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Polaritons in two-dimensional (2D) materials provide unique opportunities for controlling light at nanoscales. Tailoring these polaritons via gradient polaritonic surfaces with space-variant response can enable versatile light-matter interaction platforms with advanced functionalities. However, experimental progress has been hampered by the optical losses and poor light confinement of conventionally used artificial nanostructures. Here, we demonstrate natural gradient polaritonic surfaces based on superlattices of solitons-localized structural deformations-in a prototypical moire system, twisted bilayer graphene on boron nitride. We demonstrate on-off switching and continuous modulation of local polariton-soliton interactions, which results from marked modifications of topological and conventional soliton states through variation of local strain direction. Furthermore, we reveal the capability of these structures to spatially modify the near-field profile, phase, and propagation direction of polaritons in record-small footprints, enabling generation and electrical switching of directional polaritons. Our findings open up new avenues toward nanoscale manipulation of light-matter interactions and spatial polariton engineering through gradient moire superlattices., Comment: 18 pages, 4 figures

Electrifying heavy-duty trucks offers a substantial opportunity to curtail carbon emissions, advancing toward a carbon-neutral future. However, the inherent challenges of limited battery energy and the sheer weight of heavy-duty trucks lead to reduced mileage and prolonged charging durations. Consequently, battery-swapping services emerge as an attractive solution for these trucks. This paper employs a two-fold approach to investigate the potential and enhance the efficacy of such services. Firstly, spatial-temporal demand prediction models are adopted to predict the traffic patterns for the upcoming hours. Subsequently, the prediction guides an optimization module for efficient battery allocation and deployment. Analyzing the heavy-duty truck data on a highway network spanning over 2,500 miles, our model and analysis underscore the value of prediction/machine learning in facilitating future decision-makings. In particular, we find that the initial phase of implementing battery-swapping services favors mobile battery-swapping stations, but as the system matures, fixed-location stations are preferred., Comment: 9 pages, 6 figures

We establish generic uniform convergence guarantees for Gaussian data in terms of the Rademacher complexity of the hypothesis class and the Lipschitz constant of the square root of the scalar loss function. We show how these guarantees substantially generalize previous results based on smoothness (Lipschitz constant of the derivative), and allow us to handle the broader class of square-root-Lipschitz losses, which includes also non-smooth loss functions appropriate for studying phase retrieval and ReLU regression, as well as rederive and better understand "optimistic rate" and interpolation learning guarantees.

We study the inversion analog of the well-known Gauss algorithm for multiplying complex matrices. A simple version is $(A + iB)^{-1} = (A + BA^{-1}B)^{-1} - i A^{-1}B(A+BA^{-1} B)^{-1}$ when $A$ is invertible, which may be traced back to Frobenius but has received scant attention. We prove that it is optimal, requiring fewest matrix multiplications and inversions over the base field, and we extend it in three ways: (i) to any invertible $A + iB$ without requiring $A$ or $B$ be invertible; (ii) to any iterated quadratic extension fields, with $\mathbb{C}$ over $\mathbb{R}$ a special case; (iii) to Hermitian positive definite matrices $A + iB$ by exploiting symmetric positive definiteness of $A$ and $A + BA^{-1}B$. We call all such algorithms Frobenius inversions, which we will see do not follow from Sherman--Morrison--Woodbury type identities and cannot be extended to Moore--Penrose pseudoinverse. We show that a complex matrix with well-conditioned real and imaginary parts can be arbitrarily ill-conditioned, a situation tailor-made for Frobenius inversion. We prove that Frobenius inversion for complex matrices is faster than standard inversion by LU decomposition and Frobenius inversion for Hermitian positive definite matrices is faster than standard inversion by Cholesky decomposition. We provide extensive numerical experiments, applying Frobenius inversion to solve linear systems, evaluate matrix sign function, solve Sylvester equation, and compute polar decomposition, showing that Frobenius inversion can be more efficient than LU/Cholesky decomposition with negligible loss in accuracy. A side result is a generalization of Gauss multiplication to iterated quadratic extensions, which we show is intimately related to the Karatsuba algorithm for fast integer multiplication and multidimensional fast Fourier transform., Comment: 29 pages, 8 figures

We extend rank-constrained optimization to general hyperbolic programs (HP) using the notion of matroid rank. For LP and SDP respectively, this reduces to sparsity-constrained LP and rank-constrained SDP that are already well-studied. But for QCQP and SOCP, we obtain new interesting optimization problems. For example, rank-constrained SOCP includes weighted Max-Cut and nonconvex QP as special cases, and dropping the rank constraints yield the standard SOCP-relaxations of these problems. We will show (i) how to do rank reduction for SOCP and QCQP, (ii) that rank-constrained SOCP and rank-constrained QCQP are NP-hard, and (iii) an improved result for rank-constrained SDP showing that if the number of constraints is $m$ and the rank constraint is less than $2^{1/2-\epsilon} \sqrt{m}$ for some $\epsilon>0$, then the problem is NP-hard. We will also study sparsity-constrained HP and extend results on LP sparsification to SOCP and QCQP. In particular, we show that there always exist (a) a solution to SOCP of cardinality at most twice the number of constraints and (b) a solution to QCQP of cardinality at most the sum of the number of linear constraints and the sum of the rank of the matrices in the quadratic constraints; and both (a) and (b) can be found efficiently.

We present a notion of bilinear stability, which is to numerical stability what bilinear complexity is to time complexity. In bilinear complexity, an algorithm for evaluating a bilinear operator $\beta : \mathbb{U} \times \mathbb{V} \to \mathbb{W}$ is a decomposition $\beta = \varphi_1 \otimes \psi_1 \otimes w_1 + \dots + \varphi_r \otimes \psi_r \otimes w_r $; the number of terms $r$ captures the speed of the algorithm; and its smallest possible value, i.e., the tensor rank of $\beta$, quantifies the speed of a fastest algorithm. Bilinear stability introduces norms to the mix: The growth factor of the algorithm $\lVert \varphi_1 \rVert_* \lVert \psi_1 \rVert_* \lVert w_1 \rVert + \dots + \lVert \varphi_r \rVert_* \lVert \psi_r \rVert_* \lVert w_r \rVert$ captures the accuracy of the algorithm; and its smallest possible value, i.e., the tensor nuclear norm of $\beta$, quantifies the accuracy of a stablest algorithm. To substantiate this notion, we establish a bound for the forward error in terms of the growth factor and present numerical evidence comparing various fast algorithms for matrix and complex multiplications, showing that larger growth factors correlate with less accurate results. Compared to similar studies of numerical stability, bilinear stability is more general, applying to any bilinear operators and not just matrix or complex multiplications; is more simplistic, bounding forward error in terms of a single (growth) factor; and is truly tensorial like bilinear complexity, invariant under any orthogonal change of coordinates. As an aside, we study a new algorithm for computing complex multiplication in terms of real, much like Gauss's, but is optimally fast and stable in that it attains both tensor rank and nuclear norm., Comment: 25 pages, 7 figures

We study the problem of fair $k$-median where each cluster is required to have a fair representation of individuals from different groups. In the fair representation $k$-median problem, we are given a set of points $X$ in a metric space. Each point $x\in X$ belongs to one of $\ell$ groups. Further, we are given fair representation parameters $\alpha_j$ and $\beta_j$ for each group $j\in [\ell]$. We say that a $k$-clustering $C_1, \cdots, C_k$ fairly represents all groups if the number of points from group $j$ in cluster $C_i$ is between $\alpha_j |C_i|$ and $\beta_j |C_i|$ for every $j\in[\ell]$ and $i\in [k]$. The goal is to find a set $\mathcal{C}$ of $k$ centers and an assignment $\phi: X\rightarrow \mathcal{C}$ such that the clustering defined by $(\mathcal{C}, \phi)$ fairly represents all groups and minimizes the $\ell_1$-objective $\sum_{x\in X} d(x, \phi(x))$. We present an $O(\log k)$-approximation algorithm that runs in time $n^{O(\ell)}$. Note that the known algorithms for the problem either (i) violate the fairness constraints by an additive term or (ii) run in time that is exponential in both $k$ and $\ell$. We also consider an important special case of the problem where $\alpha_j = \beta_j = \frac{f_j}{f}$ and $f_j, f \in \mathbb{N}$ for all $j\in [\ell]$. For this special case, we present an $O(\log k)$-approximation algorithm that runs in $(kf)^{O(\ell)}\log n + poly(n)$ time.

We study the optical response of bilayer graphene with a kink potential composed of a domain wall separating two AB regions with opposite interlayer bias. The band structure and the local optical conductivity in real space are investigated in details based on a continuum model. We find that the one-dimensional chiral states localized at the domain wall contribute significantly to the local optical conductivity, which shows a clear distinction in different regions. The effects of domain wall states on optical conductivity can be detected by spatially and frequency resolved spectroscopic features. From the spectrum at various Fermi energies, important features in the band structure such as the energy separation between two chiral states can be directly measured. When the domain wall region is broad, the spatial distribution of local optical conductivity can provide important information on the bound states as well as the topological domain wall states., Comment: 8 pages, 4 figures

We present a theoretical study of the effects of heterostrain and lattice relaxation on the optical conductivity of twisted bilayer graphene near the magic angle, based on the band structures obtained from a continuum model. We find that heterostrain, lattice relaxation and their combination give rise to very distinctive spectroscopic features in the optical conductivity, which can be used to probe and distinguish these effects. From the spectrum at various Fermi energies, important features in the strain- and relaxation-modified band structure such as the bandgap, bandwidth and van Hove singularities can be directly measured. The peak associated with the transition between the flat bands in the optical conductivity are highly sensitive to the direction of the strain, which can provide direct information on the strain-modified flat bands., Comment: 8 pages, 5 figures

This paper proposes a method of identifying single line outages in power systems based on phasor measurement unit (PMU) measurements and ac power flow models. In addition to the main identification algorithm, a rejection filter is introduced so that the preliminary identified results can be further processed and categorized into three types: correctly identified, misidentified and inconclusive (including correct-filtered and misidentified-filtered). The methods are systematically tested using test systems of various sizes for various PMU placements, and the results show that the proposed identification algorithm has a high identification accuracy and the proposed rejection filter is able to reduce the misidentified rate without significantly increasing the number of inconclusive cases., Comment: 5 pages, 5 figures, accepted by 2020 IEEE PES General Meeting