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For a finite vertex set $S$ containing the origin, this quantity is the expected number of potentially open exit edges reached from the origin within $S$. If ${\phi }_p(S)<1$, the BK inequality applied to the first exit of a long open self-avoiding walk gives a contraction at the scale of $S$ and hence exponential connection decay. The threshold ${\tilde{p}}_c=\sup \{p:{\phi }_p(S)<1\text{ for some finite }S\ni 0\}$ equals the percolation critical probability by the pivotal-edge differential inequality in sharpness of the percolation transition.

A deterministic wealth level $c$ satisfying $U(c)=\mathbb{E}U(W)$. For normal wealth with exponential utility $U(x)=-e^{-\gamma x}$ it is $c=\mathbb{E}W-\gamma \mathrm{Var}(W)/2$.

An investment-consumption problem chooses portfolio holdings and consumption to maximize discounted expected utility maximization subject to a self-financing portfolio wealth equation and admissibility constraints. A Hamilton-Jacobi-Bellman equation or utility duality with martingale deflators characterizes the optimum when the value is finite.

An incentive fee above a hurdle changes a manager’s utility function into $F(y)=U(\epsilon y+\alpha (y-w_0{)}_{+})$. For constant absolute risk aversion utility, each branch is strictly concave, but the marginal reward jumps upward at the hurdle. Thus the effective terminal-wealth utility is not concave, and concavification of incentive utility can reveal optimal risk-taking lotteries.

Finite lower and upper utility bounds together with a positive discount rate bound the absolute time integral by $\max (|L|,|R|)/b$. This justifies expectation and integration in marginal-utility verification. Bounded utility alone on an undiscounted infinite horizon does not prevent both positive and negative parts from having infinite integral.

An atmospheric gas or gas combination whose abundance and environmental context can support a biological interpretation. Detectability, a plausible production flux and exclusion of significant abiotic alternatives matter together. Modern-Earth molecular oxygen and nitrous oxide have predominantly biological sources, but an oxygen biosignature false positive illustrates why extrapolation to another planet requires context.

The Spitzer Space Telescope provided infrared observations of exoplanet secondary eclipses and exoplanet thermal phase curves. Its original mission included infrared spectroscopy; its warm mission retained the 3.6 and 4.5 micrometre imaging bands.

An atmospheric retrieval fits a parametrized radiative transfer model to an exoplanet transmission spectrum or exoplanet emission spectrum. Temperature, abundances, clouds and reference radius/pressure can trade off against one another. Posterior precision within one model is not proof that its structural assumptions are unique.

A long exact sequence is a sequence of abelian groups or modules with arbitrarily many consecutive maps, for which the image of every map equals the kernel of the next. A short exact sequence of cochain complexes yields a long exact sequence of their cohomology, with connecting homomorphisms increasing degree by one. The exact sequences of relative cohomology and the Bockstein homomorphism are examples.

A phase bounded by mass-dependent lifetimes ${\tau }_a(M)$ and ${\tau }_b(M)$ occupies ${\tau }_a(M)<t<{\tau }_b(M)$, clipped to the population age range. Its boundaries can be mapped into uniform probability coordinates to evaluate evolutionary-state fractions.

Two events are disjoint when they cannot both occur. For a finite family of pairwise disjoint events, the probability of their union is the sum of their probabilities, so that sum is at most one. This condition differs from the condition defining independent events.

The event that $A$ does not occur has probability $1-P(A)$. Conditioning on an event positively associated with $A$ negatively associates its complement with $A$.

A positive one-period martingale deflator applied to $C=(S-B{)}^{+}$ gives $(s-b{)}^{+}\le c\le s$. If both strict payoff regions $S>B$ and $S<B$ have positive probability, then $c>0$ and $c-(s-b)>0$, proving the strict lower bound. No normalization of the positive pricing variable is needed.

Buying the lower-strike call and selling the higher-strike call gives a strictly negative initial cost and terminal payoff $(S-K_1{)}^{+}-(S-K_2{)}^{+}\ge 0$. Hence an arbitrage-free call-price curve is decreasing in strike. The positive initial receipt may be consumed immediately or invested in cash.

The midpoint butterfly buys half a call at each outer strike and sells one at the middle strike. Convexity of $K\mapsto (S-K{)}^{+}$ makes its terminal payoff nonnegative. A violation of the displayed price inequality makes its cost negative and creates arbitrage. The payoff is triangular between the outer strikes and zero elsewhere.

In a zero-interest one-period market with a continuous terminal stock law under a pricing measure, $C(K)={\mathbb{E}}_Q(S-K{)}^{+}$ implies $-C^{\prime }(K)=Q(S>K)$ and $C^{\prime \prime }(K)=f_Q(K)$. Thus a full differentiable call curve determines its pricing density. A finite collection of strikes generally does not. With a deterministic nonunit discount factor, divide the call curve by that factor before recovering the probability density.

Every odd prime factor of an odd square plus four is one modulo four. The integer is five modulo eight, so at least one prime factor is five modulo eight. Choosing the odd square divisible by any proposed finite list excludes the list and proves infinitude.

For a finite set of primes choose $0<g(p)<1$, put $k(p)=g(p)/(1-g(p))$, and let $W={\prod }_p(1+k(p))$. Give each squarefree integer $d$ on these primes probability $k(d)/W$. Inclusion of each prime is represented by a Bernoulli random variable with parameter $g(p)$; these are independent random variables, so $\mathbb{E}[\log d]={\sum }_{p}g(p)\log p$. If this is at most $\theta \log L$ with $0<\theta <1$, the Markov inequality givesThis turns a full Euler product into a lower bound for the truncated normalizing sum of a Selberg sieve.

Writing $q=u+iv$ in the free Schrodinger equation gives $u_t=-v_{xx}$ and $v_t=u_{xx}$, hence $u_{tt}+u_{xxxx}=0$. To encode initial velocity $u_1$, choose the decaying primitive $v_0(x)=-{\int }_x^{\infty }(s-x)u_1(s)ds$. A prescribed endpoint curvature becomes the time derivative of the imaginary boundary trace. This turns the normalized Euler-Bernoulli beam equation into a complex second-order boundary problem.

For a real vector $v$, the square of its Euclidean norm is $\Vert v{\Vert }_2^2=v\cdot v={\sum }_j{v}_j^2$. Unlike the norm itself it is a quadratic function, so expanding sums directly exposes diagonal and cross terms. This is useful for the mean-square displacement of an isotropic planar random walk.