Analysis and extraction of strongly frequency modulated signals have been a challenging problem for Adaptive Data Analysis methods; e.g., Empirical Mode Decomposition (EMD) [13]. In fact, many of the Newtonian dynamical systems, including conservative mechanical systems, are sources of signals with low to strong levels of frequency modulation. Analysis of such signals is an important issue in system identification problems. In this paper, we present a novel method to accurately extract Intrawave Signals. This method is a descendant of Sparse Time-Frequency Representation (STFR) methods [8, 7]. We will present numerical examples to show the performance of this new algorithm. Theoretical analysis of convergence of the algorithm is also presented as a support for the method. We will show that the algorithm is stable to noise perturbation, as well.

We discuss the problem of deciding when a metrisable topological group G has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on G, that we characterise intrinsically in terms of a linear growth condition on powers of group elements. Combining this with work on the large scale geometry of topological groups, we also identify the class of metrisable groups admitting a canonical global Lipschitz geometry. In turn, minimal metrics connect with Hilbert’s fifth problem for completely metrisable groups and we show, assuming that the set of squares is sufficiently rich, that every element of some identity neighbourhood belongs to a 1-parameter subgroup.

We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two.

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We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $langle v, (X^* X - z)^{-1} w rangle - langle v,w rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v, w in mathbb C^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $Im z geq N^{-1+epsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.