We review first Azumaya geometry and D-branes in the realm of algebraic geometry along the line of Polchinski-Grothendieck Ansatz from our earlier work and then use it as background to introduce Azumaya C^{\infty} -manifolds with a fundamental module and morphisms therefrom to a projective complex manifold. This gives us a description of D-branes of A-type. Donaldson's picture of Lagrangian and special Lagrangian submanifolds as selected from the zero-locus of a moment map on a related space of maps can be merged into the setting. As a pedagogical toy model, we study D-branes of A-type in a Calabi-Yau torus. Simple as it is, it reveals several features of D-branes, including their assembling/disassembling. The 4th theme of Sec. 2.4, the 2nd theme of Sec. 4.2, and Sec. 4.3 are to be read respectively with Gmez-Sharpe (arXiv: hep-th/0008150), Donagi-Katz-Sharpe (arXiv: hep-th/0309270), and Denef (arXiv: hep-th/0107152). Some string-theoretical remarks are given at the end of each section.

In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.

Wave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic media. Starting with Maxwell's equations with a honeycomb material weight as well as the nonlinear Kerr effect, we first prove the existence of Dirac points in the dispersion surfaces of transverse electric and magnetic Maxwell operators under very general assumptions of the material weight. Our assumptions on the material weight are almost the minimal requirements to ensure the existence of Dirac points in a general hexagonal photonic crystal. We then derive the associated wave packet dynamics in the scenario where the honeycomb structure is weakly modulated. It turns out the reduced envelope equation is generally a two-dimensional nonlinear Dirac equation with a spatially varying mass. By studying the reduced envelope equation with a domain-wall-like mass term, we realize the subtle wave motions which are chiral and immune to local defects. The underlying mechanism is the existence of topologically protected linear line modes, also referred to as edge states. However, we show that these robust linear modes do not survive with nonlinearity. We demonstrate the existence of nonlinear line modes, which can propagate in the nonlinear media based on high-accuracy numerical computations. Moreover, we also report a new type of nonlinear modes which are localized in both directions.

We report a numerical study of Rayleigh-Benard convection through random porous media using pore-scale modelling, focusing on the Lagrangian dynamics of fluid particles and heat transfer for varied porosities . Due to the interaction between the porous medium and the coherent flow structures, the flow is found to be highly heterogeneous, consisting of convection channels with strong flow strength and stagnant regions with low velocities. The modifications of flow field due to porous structure have a significant influence on the dynamics of fluid particles. Evaluation of the particle displacement along the trajectory reveals the emergence of anomalous transport for long times as is decreased, which is associated with the long-time correlation of Lagrangian velocity of the fluid. As porosity is decreased, the cross-correlation between the vertical velocity and temperature fluctuation is enhanced, which reveals a mechanism to enhance the heat transfer in porous-media convection.

Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group G, a parabolic subgroup P, and a topological surface M, the (enhanced) spectral Eisenstein series category of M is the factorization homology over M of the E2-Hecke category H_{G,P} =IndCoh(LS_{G,P}(D^2, S^1)), where LS_{G,P}(D^2, S^1) denotes the moduli stack of G-local systems on a disk together with a P-reduction on the boundary circle. More generally, for any pair of stacks Y → Z satisfying some mild conditions and any map between topological spaces N → M, we define (Y, Z)^{N,M} = Y^N ×_{Z^N} {Z^M} to be the space of maps from M to Z along with a lift to Y of its restriction to N. Using the pair of pants construction, we define an E_n-category H_n(Y, Z) = IndCoh_0((Y, Z)^{S^n−1,D^n})_{Y}), and compute its factorization homology on any d-dimensional manifold M with d ≤ n, \int_M H_n(Y, Z) = IndCoh_0(Y, Z)^{∂(M×D^n−d),M}_{Y^M}, where IndCoh_0 is the sheaf theory introduced by ArinkinGaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi–Francis–Nadler and Beraldo.

Remeshing aims to produce a more regular mesh from a given input mesh, while representing the original geometry as accurately as possible. Many existing remeshing methods focus on where to place new mesh vertices; these samples are placed exactly on the input mesh. However, considering the output mesh as a piecewise linear approximation of some geometry, this simple scheme leads to significant systematic error in non-planar regions. Here, we use parameterised meshes and the recent mathematical development of orthogonal approximation using Sobolev-type inner products to develop a novel sampling scheme which allows vertices to lie in space near the input surface, rather than exactly on it. The algorithm requires little extra computational effort and can be readily incorporated into many remeshing approaches. Experimental results show that on average, approximation error can be reduced by 40% with the same number of vertices.

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!