For a circular cone with intrinsic metric $d{\rho }^2+{\lambda }^2{\rho }^{2}d{\theta }^2$, $0<\lambda \le 1$, local development uses plane angle $\varphi =\lambda \theta$. A plane straight segment that avoids the origin lifts to a geodesic. Choosing an angular difference at most $\pi$ gives plane difference at most $\pi \lambda$, so for $\lambda <1$ this constructs a joining geodesic between any two distinct points. Different permitted angle lifts can yield distinct joining geodesics; the missing apex cannot be used to join straight segments through the plane origin.

For a statistical scalar theory with gradient coefficient $K/2>0$, this rescaling gives a canonical $(\nabla \psi {)}^2/2$ term. A mass coefficient $r$ becomes $r/K$, a quartic coefficient $g$ becomes $g/K^2$, and a source $h$ becomes $h/\sqrt{K}$. Thus the simple propagator denominator $p^2+R$ and loop-coupling formulas assume these normalized coefficients. The associated measure Jacobian is field independent and belongs to the free-energy normalization.

The stabilizer of a timelike four-momentum in $D$-dimensional Minkowski spacetime has connected group $SO(D-1)$. Massive particle polarizations form representations of this group.

The stabilizer of a nonzero null four-momentum has connected group $ISO(D-2)$. For ordinary finite-polarization particles its translation part acts trivially, leaving a representation of $SO(D-2)$.

A particle polarization specifies a physical spin state at fixed four-momentum. Massive spin states transform under the rotation little group $SO(D-1)$; finite-spin massless states transform under its transverse rotation part $SO(D-2)$. A massive two-form field has $\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$ such states, while a two-form gauge field has $\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)$. Polarization components must be counted after field equations and any gauge equivalence are imposed.

At a regular point of a second-order linear ordinary differential equation, a nontrivial solution cannot have both $x$ and $x^{\prime }$ zero, by initial-value uniqueness. Its zeros are therefore simple. Under the linearization of a Riccati equation, such a zero $t_{\ast }$ produces $y=x^{\prime }/(cx)\sim 1/[c(t-t_{\ast })]$. Zeros separate the intervals on which the Riccati equation solution is finite.

With linear growth normalized to unity today, a collapse threshold ${\delta }_{\mathrm{sc}}$ at epoch $z$ corresponds to a present-extrapolated initial barrier ${\delta }_c(z)={\delta }_{\mathrm{sc}}/D(z)$. Comparing this barrier to present variance is equivalent to comparing ${\delta }_{\mathrm{sc}}$ to variance evolved to the collapse epoch. Mixing the two conventions counts growth twice.

In an ordinary least squares regression with an intercept and a nonconstant response, $R^2=1-\mathrm{RSS}/{\sum }_i(Y_i-\overline{Y}{)}^2$. It measures the fitted proportion of sample variation relative to an intercept-only baseline. It increases when regressors are added, even if those regressors are unhelpful for prediction. A high value does not establish the assumptions of a normal linear model or rule out a systematic regression residual pattern.

For the five-point example in linear regression, the total centered sum of squares is $10.8$ and the residual sum of squares is $0.8$, so $R^2=25/27\approx 0.926$. This summarizes the fitted reduction in squared error. It does not record the order or shape of the regression residuals, which are worth examining in a residual-versus-fitted plot.

Here is a fully constructed illustration. Use $x=(-2,-1,0,1,2)$ and the same fitted line $3+x$ for both datasets. Add respectively the residual vectorsBoth sum to zero, both have zero inner product with $x$, and both have squared norm $0.56$. Their ordinary least squares fit is therefore exactly the same line, with the same $R^2=10/(10+0.56)=125/132\approx 0.947$. But the first residual is $0.2(x^2-2)$, a systematic quadratic shape, while the second changes direction across the predictor values. These examples show directly which information the scalar $R^2$ discards; neither constructed pattern proves a random-error model.

A linear regression in the coefficients with a polynomial mean function $\mathbb{E}(Y\mid t)={\sum }_{j=0}^d{\beta }_j{t}^j$. Successive degrees define nested models that can be compared by an F-test.

The intercept is the coefficient of the constant design column. Adding a constant to a predictor can be absorbed by changing the intercept, leaving fitted values and all slope coefficients unchanged.

A reciprocal-predictor linear regression uses $\mathbb{E}(Y\mid v)=\alpha +\beta /v$ for nonzero predictor $v$. It remains linear in its coefficients. Its flattening shape can suit a saturating response over a positive predictor range; residual checks and cross-validation must assess adequacy, and extrapolation toward zero is hazardous.

A piecewise linear regression-path algorithm which starts at zero, activates a predictor with maximal absolute residual score, and moves in a direction that decreases all active absolute scores equally until another predictor ties them. With $G=X^{\mathsf{T}}X/n$ and active correlation signs $s_A$, the coefficient direction is $G_{AA}^{-1}{s}_A$. Ordinary LAR does not drop a predictor when its coefficient crosses zero; sign compatibility of LAR and Lasso characterizes when its path also obeys the Lasso KKT conditions.

Regression diagnostics check assumptions and influential observations in a fitted regression function. Residual-versus-fitted plots can reveal omitted mean structure or nonconstant variance; a quantile-quantile plot checks a specified error distribution. Regression leverage measures unusual predictor configurations, and Cook's distance combines leverage and residual size to measure coefficient sensitivity. Independence may require checking collection order and the study design in addition to residual plots.

A linear projection is an idempotent linear operator: $P^2=P$. Every vector decomposes as $x=Px+(I-P)x$, into the image of a linear map and the kernel of a linear map. Those two vector subspaces form a direct sum. An orthogonal projection additionally uses the orthogonal complement in an inner product space.

For a finite almost-everywhere measurable function $f$, the multiplication operator on L2 space has maximal operator domain $D(M_f)=\{u\in L^2:fu\in L^2\}$. It is a closed operator: if $u_n\to u$ and $f{u}_n\to v$ in L2 space, subsequences converge almost everywhere, giving $v=fu$. Its domain is dense, since truncating a function to the sets $|f|\le n$ approximates it in L2 space. If $f$ is bounded, this is a bounded linear operator of norm equal to the essential supremum of $|f|$.

For $t>0$ this Green function is the response to a point impulse on the infinite line, using the continuous square root of $1+i{c}_d$. The Fourier transform multiplies the initial transform by $e^{\mu t-ikUt-(1+i{c}_d)k^{2}t}$; its Gaussian inverse gives the displayed expression. At fixed $x$, its exponential rate is $\mu -U^2/[4(1+c_d^2)]$, whereas along $x=Ut$ it is $\mu$. Thus a temporally growing flow with $0<\mu <U^2/[4(1+c_d^2)]$ is convectively rather than absolutely unstable.

For the bounded rate diffusion of linear bond pricing in a bounded short-rate diffusion, $D_t{e}^{-(T-t)}{r}_t$ is a bounded martingale with terminal value $D_T{r}_T$. Dividing its conditional expectation by $D_{t}P(t,T)$ through the Bayes formula for conditional expectation gives the displayed forward-measure expectation. It lies between zero and one.

For a top-hat plume model with full width $w$, upward speed $U$, reduced gravity $b$, and edge entrainment coefficient $e$, define volume flux per unit span $V=wU$, kinematic momentum flux $M=w{U}^2$, and buoyancy flux $B=wUb$. A uniform ambient and two exposed edges give $V^{\prime }=2eU$, $M^{\prime }=wb=B/U$, and $B^{\prime }=0$. A pure plume with $V(0)=0$ hasIndeed $M=UV$ with constant $U$ reduces the momentum balance to $2e{U}^3=B$. If width means half-width, its growth coefficient is $e$, not $2e$; profile shape factors also alter the coefficients.

A two-sided line plume whose axial velocity and density anomaly both have triangular transverse profiles. For half-width $b$, ${\int }_{-b}^{b}f(x/b{)}^{n}dx=2b/(n+1)$. These profile moments set the different coefficients of stored mass, momentum, and their fluxes. It differs from a wall plume because both edges entrain ambient fluid.

Conjugation by every element of $G$ preserves a normal Lie subgroup $H$. Its differential therefore preserves $\mathfrak{h}=T_{e}H$. Differentiate that invariance along $\exp (tX)$ to obtain $\mathrm{ad}(X)\mathfrak{h}\subseteq \mathfrak{h}$, proving that $\mathfrak{h}$ is an ideal of a Lie algebra. Closed subgroups have a canonical Lie-subgroup structure; an arbitrary abstract subgroup need not.