The fixed point zero has multiplier $F^{\prime }(0)=\mu$, so it is attracting for $|\mu |<1$ and repelling for $|\mu |>1$. Other real fixed points satisfyTheir multiplier is $M=1+bx+2a{x}^2$, and they are attracting exactly when $-2<bx+2a{x}^2<0$. At equality $M=\pm 1$ the linear test is inconclusive. For $b\ne 0$ two branches meet at $x_s=-b/(2a)$, ${\mu }_s=1+b^2/(4a)$, a generic saddle-node bifurcation. At $\mu =1$, the nonzero branch crossing zero gives a transcritical bifurcation when $b\ne 0$.

The three sketches use $(a,b)=(1,0),(1,1/2),(-1,-1/2)$ respectively; case (iii) therefore illustrates the subcritical-flip choice.

Let $p$ be a prime number, let $n={\sum }_r{\alpha }_r{p}^r$ with $0\le {\alpha }_r<p$, and let $c_r$ be the sum of partition sizes at level $r$ of the core tower of a partition $\lambda ⊢n$. ThenIndeed $n={\sum }_r{c}_r{p}^r$. Carrying $p$ units from position $r$ to position $r+1$ decreases ${\sum }_r{c}_r$ by $p-1$. Normalization to the base-$p$ digits therefore gives ${\sum }_r{c}_r\ge {\sum }_r{\alpha }_r$, with equality exactly when no carry is needed. The P-adic valuation of a symmetric-group character degree from the core tower is the difference divided by $p-1$.

Under the hypotheses for the one-nearest-neighbour asymptotic risk, the limiting risk satisfies $R^{\ast }\le R_{\infty }\le 2R^{\ast }$. Put $a=\min (\eta ,1-\eta )$: then $a\le 2a(1-a)\le 2a$ since $0\le a\le 1/2$. Integrate, using the definition of Bayes risk. Constant $\eta =1/4$ gives limiting risk $3/8$ and Bayes risk $1/4$, so the lower comparison is not generally equality.

By dominant root-lattice highest weights have zero weight, a minuscule weight in the root lattice has zero in its Weyl group orbit. Since every element of the Weyl group is invertible, the highest weight must itself be zero. Thus equality of the weight lattice and root lattice rules out every nontrivial minuscule representation without requiring a classification of root systems.

For the Chordal Loewner equation driven by $U_t=a\sqrt{t}$, the hull is a straight slit whose endpoint is proportional to $\sqrt{t}$. Put $r_{\pm }=(a\pm \sqrt{a^2+16})/2$ and $\alpha =-r_{-}/(r_{+}-r_{-})\in (0,1)$. Its inverse map isusing upper-half-plane branches. The interval between the two real roots maps onto the two sides of a segment of argument $\pi \alpha$.

For a smooth function $f$ and vector field $X$,on every tensor field, with $X\otimes df$ interpreted as the endomorphism $Y\mapsto Y(f)X$. Both sides vanish on functions; on vector fields this follows from $[fX,Y]=f[X,Y]-Y(f)X$. Both are contraction-compatible tensor derivations, so agreement on functions and vector fields proves equality on all tensor types. In particular ${\mathcal{L}}_{fX}\omega =f{\mathcal{L}}_X\omega +\omega (X)df$.

The adjoint of $L=\omega \wedge -$ lowers real degree by two and bidegree by $(1,1)$. On a compact Kähler manifold it acts on cohomology through harmonic differential forms, because it commutes with the Hodge Laplacian. It does not generally preserve all closed forms.

In complex dimension $n$, the Lefschetz operator of a Kähler manifold and adjoint Lefschetz operator satisfy $[\Lambda ,L]=(n-k)\mathrm{id}$ on degree $k$. In middle degree, this yields $\Vert L\alpha {\Vert }^2=\Vert \Lambda \alpha {\Vert }^2$.

For degree $k\le n$, a form is primitive if $\Lambda \alpha =0$, equivalently $L^{n-k+1}\alpha =0$. In middle degree $k=n$ this is equivalent to $L\alpha =0$. On a compact Kähler manifold, the same definitions apply to cohomology via harmonic differential forms.

If two nearby inner planets couple much more strongly to each other than to a distant third planet, their slow secular eigenmode has a common complex inclination. With $L_{12}=L_1+L_2$, the effective inner-to-outer coupling isAngular momentum balance makes the reverse coupling $c{L}_{12}/L_3$. The reduced matrix has eigenvalues $0$ and $-c(1+L_{12}/L_3)$. If $L_3\gg L_{12}$, the nonzero frequency is approximately $-c$, while the fast inner differential mode has frequency $-(B_{12}+B_{21})$.

Apply the entropy solution inequality for $u(t,x)$ with level $v(s,y)$ and for $v(s,y)$ with level $u(t,x)$, integrate both in the other pair of variables, and add them. Symmetry of the Kruzhkov entropy flux produces derivatives ${\partial }_t+{\partial }_s$ and ${\partial }_x+{\partial }_y$. A nonnegative mollifier concentrated near the diagonal then yields the Kato inequality for scalar conservation laws. Boundary-time terms require an initial trace argument, not merely interior translation continuity.