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The flexible traction power supply system (FTPSS) eliminates the neutral zone but leads to increased complexity in power flow coordinated control and power mismatch. To address these challenges, the methodology for power dispatch (PD) based on traction station clusters (TSCs) in FTPSS is proposed, in which each TSC with a consistent structure performs independent local phase angle control. First, to simplify the PD problem of TSCs, the system is transformed into an equivalent model with constant topology, resulting in it can be solved by univariate numerical optimization with higher computational performance. Next, the calculation method of the feasible phase angle domain under strict and relaxed power circulation constraints are described, respectively, which ensures that power circulation can be either eliminated or precisely controlled. Finally, the PD method with three unique modes for uncertain train loads is introduced to enhance power flow flexibility: specified power distribution coefficients between traction substations (TSs), constant output power of TSs, and maximum consumption of renewable resources within TSs. In the experimental section, the performance of the TSC methodology for PD is verified through detailed train operation scenarios.

Let $\alpha, \beta$ be two relatively prime algebraic integers in a number field $K$ and $N$ be a positive integer. We show that the number of $n\in\{1,2,\dots,N\}$ such that the $\beta$-adic expansion of $\alpha^n$ omits a given digit is less than $C_1 N^{\sigma(\beta)}$, where $\sigma(\beta):=\frac{\log(|N(\beta)|-1)}{\log|N(\beta)|}$ and $C_1$ is an absolute constant, if all prime ideal factors of $\beta$ are unramified and their norms are integer primes.

Let $t \geq 2$ be an integer and $C(b,D)$ be a generalized Cantor set with $b \geq 3$. We study how close can numbers in $C(b,D)$ be approximated by rational numbers with denominators $t^n$. For any function $\psi : \mathbb{N} \to (0,\infty)$, let $W_{t} (\psi)$ be the set of numbers in $[0,1]$ such that $|x - p/t^n| < \psi(n)$ for infinitely many $(p,n) \in \mathbb{N}^2$. We correct an error in a result of Levesley, Salp and Velani (Math. Ann., 338:97-118, 2007) on the Hausdorff measure of $W_{t} (\psi) \cap C(b,D)$ when $b=t$, and also prove a generalization when $b$ and $t$ are multiplicatively dependent. If $b$ and $t$ are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed.

Let $b\geq2$ be an integer and $A=(a_{n})_{n=1}^{\infty}$ be a strictly increasing subsequence of positive integers with $\eta:=\limsup\limits_{n\to\infty}\frac{a_{n+1}}{a_{n}}<+\infty$. For each irrational real number $\xi$, we denote by $\hat{v}_{b,A}(\xi)$ the supremum of the real numbers $\hat{v}$ for which, for every sufficiently large integer $N$, the equation $\|b^{a_n}\xi\|<(b^{a_N})^{-\hat{v}}$ has a solution $n$ with $1\leq n\leq N$. For every $\hat{v}\in[0,\eta]$, let $\hat{\mathcal{V}}_{b,A}(\hat{v})$ ($\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$) be the set of all real numbers $\xi$ such that $\hat{v}_{b,A}(\xi)\geq\hat{v}$ ($\hat{v}_{b,A}(\xi)=\hat{v}$) respectively. In this paper, we give some results of the Hausdorfff dimensions of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ and $\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$. When $\eta=1$, we prove that the Hausdorfff dimensions of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ and $\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$ are equal to $\left(\frac{1-\hat{v}}{1+\hat{v}}\right)^{2}$ for any $\hat{v}\in[0,1]$. When $\eta>1$ and $\lim_{n\to\infty}\frac{a_{n+1}}{a_{n}}$ exists, we show that the Hausdorfff dimension of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ is strictly less than $\left(\frac{\eta-\hat{v}}{\eta+\hat{v}}\right)^{2}$ for some $\hat{v}$, which is different with the case $\eta=1$, and we give a lower bound of the Hausdorfff dimensions of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ and $\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$ for any $\hat{v}\in[0,\eta]$. Furthermore, we show that this lower bound can be reached for some $\hat{v}$.

Detection of vector magnetic fields at nanoscale dimensions is critical in applications ranging from basic material science, to medical diagnostic. Meanwhile, an all-electric operation is of great significance for achieving a simple and compact sensing system. Here, we propose and experimentally demonstrate a simple approach to sensing a vector magnetic field at nanoscale dimensions, by monitoring a probabilistic nanomagnet's transition probability from a metastable state, excited by a driving current due to SOT, to a settled state. We achieve sensitivities for Hx, Hy, and Hz of 1.02%/Oe, 1.09%/Oe and 3.43%/Oe, respectively, with a 200 x 200 nm^2 nanomagnet. The minimum detectable field is dependent on the driving pulse events N, and is expected to be as low as 1 uT if N = 3 x 10^6., Comment: 15 pages, 4 figures

We present room-temperature measurements of magnon spin diffusion in epitaxial ferrimagnetic insulator MgAl$_{0.5}$Fe$_{1.5}$O$_{4}$ (MAFO) thin films near zero applied magnetic field where the sample forms a multi-domain state. Due to a weak uniaxial magnetic anisotropy, the domains are separated primarily by 180$^\circ$ domain walls. We find, surprisingly, that the presence of the domain walls has very little effect on the spin diffusion -- nonlocal spin transport signals in the multi-domain state retain at least 95% of the maximum signal strength measured for the spatially-uniform magnetic state, over distances at least five times the typical domain size. This result is in conflict with simple models of interactions between magnons and static domain walls, which predict that the spin polarization carried by the magnons reverses upon passage through a 180$^\circ$ domain wall.

Let $b \geq 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any non-dense set $A \subset [0,1)$ such that $A \cap \mathbb{Q}$ is invariant under $\times b$ operation, we prove the finiteness of rational numbers in $A$ whose denominators can only be divided by primes in $S$. A quantitative result on the largest prime divisors of the denominators of rational numbers in $A$ is also obtained.

We report measurements of magnon spin transport in a spinel ferrite, magnesium aluminum ferrite $\mathrm{MgAl_{0.5}Fe_{1.5}O_4}$ (MAFO), which has a substantial in-plane four-fold magnetic anisotropy. We observe spin diffusion lengths $> 0.8$ $\mathrm{\mu m}$ at room temperature in 6 nm films, with spin diffusion length 30% longer along the easy axes compared to the hard axes. The sign of this difference is opposite to the effects just of anisotropy in the magnetic energy for a uniform magnetic state. We suggest instead that accounting for anisotropy in exchange stiffness is necessary to explain these results.

Single-crystal materials with sufficiently low crystal symmetry and strong spin-orbit interactions can be used to generate novel forms of spin-orbit torques on adjacent ferromagnets, such as the out-of-plane antidamping torque previously observed in WTe$_2$/ferromagnet heterostructures. Here, we present measurements of spin-orbit torques produced by the low-symmetry material $\beta$-MoTe$_2$, which unlike WTe$_2$ retains bulk inversion symmetry. We measure spin-orbit torques on $\beta$-MoTe$_2$/Permalloy heterostructures using spin-torque ferromagnetic resonance as a function of crystallographic alignment and MoTe$_2$ thickness down to the monolayer limit. We observe an out-of-plane antidamping torque with a spin torque conductivity as strong as 1/3 of that of WTe$_2$, demonstrating that the breaking of bulk inversion symmetry in the spin-generation material is not a necessary requirement for producing an out-of-plane antidamping torque. We also measure an unexpected dependence on the thickness of the $\beta$-MoTe$_2$ -- the out-of-plane antidamping torque is present in MoTe$_2$/Permalloy heterostructures when the $\beta$-MoTe$_2$ is a monolayer or trilayer thick, but goes to zero for devices with bilayer $\beta$-MoTe$_2$.