The disagreement point specifies each player's payoff if bargaining fails. It supplies the baseline from which the Nash product measures gains. In a game-based example it may be set to the vector of security level payoffs; this choice must be recomputed when the underlying payoff matrices change.

For an essential Nash bargaining problem, maximizing the Nash product gives a unique vector: on positive gains its logarithm is a strictly concave function. The rule satisfies the four bargaining axioms, and the supporting triangle for Nash bargaining proves their converse characterization. The nonessential case requires additional conventions, since a zero product may have many maximizers.

When the feasible payoff set and disagreement point are invariant under interchange of players, the solution must give both players the same payoff.

Transforming each utility by an independent positive affine map must transform the chosen payoff by the same map. The Nash product changes only by a positive factor, so the Nash bargaining solution has this invariance.

If an admissible smaller feasible set with the same disagreement point contains the original chosen vector, deleting the other outcomes must leave the choice unchanged. The Nash bargaining solution satisfies this axiom because its unique product maximizer remains feasible.

A payoff is individually rational for a Nash bargaining problem when neither player receives less than its disagreement point payoff. Essentiality permits strict improvement for both. Maximizing a positive Nash product therefore selects a strictly individually rational payoff.

Over a field of characteristic zero, an alternating polynomial changes sign when two variables are interchanged. It vanishes when two coordinates agree, so each factor $x_i-x_j$ divides it. These distinct linear factors are relatively prime in the polynomial ring, hence their product, the Vandermonde determinant, divides it. The quotient is a symmetric polynomial.

An algebra representation is a unital algebra homomorphism into the endomorphisms of a vector space. It is equivalent to making that vector space a left module over the algebra, with $av=\rho (a)v$.

For a prime $p$ and $k\ge 1$, write $a=b+p^{k}h$. The binomial theorem shows $a^p-b^p$ is divisible by $p^{k+1}$: the linear term contains the extra factor $p$ and every higher term contains at least $p^{2k}$. Iteration gives the displayed modular congruence for every $r\ge 0$, including $p=2$. Applied to Fermat's little theorem, it yields $a^{p^n}\equiv a^{p^{n-1}}(\mathrm{mod}{p}^n)$.

The support of a finite mixed strategy consists of the pure strategies assigned positive probability. At a Nash equilibrium, every action in the support is a best response. In a game satisfying nondegeneracy of a bimatrix game, the two equilibrium supports have equal size.

Suppose outcome collection stops at the first success or after $m$ unsuccessful attempts, and let $A$ count unsuccessful attempts. Conditional on the outcome $y$, put $q_j(y)=\mathbb{P}(A\ge j\mid A\ge j-1,Y=y)$. For $k<m$, the joint density of $A=k$ and an observed $Y=y$ isFor $A=m$, the unobserved outcome is integrated out, giving $\int f_{\beta }(y){\prod }_{j=1}^m{q}_j(y)dy$. These probabilities sum to one by telescoping and integration. Dependence of $q_j$ on $y$ models missing not at random. If all $q_j$ are independent of $y$ conditional on the regression predictors, the outcome density among responders remains $f_{\beta }$, as required for complete-case analysis of that conditional model.

For positive capacities and a feasible flow, optimality is equivalent to network dual potentials for which the network reduced cost is nonnegative at a lower bound, zero in the interior, and nonpositive at an upper bound. These signs make every feasible displacement nonnegative in total cost. They are also equivalent to absence of negative-cost cycles in the residual network. A zero-capacity edge is fixed and imposes no reduced-cost sign condition.

Compute a minimum spanning tree, then a minimum-weight perfect matching on its odd-degree vertices. Their multiset union is connected with even degrees, so it has an Euler circuit. Shortcut repeated vertices to obtain a metric tour. The tree costs at most the optimal tour, and shortcutting the optimal tour to its odd-degree subset gives a cyclic order whose alternating matchings show that the added matching costs at most half the optimum. The result is a polynomial-time $3/2$-approximation algorithm for symmetric metric instances.

The supremum of pointwise distances defines a metric on maps when it is finite, for example if the target metric space is bounded. Convergence in this metric is uniform convergence. If the target is complete, Cauchy sequences have pointwise limits and the Cauchy estimate makes convergence uniform.

Publication bias arises when the availability of studies depends on the results, such as their significance, magnitude or direction. A meta-analysis of the available studies can then differ systematically from one including all eligible evidence. Funnel plot asymmetry can motivate investigation but does not establish this mechanism; heterogeneity and other small-study effects can also cause asymmetry.

A funnel plot displays study effect estimates against a measure of their precision, often standard error increasing down the vertical axis. Under a common-effect model, imprecise estimates spread more widely around the effect. Asymmetry is evidence of a small-study effect and can be consistent with publication bias, but it has other possible explanations.

A forest plot shows each study's effect estimate and confidence interval, together with a pooled estimate when appropriate. A vertical null-effect line allows comparison on a common scale. For an odds ratio the null is one and a logarithmic axis treats reciprocal ratios symmetrically; marker areas commonly encode study weights.

Divide studies by a specified characteristic and estimate an effect in each subgroup, then test the difference using an independent subgroup contrast in meta-analysis or an appropriate interaction model. The test of a between-group difference is distinct from separate tests against no effect within each group. Exploratory splits can be affected by multiple testing and by confounding between study characteristics.

At a smooth point with nonzero gradient, rotate coordinates so $\nabla u=G{e}_2$. Write $u=u_0+Gy+\frac{1}{2}(A{x}^2+2Bxy+C{y}^2)+O(|(x,y){|}^3)$. A candidate median $u_0+d{h}^2$ has level boundary $y=(d{h}^2-A{x}^2/2)/G+O(h^3)$ in the disk. The area imbalance is $(2d-A/3)h^3/G+o(h^3)$; equal half-areas force $d=A/6$. Here $A=\Delta u-\nabla u^T(D^{2}u)\nabla u/|\nabla u{|}^2$ is the tangential second derivative, giving the formula. The disk uses uniform area measure, not its boundary measure.

A graph state with edges $13,24,34,45$ simulates $J_2(\gamma )E_{12}(J_1(\alpha )\otimes J_2(\beta ))$ on $|++⟩$. Measure vertices 1 and 2 at angles $\alpha ,\beta$, with results $s_1,s_2$, and vertex 4 at $(-1{)}^{s_2}\gamma$, with result $s_4$. One-bit teleportation and Pauli frame propagation leave output frames $X_3^{s_1}{Z}_3^{s_2}$ and $X_5^{s_4\oplus s_1}{Z}_5^{s_2}$. Computational outputs are therefore corrected by $s_1$ and $s_4\oplus s_1$ respectively.