A generalization of the flat cotorsion theory

We use the framework of Tensor-Hom-Cotensor situations to present a generalization to abelian categories of the flat cotorsion theory for modules over a ring. Using the generalized flat cotorsion theory we present a flat totally acyclic cotorsion theory in categories of chain complexes in abelian categories. Under some assumption on the generalized flat cotorsion theory we prove a K\"unneth type theorem for chain complexes in abelian categoriesComment: Strengthened some results and added more materia

Learning Homogenization for Elliptic Operators

Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context; in particular, to develop underpinning theory that establishes the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs

Spin-order-dependent magneto-elastic coupling in two dimensional antiferromagnetic MnPSe3_3 observed through Raman spectroscopy

Layered antiferromagnetic materials have emerged as a novel subset of the two-dimensional family providing a highly accessible regime with prospects for layer-number-dependent magnetism. Furthermore, transition metal phosphorous trichalcogenides, MPX3 (M = transition metal; X = chalcogen) provide a platform for investigating fundamental interactions between magnetic and lattice degrees of freedom providing new insights for developing fields of spintronics and magnonics. Here, we use a combination of temperature dependent Raman spectroscopy and density functional theory to explore magnetic-ordering-dependent interactions between the manganese spin degree of freedom and lattice vibrations of the non-magnetic sub-lattice via a Kramers-Anderson super-exchange pathway in both bulk, and few-layer, manganese phosphorous triselenide (MnPSe$_3$). We observe a nonlinear temperature dependent shift of phonon modes predominantly associated with the non-magnetic sub-lattice, revealing their non-trivial spin-phonon coupling below the N{\'e}el temperature at 74 K, allowing us to extract mode-specific spin-phonon coupling constants.Comment: 20 pages, 4 figures, 1 tabl

Beyond the classical strong maximum principle: forcing changing sign near the boundary and flat solutions

We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain $\Omega$ can be extended, under suitable conditions, to the case in which the forcing term $f(x)$ is changing sign. In addition, in the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solutions). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign is also given.Comment: 20 pages 2 Figure

Combining Decentralized IDentifiers with Proof of Membership to Enable Trust in IoT Networks

The Self-Sovereign Identity (SSI) is a decentralized paradigm enabling full control over the data used to build and prove the identity. In Internet of Things networks with security requirements, the Self-Sovereign Identity can play a key role and bring benefits with respect to centralized identity solutions. The challenge is to make the SSI compatible with resource-constraint IoT networks. In line with this objective, the paper proposes and discusses an alternative (mutual) authentication process for IoT nodes under the same administration domain. The main idea is to combine the Decentralized IDentifier (DID)-based verification of private key ownership with the verification of a proof that the DID belongs to an evolving trusted set. The solution is built around the proof of membership notion. The paper analyzes two membership solutions, a novel solution designed by the Authors based on Merkle trees and a second one based on the adaptation of Boneh, Boyen and Shacham (BBS) group signature scheme. The paper concludes with a performance estimation and a comparative analysis

Rethinking the Paradigm of Content Constraints in Unpaired Image-to-Image Translation

In an unpaired setting, lacking sufficient content constraints for image-to-image translation (I2I) tasks, GAN-based approaches are usually prone to model collapse. Current solutions can be divided into two categories, reconstruction-based and Siamese network-based. The former requires that the transformed or transforming image can be perfectly converted back to the original image, which is sometimes too strict and limits the generative performance. The latter involves feeding the original and generated images into a feature extractor and then matching their outputs. This is not efficient enough, and a universal feature extractor is not easily available. In this paper, we propose EnCo, a simple but efficient way to maintain the content by constraining the representational similarity in the latent space of patch-level features from the same stage of the \textbf{En}coder and de\textbf{Co}der of the generator. For the similarity function, we use a simple MSE loss instead of contrastive loss, which is currently widely used in I2I tasks. Benefits from the design, EnCo training is extremely efficient, while the features from the encoder produce a more positive effect on the decoding, leading to more satisfying generations. In addition, we rethink the role played by discriminators in sampling patches and propose a discriminative attention-guided (DAG) patch sampling strategy to replace random sampling. DAG is parameter-free and only requires negligible computational overhead, while significantly improving the performance of the model. Extensive experiments on multiple datasets demonstrate the effectiveness and advantages of EnCo, and we achieve multiple state-of-the-art compared to previous methods. Our code is available at https://github.com/XiudingCai/EnCo-pytorch.Comment: Accepted by AAAI 202

Petrov Type, Principal Null Directions, and Killing Tensors of Slowly-Rotating Black Holes in Quadratic Gravity

The ability to test general relativity in extreme gravity regimes using gravitational wave observations from current ground-based or future space-based detectors motivates the mathematical study of the symmetries of black holes in modified theories of gravity. In this paper we focus on spinning black hole solutions in two quadratic gravity theories: dynamical Chern-Simons and scalar Gauss-Bonnet gravity. We compute the principal null directions, Weyl scalars, and complex null tetrad in the small-coupling, slow rotation approximation for both theories, confirming that both spacetimes are Petrov type I. Additionally, we solve the Killing equation through rank 6 in dynamical Chern-Simons gravity and rank 2 in scalar Gauss-Bonnet gravity, showing that there is no nontrivial Killing tensor through those ranks for each theory. We therefore conjecture that the still-unknown, exact, quadratic-gravity, black-hole solutions do not possess a fourth constant of motion.Comment: 23 pages, 2 figures, reuploaded on 01/04/24 to correct small typo in Eq (C5

Recovering Sign Bits of DCT Coefficients in Digital Images as an Optimization Problem

Recovering unknown, missing, damaged, distorted, or lost information in DCT coefficients is a common task in multiple applications of digital image processing, including image compression, selective image encryption, and image communication. This paper investigates the recovery of sign bits in DCT coefficients of digital images, by proposing two different approximation methods to solve a mixed integer linear programming (MILP) problem, which is NP-hard in general. One method is a relaxation of the MILP problem to a linear programming (LP) problem, and the other splits the original MILP problem into some smaller MILP problems and an LP problem. We considered how the proposed methods can be applied to JPEG-encoded images and conducted extensive experiments to validate their performances. The experimental results showed that the proposed methods outperformed other existing methods by a substantial margin, both according to objective quality metrics and our subjective evaluation.Comment: 22 pages, 8 figure

Zeros of modular forms and Faber polynomials

We study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number D of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near D vertical lines, with the zeros of a weight k form lying at height approximately log(k). This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree D.Comment: Fixed section 3.2, where equation 3.1 was garble

Calibration-free online test-time adaptation for electroencephalography motor imagery decoding

Providing a promising pathway to link the human brain with external devices, Brain-Computer Interfaces (BCIs) have seen notable advancements in decoding capabilities, primarily driven by increasingly sophisticated techniques, especially deep learning. However, achieving high accuracy in real-world scenarios remains a challenge due to the distribution shift between sessions and subjects. In this paper we will explore the concept of online test-time adaptation (OTTA) to continuously adapt the model in an unsupervised fashion during inference time. Our approach guarantees the preservation of privacy by eliminating the requirement to access the source data during the adaptation process. Additionally, OTTA achieves calibration-free operation by not requiring any session- or subject-specific data. We will investigate the task of electroencephalography (EEG) motor imagery decoding using a lightweight architecture together with different OTTA techniques like alignment, adaptive batch normalization, and entropy minimization. We examine two datasets and three distinct data settings for a comprehensive analysis. Our adaptation methods produce state-of-the-art results, potentially instigating a shift in transfer learning for BCI decoding towards online adaptation.Comment: 6 pages, 4 figures, 12th International Winter Conference on Brain-Computer Interface 202