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For a nonempty subset of a hypercube graph, split it into sections of sizes $a,b$. The proof of the entropy edge bound uses $H_2(t)\ge 2t$, $t=\min (a,b)/(a+b)$. Strict concavity makes equality possible only at $t=0$ or $t=1/2$. In the first case one section is empty; in the second the sections coincide, since all possible crossing edges must be present. Induction forces a coordinate subcube. Conversely every coordinate subcube has the displayed edge count. At size $2^{n-1}$ these are exactly the two halves determined by fixing one coordinate.

For independent positive random variables $Y_1,Y_2$ with densities, finite differential entropies and finite ${\mu }_i=\mathbb{E}{\log }_2{Y}_i$, applying the entropy power inequality to $Z_i=\ln Y_i$ givesIndeed $h(Z_i)=h(Y_i)-{\mu }_i$, while $h(Y_1{Y}_2)=h(Z_1+Z_2)+{\mu }_1+{\mu }_2$, provided the sum's differential entropy is defined. Positivity alone does not ensure these expressions exist: $Y=e^T$ with $T$ having Cauchy distribution has undefined $\mathbb{E}{\log }_{2}Y$.

// Target: algebra.bigb

For a discrete memoryless source of entropy $H$, every uniquely decodable binary code has expected length at least $H$. Shannon code lengths satisfy $H\le \overline{\ell }<H+1$, and block coding can make the expected length per source symbol arbitrarily close to $H$.

For a flux that is a differentiable convex function with unique minimum at $s_{\ast }$, define $f^{+}(u)=f(u)-f(s_{\ast })$ for $u>s_{\ast }$ and zero otherwise, and $f^{-}(u)=f(u)-f(s_{\ast })$ for $u<s_{\ast }$ and zero otherwise. The Engquist-Osher flux is $F(a,b)=f(s_{\ast })+f^{+}(a)+f^{-}(b)$. Thus transsonic rarefaction wave states $a\le s_{\ast }\le b$ give $F=f(s_{\ast })$, while reversed transsonic states give $F=f(a)+f(b)-f(s_{\ast })$. This follows by integrating the positive and negative parts of $f^{\prime }$.

The endomorphism bundle connection is ${\tilde{D}}_X\alpha =D_X\circ \alpha -\alpha \circ D_X$ on sections. Expanding twice makes the two middle terms cancel, proving the displayed commutator identity. Subtracting the derivative along the Lie bracket of vector fields gives $\tilde{R}(X,Y)\alpha =[R(X,Y),\alpha ]$. The operators $D_X$ are differential operators, rather than fibrewise endomorphisms; the corrected curvature operator is fibrewise linear.

For fixed $x_1,\dots ,x_d$, the vector $({\alpha }_n(x_j){)}_{j=1}^d$ converges in distribution to $(B(G(x_j)){)}_{j=1}^d$, where $B$ is a standard Brownian bridge. To prove it, write the vector as $n^{-1/2}$ times the sum of independent centred bounded indicator vectors. Their covariance matrix is $G(\min (x_i,x_j))-G(x_i)G(x_j)$. The multivariate central limit theorem gives a centred multivariate normal distribution with this covariance, exactly the bridge vector. Singular covariance matrices, including endpoint times or repeated values of $G$, cause no problem. This is a finite-dimensional assertion; a full function-space limit additionally requires tightness.

For an independent sample with cumulative distribution function $F$, the empirical distribution process converges in the supremum norm on ${\ell }^{\infty }(\mathbb{R})$ to the F-Brownian bridge. The convergence uses outer expectation and inner expectation when finite-sample Borel measurability fails. It is a full function-space theorem, stronger than the Finite-dimensional Brownian bridge limit of an empirical distribution process. No continuity of $F$ is required. The continuous mapping theorem gives convergence of the supremum statistic; for continuous $F$, its range contains $(0,1)$ and the bridge vanishes at zero and one, giving the Kolmogorov-Smirnov theorem. The function-space statement is stated here as a theorem; its tightness assertion is an additional ingredient beyond the finite-dimensional covariance calculation.

Every complex elliptic curve, as a complex Lie group, is the quotient of $\mathbb{C}$ by a rank-two lattice. The coordinate functions are built from the Weierstrass elliptic function and its derivative. Holomorphic group homomorphisms lift to scalar multiplication on $\mathbb{C}$, and their endomorphism condition is $aL\subseteq L$. This turns complex multiplication into a statement about lattice stabilizers.

Let $E$ be a smooth plane cubic with an inflection point $O$ as identity. A point $P$ with $3(P-O)=0$ in Picard group has a line whose intersection divisor with $E$ is $3P$. Indeed, ${\mathcal{O}}_E(3P)\cong {\mathcal{O}}_E(3O)\cong {\mathcal{O}}_E(1)$. On this genus-one curve, the Riemann-Roch theorem gives $h^0({\mathcal{O}}_E(1))=3$: the dual term has negative degree. Restriction of ambient linear forms is injective and its source also has dimension three, so it is an isomorphism. The section vanishing at $3P$ therefore comes from a line. Conversely, such a line gives $3P\sim 3O$, so $P$ is a three-torsion point. This includes $O$; points of exact order three are the other such points.

Fix $O\in E(k)$. On a genus-one smooth projective curve, every degree-one divisor class has a unique effective point representative by the Riemann-Roch theorem: its section space has dimension one, since the complementary canonical-divisor degree is negative. Thus $P\mapsto [P-O]$ is a bijection onto the degree-zero Picard group of a curve. Transporting divisor-class addition proves the group axioms, with identity $O$, and characterizes $P+Q$ by $P+Q\sim R+O$.

For $d\in {\mathbb{Q}}^{\times }$, the smooth projective curve $X^3+Y^3+d{Z}^3=0$, with identity $O=(1:-1:0)$, is a genus-one elliptic curve. Its partial derivatives have no common projective zero. The invertible linear projective transformationsends it to ${\mathcal{Y}}^2\mathcal{Z}={\mathcal{X}}^3-432d^2{\mathcal{Z}}^3$, taking $O$ to $(0:1:0)$. Indeed $(X+Y{)}^3+3(X+Y)(X-Y{)}^2=4(X^3+Y^3)=-4d{Z}^3$. In affine coordinates $u=X/Z,v=Y/Z$, this gives $x=-12d/(u+v)$ and $y=-36d(u-v)/(u+v)$, with inverse $u=(y-36d)/(6x)$, $v=(-y-36d)/(6x)$. Thus diagonal sums of two cubes are naturally studied through elliptic-curve addition formula.

For an elliptic curve $E/K$, regard $E(K^{\mathrm{sep}})$ as a discrete Galois module for the absolute Galois group $G_K$. Its continuous Galois cohomology has $H^0(K,E)=E(K)$; $H^1(K,E)$ classifies torsors under $E$. For an integer $n$ invertible in $K$, the Kummer exact sequence of an elliptic curve gives $0\to E(K)/nE(K)\to H^1(K,E[n])\to H^1(K,E)[n]\to 0$. If $[n]Q=P$, the first map sends $P$ to $\sigma \mapsto \sigma Q-Q$. Changing $Q$ changes the cocycle by a coboundary; this class vanishes exactly when $P$ has a division point over $K$.

For distinct finite point classes on a complex elliptic curve, ordinary column determinants test whether a line meets the curve at those three points. Repeated columns instead vanish for a purely linear-algebraic reason. To prescribe a tangent intersection at $u$, replace the second equal column by its derivative; to prescribe a triple intersection, use $c(u),c^{\prime }(u),c^{\prime \prime }(u)$. These confluent determinants vanish exactly when the corresponding degree-three intersection divisor has point sum zero, away from the affine chart's pole. A local homogeneous lift supplies the version at the origin.

Row additions reduce an invertible matrix to diagonal form without changing determinant. A determinant-one diagonal matrix is a product of diagonal two-coordinate blocks $\mathrm{diag}(a,a^{-1})$, each expressible as a product of six elementary transvection matrices. Therefore these matrices generate the special linear group over a field.

// Target: algebra.bigb

A group of exponent two is abelian. Give such a finite group the additive notation of a vector space over ${\mathbb{F}}_2$, with zero acting as zero and one as the identity scalar. The group axioms and $g+g=0$ verify the vector-space laws. A basis with $d$ vectors then gives exactly $2^d$ elements. Thus a finite group whose elements all have order at most two must have power-of-two order.

The electroweak scale is the characteristic energy associated with electroweak symmetry breaking. With the Standard Model Higgs doublet normalized by $⟨H⟩=(0,v/\sqrt{2}{)}^T$, its vacuum expectation value satisfies $v=(\sqrt{2}{G}_F{)}^{-1/2}\simeq 246\mathrm{GeV}$ and $m_W=gv/2$. Its smallness compared with possible ultraviolet scales motivates the hierarchy problem.

Renormalization-group evolution can change a Higgs squared-mass parameter from positive at a high input scale to negative at lower scales, making a symmetry-breaking scalar expectation energetically favorable. In the Minimal supersymmetric Standard Model, the large top Yukawa coupling can drive the up-type Higgs soft squared mass downwards as the scale is lowered. Actual breaking also requires the other Higgs-potential parameters to give a stable suitable minimum; the running of one mass alone is not sufficient.

For completely degenerate, nonrelativistic electrons in fully ionized hydrogen and helium with hydrogen mass fraction $X$, $n_e=(1+X)\rho /(2m_p)$. Filling the spin-degenerate momentum states to $p_F$ gives $n_e=8\pi p_F^3/(3h^3)$ and $P_e=(1/3){\int }_0^{p_F}(p^2/m_e)(8\pi p^2/h^3)dp=8\pi p_F^5/(15m_e{h}^3)$. Eliminating $p_F$ gives $P_e=K{\rho }^{5/3}$, whereIt is a polytropic index $3/2$ relation. Complete degeneracy suppresses thermal dependence, while the nonrelativistic assumption requires $p_F\ll m_{e}c$.

In a symmetry-plane cross-section, write $\mathbf{p}=r(\theta )\mathbf{n}(\theta )$, $v=1/r$, and $\mathbf{t}=\mathbf{n}\prime$. The elastic wave surface is $\mathbf{g}=v\mathbf{n}+v\prime \mathbf{t}$. Its derivative vanishes when $v+v\prime \prime =(r^2+2r{\prime }^2-rr\prime \prime )/r^3=0$, exactly an inflection point of the regular slowness surface. At a simple zero the second and third derivatives of $\mathbf{g}$ are linearly independent, giving an ordinary cusp. Smooth nonconvex slowness surface sheets can therefore generate cusped wavefronts.

A solid-to-fluid interface can turn spherical source P waves into parallel vertical rays if its height $Z(R)$ satisfies the displayed travel-time identity, with $r=\sqrt{R^2+Z^2}$. The positive branch through $Z(0)=z_0$ is a hyperboloid when $\alpha >{\alpha }_f$. Differentiating the travel-time identity enforces Snell law for elastic and acoustic waves. Travel time above the interface is then $z_0/\alpha +(z-z_0)/{\alpha }_f$, independent of cylindrical radius. The leading wavefront is planar although its wave amplitude need not be uniform across that plane.