Let $\Sigma $ be a compact surface with boundary. For a given conformal class c on $\Sigma $ the functional $\sigma _k^*(\Sigma ,c)$ is defined as the supremum of the kth normalized Steklov eigenvalue over all metrics in c. We consider the behavior of this functional on the moduli space of conformal classes on $\Sigma $ . A precise formula for the limit of $\sigma _k^*(\Sigma ,c_n)$ when the sequence $\{c_n\}$ degenerates is obtained. We apply this formula to the study of natural analogs of the Friedlander–Nadirashvili invariants of closed manifolds defined as $\inf _{c}\sigma _k^*(\Sigma ,c)$ , where the infimum is taken over all conformal classes c on $\Sigma $ . We show that these quantities are equal to $2\pi k$ for any surface with boundary. As an application of our techniques we obtain new estimates on the kth normalized Steklov eigenvalue of a nonorientable surface in terms of its genus and the number of boundary components.

When programs feature a complex control flow, existing techniques for resource analysis produce cost relation systems (CRS) whose cost functions retain the complex flow of the program and, consequently, might not be solvable into closed-form upper bounds. This paper presents a novel approach to resource analysis that is driven by the result of a termination analysis. The fundamental idea is that the termination proof encapsulates the flows of the program which are relevant for the cost computation so that, by driving the generation of the CRS using the termination proof, we produce a linearly-bounded CRS (LB-CRS). A LB-CRS is composed of cost functions that are guaranteed to be locally bounded by linear ranking functions and thus greatly simplify the process of CRS solving. We have built a new resource analysis tool, named MaxCore, that is guided by the VeryMax termination analyzer and uses CoFloCo and PUBS as CRS solvers. Our experimental results on the set of benchmarks from the Complexity and Termination Competition 2019 for C Integer programs show that MaxCore outperforms all other resource analysis tools.

In this paper, we prove that, if $N$ is a positive odd number with $r$ distinct prime factors such that $N\mid \sigma (N)$, then $N\lt {2}^{{4}^{r} - {2}^{r} } $ and $N{\mathop{\prod }\nolimits}_{p\mid N} p\lt {2}^{{4}^{r} } $, where $\sigma (N)$ is the sum of all positive divisors of $N$. In particular, these bounds hold if $N$ is an odd perfect number.

Lower and upper bounds of degree m for the probability of the union of n not necessarily exchangeable events are established. These bounds may be constructed to improve the Bonferroni and the Sobel–Uppuluri bounds.

An application to equi-correlated multivariate normal distribution is given.

Goldstein (1974) derived an upper bound on the variance of certain non-negative functions when the first two moments of the underlying random variable are known. This bound is compared to a simple and fundamental variance bound which requires only that the range of the function be known. It is shown that Goldstein’s bound frequently exceeds the simpler bound. Finally, an interpretation of such bounds in the context of economic risk analysis is given.