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In general, it is more difficult to formulate a sufficient condition for similarity than a necessary condition. We give a sufficient condition for a Cowen-Douglas operator with a positivity condition to be similar to the backward shift operator on weighted Dirichlet space. This condition involves the holomorphic jet bundle of the eigenvector bundle of the operator.

Realistic reconstruction of 3D clothing from an image has wide applications, such as avatar creation and virtual try-on. This paper presents a novel framework that reconstructs the texture map for 3D garments from a single image with pose. Assuming that 3D garments are modeled by stitching 2D garment sewing patterns, our specific goal is to generate a texture image for the sewing patterns. A key component of our framework, the Texture Unwarper, infers the original texture image from the input clothing image, which exhibits warping and occlusion of texture due to the user's body shape and pose. The Texture Unwarper effectively transforms between the input and output images by mapping the latent spaces of the two images. By inferring the unwarped original texture of the input garment, our method helps reconstruct 3D garment models that can show high-quality texture images realistically deformed for new poses. We validate the effectiveness of our approach through a comparison with other methods and ablation studies.

We are concerned with the similarity problem for Cowen-Douglas operator tuples. The unitary equivalence counterpart was already investigated in the 1970's and geometric concepts including vector bundles and curvature appeared in the description. As the Cowen-Douglas conjecture show, the study of the similarity problem has not been so successful until quite recently. The latest results reveal the close correlation between complex geometry, the corona problem, and the similarity problem for single Cowen-Douglas operators. Without making use of the corona theorems that no longer hold in the multi-variable setting, we prove that the single operator results for similarity remain true for commuting Cowen-Douglas operator tuples as well., Comment: 42pages

Encapsulated bulk mode microresonators in the megahertz range are used in commercial timekeeping and sensing applications but their performance is limited by the current state of the art of readout methods. We demonstrate a readout using dispersive coupling between a high-Q encapsulated bulk mode micromechanical resonator and a lumped element microwave resonator that is implemented with commercially available components and standard printed circuit board fabrication methods and operates at room temperature and pressure. A frequency domain measurement of the microwave readout system yields a displacement resolution of $522 \, \mathrm{fm/\sqrt{Hz}}$, which demonstrates an improvement over the state of the art of displacement measurement in bulk-mode encapsulated microresonators. This approach can be readily implemented in cryogenic measurements, allowing for future work characterizing the thermomechanical noise of encapsulated bulk mode resonators at cryogenic temperatures.

The Kerr Parametric Oscillator (KPO) is a nonlinear resonator system that is often described as a synthetic two-level system. In the presence of noise, the system switches between two states via a fluctuating trajectory in phase space, instead of following a straight path. The presence of such fluctuating trajectories makes it hard to establish a precise count or even a useful definition, of the "lifetime" of the state. Addressing this issue, we compare several rate counting methods that allow to estimate a lifetime for the levels. In particular, we establish that a peak in the Allan variance of fluctuations can also be used to determine the levels' lifetime. Our work provides a basis for characterizing KPO networks for simulated annealing where an accurate determination of the state lifetime is of fundamental importance.

Resonant sensors based on Micro- and Nano-Electro Mechanical Systems (M/NEMS) are ubiquitous in many sensing applications due to their outstanding performance capabilities, which are directly proportional to the quality factor (Q) of the devices. We address here a recurrent question in the field: do dynamical techniques that modify the effective Q (namely parametric pumping and direct drive velocity feedback) affect the performance of said sensors? We develop analytical models of both cases, while remaining in the linear regime, and introduce noise in the system from two separate sources: thermomechanical and amplifier (read-out) noise. We observe that parametric pumping enhances the quality factor in the amplitude response, but worsens it in the phase response on the resonator. In the case of feedback, we find that Q is enhanced in both cases. Then, we establish a solution for the noisy problem with direct drive and parametric pumping simultaneously. We also find that, in the case when thermomechanical noise dominates, no benefit can be obtained from neither artificial Q-enhancement technique. However, in the case when amplifier noise dominates, we surprisingly observe that a significant advantage can only be achieved using parametric pumping in the squeezing region.

We consider a subclass of the Cowen-Douglas class in which the problem of deciding whether two operators are similar becomes more manageable. A similarity criterion for Cowen-Douglas operators is known to be dependent on the trace of the curvatures of the corresponding eigenvector bundles. Unless the given eignvector bundle is a line bundle, the computation of the curvatures, in general, is not so simple as one might hope. By using a structure theorem given in \cite{JW}, we reduce the problem of finding the trace of the curvatures to looking at the curvatures of the associated line bundles. Moreover, several questions related to the similarity problem are also taken into account., Comment: 23pages(to appear in Mathematische Nachrichten)

When the backward shift operator on a weighted space $H^2_w=\{f=\sum_{j=0} ^{\infty} a_jz^j : \sum_{j=0}^{\infty} |a_j|^2w_j < \infty\}$ is an $n$-hypercontraction, we prove that the weights must satisfy the inequality $$\frac{w_{j+1}}{w_j} \leq {\frac{1+j}{n+j}}.$$ As an application of this result, it is shown that such an operator cannot be subnormal. We also give an example to illustrate the important role that the $n$-hypercontractivity assumption plays in determining the similarity of Cowen-Douglas operators in terms of the curvatures of their eigenvector bundles., Comment: 21 pages