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Abstract. In this paper we deal with the problem of computing the sum of the k-th powers of all the elements of the matrix ring M d (R) with d > 1 and R a finite commutative ring. We completely solve the problem in the case R = Z/nZ and give some results that compute the value of this sum if R is an arbitrary finite commutative ring R for many values of k and d. Finally, based on computational evidence and using some technical results proved in the paper we conjecture that the sum of the k-th powers of all the elements of the matrix ring M d (R) is always 0 unless d = 2, card(R) ≡ 2 (mod 4), 1 < k ≡ −1, 0, 1 (mod 6) and the only element e ∈ R \ {0} such that 2e = 0 is idempotent, in which case the sum is diag(e, e).

In this paper we consider two variants of the Secretary problem: The Best-or-Worst and the Postdoc problems. We extend previous work by considering that the number of objects is not known and follows either a discrete Uniform distribution U [1, n] or a Poisson distribution P(λ). We show that in any case the optimal strategy is a threshold strategy, we provide the optimal cutoff values and the asymptotic probabilities of success. We also put our results in relation with closely related work.

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