Generalized flatness constants, spanning lattice polytopes, and the Gromov width

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.

Gearbox compound fault diagnosis based on a combined MSGMD-MOMEDA method

Weak fault detection is a complex and challenging task when two or more faults (compound fault) with discordant severity occur in different parts of a gearbox. The weak fault features are prone to be submerged by the severe fault features and strong background noise, which easily lead to a missed diagnosis. To solve this problem, a novel diagnosis method combining muti-symplectic geometry mode decomposition and multipoint optimal minimum entropy deconvolution adjusted (MSGMD-MOMEDA) is proposed for gearbox compound fault in this paper. Specifically, different fault components are separated by the improved symplectic geometry mode decomposition (SGMD), namely, multi-SGMD (MSGMD) method. The weak fault features are enhanced by the multipoint optimal minimum entropy deconvolution adjusted (MOMEDA). In the process of research, a new scheme of selecting key parameters of MOMEDA is proposed, which is a key step in applying MOMEDA. Compared with SGMD, the proposed MSGMD has two main improvements, including suppressing mode mixing and preventing the generation of the pseudo components. Compared with the original method of selecting parameters based on multipoint kurtosis, the proposed MOMEDA parameters selecting scheme has more merits of high accuracy and precision. The analysis results of two cases of simulation and experiment signal reveal that the MSGMD-MOMEDA method can accurately diagnose the gearbox compound fault even under strong background noise.

-rigidity of Lagrangian submanifolds and punctured holomorphic disks in the cotangent bundle

Our main result is the $\mathbb {\mathcal {C}}^{0}$ -rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic disks in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured disks in symplectic geometry.

Symplectic Geometry of the Koopman Operator

We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.

Systolic inequalities for K3 surfaces via stability conditions

We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface X, there exists a constant C depending only on the rank and discriminant of NS(X), such that $$\begin{aligned} \mathrm {sys}(\sigma )^2\le C\cdot \mathrm {vol}(\sigma ) \end{aligned}$$ sys ( σ ) 2 ≤ C · vol ( σ ) holds for any stability condition on $$\mathcal {D}^b\mathrm {Coh}(X)$$ D b Coh ( X ) . This is an algebro-geometric generalization of a classical systolic inequality on two-tori. We also discuss applications of this inequality in symplectic geometry.