The modules of principal parts$P$^{$k$}($E$) of a locally free sheaf ε on a smooth scheme$X$is a sheaf of$O$_{$X$}-bimodules which is locally free as left and right$O$_{$X$}-module. We explicitly split the modules of principal parts$P$^{$k$}($O$($n$)) on the projective line in arbitrary characteristic, as left and right$O$_{p1}-modules. We get examples when the splitting-type as left module differs from the splitting-type as right module. We also give examples showing that the splitting-type of the principal parts changes with the characteristic of the base field.

Classification of DNA sequences is an important issue in the bioinformatics study, yet most existing methods for phylogenetic analysis including Multiple Sequence Alignment (MSA) are time-consuming and computationally expensive. The alignment-free methods are popular nowadays, whereas the manual intervention in those methods usually decreases the accuracy. Also, the interactions among nucleotides are neglected in most methods. Here we propose a new Accumulated Natural Vector (ANV) method which represents each DNA sequence by a point in R18. By calculating the Accumulated Indicator Functions of nucleotides, we can further find an Accumulated Natural Vector for each sequence. This new Accumulated Natural Vector not only can capture the distribution of each nucleotide, but also provide the covariance among nucleotides. Thus global comparison of DNA sequences or genomes can be done easily in R18. The tests of ANV of datasets of different sizes and types have proved the accuracy and time-efficiency of the new proposed ANV method.

We study front speeds of curvature and strain G-equations arising in turbulent combustion. These G-equations are HamiltonJacobi type level set partial differential equations (PDEs) with non-coercive Hamiltonians and degenerate nonlinear second order diffusion. The Hamiltonian of a strain G-equation is also non-convex. Numerical computation is performed based on monotone discretization and weighted essentially nonoscillatory (WENO) approximation of transformed G-equations on a fixed periodic domain. The advection field in the computation is a two dimensional Hamiltonian flow consisting of a periodic array of counter-rotating vortices, or cellular flows. Depending on whether the evolution is predominantly in the hyperbolic or parabolic regimes, suitable explicit and semi-implicit time stepping methods are chosen. The turbulent flame speeds are computed as the linear growth rates of large time solutions. A

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.

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge- Amp`ere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R3. We prove a general local existence result for a large class of Monge-Amp`ere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth curve.