The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.

The nonlinear and non-stationary nature of Navier-Stokes equations produces fluid flows that can be noticeably different in appearance with subtle changes. In this paper we introduce a method that can analyze the intrinsic multiscale features of flow fields from a decomposition point of view, by using the Hilbert-Huang transform method on 3D fluid simulation. We show how this method can provide insights to flow styles and help modulate the fluid simulation with its internal physical information. We provide easy-toimplement algorithms that can be integrated with standard grid-based fluid simulation methods, and demonstrate how this approach can modulate the flow field and guide the simulation with different flow styles. The modulation is straightforward and relates directly to the flow¡¯s visual effect, with moderate computational overhead.

We study a robust and efficient eigensolver for computing the positive dense spectrum of the two-dimensional transmission eigenvalue problem (TEP) which is derived from the Maxwell’s equation with complex media in pseudo-chiral model and the transverse magnetic mode. The discretized governing equations by the N ́ed ́elec edge element result in a large-scale quadratic eigenvalue problem (QEP). We estimate half of the positive eigenvalues of the QEP are on some interval which forms a dense spectrum of the QEP. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Intensive numerical experiments show that our proposed method makes the convergence efficiently and robustly even it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by the non- linear functions which can be applied to estimate the density of the eigenvalues for the TEP.

Let O_c be the category of finite-length central-charge-c modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that O_c admits vertex algebraic tensor category structure for any c∈C. Here, we determine the structure of this tensor category when c=13−6p−6/p for an integer p>1. For such c, we prove that O_c is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory O_c^0. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that O_c has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine sl_2 at levels −2+p^±1. Next, as a straightforward consequence of the braided tensor category structure on O_c together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras W(p), including rigidity, fusion rules, and construction of projective covers. Finally, we prove a recent conjecture of Negron that O_c^0 is braided tensor equivalent to the PSL(2,C)-equivariantization of the category of W(p)-modules.

Loewner’s equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the present paper is that the Dirichlet energy of this driving function (also known as the Loewner energy) is equal to the Dirichlet energy of the log-derivative of the (appropriately defined) uniformizing conformal map. This description of the Loewner energy then enables to tie direct links with regularized determinants and Teichmüller theory: We show that for smooth simple loops, the Loewner energy can be expressed in terms of the zeta-regularized determinants of a certain Neumann jump operator. We also show that the family of finite Loewner energy loops coincides with the Weil–Petersson class of quasicircles, and that the Loewner energy equals to a multiple of the universal Liouville action introduced by Takhtajan and Teo, which is a Kähler potential for the Weil–Petersson metric on the Weil–Petersson Teichmüller space.