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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.

An orthonormal multiresolution analysis separates coarse scaling functions from progressively finer wavelet details. On a bounded interval, the approximation space at level $J$ has dimension of order $2^J$. Higher resolution reduces projection bias but estimates more noisy coefficients. In an orthonormal Gaussian sequence model with coefficient noise variance ${\sigma }^2/n$, risk is the omitted squared coefficient sum plus ${\sigma }^2{d}_J/n$. Selecting or shrinking individual detail coefficients yields spatially adaptive complexity; a data-selected subset need not have effective degrees of freedom equal to its raw selected count.

A group action on a set of at least three points is three-transitive when it is transitive on ordered triples of distinct points. It implies a two-transitive group action, since a distinct pair can be completed to a distinct triple. The implication is strict: the projective general linear group in projective dimension at least two is two-transitive on projective points, but collinear and noncollinear triples belong to different group orbits.

The groups $U_m$ form a compact-open neighbourhood basis at one inside the units of a non-Archimedean local field. A smooth character trivial on $U_{i+1}$ but nontrivial on $U_i$ cannot have a nonzero fixed vector under $U_m$ for $m\le i$. This elementary conductor comparison controls admissibility and smooth duals of direct sums.

A bouquet graph consists of one vertex and $j$ self-loops. Its vertex has degree $2j$, and its graph excess is $j-1$. The two-loop bouquet is one of the three possible kernels of a bicyclic graph; in a simple subdivision, each loop must expand to a cycle of length at least three.

The endpoint expansion of this multiderivative multistep method has unscaled defect $h^4{y}^{(4)}/3+O(h^5)$ and zero-stability roots $1,1/7$, giving third-order convergence with suitably accurate starting values. On the Dahlquist test equation its amplification roots obey $(7-6z+2z^2){\zeta }^2-8\zeta +1=0$. The complex quadratic Schur criterion puts both roots in the open unit disk for $\mathrm{Re}z<0$, and only the simple root one reaches the boundary at $z=0$. Thus it is A-stable; its higher derivative places it outside the hypotheses of the Second Dahlquist barrier.

For the multiderivative multistep method $y_{n+1}-\alpha h{f}_{n+1}+\beta h^2{g}_{n+1}=\gamma h{f}_n+y_{n-1}+\alpha h{f}_{n-1}+\beta h^2{g}_{n-1}$, with $g=f^{\prime }f$, centered expansion gives defect coefficients $2-2\alpha -\gamma$, $1/3-\alpha +2\beta$, and $1/60-\alpha /12+\beta /3$ at powers $h,h^3,h^5$. Their simultaneous vanishing gives the displayed parameters. The next coefficient is $1/4725$ at $h^7$, so the formal order is six. Reflection symmetry removes every even defect power. The method satisfies fifth-order conditions but cannot have exact order five.

For $y^{\prime }=f(y)$ and $g=f^{\prime }f$, inserting a smooth exact solution into the displayed recurrence gives defect $h^7{y}^{(7)}(t_n)/4725+O(h^9)$. This follows by expanding $2\sinh z-(14/15)z\cosh z-(16/15)z+(2/15)z^2\sinh z$. The terms through $z^6$ vanish and the $z^7$ coefficient is nonzero. Its zero-step polynomial is ${\zeta }^2-1$, whose roots are simple and of unit modulus. Thus it is zero-stable and has order six, provided starting values and derivative evaluations have corresponding accuracy.

A two-class queue has one total buffer but serves the high-priority class before the other. A preemptive-resume service discipline interrupts low-priority work immediately and later resumes it without wasting completed work. Under first come first served within the high-priority class, a new high-priority packet waits $V/C$ when $V$ is the high-priority workload immediately before its arrival. The total workload determines shared-buffer overflow, while $V$ determines the high-priority waiting time. An additional admission gate rejects high-priority packets when $V/C$ exceeds a prescribed delay limit. Finite packets require a fit test against the remaining buffer space, and nonpreemptive service adds residual low-priority service time.

The actual Sv39 page table geometry uses three nine-bit indices and a twelve-bit offset for base 4096-byte memory pages. Each table contains 512 eight-byte page table entries, exactly one 4096-byte memory page; the satp processor register identifies the root. A 64-bit virtual address must repeat bit 38 in bits 63 through 39. Higher-level terminal entries can also map aligned larger memory pages, shortening the page table walk. These rules differ from the 16-bit, four-byte-entry two-level address translation model. ^[ref]

For ordinary Sv39 page tables, valid nonterminal page table entries reference the next table with read, write, and execute flags clear; those clear flags do not prohibit reading the eventual data. The terminal entry supplies access permissions, with additional processor privilege level and status checks. An invalid entry, malformed encoding, or denied access can cause a page fault. Thus the model's set intersection of read/write permissions at both levels must not be applied mechanically to Sv39 page tables. ^[ref]

In xv6 RISC-V source code, root allocation is separate from lower-level allocation: uvmcreate() allocates and clears a root page table, while walk() with allocation enabled allocates and clears missing lower-level tables and installs valid references to them. With allocation disabled, a missing table makes walk() return failure. This software routine finds a page table entry for the operating system; it is not the hardware's complete permission-checking page table walk. ^[ref]