For a Serre fibration, this first-quadrant spectral sequence has differential of bidegree $(r,1-r)$ and a fiber-cohomology local coefficient system on the base. A simply connected base makes that system constant. Its edge map is restriction to the fiber; a surviving fiber class is the restriction of a total-space class. For commutative ring coefficients it is multiplicative and its differentials are graded derivations.

For a series ${\sum }_n{a}_n$, let $r={\mathrm{limsup}}_{n\to \infty }|a_n{|}^{1/n}$. If $r<1$, choose $r<\rho <1$; eventually $|a_n|\le {\rho }^n$, so geometric comparison gives absolute convergence. If $r>1$, the terms fail to tend to zero along a subsequence and the series diverges. At $r=1$, the test is inconclusive.

For a leader with nonnegative value $v$ and a continuous nonnegative follower-value cumulative distribution function $F$, a bid $b\ge 0$ gives expected payoff $vF(b)-b$. Bidding zero guarantees nonnegative payoff, whereas a bid $b>v$ gives $vF(b)-b\le v-b<0$. Every best response therefore lies in $[0,v]$, and the optimization isA fixed upper bound $1$ is justified only under an additional normalization of the follower-value support; it is not a general restriction on effort.

If $F$ is a concave function, this objective function is concave. At a differentiable interior optimum it satisfies $v{F}^{\prime }(b)=1$; one-sided derivatives handle the endpoints. Flat intervals can give several best responses, as described in best-response selection at a flat leader objective. Every optimal bid also obeys the sharper inequality $b\le vF(b)\le v$, by comparison with zero effort.

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.

An admissible trading strategy is a permitted stochastically integrable trading rule satisfying the model's wealth restriction. A common finite-horizon convention, relative to a fixed positive numéraire $B$, requires a deterministic $a\ge 0$ such thatThe bound may depend on the strategy. Nonnegative wealth is sufficient. The choice of numéraire and whether the restriction is imposed on nominal or discounted wealth must be stated; the corresponding conditions need not agree for an arbitrary random account.

Under an equivalent local martingale measure, discounted self-financing gains are local martingales. The deterministic lower bound makes the discounted wealth a supermartingale, supporting the usual no-arbitrage and minimal-replication-cost arguments. Restrictions only on terminal wealth do not exclude all doubling strategies.