ErdH{o}s determined the maximum size of a nonhamiltonian graph of order $n$ and minimum degree at least $k$ in 1962. Recently, Ning and Peng generalized. ErdH{o}s work and gave the maximum size $h(n,c,k)$ of graphs with prescribed order $n$, circumfe
rence $c$ and minimum degree at least $k.$ But for some triples $n,c,k,$ the maximum size is not attained by a graph of minimum degree $k.$ For example, $h(15,14,3)=77$ is attained by a unique graph of minimum degree $7,$ not $3.$ In this paper we obtain more precise information by determining the maximum size of a graph with prescribed order, circumference and minimum degree. Consequently we solve the corresponding problem for longest paths. All these results on the size of graphs have cliq
Motivated by work of ErdH{o}s, Ota determined the maximum size $g(n,k)$ of a $k$-connected nonhamiltonian graph of order $n$ in 1995. But for some pairs $n,k,$ the maximum size is not attained by a graph of connectivity $k.$ For example, $g(15,3)=77$
is attained by a unique graph of connectivity $7,$ not $3.$ In this paper we obtain more precise information by determining the maximum size of a nonhamiltonian graph of order $n$ and connectivity $k,$ and determining the extremal graphs. Consequently we solve the corresponding problem for nontraceable graphs.
Electrides are an emerging class of materials with excess electrons localized in interstices and acting as anionic interstitial quasi-atoms (ISQs). The spatial ion-electron separation means that electrides can be treated physically as ionic crystals,
and this unusual behavior leads to extraordinary physical and chemical phenomena. Here, a completely different effect in electrides is predicted. By recognizing the long-range Coulomb interactions between matrix atoms and ISQs that are unique in electrides, a nonanalytic correction to the forces exerted on matrix atoms is proposed. This correction gives rise to an LA-TA splitting in the acoustic branch of lattice phonons near the zone center, similar to the well-known LO-TO splitting in the phonon spectra of ionic compounds. The factors that govern this splitting are investigated, with isotropic fcc-Li and anisotropic hP4-Na as the typical examples. It is found that not all electrides can induce a detectable splitting, and criteria are given for this type of splitting. The present prediction unveils the rich phenomena in electrides and could lead to unprecedented applications.
We propose a dynamic boosted ensemble learning method based on random forest (DBRF), a novel ensemble algorithm that incorporates the notion of hard example mining into Random Forest (RF) and thus combines the high accuracy of Boosting algorithm with
the strong generalization of Bagging algorithm. Specifically, we propose to measure the quality of each leaf node of every decision tree in the random forest to determine hard examples. By iteratively training and then removing easy examples from training data, we evolve the random forest to focus on hard examples dynamically so as to learn decision boundaries better. Data can be cascaded through these random forests learned in each iteration in sequence to generate predictions, thus making RF deep. We also propose to use evolution mechanism and smart iteration mechanism to improve the performance of the model. DBRF outperforms RF on three UCI datasets and achieved state-of-the-art results compared to other deep models. Moreover, we show that DBRF is also a new way of sampling and can be very useful when learning from imbalanced data.
