Applying the contravariant Hom functor ${\mathrm{Hom}}_R(-,W)$ to $0\to I\to X\to C\to 0$ givesThe connecting map sends $I\to W$ to its pushout of a module extension. Its vanishing means that map extends to $X\to W$. For a hereditary ring, ${\mathrm{Ext}}^2(C,W)=0$, making the indicated restriction on first extension groups surjective. The covariant variable has its corresponding long exact sequence.

A free resolution of $\mathbb{Q}$ has generators $a_n=1/n!$ and independent relations $a_n-(n+1)a_{n+1}$. Thus the Ext functor gives the cokernel of the map on integer sequence productsThe constant sequence one is not in its image. A preimage would require $u_1\equiv {\sum }_{j=1}^{N}j!(\mathrm{mod}(N+1)!)$ for every $N$. For $N\ge 3$, the sum is below $2N!$, tends to infinity, and its distance from $(N+1)!$ also tends to infinity. No fixed integer satisfies all these congruences. Hence ${\mathrm{Ext}}_{\mathbb{Z}}^1(\mathbb{Q},\mathbb{Z})\ne 0$.

For an exponential galactic disk, the first nonspherical term in the Legendre expansion of thin-disk gravity is positive and proportional to $R^{-3}$ in $v_c^2$. The missing enclosed mass and exterior-ring terms are exponentially small at large radius. Hence the galaxy rotation curve approaches the point-mass Keplerian curve from above.

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 complement of an event $A$ is $A^c=\Omega \setminus A$: it occurs exactly when $A$ does not. Since $A$ and $A^c$ are disjoint events with union $\Omega$, finite additivity gives $\mathbb{P}(A^c)=1-\mathbb{P}(A)$. For a die roll, the complement of an even result is an odd result. Under conditioning on an event $B$ with positive probability, the same rule is $\mathbb{P}(A^c\mid B)=1-\mathbb{P}(A\mid B)$; the complement is still relative to the original sample space.

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.

For an integrable terminal stock price and a deterministic positive discount factor $D$, a European call option curve has the form $C(K)=D{\mathbb{E}}_Q[(S_T-K{)}^{+}]$. Bounded difference quotients and dominated convergence give its one-sided derivatives:Thus the slope jump at a strike is $DQ(S_T=K)$. If the terminal probability distribution has a continuous probability density function $f_Q$, differentiation givesThe call curve's convexity makes this curvature nonnegative. The original construction obtains Arrow state prices from increasingly narrow call spreads.^[ref]

A continuous distribution need not have a density: a singular continuous law is a counterexample. In general the distributional second derivative is the measure $DQ(S_T\in dK)$. For a deterministic terminal value $s_0$, the curve $D(s_0-K{)}^{+}$ has a kink and second derivative $D{\delta }_{s_0}$, rather than an ordinary density function. The entire strike curve therefore records atoms as well as any density. Finitely many observed strikes usually leave many compatible terminal laws.

There are infinitely many prime numbers congruent to five modulo eight. Suppose instead that they form a finite list $q_1,\dots ,q_r$, and put $m=q_1\cdots q_r$, with $m=1$ if the list is empty. Then $m$ is an odd number andFor every prime factor $\ell$ of $N$, the integer $m$ is nonzero modulo $\ell$ and $(m/2{)}^2\equiv -1(\mathrm{mod}\ell )$. Thus a square root of minus one modulo a prime exists for $\ell$, which forces $\ell \equiv 1(\mathrm{mod}4)$. All prime factors are therefore one or five modulo eight. Their product is five modulo eight, so at least one prime factor is five modulo eight. Yet none of the $q_i$ divides $N$, since $N\equiv 4(\mathrm{mod}{q}_i)$. This contradicts completeness of the list.