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Recent developments have introduced a groundbreaking form of collinear magnetism known as "altermagnetism". This emerging magnetic phase is characterized by robust time-reversal symmetry breaking, antiparallel magnetic order, and alternating spin-splitting band structures, yet it exhibits vanishing net magnetization constrained by symmetry. Altermagnetism uniquely integrates traits previously considered mutually exclusive to conventional collinear ferromagnetism and antiferromagnetism, thereby facilitating phenomena and functionalities previously not achievable within these traditional categories of magnetism. Initially proposed theoretically, the existence of the altermagnetic phase has since been corroborated by a range of experimental studies, which have confirmed its unique properties and potential for applications. This review explores the rapidly expanding research on altermagnets, emphasizing the novel physical phenomena they manifest, methodologies for inducing altermagnetism, and promising altermagnetic materials. The goal of this review is to furnish readers with a comprehensive overview of altermagnetism and to inspire further innovative studies on altermagnetic materials which could potentially revolutionize applications in technology and materials science., Comment: Submitted to Adv. Funct. Mater

The notion of twisted sectors play a crucial role in orbifold Gromov-Witten theory. We introduce the notion of dihedral twisted sectors in order to construct Lagrangian Floer theory on symplectic orbifolds and discuss related issues., Comment: In honour of Professor Jean-Pierre Bourguignon on the occasion of his 75th birthday

For any compact connected Lie group $G$, we study the Hamiltonian sum of two compact Hamiltonian group $G$-manifolds $(X^+,\omega^+,\mu^+)$ and $(X^-,\omega^-,\mu^-)$ with a common codimension 2 Hamiltonian submanifold $Z$ of the opposite equivariant Euler classes of the normal bundles. We establish that the symplectic reduction of the Hamiltonian sum agrees with the symplectic sum of the reduced symplectic manifolds. We also compare the equivariant first Chern class of the Hamiltonian sum with the equivariant first Chern classes of $X^\pm$., Comment: 25 pages

Accurately identifying patient's status through medical images plays an important role in diagnosis and treatment. Artificial intelligence (AI), especially the deep learning, has achieved great success in many fields. However, more reliable AI model is needed in image guided diagnosis and therapy. To achieve this goal, developing a balanced, safe and robust model with a unified framework is desirable. In this study, a new unified model termed as automated multi-objective Mixer (AutoMO-Mixer) model was developed, which utilized a recent developed multiple layer perceptron Mixer (MLP-Mixer) as base. To build a balanced model, sensitivity and specificity were considered as the objective functions simultaneously in training stage. Meanwhile, a new evidential reasoning based on entropy was developed to achieve a safe and robust model in testing stage. The experiment on an optical coherence tomography dataset demonstrated that AutoMO-Mixer can obtain safer, more balanced, and robust results compared with MLP-Mixer and other available models.

We give a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions on manifolds, and cyclic periodic cohomology of the group algebra. Indeed, for any such group $\Gamma$ (without any further assumptions on it) we construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map to the periodic cyclic homology of the group algebra. This morphism, called here the Chern-Baum-Connes assembly map, allows to give a proper and explicit formulation for a Chern-Connes pairing with the periodic cyclic cohomology of the group algebra. Several theorems are needed to formulate the Chern-Baum-Connes assembly map. In particular we establish a delocalised Riemann-Roch theorem, the wrong way functoriality for periodic delocalised cohomology for $\Gamma$-proper actions, the construction of a Chern morphism between the Left-Hand side of Baum-Connes and a delocalised cohomology group associated to $\Gamma$ which is an isomorphism once tensoring with $\mathbb{C}$, and the construction of an explicit cohomological assembly map between the delocalised cohomology group associated to $\Gamma$ and the homology group $H_*(\Gamma,F\Gamma)$. We then give an index theoretical formula for the above mentioned pairing (for any $\Gamma$) in terms of pairings of invariant forms, associated to geometric cycles and given in terms of delocalized Chern and Todd classes, and currents naturally associated to group cocycles using Burghelea's computation. As part of our results we prove that left-Hand side group used in this paper is isomorphic to the usual analytic model for the left-hand side of the assembly map., Comment: This article replaces a previous version whose title was the same plus part I. Originally we planned to work out a second part but we finally extended and completed the first part in a single paper. We thank the colleague that encouraged us to complete part I into a single piece and gave us invaluable references that were key to the present work