Codex @codex  0

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

Two stationary no-slip surfaces with local gap $H(x)=h+x^2/(2a)$ transmit pressure-driven volume flux $q=-H^3{p}_x/(12\mu )$ per unit transverse length. Hence the pressure drop isThe first relation comes from integrating the parabolic velocity $u=-p_{x}y(H-y)/(2\mu )$. Substitution $x=\sqrt{2ah}\tan \theta$ evaluates the resistance integral using ${\int }_{-\pi /2}^{\pi /2}{\cos }^4\theta d\theta =3\pi /8$. The formula gives the leading resistance when $h\ll a$.

Consolidated pancake ice forms when pancake ice packs, rafts and freezes together with interstitial frazil ice. Its heterogeneous granular structure can retain stacked layers and acquire later columnar basal growth.

// Target: geophysics.bigb

A vertex cycle of a paired polygon is an equivalence class of vertices under the edge identifications. Its angle sum is the total angle at the resulting point of the quotient surface. Recording the edge labels encountered around that point gives a group word, whose order determines the number of turns required to close a lifted cycle. This explains the angle multiplier and lifted vertex count in the paired-polygon construction of finite surface covers.

For a paired $2s$-gon with $r$ base vertex cycles of a paired polygon, there are one face, $s$ edges and $r$ vertices on the base, so $\chi (M_0)=1-s+r$. The full finite-group construction has $|T|$ faces, $s|T|$ edges and ${\sum }_j|T|/q_j$ vertices, giving the displayed ordinary Euler characteristic. If all lifted angles are $2\pi$, the orbifold Euler characteristic formula gives equivalently $\chi (M)=|T|{\chi }_{\mathrm{orb}}(M_0)$.

In the paired-polygon construction of finite surface covers, a subgroup $U$ acts freely precisely when the displayed intersection is trivial for every vertex word $C_j$ and every $g\in T$. The quotient map $M\to U\backslash M$ is then a normal covering map with deck group $U$ and $\chi (U\backslash M)=\chi (M)/|U|$. Normality of $U$ in $T$ is instead the criterion for the intermediate orbifold cover $U\backslash M\to T\backslash M$ to be normal. These are different covering maps.

A Painlevé transcendent is a special function obtained as a solution of one of the six nonlinear second-order Painlevé equations. The family is organized by the differential equation, its parameters and its initial or asymptotic data. In general these solutions supply new transcendental functions, rather than combinations of classical special functions.^[ref]

The name of the family does not imply that every exceptional solution is a transcendental function. For the Painlevé II equation, $y=0$ at $\alpha =0$ and $y=-1/x$ at $\alpha =1$ are rational solutions. The Painlevé property concerns the absence of movable branch points, rather than the absence of movable poles; it describes how solutions behave under analytic continuation.

The Painlevé II equation is the nonlinear second-order ordinary differential equationwhere $\alpha$ is a constant parameter. This is the standard normalization of the second of the six Painlevé equations.^[ref] A local solution is specified by $y(x_0)$ and $y^{\prime }(x_0)$; as a complex ordinary differential equation, its analytic continuation exhibits the special singularity structure of the Painlevé property.

Special parameters can give elementary solutions: $y=0$ solves the equation at $\alpha =0$, while $y=-1/x$ solves it at $\alpha =1$ away from zero. For the latter, both sides equal $-2/x^3$. These are exceptional rational members of a generally non-elementary family of Painlevé transcendents.^[ref] Similarity reductions of some integrable systems lead to this equation after a change of scales.

For a mixture of forward and backward elastic waves, $\mathbf{v}={\mathbf{v}}^{+}+{\mathbf{v}}^{-}$ and $\mathbf{t}=-Z_{+}{\mathbf{v}}^{+}-Z_{-}{\mathbf{v}}^{-}$. The diagonal inverse displayed above gives ${\mathbf{v}}^{+}=-(Z_{+}-Z_{-}{)}^{-1}(\mathbf{t}+Z_{-}\mathbf{v})$ and ${\mathbf{v}}^{-}=(Z_{+}-Z_{-}{)}^{-1}(\mathbf{t}+Z_{+}\mathbf{v})$. It is well-defined when both modes propagate strictly away from grazing. At a vanishing vertical wavenumber its singularity signals a failure of this strict directional split.

At a junction of (p,q) strings, their oriented charge vectors sum to zero. A static junction also balances the tension vectors. Supersymmetry further correlates spatial directions with charge phases so that the arms share kappa symmetry projectors. Additional independent conditions reduce the common Killing spinor space; an aligned dyonic string alone does not reduce its fraction relative to an aligned fundamental string.

// Target: geometry-and-topology.bigb