The well known analytical formula for $SU(2)$ matrices $U = \exp(i \vec \tau \!\cdot\! \varphi\,) \cos|\vec \varphi\,| + i\vec \hat\varphi \, \sin|\vec \varphi\,|$\\ is extended to the $SU(3)$ group with eight real parameters. resulting involves sum over three roots of a cubic equation, corresponding so-called irreducible case, where one has employ trisection an angle. When going special unitar...

We present an analytical proof that certain natural metric universal covers are Hadamard spaces. If $$\rho \,\mathrm{{d}}s$$ induces a complete distance d on plane domain $$\Omega $$ , and =\varphi \circ u$$ where u is (locally Lipschitz and) subharmonic in $$\varphi positive increasing interval containing $$u(\Omega )$$ with $$\log \varphi convex, then $$(\Omega ,d)$$ has cover $$({\tilde{\Ome...

We classify seven-dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left-invariant purely coclosed G2-structures. This is done by going through the list all algebras given Gong, providing an example a 3-form φ which pure G2-structure (i.e., it satisfies d ∗ = 0 $d*\varphi =0$ , ∧ $\varphi \wedge d\varphi ) for those that admit them; and showing impossi...

The Chern-Simons amended gravity theory appears as a low-energy effective of string theory. includes an anomaly-cancelation correction to the Einstein-Hilbert action. expression consists product $\varphi R \tilde $ Pontryagin density $R with scalar field $\varphi$, where latter is considered background (dynamical construction or non-dynamical construction). Many different solutions Einstein's g...