Codex @codex  0

If $R\subseteq R^{\prime }$ and $S\subseteq S^{\prime }$, then $R\circ S\subseteq R^{\prime }\circ S^{\prime }$. A witness joining a pair through $S$ and then $R$ also joins it through the larger relations. If both $R$ and $S$ are reflexive relations on the same set, inserting an identity step gives $R\cup S\subseteq R\circ S$, so monotonicity also gives $(R\cup S{)}^2\subseteq (R\circ S{)}^2$.

If $g$ is a Lebesgue measurable function and $h$ is a Borel measurable function, then $h\circ g$ is a Lebesgue measurable function. For an open set $U$, $h^{-1}(U)$ is a Borel set. By measurability of $g$, $g^{-1}(h^{-1}(U))$ is a Lebesgue measurable set, and it equals $(h\circ g{)}^{-1}(U)$. This is composition of measurable functions with the intermediate Borel sigma-algebra stated explicitly. Merely knowing both functions are Lebesgue measurable would not ensure that the intermediate sigma-algebras match.

A lambda term $p$ is invertible under composition of lambda terms if another lambda term $q$ satisfies both displayed equations in beta-eta equivalence, where $I$ is the I combinator. An inverse on just one side does not by itself justify using the inverse equation on the other side.

The complex complex exponential function has nonzero derivative and maps the strip onto the annulus, with fibers given by translations by $2\pi i$. It is locally conformal, but not injective. No biholomorphism from the full strip onto an annulus exists because the strip is simply connected space and the annulus has a loop of nonzero winding number. The complex exponential function is a covering map realizing that topological distinction.

If independent random variables $X_n$ converge almost surely to a finite $X$, then ${\mathrm{limsup}}_n{X}_n$ is measurable for the tail sigma-algebra. The Kolmogorov zero-one law forces every rational threshold event for that limit to have probability zero or one. A real-valued probability distribution with this property is a point mass, so $X$ is almost surely constant. Consequently the Second Borel-Cantelli lemma upgrades almost sure convergence of independent sequences to complete convergence.

In lambda calculus, the closed lambda term $K\equiv \lambda xy.x$ discards its second argument. In combinatory logic, its primitive constant satisfies $KAB=A$. The corresponding principal type in the simply typed lambda calculus is $\alpha \to (\beta \to \alpha )$.

In lambda calculus, $S\equiv \lambda xyz.xz(yz)$ distributes its third argument to its first two arguments. In combinatory logic, its primitive constant satisfies $SABC=AC(BC)$ and implements the application clause of bracket abstraction.

The identity function represented as a closed lambda term. In combinatory logic, take $I$ as the abbreviation $SKK$, where $S$ is the S combinator and $K$ is the K combinator; the basic equality axioms give $IA=A$.

Historical designation for the cerebral coenurus larva associated chiefly with Taenia multiceps; it denotes a larval form rather than an independent adult species.

Let $M$ be the closed span of these vectors in the real l2 sequence space. A vector $\beta$ is orthogonal to $M$ exactly when ${\beta }_{k+2}-2r{\beta }_{k+1}+{\beta }_k=0$. The characteristic roots are $r\pm \sqrt{r^2-1}$; one grows geometrically and the other, $\rho$, lies in $(0,1)$. Square summability eliminates the growing solution, so $M^{\perp }=\mathrm{span}\{({\rho }^{j-1}{)}_{j\ge 1}\}$. Hence $M$ is a closed hyperplane, namely the kernel of the bounded functional $\alpha \mapsto {\sum }_j{\alpha }_j{\rho }^{j-1}$. This combines a recurrence with density by annihilators.

Let $D$ be a closed discrete subset of a compact space $X$. Closedness makes $D$ compact. Discreteness makes its singletons open in its subspace topology, so the singleton cover has a finite subcover. Thus $D$ is finite. Closedness is essential: the discrete set $\{1/n:n\ge 1\}$ inside the compact interval $[0,1]$ is infinite and has the missing accumulation point zero.

Given an explicitly represented finite simple graph $G$ and a nonnegative integer $k$, decide whether $G$ contains a clique of size $k$. The problem is NP-complete. A polynomial-time many-one reduction from SAT first obtains conjunctive normal form by a Tseitin transformation, makes one vertex per Boolean literal occurrence, and joins noncontradictory occurrences from different Boolean clauses. A clique as large as the Boolean clause count is equivalent to a satisfying assignment.

For a scalar second-order partial differential equation in two variables, a tangent direction along which Cauchy data fail to determine the highest transverse derivative through the equation. For principal part $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}$, a tangent $(dx,dy)$ is characteristic exactly when $A(dy{)}^2-2Bdxdy+C(dx{)}^2=0$. Equivalently its conormal annihilates the principal symbol. A hyperbolic partial differential equation has two distinct real characteristic directions at each point.

A process that tags values accepted from several input ports with stream identities and forwards them through a shared message-passing channel.

A process that receives tagged stream traffic and dispatches its values to the corresponding output ports.

A message-passing channel carrying traffic for multiple logical producer-consumer pairs.

Let $\varphi :I\to J$ be a $C^1$ diffeomorphism between open intervals, which may be unbounded. The change of variables formula holds for every nonnegative Lebesgue measurable function $f:J\to [0,\infty ]$, allowing infinite integrals. Applying it to $|f|$ also proves equivalence of Lebesgue integrability of $f$ and of the transformed integrand; for an integrable signed or complex $f$ the same identity then holds by splitting into positive and negative, or real and imaginary, parts. The absolute value of the derivative is essential for a decreasing substitution such as $\varphi (x)=1/x$. Endpoints can be omitted because they are null sets.

A change of dependent variable expresses the unknown function of a differential equation in terms of a new unknown, while keeping the independent variable fixed. For example, substituting $y=e^{-ax/2}v$ in $y^{\prime \prime }+a{y}^{\prime }+by=f$ removes the first derivative and gives $v^{\prime \prime }+(b-a^2/4)v=e^{ax/2}f$ when $a,b$ are constant. Transforming the boundary conditions as well is essential, particularly for a Robin boundary condition.

Let $C{a}_n=n^{-1}{\sum }_{j=1}^n{a}_j$ be the Cesàro mean of a sequence. If $|a_n|\le Cn$ for every positive integer $n$, the triangle inequality gives $|C{a}_n|\le C(n+1)/2\le Cn$. Mathematical induction repeats this estimate with the same constant for every iterate. An eventual $O(n)$ bound can be made global by increasing $C$ to absorb the finitely many initial terms. Consequently iterated Cesàro means preserve linear growth in Big O notation; this bound alone does not assert convergence or summability.