In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and nonstationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t) cos(θ(t))}, where a ∈ V (θ), V (θ) consists of the functions smoother than cos(θ(t)) and θ   0. This problem can be formulated as a nonlinear l0 optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the l0 optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide an error analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the state-of-the-art methods. We also apply our method to study data without scale separation, and data with incomplete or under-sampled data.

Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : (Cn , 0) → (C, 0). The Yau algebra L(V ) is defined to be the Lie algebra of derivations of the moduli algebra A(V ) := On/(f, ∂f , · · · , ∂f ), ∂x1 ∂xn i.e., L(V ) = Der(A(V ), A(V )) and plays an important role in singularity theory. It is known that L(V ) is a finite dimensional Lie algebra and its dimension λ(V ) is called Yau number. In this article, we generalize the Yau algebra and introduce a new series of k-th Yau algebras Lk(V) which are defined to be the Lie algebras of derivations of the moduli algebras Ak(V) = On/(f,mkJ(f)),k ≥ 0, i.e., Lk(V) = Der(Ak(V),Ak(V)) and where m is the maximal ideal of On. In particular, it is Yau algebra when k = 0. The dimension of Lk(V ) is denoted by λk(V ). These numbers i.e., k-th Yau numbers λk(V ), are new numerical analytic invariants of an isolated singularity. In this paper we studied these new series of Lie algebras Lk(V ) and also compute the Lie algebras L1(V ) for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the λk(V ) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of k = 1 for large class of singularities.

In this paper, we consider local multiscale model reduction for problems with multiple scales in space and time. We developed our approaches within the framework of the Generalized Multiscale Finite Element Method (GMsFEM) using spacetime coarse cells. The main idea of GMsFEM is to construct a local snapshot space and a local spectral decomposition in the snapshot space. Previous research in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. In this paper, our main objective is to develop a multiscale model reduction framework within GMsFEM that uses spacetime coarse cells. We construct spacetime snapshot and offline spaces. We compute these snapshot solutions by solving local problems. A complete snapshot space will use all possible boundary conditions; however, this can be very expensive. We propose using

We prove that a general miraculous cancellation formula, the divisibility of certain characteristic numbers, and some other topological results related to the generalized Rochlin invariant, the -invariant and the holonomies of certain determinant line bundles, are consequences of the modular invariance of elliptic operators on loop space.

Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of Q-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincar´e transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GVsubschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.