For irrational $\alpha$, the circle rotation $T(x)=x+\alpha (\mathrm{mod}1)$ preserves Lebesgue measure and is an ergodic transformation.

For nonzero coefficients of $A^{\ast }V$ in $\dot{B}$ and $AB$ in $\dot{V}$, a relative phase rotation makes $c_1$ real positive. Positive rescalings of $A,V,B$ then normalize both magnitudes to one. The phase of $c_1{c}_3$ is invariant under these rescalings and remains as the phase of normalized $c_3$. Zero couplings cannot be normalized to one; they are degenerate parameter cases.

For real $c_2,d_4$ and the simplified magnetic amplitude equations, the pure-velocity branch is $V=s{e}^{i{\omega }_{3}t}$ with $A=B=0$. In its rotating frame the magnetic linearization on $(B,\mathrm{Re}A,\mathrm{Im}A)$ has rows $({\mu }_1-e{s}^2,2s,0)$, $(c_{2}s,{\mu }_2,-\Delta )$ and $(0,\Delta ,{\mu }_2)$, where $\Delta ={\omega }_2-{\omega }_3$. Its zero determinant gives the displayed onset relation. A nonzero magnetic branch needs a kernel involving both magnetic amplitudes, a transverse parameter crossing and nondegenerate nonlinear saturation; determinant zero alone is insufficient in degenerate cases.

For complex velocity amplitudes $v_x=v_y=V$ and magnetic amplitudes $b_x=-b_y=B$ in Alfvén-speed units, phase averaging gives the displayed stress. For fixed horizontal velocity energy, equal radial and azimuthal components maximize their positive Reynolds stress. Opposite equal magnetic components maximize the outward tension contribution of the Maxwell stress tensor, since $-B_x{B}_y$ is then positive. These conclusions follow from $2|ab|\le |a{|}^2+|b{|}^2$. The vertical channel transports angular momentum outward and extracts energy from decreasing angular velocity.

For $f(x)=x^2$ at any real $a$, polynomial factorization givesIf $|x-a|<1$, the triangle inequality yields $|x+a|\le |x-a|+2|a|<2|a|+1$. Therefore the displayed choice of $\delta$ makes $|x-a|<\delta$ imply $|f(x)-f(a)|<\epsilon$. This proves continuity and gives an explicit local error bound, rather than an inference from a drawing.

At $a=1$ and $\epsilon =0.6$, this sufficient choice is $\delta =0.2$. The permitted function inputs satisfy $0.8<x<1.2$; their actual function outputs lie in $(0.64,1.44)$, inside the requested band $(0.4,1.6)$. A sufficient input window need not be the largest possible one.

Fix the second moment $v>0$. Put $B=\sqrt{5v}$ and $E(u)=3(B^2-u^2{)}_{+}/(4B^3)$. For every nonnegative unit-integral kernel for density estimation $K$ with the same second moment and finite $R(K)$, the mass and second-moment constraints give $\int (B^2-u^2)(K-E)=0$. Outside $[-B,B]$, $E=0$ and $(B^2-u^2)K\le 0$, so $\int E(K-E)\ge 0$. Expanding the square gives $R(K)-R(E)=\int (K-E{)}^2+2\int E(K-E)\ge 0$, with equality only when $K=E$ almost everywhere. Since $R(K{)}^2{\mu }_2(K)$ is scale invariant, rescaled Epanechnikov kernels minimize the optimized asymptotic mean integrated squared error over all such second moments. This conclusion concerns nonnegative second-order kernels; signed higher-order kernels obey a different bias expansion.

For a scalar conservation law law $u_t+f(u{)}_x=0$, the Kruzhkov entropy inequalities requirein distributions for every real constant $c$, together with the initial trace. This means the integral against every nonnegative compactly supported test function is nonpositive after applying the distributional derivatives. For a smooth solution the entropy balance is an equality away from $u=c$. For a viscous approximation $u_t+f(u{)}_x=\epsilon u_{xx}$, smooth convex approximations $\eta$ of $|u-c|$ give ${\partial }_t\eta (u)+{\partial }_{x}q(u)=\epsilon {\partial }_{xx}\eta (u)-\epsilon {\eta }^{\prime \prime }(u)u_x^2\le \epsilon {\partial }_{xx}\eta (u)$, with $q^{\prime }={\eta }^{\prime }{f}^{\prime }$. Letting the entropy smoothing vanish and then taking a bounded, strongly locally convergent zero-viscosity limit proves the inequality. The discrete entropy inequality of a monotone conservative scheme is its numerical analogue.

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 finite-alphabet memoryless classical information source with information entropy $H$, every uniquely decodable code for blocks of $m$ source symbols has expected binary length at least $mH$. This follows from Kraft inequality for uniquely decodable codes and the entropy lower bound for prefix codes.

If at least two source symbols have positive probability, give each positive-probability block $w$ Shannon length $\ell (w)=\lceil -{\log }_{2}p(w)\rceil$. These lengths satisfy Kraft inequality, so a prefix code exists, and independence givesThus coding longer blocks approaches the information entropy in expected bits per source symbol. This is an average-length statement: rare blocks can have long codewords.

For a source with only one possible symbol, a one-bit codeword for each length-$m$ block gives rate $1/m\to 0=H$ when positive codeword lengths are required. The limiting theorem therefore remains true, although the strict one-block bound $\overline{\ell }<H+1$ needs the nontrivial-alphabet assumption. Countably infinite alphabets require additional care about finite information entropy and the chosen code construction.

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.