Subquadratic harmonic functions on Calabi-Yau manifolds with Euclidean volume growth

We prove that on a complete Calabi-Yau manifold $M$ with Euclidean volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon-Hein. We prove this result by proving a Liouville type theorem for harmonic $1$-forms, which follows from a new local $L^2$ estimate of the differential. We also give another proof based on the construction of harmonic functions with polynomial growth in Ding, and the algebraicity of tangent cones in Liu-Sz\'ekelyhidi.Comment: 30 pages. Comments are welcom

Compatibility Family Learning for Item Recommendation and Generation

Compatibility between items, such as clothes and shoes, is a major factor among customer's purchasing decisions. However, learning "compatibility" is challenging due to (1) broader notions of compatibility than those of similarity, (2) the asymmetric nature of compatibility, and (3) only a small set of compatible and incompatible items are observed. We propose an end-to-end trainable system to embed each item into a latent vector and project a query item into K compatible prototypes in the same space. These prototypes reflect the broad notions of compatibility. We refer to both the embedding and prototypes as "Compatibility Family". In our learned space, we introduce a novel Projected Compatibility Distance (PCD) function which is differentiable and ensures diversity by aiming for at least one prototype to be close to a compatible item, whereas none of the prototypes are close to an incompatible item. We evaluate our system on a toy dataset, two Amazon product datasets, and Polyvore outfit dataset. Our method consistently achieves state-of-the-art performance. Finally, we show that we can visualize the candidate compatible prototypes using a Metric-regularized Conditional Generative Adversarial Network (MrCGAN), where the input is a projected prototype and the output is a generated image of a compatible item. We ask human evaluators to judge the relative compatibility between our generated images and images generated by CGANs conditioned directly on query items. Our generated images are significantly preferred, with roughly twice the number of votes as others.Comment: 9 pages, accepted to AAAI 201

The role of initial geometry in experimental models of wound closing

Wound healing assays are commonly used to study how populations of cells, initialised on a two-dimensional surface, act to close an artificial wound space. While real wounds have different shapes, standard wound healing assays often deal with just one simple wound shape, and it is unclear whether varying the wound shape might impact how we interpret results from these experiments. In this work, we describe a new kind of wound healing assay, called a sticker assay, that allows us to examine the role of wound shape in a series of wound healing assays performed with fibroblast cells. In particular, we show how to use the sticker assay to examine wound healing with square, circular and triangular shaped wounds. We take a standard approach and report measurements of the size of the wound as a function of time. This shows that the rate of wound closure depends on the initial wound shape. This result is interesting because the only aspect of the assay that we change is the initial wound shape, and the reason for the different rate of wound closure is unclear. To provide more insight into the experimental observations we describe our results quantitatively by calibrating a mathematical model, describing the relevant transport phenomena, to match our experimental data. Overall, our results suggest that the rates of cell motility and cell proliferation from different initial wound shapes are approximately the same, implying that the differences we observe in the wound closure rate are consistent with a fairly typical mathematical model of wound healing. Our results imply that parameter estimates obtained from an experiment performed with one particular wound shape could be used to describe an experiment performed with a different shape. This fundamental result is important because this assumption is often invoked, but never tested

MIMO Evolution toward 6G: End-User-Centric Collaborative MIMO

In 6G, the trend of transitioning from massive antenna elements to even more massive ones is continued. However, installing additional antennas in the limited space of user equipment (UE) is challenging, resulting in limited capacity scaling gain for end users, despite network side support for increasing numbers of antennas. To address this issue, we propose an end-user-centric collaborative MIMO (UE-CoMIMO) framework that groups several fixed or portable devices to provide a virtual abundance of antennas. This article outlines how advanced L1 relays and conventional relays enable device collaboration to offer diversity, rank, and localization enhancements. We demonstrate through system-level simulations how the UE-CoMIMO approaches lead to significant performance gains. Lastly, we discuss necessary research efforts to make UE-CoMIMO available for 6G and future research directions.Comment: 7 pages, 5 figures, 1 table. This work has been accepted in IEEE Communications Magazin