Lie nilpotency indices of symmetric elements under oriented involutions in group algebras

Let $G$ be a group and let $F$ be a field of characteristic different from 2. Denote by $(FG)^+$ the set of symmetric elements and by $\mathcal{U}^+(FG)$ the set of symmetric units, under an oriented classical involution of the group algebra $FG$. We give some lower and upper bounds on the Lie nilpotency index of $(FG)^+$ and the nilpotency class of $\mathcal{U}^+(FG)$.Comment: Some corrected typos from version v2 and problems with the bibliograph

Academic Resilience and Achievement

This study uses narrative analysis to understand the ways in which Mexican university faculty members used their self-motivational resources to persist in an instructional technology training program within adverse work conditions. The methodology included interviews and participant observation. Findings suggest that faculty’s academic resilience was based on faculty’s self-definition as permanent learners and innovators, the perception of instruction as a field of reflection-in-action and systematization, and the caring instructional approach used by training instructors

The reform of mechanisms for foreign exchange allocation : theory and lessons from sub-Saharan Africa

Administrative exchange allocation has been common in developing countries, especially in sub-Saharan Africa. Steps to dismantle or modify these control mechanisms have been carried out through traditional schemes. The authors draw lessons from sub-Saharan Africa's historical experience useful both to African former socialist economies. Exchange regime reform should be given highest priority for its role in reducing anti-export bias. Although many sub-Saharan countries have attempted to reform their allocation mechanisms, only a few have made the transition to market allocation (virtually convertible currency, at least on the current account.) Failure to do so is the major shortcoming of most adjustment packages. Both gradual and rapid approaches have succeeded. On purely economic grounds (given the problems of such intermediate steps as auctions), speed is preferable but it is not always politically or institutionally feasible. The transition must be accompanied by a coherent set of fiscal and monetary policies and a willingness to allow the exchange rate to seek a true market-clearing level. Some lessons regarding the specific mechanisms, discussed in approximate order of their proximity to convertibility, are as follows. The most rudimentary transition mechanism is the own-funds scheme, which is no more than a beginning of reform. Own-funds schemes should be accompanied by liberalization of the rules governing exports, or illegal exports and the black market premium may increase. Export retention schemes can minimize the adverse effects on exporters of foreign exchange shortages, reduce the implicit export tax, and fund a legal private exchange market. But the retained funds must be saleable, the retention rates substantial, and traditional exports must be included to adequately fund the legal private exchange market. Open general licensing (OGL) and similar schemes can be a useful intermediate step in liberalizing import and exchange allocation regimes. But in practice the benefits are limited by two features. First, consumer goods competing with local production, whose imports were restricted the most, have usually been excluded, at least initially. Moreover, OGL has no endogenous price-setting mechanism for the exchange rate. The OGL rate should generally be connected to, but lower than, the parallel rate. An auction incorporates a pricing mechanism, which is an important advantage. But the pricing mechanism must be allowed to work, which has not always been the case. Auction rules should be clear (should not allow discretionary disqualification of bids, for example), should minimize participation costs, and allow wide participation. Marginal, rather than the more common Dutch, pricing system is preferred. The use of a reservation price may reduce volatility but may also impede the full disbursement of funds. The shortcomings of transitional schemes to dismantle or modify foreign exchange controls become more important the longer they are in place. A strong case can be made for avoiding delay in moving to full currency convertibility.Environmental Economics&Policies,Economic Theory&Research,Economic Stabilization,Access to Markets,Markets and Market Access

The set of kk-units modulo nn

Let $R$ be a ring with identity, $\mathcal{U}(R)$ the group of units of $R$ and $k$ a positive integer. We say that $a\in \mathcal{U}(R)$ is $k$-unit if $a^k=1$. Particularly, if the ring $R$ is $\mathbb{Z}_n$, for a positive integer $n$, we will say that $a$ is a $k$-unit modulo $n$. We denote with $\mathcal{U}_k(n)$ the set of $k$-units modulo $n$. By $\text{du}_k(n)$ we represent the number of $k$-units modulo $n$ and with $\text{rdu}_k(n)=\frac{\phi(n)}{\text{du}_k(n)}$ the ratio of $k$-units modulo $n$, where $\phi$ is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation $\text{rdu}_2(n)=1$ are the divisors of $24$. The main result of this work, is that for a given $k$, we find the positive integers $n$ such that $\text{rdu}_k(n)=1$. Finally, we give some connections of this equation with Carmichael's numbers and two of its generalizations: Kn\"odel numbers and generalized Carmichael numbers

Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers

We introduce a new class of pseudoprimes-so called "overpseudoprimes to base $b$", which is a subclass of strong pseudoprimes to base $b$. Denoting via $|b|_n$ the multiplicative order of $b$ modulo $n$, we show that a composite $n$ is overpseudoprime if and only if $|b|_d$ is invariant for all divisors $d>1$ of $n$. In particular, we prove that all composite Mersenne numbers $2^{p}-1$, where $p$ is prime, are overpseudoprime to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well known numbers are overpseudoprime to a base $b$.Comment: 9 page