Profinite groups with a rational probabilistic zeta function

In this thesis, we investigate the connection between finitely generated profinite groups $G$ and the associated Dirichlet series $P_G(s)$ of which the reciprocal is called the probabilistic zeta function of $G$. In particular, we consider the conjecture of Lucchini saying that given a finitely generated profinite group $G$, the associated Dirichlet series $P_G(s)$ is rational if and only if the quotient group $G/\textrm{Frat}(G)$ is finite. Detomi and Lucchini first showed that the conjecture holds when $G$ is prosoluble. For non-prosoluble groups, they later showed that the conjecture also holds when almost every nonabelian composition factor of $G$ is an alternating group. In this thesis, we prove the conjecture in several other cases. We first show that it holds when almost every nonabelian composition factor of $G$ is isomorphic to a simple group of Lie type over a field of characteristic $p$, where $p$ is a fixed prime. When there are different characteristics, the problem becomes quite difficult and we do not have any answer yet. However, we obtain that the conjecture holds when almost every nonabelian composition factor of $G$ is $\textrm{PSL}(2,p)$ for some prime $p\geq 5$. This is also the case when we replace $\textrm{PSL}(2,p)$ by a sporadic simple group. The conjecture is still open in general and it cannot be proved by our techniques. We give some examples supporting this. Nevertheless, we also obtain a partial result by showing that the conjecture holds when almost every nonabelian composition factor is isomorphic to either $\textrm{PSL}(2,p)$ for some prime $p\geq 5$, or a sporadic simple group, or an alternating group $\textrm{Alt}(n)$ where $n$ is either a prime or a power of $2

Self-Centering Prestressed Concrete Pier Considering the Effect of Vertical Earthquake Motions with External Aluminum Dissipators

A prototype self-centering prestressed concrete (SCPC) bridge pier and conventional reinforced concrete (RC) pier with similar backbone curves are designed and modeled. The objective of this study is to investigate the seismic performance of the SCPC bridge pier considering the effect of vertical earthquake ground motions. Under a severe vertical earthquake motion, the RC pier damage is much greater than the SCPC pier. The SCPC bridge pier shows a great capability in reducing residual drift at the top of the structure, therefore, the probability of bridge survival is increased. In this study, the external aluminum bars were used to reduce the seismic energy impacts on the bridge pier structure. In addition, The averages of the maximum and residual drifts of bridge piers under a set of 20 earthquake ground motion records impact with different vertical-to-horizontal peak acceleration ratios on the bridge seismic response are presented. The results are compared with the case of horizontal-only excitations, to clarify the effect of the vertical earthquake on the SCPC and RC bridge pier. Hence, the designers can find good solutions for structures in earthquakes resistance

A New Approach to Keep the Privacy Information of the Signer in a Digital Signature Scheme

In modern applications, such as Electronic Voting, e-Health, e-Cash, there is a need that the validity of a signature should be verified by only one responsible person. This is opposite to the traditional digital signature scheme where anybody can verify a signature. There have been several solutions for this problem, the first one is we combine a signature scheme with an encryption scheme; the second one is to use the group signature; and the last one is to use the strong designated verifier signature scheme with the undeniable property. In this paper, we extend the traditional digital signature scheme to propose a new solution for the aforementioned problem. Our extension is in the sense that only a designated verifier (responsible person) can verify a signer’s signature, and if necessary (in case the signer refuses to admit his/her signature) the designated verifier without revealing his/her secret key is able to prove to anybody that the signer has actually generated the signature. The comparison between our proposed solution and the three existing solutions shows that our proposed solution is the best one in terms of both security and efficiency