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Let $\Lambda_X(s)=\det(I-sX^{\dagger})$ be the characteristic polynomial of the unitary matrix $X$. It is believed that the distribution of values of $\Lambda_X(s)$ model the distribution of values of the Riemann zeta-function $\zeta(s)$. This principle motivates many avenues of study. Of particular interest is the behavior of $\Lambda_X'(s)$ and the distribution of its zeros (all of which lie inside or on the unit circle). In this article we present several identities for the moments of $\Lambda_X'(s)$ averaged over $U(N)$, for $x \in \mathbb{C}$ as well as specialized to $|x|=1$. Additionally, we prove, for positive integer $k$, that the polynomial $\int_{U(N)} |\Lambda_X(1)|^{2k} dX$ of degree $k^2$ in $N$ divides the polynomial $\int_{U(N)} |\Lambda_X'(1)|^{2k} dX$ which is of degree $k^2+2k$ in $N$ and that the ratio, $f(N,k)$, of these moments factors into linear factors modulo $4k-1$ if $4k-1$ is prime. We also discuss the relationship of these moments to a solution of a second order non-linear Painl\'{e}ve differential equation. Finally we give some formulas in terms of the $_3F_2$ hypergeometric series for the moments in the simplest case when $N=2$, and also study the radial distribution of the zeros of $\Lambda_X'(s)$ in that case.

Customizing text-to-image (T2I) models has seen tremendous progress recently, particularly in areas such as personalization, stylization, and conditional generation. However, expanding this progress to video generation is still in its infancy, primarily due to the lack of customized video data. In this work, we introduce Still-Moving, a novel generic framework for customizing a text-to-video (T2V) model, without requiring any customized video data. The framework applies to the prominent T2V design where the video model is built over a text-to-image (T2I) model (e.g., via inflation). We assume access to a customized version of the T2I model, trained only on still image data (e.g., using DreamBooth or StyleDrop). Naively plugging in the weights of the customized T2I model into the T2V model often leads to significant artifacts or insufficient adherence to the customization data. To overcome this issue, we train lightweight $\textit{Spatial Adapters}$ that adjust the features produced by the injected T2I layers. Importantly, our adapters are trained on $\textit{"frozen videos"}$ (i.e., repeated images), constructed from image samples generated by the customized T2I model. This training is facilitated by a novel $\textit{Motion Adapter}$ module, which allows us to train on such static videos while preserving the motion prior of the video model. At test time, we remove the Motion Adapter modules and leave in only the trained Spatial Adapters. This restores the motion prior of the T2V model while adhering to the spatial prior of the customized T2I model. We demonstrate the effectiveness of our approach on diverse tasks including personalized, stylized, and conditional generation. In all evaluated scenarios, our method seamlessly integrates the spatial prior of the customized T2I model with a motion prior supplied by the T2V model., Comment: Webpage: https://still-moving.github.io/ | Video: https://www.youtube.com/watch?v=U7UuV_VIjnA

We present Magic Insert, a method for dragging-and-dropping subjects from a user-provided image into a target image of a different style in a physically plausible manner while matching the style of the target image. This work formalizes the problem of style-aware drag-and-drop and presents a method for tackling it by addressing two sub-problems: style-aware personalization and realistic object insertion in stylized images. For style-aware personalization, our method first fine-tunes a pretrained text-to-image diffusion model using LoRA and learned text tokens on the subject image, and then infuses it with a CLIP representation of the target style. For object insertion, we use Bootstrapped Domain Adaption to adapt a domain-specific photorealistic object insertion model to the domain of diverse artistic styles. Overall, the method significantly outperforms traditional approaches such as inpainting. Finally, we present a dataset, SubjectPlop, to facilitate evaluation and future progress in this area. Project page: https://magicinsert.github.io/, Comment: Project page: https://magicinsert.github.io/

Polymers are an effective test-bed for studying topological constraints in condensed matter due to a wide array of synthetically-available chain topologies. When linear and ring polymers are blended together, emergent rheological properties are observed as the blend can be more viscous than either of the individual components. This emergent behavior arises since ring-linear blends can form long-lived topological constraints as the linear polymers thread the ring polymers. Here, we demonstrate how the Gauss linking integral can be used to efficiently evaluate the relaxation of topological constraints in ring-linear polymer blends. For majority-linear blends, the relaxation rate of topological constraints depends primarily on reptation of the linear polymers, resulting in the diffusive time $\tau_{d,R}$ for rings of length $N_R$ blended with linear chains of length $N_l$ to scale as $\tau_{d,R}\sim N_R^2N_L^{3.4}$., Comment: 6 pages, 6 figures

Recent advances in large-scale pretraining have yielded visual foundation models with strong capabilities. Not only can recent models generalize to arbitrary images for their training task, their intermediate representations are useful for other visual tasks such as detection and segmentation. Given that such models can classify, delineate, and localize objects in 2D, we ask whether they also represent their 3D structure? In this work, we analyze the 3D awareness of visual foundation models. We posit that 3D awareness implies that representations (1) encode the 3D structure of the scene and (2) consistently represent the surface across views. We conduct a series of experiments using task-specific probes and zero-shot inference procedures on frozen features. Our experiments reveal several limitations of the current models. Our code and analysis can be found at https://github.com/mbanani/probe3d., Comment: Accepted to CVPR 2024. Project page: https://github.com/mbanani/probe3d

We introduce Lumiere -- a text-to-video diffusion model designed for synthesizing videos that portray realistic, diverse and coherent motion -- a pivotal challenge in video synthesis. To this end, we introduce a Space-Time U-Net architecture that generates the entire temporal duration of the video at once, through a single pass in the model. This is in contrast to existing video models which synthesize distant keyframes followed by temporal super-resolution -- an approach that inherently makes global temporal consistency difficult to achieve. By deploying both spatial and (importantly) temporal down- and up-sampling and leveraging a pre-trained text-to-image diffusion model, our model learns to directly generate a full-frame-rate, low-resolution video by processing it in multiple space-time scales. We demonstrate state-of-the-art text-to-video generation results, and show that our design easily facilitates a wide range of content creation tasks and video editing applications, including image-to-video, video inpainting, and stylized generation., Comment: Webpage: https://lumiere-video.github.io/ | Video: https://www.youtube.com/watch?v=wxLr02Dz2Sc