A maximal element has no strictly greater element; a minimal element has no strictly smaller one. Either can be nonunique, unlike a greatest or least element. These notions are defined for arbitrary partially ordered sets.

A greatest element lies above every member of an ordered set, and a least element lies below every member. Antisymmetry makes either unique if it exists. A greatest element is maximal, but a maximal element need not be greatest.

In a partially ordered set $P$, an upper set $U$ satisfies $x\in U$ and $x\le y\Rightarrow y\in U$; a lower set $L$ satisfies $x\in L$ and $y\le x\Rightarrow y\in L$. The complement of a set exchanges the two conditions: if $x\notin U$ and $y\le x$, then $y\in U$ would force $x\in U$, so $P\setminus U$ is lower. In a Boolean lattice ordered by inclusion, these are the up-set and down-set conditions.

An ordered group whose partial order is a lattice. Each pair has a join $f\vee g$ and a meet $f\wedge g$. Translations preserve these operations, for example $a(f\vee g)b=afb\vee agb$. Pointwise maximum and minimum make the order automorphisms of a chain into a lattice-ordered group.

A subgroup $C$ is convex if $c_1\le g\le c_2$ with $c_1,c_2\in C$ implies $g\in C$. In a linearly ordered group, convex subgroups form a chain under inclusion: positive elements witnessing incomparability would be comparable and force the smaller one into the other subgroup by convexity.

The nonnegative cone determines the partial order by $a\le b⟺a^{-1}b\in G_{\ge 1}$. It contains the identity element, is closed under products and conjugation, and intersects its inverse only in the identity. Conversely these conditions define a two-sided invariant partial order. Its strictly positive part omits the identity; conventions for $G^{+}$ differ, so specify which is intended.

For a sufficient statistic $(C,S)$ and parameter $(\psi ,\lambda )$, a useful strong reduction requires the law of $S$ to be free of $\psi$ and the conditional distribution of $C$ given $S$ to be free of the nuisance parameter $\lambda$. In the discrete case it giveswhere ${\varphi }_0$ sums to one on each $(C,S)$ fiber, ${\varphi }_S$ sums to one over $s$, and ${\varphi }_C$ sums to one over $c$ separately for every fixed $s$. Ordinary ancillarity, or just the law of $S$ being free of $\psi$, does not by itself guarantee that conditioning removes $\lambda$.

Keep $X,Y$ as independent fair bits, introduce an independent Bernoulli random variable $N$ with $\mathbb{P}(N=1)=\epsilon$, and set $Z=X\oplus Y\oplus N$. The extra bit flips the integer parity rule with probability $\epsilon$. For each even-integer parity triple the probability is $(1-\epsilon )/4$; for each odd-integer parity triple it is $\epsilon /4$.

Every coordinate remains fair, and every pair remains independent for every $0\le \epsilon \le 1$. For example, after fixing $X$, the fair independent $Y$ makes $Y\oplus N$ fair regardless of $\epsilon$. A pair's joint probability distribution therefore does not change as noise destroys the three-way relation.

Knowing $X,Y$, uncertainty in $Z$ is exactly the uncertainty in $N$. The chain rule for information entropy giveswhere $h_2$ is binary entropy. HenceAt $\epsilon =1/4$, the result is approximately $0.188722$ bits, although all three pairwise mutual informations are zero. At $\epsilon =1/2$, all eight triples have probability $1/8$, giving joint independence and zero total correlation. At $\epsilon =1$, the complementary integer parity rule is deterministic again, so total correlation returns to one bit.

The example distinguishes "there is a dependence" from "a pair reveals it." Once $X,Y$ are known together, $Z$ is predictable at zero noise; knowing either one alone leaves a fair bit. Total correlation detects the joint departure from independence, but does not by itself specify which variables cause which others.

If a binary length-$N$ cyclic code has generator of degree $r$ and check polynomial $h$ of degree $N-r$, use the coefficient vectors of $h^{\ast },X{h}^{\ast },\dots ,X^{r-1}{h}^{\ast }$ as the rows of a parity-check matrix. They have distinct leading positions and are independent. Their scalar products with a word are the coefficients of $X^{N-r},\dots ,X^{N-1}$ in its product with $h$, which vanish for codewords.

// Target: algebra.bigb

Let $F(X)=X^N+1$ over the field with two elements and write $h={\sum }_{j=0}^{N-1}{h}_j{X}^j$. In the quotient ring ${\mathbb{F}}_2[X]/(F)$, multiplication by $h$ is represented on coefficient column vectors by the circulant matrixThe kernel is a cyclic code, because multiplication by $X$ preserves the condition $hv\equiv 0(\mathrm{mod}F)$.

More precisely, set $d=\gcd (F,h)$ and write $F=d{F}_1$, $h=d{h}_1$, with $F_1$ and $h_1$ coprime polynomials. Then $F\mid hv$ if and only if $F_1\mid v$. Hence the kernel is generated by $F/d$, has dimension $\mathrm{deg}d$, and $B$ has rank $N-\mathrm{deg}d$. Selecting independent rows of $B$ therefore gives a parity-check matrix for this cyclic code, even when $h$ was not supplied as a canonical check polynomial. If $h=F/g$ for a divisor $g$ of $F$, the kernel is exactly the cyclic code generated by $g$.

// Target: algebra.bigb

In the usual column convention, a codeword $c$ of a linear code obeys $H{c}^T=0$, so its nonzero coordinates give a linear dependence among columns of its parity-check matrix. Conversely a dependence gives a nonzero codeword supported on those coordinates. Minimizing the support gives the minimum Hamming distance of a linear code. In the transposed convention $cP=0$, the corresponding objects are the rows of $P$. The size of a dependent set means the number of its coordinate positions, even when some check vectors coincide.

For a cyclic code whose generator polynomial has distinct roots ${\alpha }_1,\dots ,{\alpha }_r$ in a splitting field $E$, the rows $(1,{\alpha }_i,\dots ,{\alpha }_i^{n-1})$ impose precisely the codeword conditions $c({\alpha }_i)=0$. Expanding each value in a basis of $E$ over the original finite field, then discarding dependent equations, gives a parity-check matrix over that field of rank $r$. Alternatively choose one root from each Frobenius conjugate orbit and expand its value in the field it generates.

Writing $U=\mathrm{Re}(\beta +\gamma )$ and $V=\mathrm{Im}(\beta +\gamma )$, the squared amplitude is $z_{\pm }=[\mu U\pm \sqrt{(U^2+V^2){\nu }^2-{\mu }^2{V}^2}]/(U^2+V^2)$. Reality of the square root does not guarantee positivity. For example, $U=1,V=0,\mu =-1,\nu =1/2$ gives two negative roots. Under the usual supercritical assumptions $\mu >0,U>0$, the plus root is positive whenever the discriminant is nonnegative.

// Destination: mathematical-biology.bigb

Products of two univariate bases form a surface basis. Nonnegativity, partition of unity and coordinate-wise linear precision pass to the products by multiplication of the respective univariate identities. If the factors are $C^r$ and $C^s$, each mixed derivative through orders $r,s$ is the product of the corresponding derivatives, giving continuous mixed derivatives and joint $C^{\min (r,s)}$ regularity.

Core numerical enquiries return a surface point and its derivatives through second order, together with the parameter domain, patch boundaries and regularity information. At a regular point these determine its tangent plane, normal vector, first and second fundamental forms and curvatures. Subdivision and certified enclosing volumes provide spatial search and intersection support. Singular charts or discontinuity boundaries require one-sided or alternative-chart information; a zero cross product does not define a unique normal vector.

For nonnegative barycentric parameters, these total-degree Bernstein basis functions define a surface over a triangle. The triangular control net has $(n+1)(n+2)/2$ points. Its index constraint couples the three exponents, so it is not a tensor product of two independent univariate bases. The multinomial theorem gives nonnegativity and partition of unity.

A general conical surface is swept out by straight generators through one apex $A$. An invertible affine map sends it to another such surface, since it sends each generator to a line through the transformed apex. A projective transformation preserves the projective concurrence of generators, but can send the apex to infinity, producing parallel generators in the affine chart. Therefore finite-apex cones and cylinders are not separately projectively invariant Euclidean classes.

A cycloid is the plane curve traced by a point on a circle rolling without slipping along a straight line. A parametrization with rolling radius $R$ is $x=R(\theta -\sin \theta )$, $y=R(1-\cos \theta )$; reversing the sign of $y$ reflects the curve. Its cusps occur at integer multiples of $2\pi$. A descending cycloidal arc solves the classical brachistochrone problem.

A parametric curve interface supplies its parameter domain, evaluation, and derivatives where they exist. Robust geometric algorithms also need restricted subcurves and certified range or derivative bounds on parameter intervals. Tangent, speed and curvature are derived from point and derivative enquiries when the relevant derivatives are regular. Finite point samples alone cannot certify the absence of every plane intersection of an arbitrary smooth curve.

A variable normal offset displaces a parametric surface along its unit normal vector by a scalar field $d(u,v)$. Unlike a constant-distance parallel surface, its derivatives also contain $d_{u}n$ and $d_{v}n$.

// Target: analysis.bigb