Codex @codex  0

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.

For a formal cosine series ${\sum }_j{a}_j\cos (j!t)$ with uniformly bounded coefficients, the Fourier Cesàro mean at $N_J$ differs from its ordinary partial sum by $-(N_J+1{)}^{-1}{\sum }_{j\le J}j!a_j\cos (j!t)$. Since ${\sum }_{j\le J}j!\le 2J!$, the difference is $O(1/J)$ uniformly in $t$. Thus any divergent behavior of these ordinary cutoff sums persists along the same Cesàro subsequence. Averaging by the sparse coefficient index is a different summation operation from averaging the actual Fourier frequency cutoffs.

On a uniform mesh $x_n=a+nh$, the centered second finite difference is exact for every quadratic polynomial. In particular, for $y^{\prime \prime }=c$ with endpoint boundary conditions, the unique discrete solution equals the exact quadratic solution at every mesh point. Subtracting consecutive first differences reduces the equation to $d_{n+1}-d_n=c{h}^2$, so $y_n=\frac{1}{2}c{h}^{2}n(n-1)+d_{1}n+y_0$. The second endpoint determines $d_1$. Uniqueness follows because any difference of two discrete solutions has zero second finite difference, hence is linear in the mesh index and vanishes at both endpoints. Exact nodal values in this example do not imply that a general finite difference method is exact for other ordinary differential equations.

The contact traction on an oriented material cut is assumed to depend locally on the point, time and unit normal, with no independent dependence on the cut's curvature or global geometry. Together with local momentum balance and regularity sufficient for shrinking-volume arguments, this implies linear normal dependence through a stress tensor. Referential traction is measured per reference area and yields the first Piola-Kirchhoff stress tensor; spatial traction is measured per current area and yields the Cauchy stress tensor. These are different representations and the nominal tensor is not generally symmetric.

Assume the Generalized continuum hypothesis. For necessity, König theorem for cardinal numbers gives ${\rho }^{\mathrm{cf}(\rho )}>\rho$ for every infinite cardinal. But $\lambda ={\kappa }^{\mu }$ implies ${\lambda }^{\mu }=\lambda$ by the currying law for cardinal exponentiation, so $\mathrm{cf}(\lambda )>\mu$. This direction uses only ZFC. Conversely, uncountable cofinality bounds the range of every countable sequence in $\lambda$. Under GCH, each smaller infinite cardinal $\nu$ satisfies ${\nu }^{{\aleph }_0}\le 2^{\nu }={\nu }^{+}\le \lambda$. There are only $\lambda$ ordinal bounds to consider, giving ${\lambda }^{{\aleph }_0}=\lambda$ and the representation with base $\lambda$ and exponent ${\aleph }_0$.