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Every optimal leader bid satisfies $b(v)\le v$, because bidding zero guarantees payoff zero and winning probability is at most one. Hence $F(b(v))\le F(v)$. With identically distributed independent continuous valuations, the probability integral transform gives $\mathbb{E}F(V)=1/2$. The leader's unconditional winning probability is therefore at most $1/2$, even when some high leader types have a conditional advantage. This argument does not depend on selecting a unique optimum.

If $J_i(S,r)$ is a player's net expected utility with remaining players $S$ and $r$ prizes, losing the current stage to player $j$ gives baseline $\delta J_i(S\setminus \{j\},r-1)$. This baseline includes later prize values and later effort costs. Subtracting it from the immediate prize value gives an effective prize in a sequential contest. Discounting must be applied to utilities, rather than to winning probabilities.

With ordered values $v_1>\cdots >v_k$ and $r<k$ prizes remaining, define $C_r=(1-\delta ){\sum }_{j=2}^r{\delta }^{j-2}{v}_j+{\delta }^{r-1}{v}_{r+1}$. In the recursively constructed discounted subgame perfect equilibrium, it is the second active player's effective prize and the effort-support upper endpoint. The highest player's utility is $v_1-C_r$. The coefficients form a convex combination, so $C_r\ge v_{r+1}$, allowing lower-player deviations to be bounded.

A sequence is Fejér monotone with respect to a nonempty set $C$ if $\Vert z^{k+1}-p\Vert \le \Vert z^k-p\Vert$ for every $p\in C$. It is bounded. If it has a cluster point $\bar{z}\in C$, its nonincreasing distance to $\bar{z}$ and that convergent subsequence force the entire sequence to converge to $\bar{z}$.

A topological space is T1 when every point is a closed point. Equivalently, for distinct $x,y$ there is an open set containing $x$ but not $y$, and an open set containing $y$ but not $x$. These open sets need not be disjoint. Every Hausdorff space is T1, but an infinite set with its cofinite topology is T1 without being Hausdorff.

An inclusion-maximal separated set is one to which no further point of the ambient metric space can be added while keeping the same separation. It need not have the largest possible cardinality. In a finite space it can be obtained by repeatedly adding an admissible point.

The substitution $z=y^{-2}$ turns the nonzero solutions into $\dot{z}=1-z$. Thus $y=\pm (1+K{e}^{-t}{)}^{-1/2}$ where the denominator is positive, together with $y=0$. The nonzero equilibrium points $\pm 1$ are attracting and $0$ is unstable. The sign of nonzero initial data is preserved.

Almost global existence means a lifespan which grows exponentially in the inverse size of small initial data, typically $T_{\epsilon }\ge \exp (c/\epsilon )-1$ for derivative-quadratic semilinear wave equations in three dimensions. Commuted wave energy and the Klainerman-Sobolev inequality lead to a logarithmic accumulation $\epsilon \log (1+T)$ in a bootstrap argument. This implies existence up to every fixed inverse power ${\epsilon }^{-N}$ when the data are small enough depending on $N$.

For a smooth semilinear wave equation whose nonlinearity is a smooth function of the solution and its first derivatives, local Sobolev space theory gives a continuation criterion: a finite maximal existence time requires divergence of a sufficiently high solution norm. A bounded base norm $H^{s+1}\times H^s$, with $s>n/2$, allows restarting local existence. Higher regularity persists by differentiated energy estimates. Precise norms can be weakened for particular nonlinearities.

The focusing power wave equation is $u_{tt}-\Delta u=u|u{|}^{p-1}$. Its conserved energy has the negative potential term $-|u{|}^{p+1}/(p+1)$ and is not positive definite. Localized ordinary differential equation blowup for a wave equation produces smooth compactly supported data with finite maximal lifespan for suitable focusing powers.

A quadratic derivative expression is a null form when its symbol vanishes on parallel null covectors. The Lorentz contraction $Q_0(f,g)=-f_t{g}_t+\nabla f\cdot \nabla g$ is the basic example. Its cancellation suppresses interactions of derivatives pointing along the same light ray. It is the algebraic structure behind the classical null condition for wave equations.

A wave map is a harmonic map with Lorentzian domain: a critical point of the action obtained by contracting the pullback target metric with the domain Lorentzian metric. For target sphere $S^2$ in Euclidean space and $\square =-{\partial }_t^2+\Delta$, its extrinsic equation is $\square \varphi =\varphi (|{\varphi }_t{|}^2-|\nabla \varphi {|}^2)$, subject to $|\varphi |=1$. The derivative contractions are null forms for wave equations. Wave map Cauchy data must include a tangent initial velocity.

RT Lac is a named interacting binary used in studies of binary star evolution. Conservative-progenitor tests and nonconservative mass transfer provide ways to interpret a stripped, mass-losing component.

A lower-boundary Eady edge wave in the semi-infinite Eady model has a smooth exponentially decaying vertical structure and a real frequency. It is supported by the horizontal boundary buoyancy gradient while its interior potential vorticity anomaly vanishes. Its speed equals the basic velocity at one penetration depth.

A growing complex phase speed cannot equal the real basic velocity. The linear interior potential-vorticity equation then forces zero anomaly everywhere, decay fixes one exponential, and the wall fixes the real speed $c=\Lambda /m$. Thus there is no growing smooth decaying edge-wave eigenmode. This excludes exponential normal-mode instability without claiming that every initial disturbance lacks transient amplification or a singular neutral component.

A constant-density gas sphere in hydrostatic equilibrium has $g={\omega }_d^{2}r$ and $p=\rho {\omega }_d^2(R^2-r^2)/2$. Its stellar oscillations can be expanded using regular solid harmonics and radial displacement amplitudes. Uniform density does not imply uniform specific entropy, so this equilibrium can be radially stable and still exhibit convective instability.

A self-financing portfolio is admissible when its wealth satisfies the prescribed lower-bound restriction, typically a deterministic bound below. Nonnegative wealth is sufficient. This condition permits supermartingale bounds on discounted gains and excludes doubling strategies.

Suppose a self-dual set family has level proportions $a_j\le j/n$ for $j<n/2$. Subtract the binomial identity $p={\sum }_j\left(\genfrac{}{}{0ex}{}{n}{j}\right)(j/n)p^j(1-p{)}^{n-j}$ from its biased measure of a set family and pair levels $j,n-j$. The difference isEvery term is nonnegative for $p\ge 1/2$. A self-dual intersecting family has the required level bounds by the Erdős-Ko-Rado theorem.

For $w=x^1+i{x}^2$, $z=x^3+i{x}^4$ and the standard oriented Euclidean metric, the real and imaginary parts of $dw\wedge dz$, together with $i(dw\wedge d\bar{w}+dz\wedge d\bar{z})$, span the self-dual two-forms. The printed complex four-form $dw\wedge dz\wedge d\bar{w}\wedge d\bar{z}$ is four times the metric volume form and gives the same orientation.

A densely defined linear operator on a complex Hilbert space is self-adjoint when its adjoint operator has exactly the same domain and agrees there. Symmetry only requires $A\subseteq A^{\ast }$ and does not guarantee self-adjointness. A self-adjoint operator is a closed operator with real spectrum. Domains matter for differential operators and unbounded multiplication operators.