Research proof index

Let $f$ be a twice differentiable complex-valued function whose first two derivatives are integrable, and let $g$ be a twice differentiable compactly supported cutoff that equals $1$ on $[-1,1]$ and vanishes outside $(-2,2)$. Then $g(x/R)f(x)$ converges to $f$ as $R\to\infty$ in the project norm $\|h\|=\int_{\mathbb R}|h(x)|\,dx+\frac{1}{4\pi^2}\int_{\mathbb R}|h''(x)|\,dx.$

An asymmetric weak-PFR estimate. Let $A,B$ be nonempty finite subsets of a rank-$n$ free $\mathbb Z$-module $G$. There are a subgroup $N\le G$, cosets $x,y\in G/N$, and nonempty fibers $A_x=\{a\in A:a+N=x\}$ and $B_y=\{b\in B:b+N=y\}$ such that $n\log2\le\log|G/N|+40d[U_A;U_B]$ and $\log|A|+\log|B|-\log|A_x|-\log|B_y|\le34\bigl(d[U_A;U_B]-d[U_{A_x};U_{B_y}]\bigr).$

Uniqueness of weak multi-derivatives on U: any two candidates must agree almost everywhere on U. The proof reduces to the du Bois-Reymond lemma via the defining identity.

The Chebyshev function is asymptotic to the identity: $\psi(x)\sim x$ as $x\to\infty$. Equivalently, $\psi(x)/x\to1$.

The rational points of the quadratic twist, viewed inside E(L) via the isomorphism over L, are exactly the points of E(L) on which the nontrivial element of Gal(L/K) acts as -1 (just as E(K) consists of the points on which it acts as +1). One inclusion is Affine.Point.map_baseChange (the base change of a K-point is σ-fixed) together wi...

An explicit L-isomorphism (Eᶿ)ᴸ ≅ Eᴸ (the change of variables of the module docstring) which moreover is anti-equivariant for the Galois action: its conjugate by the nontrivial σ ∈ Gal(L/K) differs from it by the automorphism [-1] of E. This nontrivial cocycle is the origin of the twist being a nontrivial form of E.