For a smooth genus one curve with origin $O$, the map $P\mapsto [(P)-(O)]$ identifies geometric points with the degree-zero Picard group of a curve. Riemann-Roch theorem gives a unique effective representative for each divisor class of degree one: its Riemann-Roch space has dimension one. A line cuts a plane cubic in a divisor linearly equivalent to $3(O)$ when $O$ is the Weierstrass flex. The two successive line intersections in the chord-and-tangent group law therefore give addition of the corresponding divisor classes. Associativity follows by injectivity of the identification and associativity of divisor-class addition, including tangent intersection multiplicities.

Assume the elliptic-curve discriminant is nonzero. Then $b\ne 0$, and tangency at $P$ gives $2P=(-b,b(a-1))$ and $-2P=(-b,0)$. The line through $P,2P$ has slope $1-a$ and gives $3P=(1-a,a-b-1)$, $-3P=(1-a,(1-a{)}^2)$. Consequently $P$ has exact order four when $a=1$, exact order five when $a=b+1$, and exact order six when $b=(a-1)(2-a)$. The nonsingularity condition excludes the zero values of $b$ that would otherwise invalidate these torsion conclusions.

The equality holds when an alternating unit-norm spline $s_{\ast }$ has as many ordered extremal sites as the dimension, and those sites satisfy the Schoenberg–Whitney theorem. For every admissible set, sampling has operator norm at most one, giving $\kappa (\mathcal{S})\le \Vert A_{\mathbf{x}}^{-1}{\Vert }_{\infty }$. At the alternating sites the checkerboard inverse of a totally nonnegative matrix identifies each inverse absolute row sum with the magnitude of the corresponding coefficient of $s_{\ast }$. The Chebyshev spline coefficient and dual norm equality therefore gives equality. This minimizes amplification into B-spline coefficients; it does not by itself prove optimality for the distinct Lebesgue constant of interpolation.

Fix the Bernstein filtration on the Weyl algebra. Two good filtrations of a module are bounded shifts of one another: $F_{i}M\subseteq G_{i+c}M$ and conversely. If an operator of degree $d$ has symbol annihilating ${\mathrm{gr}}_{F}M$, its $r$th power lowers the expected filtration degree by $r$. For $r>2c$, comparison of filtrations forces the same power of its symbol to annihilate ${\mathrm{gr}}_{G}M$. Reversing the roles gives equality of the radical annihilator ideals and hence equality of support of a module. The algebra filtration is held fixed; changing between order and Bernstein filtrations need not preserve the same variety.

Let $H\le G$ be finite abelian and let $\chi :H\to \mathbb{T}$ be a character of a finite abelian group. To adjoin $x\in G$, let $d$ be the least positive integer with $dx\in H$ and choose a complex number $z$ of modulus one with $z^d=\chi (dx)$. Then $\tilde{\chi }(h+jx)=\chi (h)z^j$ is well-defined and extends $\chi$ to $H+⟨x⟩$. There are exactly $d$ choices, corresponding to the $d$ roots. Iterating proves character extension to $G$, proves that $G$ has exactly $|G|$ characters, and proves separation of points by extending a primitive character of the cyclic subgroup generated by a nonzero point. No structure theorem is needed for this argument.

A multiplicative character is a group homomorphism from a finite field's multiplicative group of a finite field to the complex unit circle. A generator of the cyclic multiplicative group of a finite field produces a character of every order dividing $q-1$. Extend every such character to zero by value zero, including the trivial multiplicative character, when using cyclotomic indicator formulas.

For three independent uniform points on a sphere, condition on the central angle $\theta$ between the first two. The third point makes acute angles with both exactly when it lies in the intersection of their positive hemispheres. That intersection is a spherical lune of opening $\pi -\theta$, hence has probability $(\pi -\theta )/(2\pi )$. The first two points also need $\theta <\pi /2$. Integrating against the central angle between independent uniform sphere points density gives $(4\pi {)}^{-1}{\int }_0^{\pi /2}(\pi -\theta )\sin \theta d\theta =(\pi -1)/(4\pi )$.

Join three independent uniform points on a sphere when their central angle is acute. Given angle $\gamma$ between the first pair, the third point's two favorable hemispheres overlap in a spherical lune of area fraction $(\pi -\gamma )/(2\pi )$. If $\gamma <\pi /2$, connection requires membership in their union, of probability $1/2+\gamma /(2\pi )$. If $\gamma >\pi /2$, it requires membership in their intersection, of probability $1/2-\gamma /(2\pi )$. Integrating these against the central-angle density $\sin \gamma /2$ proves the displayed probability. Connectivity is weaker than all three pairwise angles being acute.

The cellular boundary ${\mathbb{Z}}^2\to \mathbb{Z}$ is $(m,n)$, with kernel $\mathbb{Z}$ and zero cokernel when the degrees are coprime. The van Kampen theorem gives trivial fundamental group. The Hurewicz theorem supplies a map from the sphere representing a homology generator, and the homological Whitehead theorem makes it a homotopy equivalence. For degrees $2,3$ the space cannot be homeomorphic to a sphere: an embedded circle in a sphere bounds its two complementary regions, so both characteristic boundary maps would have degree of absolute value one.

If an $n$-dimensional CW complex is extended by exactly one cell in dimension $n+1$, then its cellular cochain group in that degree is the coefficient field $k$. Regardless of how many higher cells are attached, the cohomology is a quotient of a subspace of that one-dimensional group. This gives the displayed bound. Finiteness of the higher cell sets is unnecessary for this particular conclusion.

For analytic holomorphic sections, the two-chart Čech cohomology uses convergent Laurent series, rather than only Laurent polynomials. A frame on the infinity chart equals $z^n$ times a finite-chart frame. Entire finite-chart functions remove Laurent powers $k\ge 0$, and infinity-chart functions remove powers $k\le n$. The positive and negative tails converge to entire chart functions, so only the finite gap $n<k<0$ remains. Thus the first cohomology has basis $z^{n+1},\dots ,z^{-1}$ when $n\le -2$ and is zero when $n\ge -1$. The same transition relation gives global sections of dimension $\max (n+1,0)$.

The Čech differential sends a cochain on $(p+1)$-fold intersections to the alternating sum of its restrictions to $(p+2)$-fold intersections. Each pair of omitted indices occurs twice with opposite signs in the composite, so ${\delta }^2=0$. It defines the Čech cohomology of the cover.

For real $C$, this analytic homogeneous field has real boundary values on both faces of the negative-axis crack, so it adds no face shear traction. Ahead of the tip it adds stress $-\mu C/\sqrt{x}$. Its stress tends to zero at infinity, but its primitive grows like $\sqrt{z}$, producing unbounded remote displacement and infinite total elastic energy. Thus merely requiring vanishing far stress does not uniquely fix the face-loaded crack solution. Excluding an added remote stress-intensity field, for example through the localized-load bounded-displacement far condition, removes this homogeneous ambiguity. Ordinary inverse-square-root tip stress remains allowed and has finite energy in any bounded neighbourhood of the tip.

Sampling consecutive Cardinal cubic B-splines at the centers of their supports gives diagonal values $2/3$ and adjacent values $1/6$, with all other values zero. Finite truncation gives the displayed B-spline collocation matrix, including its endpoint rows without wrapping or adding boundary conditions. Its inverse has alternating signs. The exact inverse norm of midpoint cubic spline collocation is less than three for each finite dimension and approaches three as the dimension grows.

For the unit-spaced cubic cardinal B-spline on $[0,4]$, the four scalar cubic Bernstein basis coefficient rows are $(0,0,0,1/6)$, $(1/6,1/3,2/3,2/3)$, $(2/3,2/3,1/3,1/6)$ and $(1/6,0,0,0)$. Each row applies on its interval with local parameter in $[0,1]$. They result from the compactly supported $C^2$ truncated-cube combination $\frac{1}{6}{\sum }_{j=0}^4(-1{)}^j(\genfrac{}{}{0ex}{}{4}{j})(x-j{)}_{+}^3$. Each graph piece is a cubic Bézier curve with abscissa controls $j,j+1/3,j+2/3,j+1$.

The positive Liouville volume for the base-orientation/positive-fibre orientation is $\mu ={\pi }^{\ast }{\Omega }_a\wedge \omega$. Its fibre integral is $2\pi {\Omega }_a$, so its total volume is $2\pi \mathrm{Area}(M)$. With the stated coframe convention, $\alpha \wedge d\alpha =-\mu$; the signed contact volume and the positive measure should not be confused. The fields $X,H,V$ preserve this volume by their structure equations.

On an oriented Riemannian surface, the vertical vector field generates positive rotation in each fibre of its unit tangent bundle: $Vf(x,v)={{\partial }_{t}f(x,\cos tv+\sin tiv)|}_{t=0}$. Hence $V(v)=iv$ and $V(iv)=-v$. In the canonical frame, $[V,X]=H$, $[V,H]=-X$ and $[X,H]=KV$.

If $h(t)={\int }_0^t{g}_{s}ds$ with deterministic $g\in L^2(0,\infty )$, the finite-time Cameron-Martin theorem densities have second moments $e^{{\int }_0^t{g}_s^{2}ds}$ bounded uniformly in time. Their uniformly integrable martingale limit is the strictly positive displayed density, giving equivalence on the full infinite continuous-path space. Finite-time densities alone do not guarantee such a global density: a nonzero constant drift has infinite energy and its infinite-horizon path law is singular, distinguished by the limiting ratio $B_t/t$.

For a minimally coupled massless scalar field in flat de Sitter spacetime, $a=-1/(H\tau )$ and the Canonically rescaled de Sitter scalar mode is $u_k=a{f}_k$. The equation $u_k^{\prime \prime }+(k^2-2/{\tau }^2)u_k=0$ and the positive-frequency condition $u_k\sim e^{-ik\tau }/\sqrt{2k\mathcal{V}}$ select the displayed mode. Direct differentiation verifies its Wronskian normalization, which ensures the equal-time canonical commutation relations with box-normalized creation and annihilation operators. A conformally coupled massless scalar has a different mode equation.

For Brownian motion started at zero, let $f_n$ equal $|x|$ outside $(-1/n,1/n)$ and $n{x}^2/2+1/(2n)$ inside. Its derivative is the clipped sign function $g_n(x)=\max \{-1,\min \{1,nx\}\}$. The Itô integral $M_t^{(n)}={\int }_0^t{g}_n(B_s)d{B}_s$ converges uniformly on bounded time intervals in $L^2$ to a continuous martingale $M$, by the Itô isometry and Doob L2 maximal inequality: the squared maximal error is at most $8\sqrt{2T/\pi }/n$. The Itô formula gives $\frac{n}{2}{\int }_0^t{1}_{\{|B_s|<1/n\}}ds=f_n(B_t)-f_n(0)-M_t^{(n)}$. Along $n=k^2$, the maximal errors are summable, so First Borel-Cantelli lemma gives uniform almost-sure convergence. The ratio-one interpolation for shrinking-window occupation integrals extends it to every integer $n$. The limit is $|B_t|-M_t=L_t^0$, the local time of a semimartingale in the Tanaka formula convention. The unnormalized factor $n$ gives twice this limit.