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Register a new userOn the last digits of tetrations of base $2^{k}$ and $5^{k}$
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Luca Onnis
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In this paper will be proved the existence of a formula to reduce a tetration of base $2^{k}$ and $5^{k}$ $mod 10^{n}$. Indeed, last digits of a tetration are the same starting from a certain hyper-exponent; In order to compute the last digits of those expressions we reduce them $mod 10^{n}$. Lots of different formulas will be derived, for different cases of $k$ (where $k$ is the exponent of the base of the tetration). This kind of operation is fascinating, because the tetration grows very fast. But using these formulas we can actually have informations about the last digits of those expressions. Its possible to use these results on a software in order to reduce tetrations $mod 10^{n}$ faster.
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In 1997, Bousquet-Melou and Eriksson stated a broad generalization of Eulers distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual difference conditions of partitions in Rogers-Ramanujan type identities, satisfy some ratio constraints. In a 2008 paper, in response to a question suggested by Richard Stanley, Savage and Yee provided a simple bijection for the $l$-lecture-hall partitions (the case $k=l$), whose specialization in $l=2$ corresponds to Sylvesters bijection. Subsequently, as an open question, a generalization of their bijection was suggested for the case $k,lgeq 2$. In the spirit of Savage and Yees work, we provide and prove in this paper slight variations of the suggested bijection, not only for the case $k,lgeq 2$ but also for the cases $(k,1)$ and $(1,k)$ with $kgeq 4$. Furthermore, we show that our bijections equal the recursive bijections given by Bousquet-Melou and Eriksson in their recursive proof of the $(k,l)$-lecture hall and finally provide the analogous recursive bijection for the $(k,l)$-Euler theorem.
The two-channel photoproductions of $gamma p to K^{*+} Sigma^{0}$ and $gamma p to K^{*0} Sigma^{+}$ are investigated based on an effective Lagrangian approach at the tree-level Born approximation. In addition to the $t$-channel $K$, $kappa$, $K^*$ exchanges, the $s$-channel nucleon ($N$) and $Delta$ exchanges, the $u$-channel $Lambda$, $Sigma$, $Sigma^*$ exchanges, and the generalized contact term, we try to take into account the minimum number of baryon resonances in constructing the reaction amplitudes to describe the experimental data. It is found that by including the $Delta(1905)5/2^+$ resonance with its mass, width, and helicity amplitudes taken from the Review of Particle Physics [Particle Data Group, C. Patrignani {it et al.}, Chin. Phys. C {bf 40}, 100001 (2016)], the calculated differential and total cross sections for these two reactions are in good agreement with the experimental data. An analysis of the reaction mechanisms shows that the cross sections of $gamma p to K^{*+}Sigma^{0}$ are dominated by the $s$-channel $Delta(1905)5/2^+$ exchange at low energies and $t$-channel $K^*$ exchange at high energies, with the $s$-channel $Delta$ exchange providing significant contributions in the near-threshold region. For $gamma p to K^{*0}Sigma^{+}$, the angular dependences are dominated by the $t$-channel $K$ exchange at forward angles and the $u$-channel $Sigma^*$ exchange at backward angles, with the $s$-channel $Delta$ and $Delta(1905)5/2^+$ exchanges making considerable contributions at low energies. Predictions are given for the beam, target, and recoil asymmetries for both reactions.
We give a process to construct non-split, three-dimensional simple Lie algebras from involutions of sl(2,k), where k is a field of characteristic not two. Up to equivalence, non-split three-dimensional simple Lie algebras obtained in this way are parametrised by a subgroup of the Brauer group of k and are characterised by the fact that their Killing form represents -2. Over local and global fields we re-express this condition in terms of Hilbert and Legendre Symbols and give examples of three-dimensional simple Lie algebras which can and cannot be obtained by this construction over the field of rationals.
The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. In the case of a Galois extension of a number field F/L with rings of integers A/B respectively, this K-theory of the norm functor is an extension of a subgroup of the ideal class group Cl(A) by the 0-Tate cohomology group with coefficients in A*. The Mayer-Vietoris exact sequence enables us to describe quite explicitly this extension which is related to the coinvariants of Cl(A) under the action of the Galois group. We apply these ideas to find results in Number Theory, which are known for some of them with different methods.
We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2} and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is accomplished by showing that the equivariant homotopy groups of the topological Hochschild spectrum THH(Z) are finite, in odd degrees, and free abelian, in even degrees, and by evaluating their orders and ranks, respectively.
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