In a canonical Wess–Zumino model with cubic superpotential $g{\Phi }^3/3$, auxiliary elimination gives a complex-scalar quartic interaction with two legs of each field type. With all momenta incoming and the displayed coefficient, its Feynman rule is $-i(2!2!)|g{|}^2$. For $\phi =(A+iB)/\sqrt{2}$, the same quartic scalar potential is $|g{|}^2(A^2+B^2{)}^2/4$, with real-field rules $-6i|g{|}^2$ for four identical legs and $-2i|g{|}^2$ for two of each. Factorials depend on how the coefficient is defined, not on a new physical coupling.

Choose a fixed sample size $m$ for which the normalized Turan theorem bound lies below the given density by a positive margin. The expected edge count in a random $m$-vertex subset then forces a positive proportion of subsets to contain a forbidden-size clique. Each clique belongs to a known number of subsets, so double counting gives a positive constant times $n^{r+1}$ cliques.

A local supermartingale becomes a supermartingale after stopping along an increasing sequence of stopping times tending to infinity. Localization allows an Itô formula comparison before global integrability is established. Additional lower bounds or uniform integrability are needed to pass from the local comparison to an unrestricted expectation inequality.

The subobject classifier $\Omega$ of a topos carries internal truth, falsity, meet, join and implication. Its generalized elements over $X$ are subobjects of $X$; implication satisfies $R\le (P\Rightarrow Q)$ exactly when $R\cap P\le Q$. Pullback-compatible operations make this an internal Heyting algebra, not merely a structure on global truth values.

The union of two subgroups is a subgroup exactly when one contains the other. Otherwise choose $h\in H\setminus K$ and $k\in K\setminus H$. Their product cannot lie in $H$, since $h^{-1}(hk)=k$ would then lie in $H$, and cannot lie in $K$, since $(hk)k^{-1}=h$ would then lie in $K$. The union therefore fails closure. Unlike the union, the subgroup generated by both subgroups always exists.

A Hall subgroup has order coprime to its index. For a set of primes $\pi$, a Hall $\pi$-subgroup has order using only primes of $\pi$ and index using only primes outside $\pi$. Thus its order contains the full prime-power contributions belonging to $\pi$. A Sylow subgroup is the single-prime instance. Hall subgroup existence in soluble groups generalizes Sylow existence to every prime set.

Here $r+1$ is the least chromatic number of an excluded graph, for a proper unbounded subgraph-closed property. All subgraphs of a balanced Turán graph give the lower bound. For the upper bound, a regularity reduced graph is clique-free, so its dense pairs have the Turan theorem edge bound; sparse pairs contribute only small binary entropy and the bounded partition description costs only a linear number of bits.

A shallow current receiving meltwater over its wetted area obeys $h_t+q_x=s_m$, with $q=-D{h}^2{h}_x$. The source is zero ahead of the advancing front. The Stefan condition determines normal ice-retreat speed $v_m$; the water-volume source is $s_m=({\rho }_i/{\rho }_w)v_m$, or simply $v_m$ in the equal-density approximation. Water volume and geometric excavation must be distinguished if densities differ.

For fixed $0<\lambda <1$ in $G(n,\lambda /n)$, the largest component of a graph has the displayed order with high probability. A dominating Galton-Watson process gives the tail bound $\mathbb{P}(T\ge j)\le e^{-(\lambda -1-\log \lambda )(j-1)}$ by the exponential Markov inequality. For the lower bound, the tree-component expectation in the Erdős-Rényi model diverges at $j=\lfloor (1-\eta )\log n/(\lambda -1-\log \lambda )\rfloor$. The disjoint-set covariance factor and the exclusion of overlaps give relative variance tending to zero, so the second moment method supplies a tree component of that order.

Separate first-event episodes from all later episodes with a stratum indicator. A Stratified Cox model permits one baseline hazard for first events and another for recurrent events while retaining a common treatment coefficient. Later episodes can share their baseline; stratifying each recurrence number separately is a stronger model than the first-versus-later distinction alone.

The stopped stochastic process follows $X$ until the stopping time $\tau$ and thereafter keeps the value $X_{\tau }$ when $\tau$ is finite. For a pathwise right-continuous adapted process, the stopped process is adapted, by progressive measurability and measurable evaluation at $t\wedge \tau$. When $X$ is a martingale, further stopping and integrability conditions determine whether its stopped process is a martingale.

For independent trials of success probability $p\in (0,1]$, let $T$ be the first index completing two consecutive successes. If $e_0,e_1$ are the mean remaining times with no trailing success and with one trailing success, conditioning gives $e_0=1+(1-p)e_0+p{e}_1$ and $e_1=1+(1-p)e_0$. Solving yields $e_0=(1+p)/p^2$, so fair trials have mean waiting time six. Disjoint pairs give a geometric tail bound and justify finiteness before solving the equations. Overlapping pairs are not independent trials.

A fluid displacement measured from an initial position must vanish at the initial time. Integrating a harmonic velocity therefore introduces phase-dependent constants. Mean-orbit labels absorb those constants and leave purely oscillatory displacements. The instantaneous displacement correction to velocity can differ under the two reference conventions, but its period mean gives the same leading Stokes drift.

For a step-response kernel with $K(0)=0$, differentiating the start time produces no delta term from the response itself, because its coefficient is zero. The negative shift derivative is the impulse response. More generally an additional $\delta (t-b)K(0)$ term appears if the kernel has a nonzero initial value. The identity holds distributionally as well as for the ordinary smooth pieces.

During a spherical stellar homology contraction at fixed mass and angular momentum, a constant inertia coefficient gives $I=\alpha M{R}^2$ and solid-body rotation gives $\Omega =J/I\propto R^{-2}$. A fixed fluid label $s=r/R$ retains its enclosed mass and polar angle, so centrifugal acceleration scales as $R^{-3}$ and gravity as $R^{-2}$. Their ratio increases as $R^{-1}$. This assumes spin conservation, not continuous synchronization to an external orbit.

The statistical tangent space is the closed linear span in L2 space of a statistical tangent set. Taking this closure makes orthogonal projection available and ensures that a continuous derivative specified on attainable score functions extends to the whole space.

For bounded measurable $g$ with $Pg=0$, the formula $p_t=p(1+tg)$ defines a normalized positive probability density function for $|t|\Vert g{\Vert }_{\infty }<1$. A uniform Taylor expansion of the square root proves differentiability in quadratic mean with score function $g$: the squared remainder is $O(t^{4}P{g}^4)$. Such paths realize all bounded centered directions of an unrestricted density model.

A measurable summary of a possible dataset is chosen to detect disagreement with a specified predictive model. Compare its observed value with its distribution under replicated datasets. Large-discrepancy statistics can be checked with $\mathbb{P}(T(X^{\mathrm{rep}})\ge T(x))$. A posterior predictive check integrates over posterior parameter uncertainty; a fully specified null instead supplies a fixed reference law.

Generate independent replicated datasets under a fully specified null hypothesis and calculate the same predictive discrepancy statistic for them and the observation. Their null exchangeability gives a finite-simulation rank test. The displayed add-one tail probability is valid with conservative treatment of ties, and avoids reporting zero solely because no replicate exceeds the observation.

For independent effect estimates $1.01$ with standard error $0.50$, and $0.99$ with standard error $0.51$, one two-sided Wald test is significant at five percent and the other is not. But their difference is $0.02$ with standard error $\sqrt{0.50^2+0.51^2}\approx 0.714$, giving a negligible test statistic. Hence different significance labels do not establish an interaction or between-study heterogeneity; compare the effects directly.^[ref]