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.

Suppose $g$ is bounded and analytic on the complex torus strip $|\mathrm{Im}{x}_j|<\sigma$, has zero mean, andFor $0<\delta <\sigma$, the normalized solution of the cohomological equation on a Diophantine torus obeysContour shifting bounds $|g_k|\le \Vert g{\Vert }_{\sigma }{e}^{-\sigma |k{|}_1}$. On the smaller strip the Fourier solution is consequently bounded byThere are $O_n(m^{n-1})$ integer vectors with $|k{|}_1=m$. Comparing the resulting sum with its exponential integral bounds it by $C_{n,\tau }{\delta }^{-(n+\tau )}$. This gives both convergence and the stated loss-of-strip estimate. It is sufficient for analytic conjugacy iterations even when sharper small-divisor 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.

In a variational Hilbert space setting, a bounded operator $L$ is elliptic or uniformly coercive when $\mathrm{Re}⟨Lv,v⟩\ge \gamma \Vert v{\Vert }^2$ for a constant $\gamma >0$. On a real Hilbert space this condition depends on the symmetric part of $L$ and does not imply self-adjointness. It differs from the principal-symbol condition defining an elliptic differential operator.

Replacing bright ice or snow with darker water or ground decreases surface albedo, increases absorbed sunlight and can cause further melting. Locally the extra absorbed heat flux is $({\alpha }_{\mathrm{ice}}-{\alpha }_{\mathrm{water}})S_{\downarrow }$ on the newly exposed surface. The effect depends on season, cloud cover and incident sunlight.

Warming can mobilize frozen organic material or change sediment conditions, increasing methane emissions and strengthening the greenhouse effect. Production, transport and atmospheric release must be distinguished: methane can dissolve or be oxidized before reaching the atmosphere. Observed seabed bubbles alone do not establish the magnitude or timing of a global atmospheric feedback.

For primitive real nonprincipal Dirichlet characters, there is an absolute $c>0$ with at most one zero in this interval, counted with multiplicity. Positivity of $-{\zeta }^{\prime }/\zeta -L^{\prime }/L$ and the local partial fractions give $0\le 1/(\sigma -1)+C\log q-\sum (\sigma -\beta {)}^{-1}$. Testing $\sigma =1+a/\log q$ with small fixed $a$ rules out two nearby real zeros. Any possible zero is consequently simple; existence is not asserted.

The probability that the insurance surplus in a classical risk model becomes negative at some finite time, starting from capital $u\ge 0$. Ultimate survival has probability $1-\psi (u)$. A finite-horizon ruin probability instead restricts the crossing time to a specified time interval.

For unit mass with specific angular momentum $l$ in $V(r)=-e^{-r}$, the effective potential is $-e^{-r}+l^2/(2r^2)$. A positive-radius circular orbit satisfies the displayed equation. Since $r^3{e}^{-r}$ has its unique maximum $27/e^3$ at $r=3$, exactly two radii occur for $0<l^2<27/e^3$. Their radial stability follows from $V_{\mathrm{eff}}^{\prime \prime }=e^{-r}(3/r-1)$: the inner orbit is stable at fixed angular momentum, the outer unstable. The case $l=0$ has no finite-radius circular orbit despite satisfying the upper bound.

The languages whose length-$n$ indicator functions have Boolean circuits with at most $T(n)$ gates for sufficiently large $n$, over a fixed bounded-fan-in complete basis. The family need not be constructible by one algorithm. Polynomial-size bounds give P/poly; the truth-table upper bound for circuit size applies even to undecidable languages.

A $C_1$ field is a Ci field with the bound $N>d$. The Chevalley-Warning theorem gives finite-field examples; the Tsen theorem gives function fields of curves over algebraically closed fields. A $C_1$ field has trivial Brauer group.

A $C_2$ field is a Ci field with the bound $N>d^2$. Function fields of curves over finite fields are examples. The extra-variable construction in surjectivity of reduced norms over C2 fields makes this polynomial-solvability property useful for division algebras.

A finitely generated field extension of transcendence degree $s$ of a $C_i$ field is $C_{i+s}$. In particular a function field of a curve over a C1 field is $C_2$. The theorem includes stability under finite algebraic extensions when $s=0$.