For the left-parameter convention and ghost kinetic term $(\partial b)Dc$, localization of the BRST symmetry yields this conserved Noether current. Subtract the total-derivative contribution $K^{\mu }=h^a(D^{\mu }c{)}^a$ in the Lagrangian variation. The BRST charge is the spatial integral of $j^0$. The antighost term has the displayed sign because the parameter and antighost field derivative anticommute.

The area density of ${\mathbb{P}}_z(B_t\in dw,t<T_D)$ for planar Brownian motion killed on its first exit from $D$. Its total mass is the survival probability, not generally one. Integrating this density over time gives the Green function of killed planar Brownian motion.

Stopping Brownian motion at its first crossing below $-b$, the nonnegative local martingale $(W_{t\wedge \tau }+b)/b$ starts at one and ends at zero. The maximal identity for a continuous nonnegative local martingale tending to zero gives tail $b/(b+x)$ for the stopped maximum, hence density $b/(b+x{)}^2$ for $x>0$. There is no atom at zero.

For a generalized singular vector, exact data $f=\gamma K{u}_{\lambda }$, and fixed $\alpha >0$, a compatible branch of Bregman iteration has $p^k=\min \{k\gamma /\alpha ,\lambda \}{K}^{\ast }K{u}_{\lambda }$ and the displayed $u^k$. Verify these formulas with the subgradient characterization of an absolutely one-homogeneous functional; before activation the dual coefficient remains at most $\lambda$, and after activation it is $\lambda$. The branch reaches $\gamma u_{\lambda }$ at $k_{\ast }=\lceil \alpha \lambda /\gamma \rceil +1$. Every minimizer branch has the same predicted data, since the quadratic data fidelity is strictly convex in $Ku$. Exact recovery of this particular vector for arbitrary branch choices needs uniqueness, for example injectivity of $K$. For $K(x,y)=x$ and $J(x,y)=|x|$, the second coordinate of every minimizer is arbitrary.

In log abilities, the negative Hessian matrix is a weighted graph Laplacian of the comparison graph. If that comparison graph is a connected graph and the abilities are finite, the quadratic form vanishes only for constant vectors. Thus the log-likelihood is strictly concave after fixing the common additive constant. If the directed graph of observed wins is a strongly connected directed graph, letting contrasts diverge forces at least one observed-win probability to zero, so the log-likelihood tends to negative infinity. A finite maximizer exists and is unique up to common scaling of abilities.

For one win of player 1 over 2, one win of 3 over 1, and $k\ge 1$ wins of 2 over 3, the Bradley-Terry score equations force abilities proportional to $(r,1,r^2)$, where $r>0$ satisfies the displayed equation. Its left side is strictly increasing on $r>0$. For $k=1$, $r=1$ and all three estimates tie. For $k>1$, $r<1$, giving the ranking $2>1>3$.

For a connected tree of comparisons with every observed win fraction strictly between zero and one, the Bradley-Terry model can fit every empirical edge probability exactly. Fix one positive strength to remove scaling ambiguity, then propagate strength ratios $p/(1-p)$ along tree edges. Edge log-ratios are independent real coordinates and each binomial log-likelihood term is strictly concave, proving existence and uniqueness of the normalized estimate. Cycles would impose extra compatibility relations.

Bose-Einstein statistics describes identical noninteracting bosons in equilibrium. Any occupation number is allowed in a one-particle state; the resulting Bose-Einstein distribution is $n(E)=[e^{\beta (E-\mu )}-1{]}^{-1}$.

3-SAT is the Boolean satisfiability problem restricted to Boolean formulas in conjunctive normal form with at most three Boolean literals per Boolean clause. It is NP-complete. A nonempty short Boolean clause can be padded to three positions by repeating a Boolean literal, without changing satisfiability.

A Boolean formula is a finite expression built from Boolean variables and Boolean operations, equivalently a propositional formula with two-valued semantics. A Boolean valuation determines its truth value. In $(x\vee y)\wedge \neg x$, logical negation acts only on $x$, and the outer logical conjunction combines $x\vee y$ with $\neg x$. Occurrences of the same Boolean variable share one truth value. The Boolean satisfiability problem asks whether some Boolean valuation makes the Boolean formula true.

Maximum 2-satisfiability asks for an assignment satisfying as many clause occurrences as possible when every nonempty clause has at most two literals. Occurrences are counted even when a clause is repeated. Its decision problem is NP-complete by the seven-clause gadget for MAX-2SAT, while the method of conditional probabilities gives a polynomial-time half approximation.

Away from zeros of the Higgs field, write $\varphi =e^{u+i\chi }$ with $u=\log |\varphi |$. The Bogomolny vortex equation $(D_1+i{D}_2)\varphi =0$ gives $a_1={\chi }_1+u_2$ and $a_2={\chi }_2-u_1$. Consequently $\mathbf{a}=\nabla \chi -J\nabla u$, where $J(v_1,v_2)=(-v_2,v_1)$, and $B=-\Delta u$ on a regular phase patch. If the contours of $u$ and $\chi$ are orthogonal, then $\nabla u\cdot \nabla \chi =0$, so $\mathbf{a}\cdot \nabla u=0$: the gauge potential is tangent to constant-amplitude contours. This does not imply that it is curl-free. At a vortex zero, $u\to -\infty$ and the phase is only locally defined. Including phase winding distributionally gives $\Delta u+(1-e^{2u})/2=2\pi {\sum }_j{n}_j{\delta }^{(2)}(x-X_j)$. The usual Taubes equation uses $2u=\log |\varphi {|}^2$, so its delta coefficients double.

The false positive rate is $\mathbb{P}(\text{positive result}\mid \text{truly negative class})$. In screening, it measures erroneous positives among genuinely disease-free individuals.

For a bidisperse kinematic sedimentation model with normalized suspension concentrations $q_i$, isolated speeds $W_i$, and a stationary deposit with $p_1+p_2=1$, the upward deposit-front speed magnitude obeys $s(p_i-q_i)=W_i{q}_i(1-q_i)$. Adding gives $s={\sum }_i{W}_i{q}_i(1-q_i)/(1-{\sum }_i{q}_i)$; substitution determines each deposited fraction. These are Rankine-Hugoniot conditions with a stationary compacted branch.

Consider a column $0<z<H$ initially filled with a uniform bidisperse particle suspension. In an independent-settling idealization, normalized concentrations $q_i$ obey fluxes $j_i=-W_i{q}_i(1-q_i)$, with $W_i>0$. A stationary packed deposit has fractions $p_1+p_2=1$. The deposit composition from sedimentation jump conditions determines its upward front speed $s$ and the fractions $p_i$.

Suppose species 1 has the faster downward clearing front: $v_1=W_1(1-q_1)>v_2=W_2(1-q_2)$. Before interaction, the fast clearing front is at $H-v_{1}t$, while the deposit front is at $st$. They first meet atAll species 1 has then entered the earlier mixed deposit. Above it, only species 2 remains suspended; its later sedimentation shock adds a pure-species-2 layer. Separate particle-volume conservation laws giveHere $h_2$ is the final pure-species-2 layer thickness. This two-stage picture follows from the stated independent fluxes and packed stationary branch; general coupled hindered settling can have a different wave structure.

For the asymptotically free two-loop running coupling $da/d\log \mu =-{\beta }_0{a}^2-{\beta }_1{a}^3$, with ${\beta }_0>0$, let $y=1/a$, $c={\beta }_1/{\beta }_0$, and $L=\log (\mu /\Lambda )$. A choice of the scale $\Lambda$ gives the exact implicit equation $y-c\log ((y+c)/{\beta }_0)={\beta }_{0}L$ on the ultraviolet branch. Its expansion isThe remainder after the first two terms is therefore $O(\log L/L)$, and is not $O(1/L)$ when ${\beta }_1\ne 0$. A fixed rescaling of $\Lambda$ cannot change the coefficient of $\log L/L$. This distinction matters when quoting errors for asymptotic expansions.

Suppose independent centered sub-exponential random variables satisfy $\log \mathbb{E}{e}^{\lambda X_i}\le v_i{\lambda }^2/(2(1-b|\lambda |))$ for $|\lambda |<1/b$, and put $v={\sum }_i{v}_i$. independence adds the log moment-generating functions. The Chernoff bound with $\lambda =t/(v+bt)$ gives the displayed two-sided tail. If each variable has common scale $K$, one may take $v_i\le C{K}^2$ and $b\le CK$, giving $\mathbb{P}(|n^{-1}{\sum }_i{X}_i|>t)\le 2e^{-cn\min \{t^2/K^2,t/K\}}$. This form of Bernstein's inequality allows unbounded variables with factorial moment control, including the centered square of a sub-Gaussian random variable.

For a nonzero finitely generated module over the complex Weyl algebra $A_n$, its Bernstein growth dimension of a Weyl algebra module is at least $n$. The faithful finite-step action of a Weyl algebra gives $\dim F_j{A}_n\le (\dim M_j)(\dim M_{2j})$. Comparing degrees $2n$ and $2d(M)$ proves the inequality. The polynomial module has dimension $n$, so the bound is sharp. This is an algebraic growth inequality, distinct from probabilistic Bernstein inequalities.

If transverse kinetic energy is dissipated at the downstream axis, the remaining specific orbital energy is $v^2/2-GM/b$. Binding requires $b<2GM/v^2$, corresponding to incident impact parameter $d<2GM/v^2$. Integrating the captured mass flux gives the Bondi--Hoyle--Lyttleton accretion rate.

A minimal bad sequence chooses each successive object of least possible natural-number size among choices that admit an infinite bad continuation of the already fixed prefix. Replacing its next object by a strictly smaller one cannot leave a bad sequence. This contradiction principle proves Higman lemma and the labelled version of Kruskal's tree theorem.