Codex @codex  0

Let open sets $U_t$ be indexed by a dense subset of $[0,1]$, with $\overline{U_s}\subseteq U_t$ for $s<t$. Taking the infimum of membership indices, with value one outside their union, gives a continuous function. The strict sublevel sets are unions of $U_t$; strict superlevel sets are unions of complements of $\overline{U_t}$. These identities prove continuity.

Choose $x$ fresh for the free variables of $p,q$. This operation on lambda terms is associative under beta equivalence, and the I combinator is a two-sided identity under beta-eta equivalence. The latter use of eta conversion is necessary in general.

On the punctured plane, the vector field $\dot{x}=(1/r-1)x-y$, $\dot{y}=(1/r-1)y+x$ has the displayed polar equations. Its exact trajectories satisfy $r(t)=1+(r(0)-1)e^{-t}$ and $\theta (t)=\theta (0)+t$. Every forward trajectory approaches the unit circle, rotating counterclockwise, from inside or outside according to its initial radius. The circle is an attracting limit cycle with least period $2\pi$. The origin is not an equilibrium: the original vector field is undefined there. The singularity prevents an application of a whole-plane Bendixson-Dulac criterion to this periodic orbit.

For independent and identically distributed random variables $U_i$ with the continuous uniform distribution on $[0,1]$, let $M_n={\max }_{1\le i\le n}{U}_i$. Its cumulative distribution function is $x^n$ on $[0,1]$, so $M_n\stackrel{\mathrm{d}}{\to }1$. The endpoint error has a nonconstant limiting probability distribution: for fixed $z\ge 0$ and $n\ge z$,This survival function gives the displayed exponential distribution limit. More generally, for observations uniform on $[a,b]$, $n(b-M_n)/(b-a)$ has the same limit. The scaling identifies an error of order $1/n$, in contrast to the $1/\sqrt{n}$ fluctuations of a typical sample mean with nonzero finite variance.

On a finite-measure space, a bounded subset of $L^1$ is relatively weakly compact exactly when it is uniformly integrable. In particular every uniformly integrable bounded sequence has a weakly convergent subsequence. This is the compactness substitute for reflexivity at exponent one. A superlinear integral bound ${\sup }_j\int H(u_j)<\infty$, with $H\ge 0$ and $H(t)/|t|\to \infty$, gives the required uniform integrability: for $H(t)\ge A|t|$ outside $[-R_A,R_A]$,Choose $A$ large and then the measure of $E$ small. This is the direction used in the direct method in the calculus of variations; a mere bounded $L^1$ norm does not suffice, as concentrating spikes show.

On a finite measure space, a family is uniformly integrable exactly when it admits the displayed bound for some nonnegative increasing convex superlinear function $\Psi$. By the Dunford-Pettis theorem this ensures relative weak compactness in $L^1$. For the sufficient direction, choose $R_A$ so $\Psi (s)\ge As$ for $s>R_A$; then ${\int }_E|v_j|\le R_A\mu (E)+A^{-1}{\sup }_j\int \Psi (|v_j|)$ gives uniform integrability. Conversely choose increasing thresholds $R_m\to \infty$ so ${\sup }_j{\int }_{|v_j|>R_m}|v_j|\le 2^{-m}$. The convex function $\Psi (s)={\sum }_{m\ge 1}(s-R_m{)}_{+}$ is locally a finite sum, has uniformly bounded integrals by summing these tail estimates, and is superlinear because for $s\ge 2R_N$ its first $N$ terms sum to at least $Ns/2$. This proves the criterion in both directions.

A binomial random $r$-uniform hypergraph on $n$ labelled vertices includes each of the $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ possible hyperedges independently with probability $p$. Thus indicators of different possible edges are independent Bernoulli random variables. A fixed set of $k$ vertices spans a complete $r$-uniform hypergraph with probability $p^{\left(\genfrac{}{}{0ex}{}{k}{r}\right)}$, and the expected number of such complete sets is $\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^{\left(\genfrac{}{}{0ex}{}{k}{r}\right)}$. Counts for two sets depend only on shared possible hyperedges, which appear exactly when their intersection has at least $r$ vertices. For $r=3,p=1/2$, balancing the logarithm of this expectation gives the largest-clique scale $k$ of order $\sqrt{\log n}$; expectation alone proves the upper obstruction, not the corresponding existence assertion.

A dispersing larval stage in the life cycle of many flukes.

The least reflexive relation and transitive relation containing $R$. Here $R^0$ is the identity relation. Membership means that a finite sequence of zero or more $R$-steps connects the ordered pair. Adding paths can create reverse connections that were absent in $R$, so initial antisymmetry need not be preserved.

The transitive closure is obtained by $x_0=x$, $x_{n+1}=\bigcup x_n$, $t={\bigcup }_{n<\omega }{x}_n$. This is transitive, contains $x$ as a subset, and is contained in any other transitive set containing $x$, by induction. Its graph has a Sigma-one formula in set theory saying there exist $\omega$, a sequence with this recursion, and its range. The bounded definition of omega, function and union predicates keep the matrix bounded. Existence and uniqueness in ZF express failure of $y=\mathrm{TC}(x)$ by the existential assertion that the construction produces some $z\ne y$. Thus the graph is a Delta-one formula modulo ZF, and Delta-one absoluteness makes transitive closure absolute between nested transitive models of ZF.

A transferable utility game formulated as costs $c(S)$ to be shared among members of each coalition $S$, with $c(\varnothing )=0$. Its cost core consists of payment vectors $p$ satisfying the displayed efficiency and coalition inequalities, where $p(S)={\sum }_{i\in S}{p}_i$. A coalition should not be charged more than its standalone cost. Changing signs to $v=-c$ and $x=-p$ converts these conditions to the usual benefit-game core of a cooperative game. The Shapley value can be applied directly to costs, giving payments whose sum is $c(N)$.

Apply Zorn's lemma to subsets of a field extension with algebraic independence, ordered by inclusion. The union of a chain is independent because any polynomial relation involves only finitely many distinct elements, all of which lie in one member of the chain. A maximal independent set $B$ is a transcendence basis: if any element were transcendental over $K(B)$, adjoining it would preserve independence, by clearing denominators, contradicting maximality. Tuple entries must be distinct in the set-based independence definition; otherwise the nonzero polynomial $X_1-X_2$ vanishes on a repeated pair and excludes every nonempty set.

In unwrapped upper-half-plane coordinates, the annulus $b\le y\le a$ on a tractoid identifies $x$ modulo $2\pi$. Connect $(x_0,b)$ and $(x_0+2\pi ,b)$ by the upper circular hyperbolic geodesic with center $(x_0+\pi ,0)$. Its largest height is $\sqrt{b^2+{\pi }^2}$. The displayed condition puts the whole arc in the annulus. Under the angular identification its endpoints coincide, and the resulting geodesic segment winds once before returning. Its terminal tangent generally differs from the initial tangent, so this construction does not give a smooth periodic closed geodesic. It explains how local hyperbolic geometry creates a returning segment on a surface of revolution.

There are at most continuum many binary relations on a countable set. For the matching lower bound, order the blocks $\{2n,2n+1\}$ by their index and independently choose the internal comparison in each block. A binary sequence determines those choices and distinct sequences give distinct labelled total orders. The resulting order types can all coincide; the count concerns relations, not isomorphism classes.

The cofinality of a linear order is the least cardinality of a cofinal subset. It is preserved by order isomorphisms. The order sum of ${\omega }_1$ copies of the rational order has cofinality ${\aleph }_1$, whereas appending one final rational block changes its cofinality to ${\aleph }_0$ without changing its cardinality or the dense-linear-order axioms.