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.

When all conditioning events have positive probability, repeatedly apply the definition of conditional probability to factor a joint event into successive conditional factors. For a sequence of discrete responses, this yields the product of conditional masses along the actual outcome path. It requires no unconditional independence assumption.

A Lévy family of graphs consists of finite connected graphs with their uniform probability measures and graph distances divided by their diameters, whose concentration of measure functions vanish at every fixed positive normalized radius. Equivalently, every family occupying at least half the vertices has a radius-$\lfloor \epsilon D_n\rfloor$ neighbourhood occupying a proportion tending to one, uniformly over such families. Harper theorem and binomial distribution tail bounds prove this for the hypercube graphs.

A free complex scalar field has independent particle and antiparticle mode operators. With $d{\Pi }_p=d^3\mathbf{p}/[(2\pi {)}^{3}2E_p]$, the expansion $\varphi =\int d{\Pi }_p(a{e}^{-ipx}+b^{†}{e}^{ipx})$ and its adjoint reproduce equal-time canonical commutation relations when $[a(\mathbf{p}),a^{†}(\mathbf{q})]=[b(\mathbf{p}),b^{†}(\mathbf{q})]=(2\pi {)}^{3}2E_p{\delta }^{(3)}(\mathbf{p}-\mathbf{q})$ and cross commutators vanish. The canonical momenta are $\pi ={\dot{\varphi }}^{†}$, ${\pi }^{†}=\dot{\varphi }$. An overall phase on $b$ can reverse the sign of the antiparticle term without changing the theory.

A nonlinear complex scalar field theory with a global phase symmetry and a curved target-space kinetic coefficient. In physical coordinates its classical density is displayed above; dimensionless coordinates measured in $M^{-1}$ put a common $M^2$ outside the reduced density. Its regular coordinate domain is ${\lambda }^2|\psi {|}^2<1$. Rotating localized solutions are charged complex sine-Gordon solitons. The local classical density does not by itself specify the treatment of the singular coordinate boundary or a global quantum completion.

With two part sizes of fixed sum and all other sizes fixed, this objective has form constant plus $(1-cB)a_i{a}_j$, where $B$ is the sum of the other sizes. A nonpositive coefficient permits merging the pair, while a positive coefficient favours balancing it. Choose an optimum with fewest nonzero parts: all remaining coefficients are positive, and the integer sizes differ by at most one. For real sizes they are exactly equal. This explains extremizers among Turán graphs.

Let $R$ indicate observation of a regression response $Y$, with fully observed predictors $X$. If $Y\perp R\mid X$ and $\mathbb{P}(R=1\mid X)>0$ on the target support, the observed conditional density equals the full conditional density. Thus a correctly specified regression model can be fitted to observed responses under conditional missing at random. This does not justify the unadjusted complete-case mean when observation rates and outcome means both vary with $X$, nor does it imply missing completely at random.

For cause-specific hazard ${\lambda }_k$, the net survival is $S_k^{\mathrm{net}}(t)=\exp [-{\int }_0^t{\lambda }_k(u)du]$. Censoring all competing events in a cause-specific Kaplan–Meier estimator targets this quantity under appropriate observation-censoring assumptions. Its complement generally exceeds the actual cumulative incidence function ${\int }_0^{t}S(u){\lambda }_k(u)du$, because all-cause survival $S(u)$ also includes the other hazards. Identifying net survival with survival after a hypothetical elimination of competing events requires extra causal assumptions.

For any proposed positive active size $d$, the displayed two-colour finite colouring of binary words gives opposite colours to the two points of every combinatorial line with exactly $d$ active coordinates. Their counts of the letter $2$ differ by $d$. Thus no dimension can guarantee a monochromatic line of one universally prescribed active size, even though the Hales-Jewett theorem guarantees some active size. The colouring is allowed to depend on $d$ because $d$ is chosen before the colouring.

If $g$ is analytic on a complex strip of width $\sigma$ and has zero mean, the normalized solution satisfies $\Vert f{\Vert }_{\sigma -\delta }\le C_{n,\tau }{\gamma }^{-1}{\delta }^{-(n+\tau )}\Vert g{\Vert }_{\sigma }$. Fourier decay supplies $e^{-\delta |k{|}_1}$ after the strip loss, and lattice-shell counting bounds the remaining sum ${\sum }_{k\ne 0}|k{|}_1^{\tau }{e}^{-\delta |k{|}_1}$. This simple bound is sufficient for analytic conjugacy iterations even when sharper estimates are available.

For a sufficiently small analytic periodic vector field $w$, a near-identity angular diffeomorphism $\chi$ and a constant vector $a$ solve $D\chi \omega =\omega +w\circ \chi +a$. They satisfy $\chi -\mathrm{id}=O(w)$, $a=O(w)$; if $⟨w⟩=0$, then $a=O(w^2)$. For the defect $e={\mathcal{D}}_{\omega }\chi -\omega -w\circ \chi -a$, set $A=D\chi$, choose $\Delta a=⟨A^{-1}{⟩}^{-1}⟨A^{-1}e⟩$, and solve ${\mathcal{D}}_{\omega }v=A^{-1}(\Delta a-e)$ with zero mean. Updating $\chi$ by $Av$ leaves a quadratic defect $(De)v-[w(\chi +Av)-w(\chi )-Dw(\chi )Av]$. The torus small-divisor estimate and geometrically decreasing strip losses yield a convergent analytic iteration.