A lower bound for the Hausdorff dimension of the set of weighted simultaneously approximable points over manifolds

Given a weight vector [Formula: see text] with each [Formula: see text] bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set [Formula: see text], where [Formula: see text] is a twice continuously differentiable manifold. From this we produce a lower bound for [Formula: see text] where [Formula: see text] is a general approximation function with certain limits. The proof is based on a technique developed by Beresnevich et al. in 2017, but we use an alternative mass transference style theorem proven by Wang, Wu and Xu (2015) to obtain our lower bound.

Characteristics of moisture release from layers of forest fuels with typical fire extinguishing agents

Typical fire extinguishing agents were considered: water; bischofite solutions (with a mass fraction of 5% and 10%); bentonite slurries (with a mass fraction of 5% and 10%); foaming agent emulsions (with a mass fraction of 5% and 10%). The heating temperature range of 150-400?? was chosen to correspond to the conditions of rapid thermal decomposition of forest fuels. The experimental research findings suggest that the rates of moisture release depend exponentially on the heating temperature. It was established that the rates of moisture release in the above temperature range may differ significantly for the forest fuels and fire extinguishing agents under study. Conditions were identified when the general approximation equations, presented in this paper, can be used to predict the vaporization characteristics of firefighting liquids.

On Subset Selection with General Cost Constraints

This paper considers the subset selection problem with a monotone objective function and a monotone cost constraint, which relaxes the submodular property of previous studies. We first show that the approximation ratio of the generalized greedy algorithm is $\frac{\alpha}{2}(1 \textendash \frac{1}{e^{\alpha}})$ (where $\alpha$ is the submodularity ratio); and then propose POMC, an anytime randomized iterative approach that can utilize more time to find better solutions than the generalized greedy algorithm. We show that POMC can obtain the same general approximation guarantee as the generalized greedy algorithm, but can achieve better solutions in cases and applications.

Optimizing Ratio of Monotone Set Functions

This paper considers the problem of minimizing the ratio of two set functions, i.e., $f/g$. Previous work assumed monotone and submodular of the two functions, while we consider a more general situation where $g$ is not necessarily submodular. We derive that the greedy approach GreedRatio, as a fixed time algorithm, achieves a $\frac{|X^*|}{(1+(|X^*| \textendash 1)(1 \textendash \kappa_f))\gamma(g)}$ approximation ratio, which also improves the previous bound for submodular $g$. If more time can be spent, we present the PORM algorithm, an anytime randomized iterative approach minimizing $f$ and $\textendash g$ simultaneously. We show that PORM using reasonable time has the same general approximation guarantee as GreedRatio, but can achieve better solutions in cases and applications.