A set $I$ is inductive if it contains the empty set and contains $x\cup \{x\}$ whenever it contains $x$. The axiom of infinity asserts that an inductive set exists. The natural numbers form the least inductive set: intersect one inductive set with the class of objects belonging to every inductive set. This use of separation gives a set, and the defining class is closed under successor.

This concerns arbitrary compact spaces, with no Hausdorff restriction. For nonempty sets $A_j$, adjoin a distinguished point ${\infty }_j$ and use the four-open-set topology $\{\varnothing ,A_j,\{{\infty }_j\},A_j\cup \{{\infty }_j\}\}$. Each resulting space is compact without choice. The product has the canonical all-infinity point. The closed coordinate conditions $x_j\in A_j$ have the finite intersection property, since finite choice is provable without the axiom of choice. If the countable product is compact, their intersection is nonempty, and any point in it supplies a choice function.

The background potential energy is the gravitational potential energy of the statically stable state obtained by adiabatically sorting a fluid's parcels. The difference $A=P-P_b$ is available potential energy. Pure rearrangement changes $A$ but not $P_b$; irreversible density mixing can increase $P_b$. Molecular exchange with internal energy must be separated when defining an irreversible mixing rate.

For independent fair signs, $|X_n|\to \infty$ almost surely from every deterministic initial value. Outside $[-L,L]$ with $L>1/(a-1)$, the sign is preserved and $|X_{n+k}|\ge a^k(|X_n|-1/(a-1))+1/(a-1)$. To reach that region, choose $m$ so $(a^m-1)/(a-1)>L$. At the beginning of each $m$-step block, choose the sign of the current value; a block of that sign pushes the absolute value past $L$ and has conditional probability $2^{-m}$. The chance of remaining inside after $k$ blocks is at most $(1-2^{-m}{)}^k$. This proof avoids any assumption about a density of the limiting random series.

For a causal autoregressive model $X_t={\sum }_{j=1}^p{\varphi }_j{X}_{t-j}+{ϵ}_t$ with centered white noise of variance ${\sigma }^2$, multiplying by past values and taking expectations yields $\gamma (h)={\sum }_j{\varphi }_j\gamma (h-j)$ for $h\ge 1$, and $\gamma (0)={\sum }_j{\varphi }_j\gamma (j)+{\sigma }^2$. These Yule-Walker equations determine the autocovariance from the coefficients and noise variance, or estimate coefficients from observed autocovariances. Causality supplies the needed orthogonality of the current noise to past observations.

For $X_t=\alpha X_{t-1}+{\alpha }^2{X}_{t-2}+{\epsilon }_t$, let $\phi =(1+\sqrt{5})/2$. The autoregressive polynomial factors as $(1-\alpha \phi z)(1+\alpha z/\phi )$. Its roots lie outside the closed unit disk exactly when $|\alpha |<1/\phi$. In that causal region,The coefficients follow from ${\psi }_0=1,{\psi }_1=\alpha$ and ${\psi }_j=\alpha {\psi }_{j-1}+{\alpha }^2{\psi }_{j-2}$. They are absolutely summable in the stated region. Causality makes future white noise orthogonal to the observation past, giving forecasts $\alpha X_T+{\alpha }^2{X}_{T-1}$ and $2{\alpha }^2{X}_T+{\alpha }^3{X}_{T-1}$ at horizons one and two.

For a self-injective finite-dimensional algebra, remove the projective vertices from its Auslander–Reiten quiver. The resulting stable quiver has vertices the nonprojective indecomposable modules. Its translation sends the end term of an almost-split exact sequence to its start term. In particular a quiver of shape $\mathbb{Z}{A}_s/⟨{\tau }^e⟩$ has $es$ vertices.

The two sides have $n$ vertices each. Given a nonperfect matching in a graph, choose unmatched vertices $u,v$ on opposite sides. An unmatched neighbour of either supplies a one-edge augmenting path in a matching. Otherwise their neighbours are matched; the matched left vertices corresponding to $N(u)$ and the set $N(v)$ have total size greater than $n$, so they intersect. This supplies the alternating path in a graph $u,b,a,v$ with $ab$ in the matching in a graph. Repeated augmentation produces a perfect matching.

Choose such an augmentation deterministically. Its reverse is specified by at most four vertices, giving at most $2n^4$ preimages for a matching in a graph of the next size. Thus the matching generating function coefficients satisfy $m_k\le 2n^4{m}_{k+1}$.

Each bidder chooses a permitted amount. The bidder with the smallest amount submitted by exactly one bidder wins and pays that amount; if no amount is unique, nobody wins. With object value $V$, the winner's payoff is $V$ minus their bid, and losers pay nothing. Unlike a first-price sealed-bid auction, repeated lowest bids can be disqualified.

In an ideal continuous-price auction with independent private values, bidders remain active up to their valuations. Without a reserve, the last remaining bidder wins at the second-highest valuation. With a reserve price $r$, there is no sale if every valuation is below $r$; otherwise the winner pays the larger of $r$ and the second-highest valuation. Strict versus weak inequalities at the reserve have no revenue effect for continuous valuation distributions. This payment rule is distinct from a first-price sealed-bid auction.

A reserve price is the seller's minimum acceptable price. In an English auction it changes both the sale rule and the payment rule: the highest valuation must exceed the reserve and the winner pays at least the reserve. Mechanisms with different reserves generally have different allocation probabilities, so revenue equivalence need not equate their expected revenues.

The highest bidder wins a single item but pays the lowest submitted bid, while losers pay nothing. This differs from a lowest unique bid auction. With at least two bidders using increasing strategies, the winning bidder pays the smallest opponent bid.

// Target: mathematical-optimization.bigb

A single-item auction in which the highest bidder wins and pays the highest competing bid. Under private values and quasilinear utility, bidding one's valuation is a dominant strategy: if the competing maximum is below one's value, winning at that price is best; if it is above one's value, losing is best; equality gives zero utility either way. Thus truthful bidding is an ex post Nash equilibrium.

A holomorphic periodic point is superattracting if its return multiplier is zero. The return map then has local expansion $G(p+u)=p+a{u}^e+O(u^{e+1})$ with $e\ge 2$. This implies a strict contraction on a sufficiently small disc and rapid convergence of returns. Infinity is superattracting for every polynomial of degree at least two: the reciprocal-coordinate return has a zero of order equal to the polynomial degree.

An atomic operation that reads a shared object's value and changes it as one indivisible action. For an increment from $0$, the two operations under mutual exclusion produce $1$ then $2$ in either software thread order. A test-and-set is another example. The operation's ordering guarantees for other data still depend on the memory model and its selected acquire-release synchronization semantics.

Take a countable structure with countably many independent unary predicates, every finite predicate pattern realized infinitely often, and a constant naming every element. Its complete theory has an atomic model: tuples of named elements have types isolated by equations to their names. But every infinite binary predicate pattern, together with inequalities to every named element, is consistent. There are uncountably many such one-types, so no countable omega-saturated model exists. Thus existence of a countable atomic model does not imply existence of a countable saturated one.

Hydrodynamic escape is atmospheric loss in which collisions couple the gas into a bulk outward flow. Heating raises the gas pressure and can drive expansion against gravity; an escaping solution often accelerates through a sonic point. The temperature and velocity profiles must satisfy energy and momentum balances, rather than being fixed by a collisionless high-speed tail alone.

The flow can carry heavier species along with the dominant light gas, provided coupling is strong enough. Energy-limited atmospheric escape estimates a mass-loss rate by assigning a fraction of absorbed stellar power to gravitational work. That estimate can fail when radiative cooling, limited heating supply or other parts of the flow's energy budget dominate. A small Jeans escape parameter is suggestive of strong escape, but does not by itself specify a unique outflow.

For two-qubit pure states with smaller squared Schmidt coefficients $s<t$, optimal stochastic conversion of a two-qubit pure state succeeds independently with probability $s/t$. The asymptotic pure-state entanglement conversion rate of collective LOCC is instead $h_2(s)/h_2(t)$, where $h_2$ is binary entropy. The difference is strict: differentiating gives $\frac{d}{dx}[h_2(x)/x]={\log }_2(1-x)/x^2<0$. Hence $h_2(s)/s>h_2(t)/t$. Independent filtering loses entanglement in its product-state failure branches, whereas collective concentration and dilution preserve entanglement asymptotically.

When the bottom of an Asymptotic giant branch star's convective envelope becomes hot enough for hydrogen burning, the CNO cycle can convert newly dredged-up carbon into nitrogen. Its onset and strength depend on mass, composition and envelope structure; it is not a universal property of all stars at one initial mass.

The fate of an Asymptotic giant branch star depends on whether its envelope is removed before its core reaches the next ignition or instability threshold. The approximate envelope-loss time is $t_{\mathrm{env}}=M_{\mathrm{env}}/|\dot{M}|$, while the time to add a required core mass $\Delta M_c$ is $t_{\mathrm{grow}}=\Delta M_c/{\dot{M}}_c$. Shell stellar nuclear fusion grows the core; third dredge-up can reduce the net growth. If $t_{\mathrm{env}}\ll t_{\mathrm{grow}}$, shell fuel disappears first and a white dwarf remnant is favored. The opposite ordering permits further core growth but does not by itself specify whether carbon burning is stable, explosive, or followed by further evolution. Both rates change with time, so the estimates guide rather than replace a stellar model.