Products and connected sums of spheres as monotone Lagrangian submanifolds

We obtain new restrictions on Maslov classes of monotone Lagrangian submanifolds of $\mathbb{C}^n$. We also construct families of new examples of monotone Lagrangian submanifolds, which show that the restrictions on Maslov classes are sharp in certain cases.Comment: 18 pages, clarified and simplified. arXiv admin note: text overlap with arXiv:1812.0500

Discrete Distribution Estimation under User-level Local Differential Privacy

We study discrete distribution estimation under user-level local differential privacy (LDP). In user-level $\varepsilon$-LDP, each user has $m\ge1$ samples and the privacy of all $m$ samples must be preserved simultaneously. We resolve the following dilemma: While on the one hand having more samples per user should provide more information about the underlying distribution, on the other hand, guaranteeing the privacy of all $m$ samples should make the estimation task more difficult. We obtain tight bounds for this problem under almost all parameter regimes. Perhaps surprisingly, we show that in suitable parameter regimes, having $m$ samples per user is equivalent to having $m$ times more users, each with only one sample. Our results demonstrate interesting phase transitions for $m$ and the privacy parameter $\varepsilon$ in the estimation risk. Finally, connecting with recent results on shuffled DP, we show that combined with random shuffling, our algorithm leads to optimal error guarantees (up to logarithmic factors) under the central model of user-level DP in certain parameter regimes. We provide several simulations to verify our theoretical findings.Comment: 26 pages, 4 figure

Solving the Problem of Silicon Dioxide Melting Based on Deviation Model

In order to reveal the dissolution behavior of iron tailings in blast furnace slag, we studied the main component of silica in iron tailings. First, edge contour features need to be established to represent the melting process of silica. We choose shape, perimeter, area and generalized radius as objects. By independently analyzing the influence of these four indexes on the melting rate, the area and shape were selected as the characteristic parameters of the edge contour of the silica particles. Then, the actual melting rate of the silica is estimated by the edge contour feature index. Finally, we can calculate the melting rate of the first second of three time periods of 0.00010312mm3/s, 0.0002399mm3/s, 0.0000538mm3/s

Best Determined Position of Vents Based on Jet Cooling Model

In some data centers, cold air is required to act on the cabinet to achieve cooling requirements, and the mixing of cold air and hot air reduces the utilization efficiency of cold air. In order to solve this problem, a jet cooling model is established to solve the optimal position of the outlet through the movement of cold air

Augmentation varieties and disk potentials II

This is the second in a sequence of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we first define the Chekanov-Eliashberg algebra and its Legendrian contact homology. For a tame Lagrangian cobordism between Legendrians, we define a chain map between their Chekanov-Eliashberg algebras.Comment: 52 pages; The original manuscript arXiv:2310.17821v1 was split into three parts: this being part I