A branched covering is a map that is a covering space away from a branch set and has a specified local covering model at that set. For a smooth two-fold covering branched over a codimension-two submanifold, the local model is $(z,u)\mapsto (z^2,u)$. A two-fold branched cover of a knot is the basic three-dimensional example.

A baseline covariate is measured before treatment assignment or the exposure being studied. A prognostic baseline covariate predicts the outcome. In a randomized controlled trial, adjusting for such a covariate can improve statistical efficiency without conditioning on a consequence of treatment. For instance, a measurement taken before treatment can explain persistent between-subject variation in a later measurement.^[ref]

Here is a linear regression example with explicit noise assumptions. Letwhere treatment $Z$ is independent of the baseline covariate $X$, all displayed variances are finite, $\mathbb{E}[\epsilon \mid X,Z]=0$ and $\mathrm{Var}(\epsilon \mid X,Z)={\sigma }^2$. ThenThe first identity uses the law of total variance. It isolates the unexplained population variation removed by adjustment for $X$. For $\gamma =2$, $\mathrm{Var}(X)=1$ and ${\sigma }^2=1$, this variation falls from $5$ to $1$. Estimating additional coefficients still uses data, so this population calculation is not a guarantee that every finite-sample reported standard error becomes smaller. Measurement before treatment also does not by itself make a covariate prognostic or establish a causal effect.

When imaginary time is divided by $\hslash$ and ranges from zero to $\beta$, integrating rotor momenta gives $I{\dot{\varphi }}^2/(2{\hslash }^2)$ but leaves the potential stiffness term $J({\partial }_x\varphi {)}^2/2$. Factoring $1/{\hslash }^2$ out of both terms requires spatial coefficient ${\hslash }^{2}J$ inside the bracket. Setting $\hslash =1$ hides this distinction, but restoring units must preserve it.

Expanding the cosine gives a wave equation $I{\varphi }_{tt}=J{\varphi }_{xx}$ and a gapless phase mode with physical frequency $\sqrt{J/I}|q|$. The exact harmonic lattice frequency is $2\sqrt{J/I}|\sin (q/2)|$. It resembles superfluid sound. In one dimension, Gaussian phase fluctuations and possible compact phase slips require care before inferring true long-range order from this branch.

For a uniform set family, the lower shadow of its shifted set family is contained in the shifted lower shadow. Check the witness in the four patterns of membership in coordinates $i,j$. A surviving shadow member containing only $j$ must have both old shadow partners; a member containing only $i$ needs either old partner. Taking complements exchanges $C_{ij}$ with $C_{ji}$ and gives the same containment for the upper shadow. Thus coordinate shifts of a set family cannot increase either shadow or their disjoint union, the external vertex boundary of a uniform set family.

A vertex of a convex polytope is an extreme point: it cannot be expressed as a nontrivial convex combination of two distinct points in that polytope. In a full-dimensional simple polytope of dimension $n$, a vertex lies on exactly $n$ incident facets.

For $f(x)=x\log x$ on $x>0$, strict convexity and the stationary condition $p=1+\log x$ give the displayed convex conjugate for every real $p$. Biconjugation recovers $f$ on its original positive domain. Its closed extended-real extension has value zero at zero and infinity on negative arguments.

For an increasing concave utility on positive wealth, its utility conjugate uses the indicated supremum and is a convex function of the positive price variable. If $F(x)=-U(x)$ on positive $x$, extended by infinity elsewhere, then $\hat{U}(y)=F^{\ast }(-y)$. Under the Inada conditions its optimizer is the inverse marginal utility, and its derivative is the negative of that optimizer. The sign convention is part of the definition.

In conic optimization, a proper cone is a closed convex cone that is a pointed cone and has nonempty interior in its ambient finite-dimensional space. These conditions permit strict cone feasibility and nondegenerate barrier geometry.

A self-dual cone equals its dual cone under the specified inner product. The nonnegative orthant, second-order cone and positive semidefinite cone are standard examples. Self-duality of a set alone does not establish primal or dual feasibility.

A sequence $Q_n$ is contiguous with respect to $P_n$ if $P_n(A_n)\to 0$ implies $Q_n(A_n)\to 0$ for every measurable sequence of events. It transfers negligible-event assertions, including consistency, between changing sampling laws. It need not imply small total variation distance.

Writing a CNOT gate as $P_0\otimes I+P_1\otimes X$ gives $I\otimes Q_{+}+Z\otimes Q_{-}$, where $Q_{\pm }=(I\pm X)/2$. In the Hadamard basis, $Z$ flips the sign label and $X$ is diagonal. Thus the same gate matrix reverses its control and target roles under the two local basis changes.

For contraction mappings $f,g$, compare their fixed points using $f$ on both arguments, then bound the change from $f$ to $g$ by the uniform metric. The resulting inequality gives the displayed estimate. Continuity at $f$ only needs its own contraction constant $q_f<1$, even if nearby maps have constants approaching one.

This continuous-time multi-state model permits progression $1\to 2$, direct death $1\to 3$, and death after progression $2\to 3$, with no recovery and death an absorbing state. For constant positive rates $a,b,c$, write $\lambda =a+b$. Integrating over the time of the progression arrow givesUse $at{e}^{-ct}$ when $\lambda =c$. The other transition probabilities are $p_{11}=e^{-\lambda t}$, $p_{13}=1-p_{11}-p_{12}$, $p_{22}=e^{-ct}$, $p_{23}=1-e^{-ct}$ and $p_{33}=1$. Unlike a purely sequential irreversible three-state disease model, the direct-death rate contributes to the first-state exit rate. A mixed panel and exact-death likelihood accounts for panel visits and exact death times with different kinds of factors.

The product of connected spaces is connected, so the product of the two connected components lies in the component of $(x,y)$. Conversely, continuous projections carry that component to connected sets in the factors, which must lie in $C_x$ and $C_y$. These two inclusions prove equality in arbitrary topological spaces.

The distance from a focus to a conic section along a line perpendicular to its symmetry axis. In focus-centred polar coordinates, an ellipse may be written $r(1-e\cos \theta )=\ell$, where $e$ is its eccentricity of a conic.

The distance from a focus of a conic section to the curve measured perpendicular to its principal axis. In the focus-based polar coordinates equation $r=p/(1+e\cos \varphi )$ it is the length $p$, where $e$ is the eccentricity. For an ellipse, $p=a(1-e^2)=b^2/a$. For a Newtonian Kepler orbit, $p=j^2/(GM)$ with $j$ the specific angular momentum.

The parameter $e\ge 0$ in a focus-centred polar coordinates equation $r=\ell /(1-e\cos \theta )$ for a conic section. It distinguishes an ellipse ($e<1$), a parabola ($e=1$), and a hyperbola ($e>1$).

For a noncircular conic section, the eccentricity of a conic is the constant ratiowhere $F$ is a focus of a conic, $L$ the corresponding directrix of a conic, and $X$ a point on the curve. For an ellipse with semi-major axis $a$ and distance $c$ from the focal midpoint to either focus of a conic, $e=c/a=\sqrt{1-b^2/a^2}$. Its circle case has $e=0$; a finite directrix does not give that circle through the displayed ratio. The circle is the limit as $c\to 0$ and its directrices of a conic move arbitrarily far away.

The eccentricity of a conic is intrinsic to the cutting plane, and is not generally its slope in a chosen three-dimensional drawing. For the family $z=1+mx$ in conic section, with $0<m<1$, completing the square givesA displacement $dx$ along this plane has Euclidean norm $\sqrt{1+m^2}|dx|$. Thus the intrinsic semiaxis ratio satisfies $b^2/a^2=(1-m^2)/(1+m^2)$, givingFor $m=3/5$, this is $3/\sqrt{17}$, rather than $3/5$. This normalization separates the curve's geometry from a distorted projection of it onto the $xy$ plane.

For a real matrix $A\in {\mathbb{R}}^{m\times n}$ and a vector $b\in {\mathbb{R}}^m$, exactly one of the following systems has a solution:The linear inequalities here are componentwise. In the first system $z$ is nonnegative; in the second system $y$ is unrestricted in sign. The second system is a certificate that the first has no feasible point. This nonnegative-equations alternative is developed in Boyd and Vandenberghe, Convex Optimization slides, §5.34.

Build the target, or separate it. Write the columns of $A$ as $a_1,\dots ,a_n$. Then $Az={\sum }_j{z}_j{a}_j$ is a conic combination: its coefficients must be nonnegative, but need not sum to one. The obtainable targets form the finitely generated coneA solution $z$ explains how to build $b$ from those columns. A certificate $y$ instead defines a linear functional $v\mapsto y^{T}v$. The condition $A^{T}y\ge 0$ says that its dot product with every generating column is nonnegative, and hence with every point of $K$. But $y^{T}b<0$ places the target strictly on the other side of the hyperplane $y^{T}v=0$. Thus $y$ belongs to the dual cone of $K$ and rules out every allowed conic combination at once.

A certificate that can be checked by eye. Take the identity matrixHere $K$ is the nonnegative orthant. The equation $Az=b$ forces $z=(-1,1{)}^T$, violating $z\ge 0$. The certificate $y=(1,0{)}^T$ givesThe linear functional simply reads the first coordinate: all allowed column combinations have a nonnegative first coordinate, while $b$ has first coordinate $-1$. For comparison, the target $(1,1{)}^T$ is obtained with $z=(1,1{)}^T$.

Why this is useful and striking. A successful search can exhibit one feasible point; an unsuccessful search alone cannot explain why every candidate fails. Farkas' lemma guarantees an equally concrete witness of impossibility for this finite linear problem. Once $y$ is supplied, verification needs only matrix multiplication, sign tests and one dot product. The same idea gives a Farkas certificate for linear inequalities and underlies linear programming duality.

Brief geometric proof. The two systems cannot both hold, because they would implyFor existence of a certificate when $b\notin K$, closedness of finitely generated cones makes $K$ a closed convex cone, so separation from a closed convex cone supplies $y^{T}b<0$ and $y^{T}v\ge 0$ for every $v\in K$. Testing each column gives $A^{T}y\ge 0$. Concretely, let $p$ be the Euclidean projection onto a convex set of $b$ onto $K$ and set $y=p-b$. The projection inequality gives $y^T(v-p)\ge 0$ for $v\in K$. Testing $v=0$ and $v=2p$ gives $y^{T}p=0$, whence $y^{T}b=-\Vert p-b{\Vert }_2^2<0$. In the example, $p=(0,1{)}^T$ and $y=(1,0{)}^T$. Alternatively, the Fenchel-Moreau theorem applied to the cone's indicator functional proves the same separation through its polar cone.

An original historical source is Julius Farkas's ^[ref]. The linked archival record verifies the author, title and publication details.

For positive curvature and zero cosmological constant, the single-fluid Friedmann solution has a bang-to-crunch conformal time interval $2\pi /(1+3w)$ for $w\ge 0$. A radial null geodesic in FLRW spacetime travels that same great-circle comoving distance. Relative to a three-sphere circumference $2\pi$, it is $1/(1+3w)$ circuits: one for dust and one half for radiation, with the final point reached only in the singular-endpoint limit.

For $H=p^2/(2m)+V(q)$, time slicing gives a Euclidean path integral over paths with the specified endpoints and action $S_E=\int [m{\dot{q}}^2/2+V(q)]dt$. Each step contributes normalization $(m/(2\pi \hslash \delta ){)}^{1/2}$. This normalization fixes the units and endpoint factors of the kernel, which differ from amplitudes between normalized localized wavepackets.