Research proof index

If G is generated by a set s, then the slash action by elements in G is uniquely determined by the slash action by elements in s. See slashaction_generators for a version where s is a set of elements in SL(2, ℤ).

For a set of 1-family SM charges satisfying all ACCs except the gravitational, if the Q charge is zero then the charges satisfy the gravitational ACCs.

Given $a<b$ and $c<d$, there is a smooth compactly supported function $\Psi:\mathbb R\to\mathbb R$ with $\mathbf 1_{[b,c]}\le\Psi\le\mathbf 1_{(a,d)}$ and support exactly $(a,d)$. In the usual ordered case $a<b\le c<d$, this is a smooth cutoff equal to $1$ on $[b,c]$, strictly supported in $(a,d)$, and taking values between $0$ and $1$.

For the smoothed Chebyshev integrand $F(s)=-\frac{\zeta'(s)}{\zeta(s)}\,\mathcal M(\widetilde{1_\varepsilon})(s)X^s,$ the central part of the normalized vertical integral on $\operatorname{Re}s=\sigma_1$ can be shifted to $\operatorname{Re}s=\sigma_2<\sigma_1$. With the project's contour-piece notation, the complete decomposition is $I_{37}=I_3-I_4+I_5+I_6+I_7,$ where $I_3$ and $I_7$ are the unchanged tails, $I_5$ is the new central vertical side, and $I_4,I_6$ are the horizontal connectors.

There exists a smooth nonnegative function $\nu:\mathbb R\to\mathbb R$ supported in $[1/2,2]$ and normalized to have multiplicative mass one: $\int_0^\infty \nu(x)\,\frac{dx}{x}=1.$

The key property from which it should be easy to construct localCenteringDensity, localCenteringDensityNhd etc below.