For a stable nonrelativistic thermal relic, compare energy densities rather than number densities. If $m_Y=m_{\mathrm{nr}}/x^3$ and $Y_Y=C{x}^{3/2}{e}^{-x}$, its density relative to baryons is $(C{m}_{\mathrm{nr}}/m_B{Y}_B)x^{-3/2}{e}^{-x}$. Requiring this ratio below an allowed dark-matter-to-baryon ratio $R$ means $x+\frac{3}{2}\log x\ge \log [C{m}_{\mathrm{nr}}/(R{m}_B{Y}_B)]$. With fixed rate parameters this excludes the weakly interacting large-mass branch, while the sufficiently Boltzmann-suppressed smaller-mass branch can satisfy the density bound. Decay or a nonthermal initial abundance requires a different constraint.

In an untied Cox proportional-hazards model with a single subject remaining after all other observed exit times, that subject's event contributes the constant factor one to Cox partial likelihood. Replacing its terminal censoring time by a later event therefore preserves the fitted coefficients and every earlier Breslow estimator increment. At its new event, the subject's fitted cumulative hazard increases by one, cancelling the increase of one in its event indicator. Every martingale residual is unchanged. This property relies on the subject being alone in the terminal risk set and the same covariate history being retained.

These are convergence results for martingales and submartingales under appropriate one-sided or absolute-integrability bounds. They distinguish almost-sure convergence from convergence in mean: uniform integrability gives the latter. The martingale convergence theorem stated for an L1-bounded martingale is one standard member of the family.

For a nonnegative submartingale $X$, the maximal tail bound is $\lambda \mathbb{P}({\max }_{k\le n}{X}_k\ge \lambda )\le \mathbb{E}[X_n;{\max }_{k\le n}{X}_k\ge \lambda ]$. Applied to the absolute value of a martingale and integrated over $\lambda$, it gives the Doob Lp maximal inequality; at $p=2$ this is the Doob L2 maximal inequality.

Every integrable discrete-time supermartingale has the form $V=M-A$, where $M$ is a martingale and $A$ is predictable, integrable, nondecreasing, and starts at zero.

More generally, an integrable adapted process $X_n$ decomposes as $X_n=M_n+A_n$ with $A_0=0$ and $A_n-A_{n-1}=\mathbb{E}[X_n-X_{n-1}\mid {\mathcal{F}}_{n-1}]$. For a supermartingale this drift is nonincreasing, so writing it as $-A$ yields the nondecreasing convention above.

If a martingale has increments with $|M_i-M_{i-1}|\le c_i$ almost surely for deterministic $c_i$, then each one-sided deviation of magnitude $t>0$ from $M_0$ has probability at most $\exp (-t^2/(2{\sum }_i{c}_i^2))$. If all $c_i$ vanish, the martingale is constant. The inequality gives concentration from bounded increments without requiring independent increments.

In a uniformly shuffled deck with $r$ red and $b$ black cards remaining, the conditional probability that the next two are red is $q(r,b)$. If at least three cards remain, $q(r,b)$ is the weighted average of $q(r-1,b)$ and $q(r,b-1)$ after one reveal. Hence these probabilities form a bounded martingale until only two cards remain. Any adapted choice of exactly one betting time before that deadline has the same winning probability by bounded optional stopping. Waiting creates no advantage when a bet must eventually be made.

In a finite population that repeatedly lets one uniformly chosen individual copy a uniformly chosen distinct individual, the fraction of either type is a bounded martingale. Conditional probabilities of an increase and decrease coincide. Absorption at one of the two unanimous states occurs almost surely, since from every state there is a uniformly positive probability of reaching unanimity in a fixed finite block of updates. Bounded convergence implies that the probability of eventual unanimity of one type equals its initial fraction.

For a nonnegative martingale, the displayed maximal probability bound follows by stopping at the first crossing of $a$. The stopped expectation is at most its initial expectation, and on the crossing event its value is at least $a$. Bound the stopping time first and pass to the limit using nonnegativity. This form is useful for the Poisson maximal concentration bound.

For a compensated jump martingale starting at zero, with jumps bounded in absolute value by $c$ and predictable quadratic variation at most $v$ on the interval, the displayed maximal bound holds. For $0<\theta c<3$, the elementary exponential boundmakes the exponential of $\theta M$ minus that multiple of its predictable bracket a nonnegative supermartingale. Stop at the first crossing of $\eta$ and use its mean bound. Taking $\theta =\eta /(v+c\eta /3)$ yields the one-sided bound; apply the same argument to $-M$ and use the union bound for the displayed two-sided version. The elementary estimate follows from the exponential series and $k!\ge 23^{k-2}$ for $k\ge 2$. If $v=0$, the compensated martingale is constant and the crossing probability is zero; the parameter choice above is needed only for $v>0$.

Let $\xi$ be a random walk, independent of centered, independent time-space signs $h$, and let $0<\epsilon <1$. Averaging over the walk gives a positive martingale in the environment filtration: each new time layer has conditional mean-one weight. Hence $\mathbb{E}{Z}_t=1$ and the martingale convergence theorem gives a finite nonnegative almost-sure limit. Preserving its mean requires an additional condition such as bounded second moments; positivity of finite-time weights alone does not suffice.

A probability measure is invariant for a Markov process with transition semigroup $P_t$ when starting with that law gives the same law at every later time. Equivalently, $\int P_{t}fd\nu =\int fd\nu$ for every bounded measurable function. Invariant laws may be constructed by weak convergence of probability measures of evolved laws when the Feller semigroup property permits passing the semigroup through the limit, as for the extremal invariant measures of the contact process.

A positive linear operator on bounded measurable functions that preserves the constant function one. A Markov semigroup is a time-indexed family of such operators with the semigroup law.

Each emitted symbol identifies the state of a stationary finite Markov chain. Here the alphabet is the state space itself. Its transition matrix $P$ and stationary distribution $\pi$ determine the entropy rate of a fully observed Markov source.

For a clean spherical gas bubble with negligible internal dynamic viscosity and thermal conductivity, a weak temperature gradient produces the displayed migration velocity, where ${\gamma }_T=d\gamma /dT$. The exterior Papkovich–Neuber representation can use only the scalar potential $\chi =-a^3\mathbf{U}\cdot \mathbf{r}/(2r^3)$. Its surface normal velocity equals $\mathbf{U}\cdot \mathbf{n}$, and its tangential traction balances minus the surface surface tension gradient from the temperature dipole around an insulating sphere. For the usual ${\gamma }_T<0$ the bubble moves toward warmer liquid. Small Reynolds number, Péclet number, capillary number, and fractional surface-tension variation justify the approximation; surfactants can oppose the temperature-driven Marangoni stress.

The mapping torus of a torus map with this monodromy has fundamental group generated by $x,y,t$ with $x$ central and $ty{t}^{-1}=x^{b}y$. Normal forms $x^m{y}^n{t}^k$ multiply byThis proves torsion-freeness and nilpotence of class at most two. For $b\ne 0$ its center is $⟨x⟩$ and commutator subgroup $⟨x^b⟩$; for $b=0$ it is ${\mathbb{Z}}^3$. These realize all noncyclic torsion-free nilpotent closed orientable three-manifold groups.

The main cardioid consists of quadratic parameters with an attracting fixed point. If that fixed point has multiplier $\lambda$, then it is $\lambda /2$ and $c=\lambda /2-{\lambda }^2/4$, $|\lambda |<1$. The map is injective on the open unit disc: equality of two parameter values forces either equal multipliers or their sum to be two, impossible for distinct interior multipliers. Its boundary is the cardioid obtained from $\lambda =e^{it}$. At $\lambda =1$ the parameter derivative vanishes, giving the cusp $c=1/4$. Parameters just to the real right of that cusp have monotonically escaping critical orbits, while the cusp's critical orbit tends to $1/2$, so it is on the Mandelbrot set boundary.

The period-two bulb is the open disc $|c+1|<1/4$ of quadratic parameters with an attracting exact two-cycle. The points in that cycle are roots of $z^2+z+c+1$, whose product is $c+1$. Its multiplier is therefore $4(c+1)$, giving precisely the disc inequality. At $c=-3/4$ the two roots merge into the fixed point $-1/2$, whose original multiplier is minus one; the second iterate has multiplier one and a triple fixed-point root. This boundary point is the attachment to the main cardioid.

Near a distinct centre of mass $M$, write $H=M/r+h+rF(r,\theta ,\varphi )$, where $h$ is the regular part evaluated at that centre and $F$ is analytic in local angular charts. Set $A=M+hr$, $Q=rH=A+r^{2}F$ and $dv=dt+(h+M/r{)}^{2}dr$. The metric becomesAll coefficients extend analytically across $r=0$, with $Q(0)=M>0$ and two-dimensional determinant $-1$. The surface is null, its generator is ${\partial }_v$, and the double zero of its squared norm gives zero surface gravity. The construction uses a constant $h$ so that the coordinate differential is integrable despite the nonspherical regular part of $H$.

The main-sequence turnoff is the location in a coeval population's Hertzsprung-Russell diagram where stars are leaving the main sequence after exhausting their core hydrogen. More massive stars generally have larger luminosity per unit mass and shorter nuclear lifetimes, so the most massive stars still on the main sequence constrain the population's age. In a simple model, equating the stellar nuclear timescale $t\sim qM/L(M)$ to the common age determines a turnoff mass; detailed calibration requires composition and stellar models.