}})=|D_E|$.

If three disjoint vertex classes have size $L$, and their three pairs are $\epsilon$-regular pairs of vertex sets of edge density of a bipartite graph at least $d$, where $0<\epsilon \le d/2$ and $\epsilon <1/2$, they contain at least the displayed number of transversal triangles in a graph. All but $2\epsilon L$ vertices in the first class have at least $(d-\epsilon )L$ neighbours in both other classes. Regularity between those two neighbour sets supplies the third edge. This strengthens a mere triangle embedding lemma for regular pairs to a cubic lower count.

A regressive function on a stationary subset of a regular uncountable cardinal is constant on a stationary subset.

The Rees module of an $R$-module $M$ with respect to an ideal $I$ is ${\mathcal{R}}_I(M)={⨁}_{j\ge 0}{I}^{j}M{t}^j$, a graded module over the Rees ring $\mathcal{R}(I)$. Generators of $M$ in degree zero generate this module. For $N\subseteq M$, the graded submodule ${⨁}_j(I^{j}M\cap N)t^j$ is finitely generated when $R$ is a Noetherian ring and $M$ is a finitely generated module; a bound on its generators' degrees yields the Artin-Rees lemma.

The integral representation $y_2=y_1\int e^{-\int p}/y_1^2$ is derived where $y_1\ne 0$. At a zero, its integral factor may have a pole while the product has a removable singularity. For regular coefficients, continue the actual second solution by its finite initial data and initial-value uniqueness; the Abel identity preserves a nonzero Wronskian. In the degree-one Legendre differential equation, $y_1=x$ and $y_2=x\mathrm{arctanh}x-1$ illustrate this cancellation at zero.

A cyclically reduced sequence has no pinch in an HNN extension either internally or across its cyclic junction. If its stable-letter length is positive, every positive power remains reduced, so Britton's lemma gives infinite order. Repeated cyclic conjugation and pinch removal shows that any finite-order element of an HNN extension is conjugate into its base group.