Let G be a graph with no isolated vertex. A semitotal forcing set of G is a (zero) forcing set S such that every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number $F_{t2}(G)$ is the minimum cardinality of a semitotal forcing set in G. In this paper, we prove that it is NP-complete to determine the semitotal forcing number of a graph. We also prove that if $G\neq K_n$ is a connected graph of order $n\geq 4$ with maximum degree $\Delta \geq 2$, then $F_{t2}(G)\leq (\Delta-1)n/\Delta$, with equality if and only if either $G=C_{4}$ or $G=P_{4}$ or $G=K_{\Delta ,\Delta }$.

Let $\mathcal {A}$ be the set of all integers of the form $\gcd (n, F_n)$ , where n is a positive integer and $F_n$ denotes the nth Fibonacci number. Leonetti and Sanna proved that $\mathcal {A}$ has natural density equal to zero, and asked for a more precise upper bound. We prove that

$$ \begin{align*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{align*} $$

Let n and k be positive integers with $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$ . In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull. 20 (1978), 117–118] confirmed a conjecture of Erdős by showing that $S_{n,1}\le 1-{1}/{2^{n-1}}$ . Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar. 160 (2020), 360–375] improved Borwein’s upper bound to $S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for $S_{n,2}$ and $S_{n,3}$ . In this paper, we present a sharp upper bound for $S_{n,4}$ and characterise the sequences $\{a_i\}_{i=1}^n$ for which the upper bound is attained.

The Chow–Robbins game is a classical, still partly unsolved, stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide whether you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads $V(n,x) = \sup_{\tau }\mathbb{E} \left [ \frac{x + S_\tau}{n+\tau}\right]$ , where S is a random walk. We give a tight upper bound for V when S has sub-Gaussian increments by using the analogous time-continuous problem with a standard Brownian motion as the driving process. For the Chow–Robbins game we also give a tight lower bound and use these to calculate, on the integers, the complete continuation and the stopping set of the problem for $n\leq 489\,241$ .

We consider coherent and mixed reliability systems composed of elements with independent and identically distributed lifetimes. We present upper bounds on variances of system lifetimes, expressed in terms of variances of single components. We also discuss attainability conditions and some special cases and examples.

Consider the random variable Ln defined as the length of a longest common subsequence of two random strings of length n and whose random characters are independent and identically distributed over a finite alphabet. Chvátal and Sankoff showed that the limit γ=limn→∞E[Ln]/n is well defined. The exact value of this constant is not known, but various methods for the computation of upper and lower bounds have been discussed in the literature. Even so, high-precision bounds are hard to come by. In this paper we discuss how large deviation theory can be used to derive a consistent sequence of upper bounds, (qm)m∈ℕ, on γ, and how Monte Carlo simulation can be used in theory to compute estimates, q̂m, of the qm such that, for given Ξ > 0 and Λ ∈ (0,1), we have P[γ < q̂ < γ + Ξ] ≥ Λ. In other words, with high probability the result is an upper bound that approximates γ to high precision. We establish O((1 − Λ)−1Ξ−(4+ε)) as a theoretical upper bound on the complexity of computing q̂m to the given level of accuracy and confidence. Finally, we discuss a practical heuristic based on our theoretical approach and discuss its empirical behavior.

Mi (2002) recently considered a two-dimensional optimization problem for the optimal age-replacement policy and the optimal work size. In order to find (y ∗,T ∗), Mi (2002) found the optimal age-replacement policy T ∗(y) for each fixed work size y, and then searched for the optimal work size y ∗. When applying this approach, for each fixed work size y, Mi (2002) obtained the bounds for T ∗(y). However, no bound for the optimal work size y ∗ was derived. In this note, the results on the upper bound for the optimal work size y ∗ are given.

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.

New bounds are found for the reliability of consecutive-k-out-of-n:F systems with equal component failure probabilities. The expressions involved are simple, thus allowing a direct use in the derivation of theoretical properties.

These bounds can also be employed in numerical computations when the value of n or k is so large that the exact calculation of the reliability is not achievable. Comparisons show that the approximation errors exhibited by these new formulas are lower than those of other widely used bounds.

Independent observations X0, X1…, XN+1 are drawn from each of N populations whose distribution functions F(x – θi) have means θ i, 0 ≦ i < N, and we wish to calculate the probability Pk;N that X0 is the k th largest observation. For normal populations an approximation is given for PK;N based on a Taylor series expansion in the θ 's. If F(x) has an increasing failure rate, as is the case for the normal, an upper bound can be given for the ‘win' probability P1;N Some moment relations for normal order statistics are also given.

Moments of the delay distribution and other measures of performance for a multi-channel queue are shown to be bounded above by corresponding quantities for one or a collection of single-channel queues.

Suppose (X1, Y1), (X2, Y2), …, (Xn, Yn) are independent random vectors such that a ≦ Xi ≦ b and a ≦ Yi ≦ b, i = 1, 2, …, n. An upper bound which exponentially converges to zero is derived for the probability Pr{Sx – nμx ≧ nt1;SY – nμY ≧ nt2} where Sx = Σ Xi, SY = Σ Yi,EYi = μY, EXi = μx and t1 > 0, t2 > 0. The bound is a function of the difference b — a, the correlation between Xi and Yi, μx and μY and t1 and t2.