Codex @codex  0

A function $f:M^n\to M$ is a definable function if its graph of a function is a definable set. Equivalently, a first-order formula $\theta (\mathbf{x},y)$ defines exactly the pairs with $y=f(\mathbf{x})$. Totality and uniqueness mean $M\models \forall \mathbf{x}\exists !y\theta (\mathbf{x},y)$. The same definition with restricted domains gives definable partial functions. A function need not be a symbol of the language to be definable; the graph can be specified through the existing relations and operations.

A deep-water gravity wave has dispersion $\omega (k)=\sqrt{g|k|}$. An observer moving with nonzero speed $V$ samples Fourier phases $kV\pm \omega (k)$. The stationary phase method selects the points where their derivatives vanish, which gives the displayed wavenumbers. Their group velocity matches the observer's velocity. For real even initial displacement with a nonzero Fourier coefficient there, the resulting signal has leading size $t^{-1/2}$ and phase $gt/(4|V|)-\pi /4$. If the initial coefficient vanishes, that leading term is absent; a stationary observer does not select a finite nonzero wavenumber by this rule.

For standard effective proof presentations, the syntactic deduction theorem and its reverse give effective transformations between proofs in a one-sentence extension and proofs of an implication in the base theory. Arithmetic can verify these transformations, yielding the displayed internal equivalence under the usual representation and proof-predicate hypotheses. It distinguishes internal proof transformations from an unlicensed reflection step taking provability to truth.

For stumps $h(x)=\mathrm{sign}(x_i-b)$ of fixed polarity, sort observations along each coordinate and sweep the threshold across whole groups of equal values. Passing a group adds its positive-label weight and subtracts its negative-label weight from the weighted misclassification risk. Include thresholds below and above all observed values. Every distinct sample prediction pattern is considered, giving exact minimization in $O(nm\log m)$ time complexity for $m$ observations in ${\mathbb{R}}^n$. Coordinate sort orders can be reused across AdaBoost rounds, reducing subsequent weighted scans to $O(nm)$.

Choose increasing $R_m\to \infty$ so $H(t)\ge m^2|t|$ whenever $|t|\ge R_m$. Put $\Psi (s)={\sum }_{m\ge 1}(s-R_m{)}_{+}$ for $s\ge 0$. This is nonnegative, increasing and convex. If $m$ terms are active, then $\Psi (s)\le ms\le H(t)$ for $s=|t|$. For every fixed $N$, $s\ge 2R_N$ gives $\Psi (s)\ge Ns/2$, proving superlinearity. Thus a nonconvex superlinear energy lower bound still supplies the convex function needed in the De la Vallée-Poussin criterion. A continuous superlinear function not initially nonnegative can first have its finite global minimum subtracted.

Historical name for the cysticercus of Taenia solium; not an independent modern species.

Parametrize the affine cuspidal cubic $x^2=y^3$ by $x=t^3,y=t^2$. Removing the zero set of $g=x-y$ removes exactly the parameter values zero and one. Its regular-function ring is the localization $k[t^2,t^3{]}_g$. To recover the normalization coordinate inside this ring, use $g^2=y(y^2-2x+y)$: inverting $g$ makes $y$ a unit, so $t=x/y$ lies in the localization. As $g=t^2(t-1)$ is a unit, so are $t$ and $t-1$, proving the displayed equality. This calculation is valid in every characteristic. At the smooth point $(1,1)$ of the original curve the local ring is instead $k[t{]}_{(t-1)}$: $y$ is a unit there, and every polynomial in $t$ nonzero at one is invertible. That point is absent from the displayed open subset, so its local ring is computed in the original curve.

Suppose the moment-generating function $M_X$ is finite on an open interval around zero. Its cumulant-generating function $K_X=\log M_X$ satisfies $K_X(0)=0$, $K_X^{\prime }(0)=\mathbb{E}X=\mu$ and $K_X^{\prime \prime }(0)=\mathrm{Var}(X)={\sigma }^2$. Indeed,and $M_X(0)=1$. The Taylor expansion gives the displayed formula, with little o notation describing the remainder. Finiteness in a neighborhood justifies differentiation: polynomial factors times $e^{tX}$ are dominated on a smaller interval by constants times two slightly larger exponential moments.

A clamped cubic spline is a cubic spline whose endpoint first derivatives are prescribed. A clamped cubic spline interpolant additionally matches prescribed values at its spline knots. Distinct spline knots, arbitrary nodal values and arbitrary endpoint slopes determine one such interpolant. The minimum roughness property of a clamped cubic spline proves uniqueness. Unlike the natural cubic spline, this construction prescribes endpoint slopes rather than zero endpoint second derivatives.

For four distinct projective points on the complex projective line, use the conventionwhere $[p_i,p_j]$ is the determinant of chosen representative vectors. Rescaling any representative cancels, and applying an invertible matrix multiplies every bracket by its determinant, which also cancels. Thus the expression is a projective invariant. The projective general linear group action on the projective line has a unique element $T$ taking the first three points to $\infty ,0,1$. In those coordinates the displayed expression is exactly $T(P_3)$. Consequently the cross-ratio completely determines the orbit of the ordered quadruple under the projective linear group. Its value lies in $\mathbb{C}\setminus \{0,1\}$.

For the convention in cross-ratio normalization of four projective points, permuting four distinct projective points produces the six displayed rational expressions. Interchanging the first pair sends $\lambda$ to $1/\lambda$; interchanging the middle pair sends it to $1-\lambda$. The four permutations consisting of the identity and the three double transpositions form a Klein four-group preserving the cross-ratio. The resulting six-element action gives every displayed expression.

For real $\lambda \notin \{0,1\}$, repeated values occur exactly for $\lambda \in \{-1,1/2,2\}$, and then the orbit has size three. To prove this, a repetition gives a nonidentity stabilizer after composing one operation with the other's inverse. Its fixed-point equations give $-1$, $1/2$, $2$, or ${\lambda }^2-\lambda +1=0$, with the last case nonreal. With one point at infinity, $\lambda =(x_1-x_3)/(x_1-x_2)$; the three real exceptional values mean that one finite point is the midpoint of the other two. Thus these harmonic conjugate points configurations have three collinear equally spaced finite points in that affine coordinate.

For mutually independent square-integrable random variables $A,B,C$, factor the joint moment as $\mathbb{E}[A{B}^{2}C]=\mathbb{E}A\mathbb{E}[B^2]\mathbb{E}C$, then subtract the product of the two means. This proves the formula. Sharing one factor can therefore produce nonzero covariance even when all original factors are independent. Products of disjoint groups of independent factors are themselves independent random variables.

If every factor is a complete metric space, the countable product metric is complete. For a product-Cauchy sequence, the inequality $2^{-i}\varphi (d_i(x_i^{(m)},x_i^{(n)}))\le D(x^{(m)},x^{(n)})$, with $\varphi (t)=t/(1+t)$, makes each coordinate sequence Cauchy. Its coordinate limits define a product point. Coordinatewise convergence and the uniformly small summable tail imply convergence to that point in $D$. It matters that the original distances are bounded before they are summed: summing arbitrary unbounded metrics directly need not be finite.

A cooperative cost game in which disjoint coalitions can merge without increasing their combined cost. Negating its cost function gives a superadditive coalitional game. Subadditivity by itself does not ensure a nonempty cost core.

Share a runway's cost among users needing different runway lengths. If user $i$ needs a facility costing $C_i\ge 0$, a nonempty coalition pays $c(S)={\max }_{i\in S}{C}_i$, and $c(\varnothing )=0$. This is a subadditive cost game because the maximum of two nonnegative costs does not exceed their sum. A cost-core allocation always exists: charge the whole cost to one longest-runway user and zero to all others. The Shapley value of an airport game gives a symmetric alternative that also lies in the cost core.

A continuous poset that is also a directed-complete partial order.

A partially ordered set in which the compact elements below each element form a directed set with that element as their least upper bound.

For a continuous function $f:X\to \mathbb{R}$, its zero set is $f^{-1}(\{0\})$. It is closed and is a G-delta set, since $f^{-1}(0)={\bigcap }_{n\ge 1}{f}^{-1}((-1/n,1/n))$. Replacing $f$ by $|f|/(1+|f|)$ gives the same zero set with values in $[0,1]$. In a normal topological space, every closed G-delta set has this form.

The set of continuous functions $\mathbb{R}\to \mathbb{R}$ has cardinality $2^{{\aleph }_0}$. Constant functions provide the lower bound. Restriction to the dense set $\mathbb{Q}$ is injective, since continuity determines a function from its values there. Real sequences have cardinality $2^{{\aleph }_0}$, by encoding a sequence using its countably many rational lower cuts. Restriction therefore supplies the upper bound and Cantor-Schröder-Bernstein gives equality.

A continuous real-valued function is a continuous map whose codomain is the real line.