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Emerging two-dimensional transition metal dichalcogenides (TMDCs) offer a promising platform for on-chip integrated photonics because of their unique optical and electronic properties. Their naturally passivated surfaces make them highly tolerant to lattice mismatch, enabling seamless heterogeneous integration by stacking different van der Waals materials, a crucial step in the development of advanced photonic devices. Here, we demonstrate the use of an inverse design wavelength demultiplexing waveguides for on-chip sorting and routing of distinct photoluminescence from the heterojunction formed by WS2 and WSe2 monolayers. The integrated nanophotonic chip splits and sorts excitonic emission into individual waveguides at room temperature. Our demonstration opens up new perspectives for integrating light sources in van der Waals materials with functional integrated photonics, offering a versatile platform for both fundamental research and practical applications.

The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the spheres can be used to create effective, stable finite difference methods based on radial basis functions (RBF-FD). For certain classes of PDEs this approach leads to precise convergence estimates for stencils which grow moderately with increasing discretization fineness.

We consider Landau-Ginzburg models stemming from groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group $G^\star$, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials., we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials., Comment: v3 fixes some errors in v2. Altogether 29 pages, 7 tables

We introduce intrinsic interpolatory bases for data structured on graphs and derive properties of those bases. Polyharmonic Lagrange functions are shown to satisfy exponential decay away from their centers. The decay depends on the density of the zeros of the Lagrange function, showing that they scale with the density of the data. These results indicate that Lagrange-type bases are ideal building blocks for analyzing data on graphs, and we illustrate their use in kernel-based machine learning applications., Comment: 17 pages

We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution., Comment: 23 pages, 2, figures, submitted to Math Comp

While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the $L_p$ norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Mat\'ern and surface spline RBFs). In this way it extends the boundary-free construction of Fuselier, Hangelbroek, Narcowich, Ward and Wright.

This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are spatially highly localized. The construction of such functions is computationally efficient and generalizes the construction given by the authors for restricted surface splines on $\mathbb{R}^d$. The kernels for which the theory applies includes the Sobolev-Mat\'ern kernels for closed, compact, connected, $C^\infty$ Riemannian manifolds., Comment: 29 pages. To appear in Festschrift for the 80th Birthday of Ian Sloan

We present a novel Galerkin method for solving partial differential equations on the sphere. The problem is discretized by a highly localized basis which is easily constructed. The stiffness matrix entries are computed by a recently developed quadrature formula unique to the localized bases we consider. We present error estimates and investigate the stability of the discrete stiffness matrix. Implementation and numerical experiments are discussed., Comment: 22 pages, 1 figure. Changes to previous version include new numerical experiments, updated figures, new styling, and re-written sections, and additional theory to address the use of local Lagrange functions for the discretization basis