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Random forest (RF) stands out as a highly favored machine learning approach for classification problems. The effectiveness of RF hinges on two key factors: the accuracy of individual trees and the diversity among them. In this study, we introduce a novel approach called heterogeneous RF (HRF), designed to enhance tree diversity in a meaningful way. This diversification is achieved by deliberately introducing heterogeneity during the tree construction. Specifically, features used for splitting near the root node of previous trees are assigned lower weights when constructing the feature sub-space of the subsequent trees. As a result, dominant features in the prior trees are less likely to be employed in the next iteration, leading to a more diverse set of splitting features at the nodes. Through simulation studies, it was confirmed that the HRF method effectively mitigates the selection bias of trees within the ensemble, increases the diversity of the ensemble, and demonstrates superior performance on datasets with fewer noise features. To assess the comparative performance of HRF against other widely adopted ensemble methods, we conducted tests on 52 datasets, comprising both real-world and synthetic data. HRF consistently outperformed other ensemble methods in terms of accuracy across the majority of datasets., Comment: 18 pages, 6 figures

Small polaron formation is dominant across a range of condensed matter systems. Small polarons are usually studied in terms of ground-state transport and thermal fluctuations, but small polarons can also be created impulsively by photoexcitation. The temporal response of the lattice and local electron correlations can then be separated, such as with transient XUV spectroscopy. To date, photoexcited small polaron formation has only been measured to be adiabatic. The reorganization energy of the polar lattice is large enough that the first electron-optical phonon scattering event creates a small polaron without significant carrier thermalization. Here, we use transient XUV spectroscopy to measure antiadiabatic polaron formation by frustrating the iron-centered octahedra in a rare-earth orthoferrite lattice. The small polaron is measured to take several picoseconds to form over multiple coherent charge hopping events between neighboring Fe3+-Fe2+ sites, a timescale that is more than an order of magnitude longer compared to previous materials. The measured interplay between optical phonons, electron correlations, and on-site lattice deformation give a clear picture of how antiadiabatic small polaron transport would occur in the material. The measurements also confirm the prediction of the Holstein and Hubbard-Holstein model that the electron hopping integral must be larger than the reorganization energy to achieve antiadiabaticity. Moreover, the measurements emphasize the importance of considering dynamical electron correlations, and not just changes in the lattice geometry, for controlling small polarons in transport or photoexcited applications.

A substitution box (S-box) in a symmetric primitive is a mapping $F$ that takes $k$ binary inputs and whose image is a binary $m$-tuple for some positive integers $k$ and $m$, which is usually the only nonlinear element of the most modern block ciphers. Therefore, employing S-boxes with good cryptographic properties to resist various attacks is significant. For power permutation $F$ over finite field $\GF{2^k}$, the multiset of values $\beta_F(1,b)=\#\{x\in \GF{2^k}\mid F^{-1}(F(x)+b)+F^{-1}(F(x+1)+b)=1\}$ for $b\in \GF{2^k}$ is called the boomerang spectrum of $F$. The maximum value in the boomerang spectrum is called boomerang uniformity. This paper determines the boomerang spectrum of the power permutation $X^{2^{3n}+2^{2n}+2^{n}-1}$ over $\GF{2^{4n}}$. The boomerang uniformity of that power permutation is $3(2^{2n}-2^n)$. However, on a large subset $\{b\in \GF{2^{4n}}\mid \mathbf{Tr}_n^{4n}(b)\neq 0\}$ of $\GF{2^{4n}}$ of cardinality $2^{4n}-2^{3n}$ (where $ \mathbf{Tr}_n^{4n}$ is the (relative) trace function from $\GF{2^{4n}}$ to $\GF{2^{n}}$), we prove that the studied function $F$ achieves the optimal boomerang uniformity $2$. It is known that obtaining such functions is a challenging problem. More importantly, the set of $b$'s giving this value is explicitly determined for any value in the boomerang spectrum.

Electron energy-loss spectroscopy (EELS) can measure similar information to X-ray, UV-Vis, and IR spectroscopies but with atomic resolution and increased scattering cross sections. Recent advances in electron monochromators have expanded EELS capabilities from chemical identification to the realms of synchrotron-level core-loss measurements and to low-loss, 10-100 meV excitations such as phonons, excitons, and valence structure. EELS measurements are easily correlated with electron diffraction and atomic-scale real-space imaging in a transmission electron microscope (TEM) to provide detailed local pictures of quasiparticle and bonding states. This perspective provides an overview of existing high-resolution EELS (HR-EELS) capabilities while also motivating the powerful next step in the field - ultrafast EELS in a TEM. Ultrafast EELS aims to combine atomic level, element specific, and correlated temporal measurements to better understand spatially specific excited state phenomena. Ultrafast EELS measurements also add to the abilities of steady-state HR-EELS by being able to image the electromagnetic field and use electrons to excite photon-forbidden and momentum-specific transitions. We discuss the technical challenges ultrafast HR-EELS currently faces, as well as how integration with in situ and cryo measurements could expand the technique to new systems of interest, especially molecular and biological samples.