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1. Electron-impact excitation of the 5$$\varvec{^2\textbf{S}_{1/2} \rightarrow 5^2\textbf{P}_{1/2}}$$ and 5$$\varvec{^2\textbf{P}_{3/2}}$$ transitions in rubidium by 40 eV electrons: theory and experiment

   https://doi.org/10.1140/epjd/s10053-022-00413-7  

   Hamilton, K. R.; Zatsarinny, O.; Bartschat, K.; Predojević, B.; Šević, D.; Marinković, B. P.; Brunger, M. J. (May 2022, The European Physical Journal D)

   AbstractWe report on a series of detailed Breit-Pauli and Dirac B-spline R-matrix (DBSR) differential cross section (DCS) calculations for excitation of the$$5\,^2\textrm{S}_{1/2} \rightarrow 5\,^2\textrm{P}_{1/2}$$ $5{}^2{\text{S}}_{1/2}\to 5{}^2{\text{P}}_{1/2}$and$$5\,^2\textrm{S}_{1/2}\rightarrow 5\,^2\textrm{P}_{3/2}$$ $5{}^2{\text{S}}_{1/2}\to 5{}^2{\text{P}}_{3/2}$states in rubidium by 40 eV incident electrons. The early BP computations shown here were carried out with both 5 states and 12 states, while the DBSR models coupled 150 and 325 states, respectively. We also report corresponding results from a limited set of DCS measurements on the unresolved$$5\,^2\textrm{P}_{1/2,3/2}$$ $5{}^2{\text{P}}_{1/2,3/2}$states, with the experimental data being restricted to the scattered electron angular range 2–$$10^\circ $$ ${10}^{\circ }$. Typically, good agreement is found between our calculated and measured DCS for excitation of the unresolved$$5\,^2\textrm{P}_{1/2,3/2}$$ $5{}^2{\text{P}}_{1/2,3/2}$states, with best accord being found between the DBSR predictions and the measured data. The present theoretical and experimental results are also compared with predictions from earlier 40 eV calculations using the nonrelativistic Distorted-Wave Born Approximation and a Relativistic Distorted-Wave model. Graphic abstract 

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2. Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

   https://doi.org/10.1007/s00454-022-00426-4  

   Greenfeld, Rachel; Tao, Terence (January 2023, Discrete & Computational Geometry)

   Abstract We construct an example of a group$$G = \mathbb {Z}^2 \times G_0$$ $G=Z^2\times G_0$for a finite abelian group $$G_0$$ $G_0$, a subsetEof $$G_0$$ $G_0$, and two finite subsets$$F_1,F_2$$ $F_1,F_2$of G, such that it is undecidable in ZFC whether$$\mathbb {Z}^2\times E$$ $Z^2\times E$can be tiled by translations of$$F_1,F_2$$ $F_1,F_2$. In particular, this implies that this tiling problem isaperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings ofEby the tiles$$F_1,F_2$$ $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ $Z^2$). A similar construction also applies for$$G=\mathbb {Z}^d$$ $G=Z^d$for sufficiently large d. If one allows the group$$G_0$$ $G_0$to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 

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3. Lower bounds for incidences

   https://doi.org/10.1007/s00222-025-01331-2  

   Cohen, Alex; Pohoata, Cosmin; Zakharov, Dmitrii (March 2025, Inventiones mathematicae)

   Abstract Let$$p_{1},\ldots ,p_{n}$$ $p_1,\dots ,p_n$be a set of points in the unit square and let$$T_{1},\ldots ,T_{n}$$ $T_1,\dots ,T_n$be a set of$$\delta $$ $\delta$-tubes such that$$T_{j}$$ $T_j$passes through$$p_{j}$$ $p_j$. We prove a lower bound for the number of incidences between the points and tubes under a natural regularity condition (similar to Frostman regularity). As a consequence, we show that in any configuration of points$$p_{1},\ldots , p_{n} \in [0,1]^{2}$$ $p_1,\dots ,p_n\in {[0,1]}^2$along with a line$$\ell _{j}$$ ${\ell }_j$through each point$$p_{j}$$ $p_j$, there exist$$j\neq k$$ $j\ne k$for which$$d(p_{j}, \ell _{k}) \lesssim n^{-2/3+o(1)}$$ $d(p_j,{\ell }_k)\lesssim n^{-2/3+o\left(1\right)}$. It follows from the latter result that any set of$$n$$ $n$points in the unit square contains three points forming a triangle of area at most$$n^{-7/6+o(1)}$$ $n^{-7/6+o\left(1\right)}$. This new upper bound for Heilbronn’s triangle problem attains the high-low limit established in our previous work arXiv:2305.18253. 

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4. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$where$$k \ge 2$$ $k\ge 2$. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$, where$$|x-y|$$ $|x-y|$is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$. From the other direction, for every$$k \ge 2$$ $k\ge 2$and$$n \ge 2$$ $n\ge 2$, there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$, such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$. For the plane, the best constant is$$c_2=2$$ $c_2=2$and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$for every$$k \ge 3$$ $k\ge 3$and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$, for every$$k \ge 2$$ $k\ge 2$. Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$. Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$, which disproves the conjecture for$$k=3$$ $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

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5. Size dependence- and induced transformations- of fractional quantum Hall effects under tilted magnetic fields

   https://doi.org/10.1038/s41598-022-22812-x  

   Wijewardena, U. Kushan; Nanayakkara, Tharanga R.; Kriisa, Annika; Reichl, Christian; Wegscheider, Werner; Mani, Ramesh G. (November 2022, Scientific Reports)

   Abstract Two-dimensional electron systems subjected to high transverse magnetic fields can exhibit Fractional Quantum Hall Effects (FQHE). In the GaAs/AlGaAs 2D electron system, a double degeneracy of Landau levels due to electron-spin, is removed by a small Zeeman spin splitting,$$g \mu _B B$$ $g{\mu }_{B}B$, comparable to the correlation energy. Then, a change of the Zeeman splitting relative to the correlation energy can lead to a re-ordering between spin polarized, partially polarized, and unpolarized many body ground states at a constant filling factor. We show here that tuning the spin energy can produce fractionally quantized Hall effect transitions that include both a change in$$\nu$$ $\nu$for the$$R_{xx}$$ $R_{\mathrm{xx}}$minimum, e.g., from$$\nu = 11/7$$ $\nu =11/7$to$$\nu = 8/5$$ $\nu =8/5$, and a corresponding change in the$$R_{xy}$$ $R_{\mathrm{xy}}$, e.g., from$$R_{xy}/R_{K} = (11/7)^{-1}$$ $R_{\mathrm{xy}}/R_K={(11/7)}^{-1}$to$$R_{xy}/R_{K} = (8/5)^{-1}$$ $R_{\mathrm{xy}}/R_K={(8/5)}^{-1}$, with increasing tilt angle. Further, we exhibit a striking size dependence in the tilt angle interval for the vanishing of the$$\nu = 4/3$$ $\nu =4/3$and$$\nu = 7/5$$ $\nu =7/5$resistance minima, including “avoided crossing” type lineshape characteristics, and observable shifts of$$R_{xy}$$ $R_{\mathrm{xy}}$at the$$R_{xx}$$ $R_{\mathrm{xx}}$minima- the latter occurring for$$\nu = 4/3, 7/5$$ $\nu =4/3,7/5$and the 10/7. The results demonstrate both size dependence and the possibility, not just of competition between different spin polarized states at the same$$\nu$$ $\nu$and$$R_{xy}$$ $R_{\mathrm{xy}}$, but also the tilt- or Zeeman-energy-dependent- crossover between distinct FQHE associated with different Hall resistances. 

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