In mostly-plus signature orient the commuting Dirac current to the future by $\overline{ϵ}=-{ϵ}^{†}{\gamma }^{\hat{0}}$. For the Nester two-form, put $b^i=n_{\mu }{B}^{\mu i}$. Its covariant divergence has a spinor-gradient term and the Einstein tensor contraction of spin curvature. Projecting onto the hypersurface gives the displayed identity with positive spinor norm. The Witten spinor equation removes the negative square, and the dominant energy condition makes the matter term nonnegative after the Einstein constraints are imposed.

For asymptotically constant spinors, the leading intrinsic spin connection term gives half the ADM metric integrand times $K_{\infty }^0$. The Sen spinor connection correction gives $-(k_{ij}-k{h}_{ij})K_{\infty }^j$. The constant-spinor derivative term integrates to zero by antisymmetry of the gamma matrices, and the remaining quadratic terms vanish under the standard decay. This identifies the Nester two-form flux with the ADM energy-momentum contraction, including momentum rather than just energy.

Condition on the Brownian values at a dyadic mesh. First-level increments and antisymmetric area increments of its polygonal lift are conditional expectations of the corresponding Brownian increments. Conditional Jensen inequalities and the homogeneous degree-two norm give the displayed bound. The Doob maximal inequality makes the right-hand side uniformly finite almost surely when the Brownian Hölder norm has sufficiently many moments. Dense-time martingale convergence and equicontinuity then yield simultaneous convergence at every time.

The neighbourhood system at a point consists of all its neighbourhoods. A neighbourhood basis is a subfamily cofinal under reverse inclusion.

For a nonrotating force-free sphere moving with speed $U$ in a stationary tube, use its moving frame: the tube wall has velocity $-U$. In the parabolic lubrication gap $h=h_0+x^2/(2a)$, $h_0=ϵa(1+\lambda \cos \theta )$, moving-boundary lubrication flux gives $q=-h^3{p}_x/(12\mu )-Uh/2$. The wall-shear and pressure-drop balance in a tube rules out an $O({ϵ}^{-3/2})$ net pressure jump. Applying this common pressure recovery to each angular strip gives $q=-2U{h}_0/3$. The next balance yields the displayed positive localized pressure drop. Meanwhile the integrated leading shear stress on the sphere vanishes in each strip; this is consistent with negligible leading torque, and does not mean the local shear or pressure force vanishes.

For an NFW profile, $x=r/r_s$ and $v_c^2=GM(r)/r$. The circular speed rises as $r^{1/2}$ at small radius and eventually decreases as $[\log (r/r_s)/r{]}^{1/2}$. Its maximum occurs at $x\simeq 2.16258$, with $v_{c,\max }^2\simeq 2.71706G{\rho }_s{r}_s^2$.

For a positive approximately constant vertical turbulent buoyancy flux, dimensional turnover scaling gives $K_b\sim B_0^{1/3}{z}^{4/3}$ and $N^2\sim -B_0^{2/3}{z}^{-4/3}$. The negative squared frequency describes unstable overturning, not a real restoring oscillation. The leading dissipation is height-independent, unlike neutral wall turbulence.

In a game invariant under permuting players, a symmetric Nash equilibrium gives every player the same strategy, which may be a mixed strategy. Independent randomization does not require realized resource loads to be equal. When resource delays are affine, a player's expected delay depends on their own certain resource use plus the expected use by the other players; replacing all loads by unconditional total expected loads loses that own-player correction.

There are $mq$ distinctly ranked students and $m$ labelled universities with $q$ seats each. Each student applies to one university; each university selects the greatest ranks, and admitted students receive their admitted cohort's average rank. At a pure Nash equilibrium, every student is admitted: any rejected student could move to a university with a vacancy. The university with the largest average must contain the top $q$ ranks, since any omitted higher rank could displace its lowest rank and improve. Repeat on the remaining students to obtain consecutive rank blocks. Assigning these blocks to the labelled universities gives exactly $m!$ pure equilibria.

For a finite two-person game, let $d_i$ be player $i$'s security level payoff, obtained by maximizing its minimum expected payoff against the other player's actions. Apply the Nash bargaining solution to the convexified feasible payoff region with this disagreement point. This is the maximin bargaining solution. “Maximin” specifies the disagreement guarantees; the arbitration objective function remains the Nash product, not the minimum of two numerically incomparable utilities.

Suppose a finite nonnegative measure has finite-level densities $Y_n$ with respect to a probability measure $\lambda$. The event $\{Y_n\le \epsilon \}$ belongs to that level's sigma-algebra, so its measure is $\int 1_{\{Y_n\le \epsilon \}}{Y}_{n}d\lambda \le \epsilon$. The set where nonnegative $Y_n$ tends to zero is contained in the eventual occurrence of every such level event; it therefore has measure at most $\epsilon$ for every $\epsilon >0$, hence zero. If the same limit set has full $\lambda$-measure, the two measures are singular.

An ordered execution obtained by mixing steps of several software threads while retaining each one's own step order. In the sequential consistency model of mutual exclusion, an indivisible step finishes before the next begins. Selecting the step boundaries is part of the model: an atomic read-modify-write increment has one boundary, whereas a separate read, local addition and write have three. Enumerating all such orders checks this specified model, not an arbitrary memory model or a real mutex implementation.

An object with acquire and release operations that provides exclusive ownership to a participating software thread. Several source-code regions can acquire the same mutex; two different mutex objects do not exclude each other's users. The caller should enter its critical section only after successful acquisition and release ownership on every exit path. A mutex may make a waiting software thread sleep or use a spinlock; the interface and the waiting mechanism are distinct.^[ref]

A region whose accesses must obey a shared synchronization protocol. For the lost update in mutual exclusion, the critical section includes the read and the local addition as well as the write. Choosing its boundary means identifying the complete operation whose intermediate state must be protected. Every conflicting access must participate in a compatible protocol; merely declaring a mutex does not protect accesses that bypass it.

In a stationary mushy layer with fixed $T=T_E+Gz$, pure solid and local liquidus $C=-T/m$, solute conservation is ${\partial }_t[(1-\varphi )C]=D{\partial }_z[(1-\varphi )C_z]$. Since $C_z=-G/m$ is constant, it reduces to $(z+T_E/G){\varphi }_t=D{\varphi }_z$. The method of characteristics makes the displayed $\eta$ constant and gives $\varphi =f(\eta )$ within the region that remains mush. For negative $T_E$, the characteristics move upward where the equilibrium concentration is positive. Extrapolating a zero-solid-fraction level is not sufficient to determine a boundary when additional flux constraints create a liquid cap or a solid-fraction jump.

A uniform-permeability mushy layer on an impermeable eutectic floor, under the quadratic roof pressure $p_0-E^2{x}^2/2$, has Darcy flux $\mathbf{q}=\alpha (x,-z)$, where $\alpha =\Pi E^2/\mu$. With pulling velocity $-V{\mathbf{e}}_z$, pure primary crystals, solid fraction $\varphi$, liquid concentration $C$ and constant phase properties, its one-dimensional balances areThe solidification product at the eutectic floor retains bulk composition $C_s=(1-\varphi (0))C_E$. Integration gives $C_s=C_0-(\alpha /V){\int }_0^h(C-C_0)dz$: horizontal strain removes solute before the material freezes. A negative predicted $C_s$ is inadmissible, and diagnoses failure of the assumed steady branch rather than a negative-composition product.

The Jacobian matrix collects all first partial derivatives of a differentiable map. When it is square, its Jacobian determinant gives the signed local volume factor.

A volume integral integrates a scalar or componentwise vector field over a three-dimensional region with its volume form. In spherical polar coordinates the Euclidean volume form is $r^2\sin \theta drd\theta d\varphi$. The divergence theorem relates a volume integral of a divergence to a boundary flux integral.

The wavelet projection is the orthogonal projection onto the approximation space $V_J$. Iterating $V_{j+1}=V_j\oplus W_j$ expresses it as the coarse part of the wavelet series plus details at levels below $J$. Orthogonality gives $\Vert m-P_{J}m{\Vert }_2^2={\sum }_{j\ge J,k}|b_{j,k}{|}^2$, which tends to zero by the Parseval identity for a Hilbertian basis. The projection can also be represented by $A_J(x,y)={\sum }_k{\varphi }_{J,k}(x){\varphi }_{J,k}(y)$ as an integral kernel in the Hilbert-space sense.