A flat quadratic quasi-Frobenius Lie superalgebra is a quadratic Lie superalgebra equipped with an additional symplectic structure that is flat with respect to the natural symplectic product. In this paper, we introduce the notion of a flat quadratic double extension of a flat quadratic quasi-Frobenius Lie superalgebra, in the cases where both the symplectic structure and the quadratic structure are either even or odd. We show that, over an algebraically closed field, any such Lie superalgebra can be constructed through a sequence of flat quadratic double extensions starting from the trivial algebra $\{0\}$. Moreover, when the quadratic and symplectic structures have different parity, we introduce the notion of a planar double extension, which constitutes the main novelty of this paper. In this case, we prove that such Lie superalgebras have total dimension $4n$. Finally, we classify flat quadratic quasi-Frobenius Lie superalgebras of dimension at most four and present explicit examples in dimensions six and eight.
We investigate symplectic nilpotent Lie groups with Lagrangian normal subgroups. We show that there exists a bijection between the isomorphism classes of nilpotent Lie groups with Lagrangian normal subgroups and the isomorphism classes of geodesically complete, flat, nilpotent Lie groups with Lagrangian extension cohomology class. Finally, we provide a complete classification of eight-dimensional symplectic nilpotent Lie groups with Lagrangian normal subgroups, identifying exactly ninety-five such groups. As a consequence, we obtain a complete classification of eight-dimensional symplectic filiform real Lie groups.
Magic-angle twisted bilayer graphene (MATBG) exhibits remarkable electronic properties under external magnetic fields, notably the emergence of flat Landau levels. In this study, we present a comprehensive analysis of MATBG's operational phase diagram under three distinct quantum thermodynamic cycles, i.e., Quantum Otto Cycle (QOC), Quantum Carnot Cycle (QCC), and Quantum Stirling Cycle (QSC). Employing the continuum eight-band model, we evaluate the thermodynamic performance of MATBG across multiple operational modes: heat engine, refrigerator, cold pump, and Joule pump, and benchmark it against other graphene systems such as monolayer graphene, AB-Bernal stacked bilayer graphene, and non-magic-angle twisted bilayer graphene. Our findings reveal that MATBG demonstrates superior heat engine performance in QSC, while achieving high efficiency albeit with reduced work output in QOC. Even though the performance of MATBG as a cold pump or refrigerator is modest in QOC and QSC, it shows notable improvement as a refrigerator in QCC. Additionally, we identify a highly reversible Joule pump mode in both QSC and QOC under strict adiabaticity, underscoring the unique thermodynamic behavior o
Flat-band lattices supporting compact localized states provide a versatile platform for exploring unconventional transport phenomena in photonic, ultracold atomic, electronic, and other systems. Here, we investigate the transport of path-entangled multi-photon NOON states in a flat-band rhombic lattice and observe intriguing localization-delocalization features that depend on both the phase and photon number of the NOON states. To experimentally emulate photon number correlations, we develop an intensity correlation measurement protocol using coherent laser light with tunable relative phases. We first apply this protocol to show spatial bunching and anti-bunching of two-photon NOON states in a one-dimensional waveguide lattice. In the rhombic lattice, we show that for an even (odd) photon number $N$, localization occurs at $0 \, (π)$ phase of the NOON state with a probability of $2^{1-N}$, which is demonstrated up to eight photons. Our results open an exciting route towards understanding the dynamics of correlated photons in complex photonic networks.
All possible variants of symmetric embedding of the metric of the spatially flat Friedman model into a ten-dimensional ambient space are analyzed. It is shown that only two such embeddings exist: the five-dimensional embedding found by Robertson in 1933 and a new eight-dimensional embedding proposed in this work. The new embedding can be utilized in developing the idea of explaining dark matter as fictitious matter arising in the description of gravity within the embedding theory approach. Some results in this direction are presented. Additionally, the possibility of constructing embeddings of the spatially flat Friedman model whose symmetry is lower than that of the metric itself is discussed. Two such embeddings are constructed.
LLM-powered Multi-Agent Systems (MAS) have demonstrated remarkable capabilities in complex domains but suffer from inherent fragility and opaque failure mechanisms. Existing failure attribution methods, whether relying on direct prompting, costly replays, or supervised fine-tuning, typically treat execution logs as flat sequences. This linear perspective fails to disentangle the intricate causal links inherent to MAS, leading to weak observability and ambiguous responsibility boundaries. To address these challenges, we propose CHIEF, a novel framework that transforms chaotic trajectories into a structured hierarchical causal graph. It then employs hierarchical oracle-guided backtracking to efficiently prune the search space via sybthesized virtual oracles. Finally, it implements counterfactual attribution via a progressive causal screening strategy to rigorously distinguish true root causes from propagated symptoms. Experiments on Who&When benchmark show that CHIEF outperforms eight strong and state-of-the-art baselines on both agent- and step-level accuracy. Ablation studies further confirm the critical role of each proposed module.
We investigate the topological phases that appear in an orbital version of the Benalcazar-Bernevig-Hughes (BBH) model in the presence of conventional spin-singlet $s$-wave superconductivity and with the possibility of tuning an in-plane magnetic field. We chart out the phase diagram by considering different boundary conditions, with the topology of the individual phases further examined by considering both the Wannier and entanglement spectra, as well as the Majorana polarization. For weak to moderate values of magnetic field and superconducting pairing amplitude, we find a second-order topological superconductor phase with eight zero-energy corner modes. Further increasing field or pairing, half of the corner states can be turned into zero-energy edge-localized modes, thus forming a type of hybrid-order phase. Then, we find two different putative first-order topological phases, a nodal and a nodeless phase, both with zero-energy flat bands localized along mirror-symmetric open edges. For the nodal phase, the flat bands are localized between the nodes in reciprocal space, while in the nodeless phase, with its a full bulk gap, the zero-energy boundary flat band spans the whole Brill
In 1964, John Milnor, using a construction of two lattices by Witt, produced the first example of two flat tori that are not globally isometric and whose Laplacians for exterior forms have the same sequence of eigenvalues. The aforementioned flat tori are sixteen-dimensional. One is reducible while the second is irreducible. In the ensuing years, pairs of non-isometric flat tori that share a common Laplace spectrum have been shown to exist in dimensions four and higher. In dimensions three and lower, Alexander Schiemann proved in 1994 that any flat tori that are isospectral are in fact isometric, so four is the lowest dimension in which such pairs exist. Using a four-dimensional such pair, one can easily construct an eight-dimensional such triplet. However, triplets of mutually non-isometric flat tori that share a common Laplace spectrum in dimensions 4, 5, 6, and 7 have eluded researchers - until now. We present here the first example.
We classify nilpotent Lie algebras with complex structures of weakly non-nilpotent type in real dimension eight, which is the lowest dimension where they arise. Our study, together with previous results on strongly non-nilpotent structures, completes the classification of 8-dimensional nilpotent Lie algebras admitting complex structures of non-nilpotent type. As an application, we identify those that support a pseudo-Kähler metric, thus providing new counterexamples to a previous conjecture and an infinite family of (Ricci-flat) non-flat neutral Calabi-Yau structures. Moreover, we arrive at the topological restriction $b_1(X)\geq 3$ for every pseudo-Kähler nilmanifold $X$ with invariant complex structure, up to complex dimension four.
At magic twisted angles, Dirac cones in twisted bilayer graphene (TBG) can evolve into flat bands, serving as a critical playground for the study of strongly correlated physics. When chiral symmetry is introduced, rigorous mathematical proof confirms that the flat bands are locked at zero energy in the entire Moiré Brillouin zone (BZ). Yet, TBG is not the sole platform that exhibits this absolute band flatness. Central to this flatness phenomenon are topological nodes and their specific locations in the BZ. In this study, considering twisted bilayer systems that preserve chiral symmetry, we classify various ordered topological nodes in base layers and all possible node locations across different BZs. Specifically, we constrain the node locations to rotational centers, such as Γ and M points, to ensure the interlayer coupling retains equal strength in all directions. Using this classification as a foundation, we systematically identify the conditions under which Moiré flat bands emerge. Additionally, through the extension of holomorphic functions, we provide proof that flat bands are locked at zero energy, shedding light on the origin of the band flatness. Remarkably, beyond Dirac c
We address the theoretical and practical problems related to the trajectory generation and tracking control of tail-sitter UAVs. Theoretically, we focus on the differential flatness property with full exploitation of actual UAV aerodynamic models, which lays a foundation for generating dynamically feasible trajectory and achieving high-performance tracking control. We have found that a tail-sitter is differentially flat with accurate aerodynamic models within the entire flight envelope, by specifying coordinate flight condition and choosing the vehicle position as the flat output. This fundamental property allows us to fully exploit the high-fidelity aerodynamic models in the trajectory planning and tracking control to achieve accurate tail-sitter flights. Particularly, an optimization-based trajectory planner for tail-sitters is proposed to design high-quality, smooth trajectories with consideration of kinodynamic constraints, singularity-free constraints and actuator saturation. The planned trajectory of flat output is transformed to state trajectory in real-time with consideration of wind in environments. To track the state trajectory, a global, singularity-free, and minimally-p
We consider the optimization problem of maximizing the $k$-th Laplacian eigenvalue, $λ_{k}$, over flat $d$-dimensional tori of fixed volume. For $k=1$, this problem is equivalent to the densest lattice sphere packing problem. For larger $k$, this is equivalent to the NP-hard problem of finding the $d$-dimensional (dual) lattice with longest $k$-th shortest lattice vector. As a result of extensive computations, for $d \leq 8$, we obtain a sequence of flat tori, $T_{k,d}$, each of volume one, such that the $k$-th Laplacian eigenvalue of $T_{k,d}$ is very large; for each (finite) $k$ the $k$-th eigenvalue exceeds the value in (the $k\to \infty$ asymptotic) Weyl's law by a factor between 1.54 and 2.01, depending on the dimension. Stationarity conditions are derived and numerically verified for $T_{k,d}$ and we describe the degeneration of the tori as $k \to \infty$.
We aim to hunt for massive binaries hosting a black hole companion (OB+BH) and establish the natal mass-ratio distribution of massive stars at the subsolar metallicity environment of the Large Magellanic Cloud (LMC). We use the shift-and-add grid disentangling technique to characterize the hidden companions in 51 SB1 O-type and evolved B-type binaries in the LMC monitored in the framework of the Tarantula Massive Binary Monitoring (TMBM). Out of the 51 SB1 systems, 43 (84%) are found to have non-degenerate stellar companions, of which 28 are confident detections, and 15 are less certain (SB1: or SB2:). Of these 43 targets, one is found to be a triple (VFTS 64), and two are found to be quadruples (VFTS 120, 702). The remaining eight targets (16%) retain an SB1 classification. Aside from the unambiguous case of VFTS 243, analysed in detailed in a separate paper, we identify two additional OB+BH candidates: VFTS 514 and VFTS 779. Additional black holes may be present in the sample but at a lower probability. Our study firmly establishes a virtually flat natal mass-ratio distribution for O-type stars at LMC metallicity, covering the entire mass-ratio range (0.05 < q < 1) and peri
Herbig stars can be classified into group I and group II depending on the shape of the far-IR excess from the spectral energy distribution. This separation may be evolutionary and related to the vertical structure of these disks. We aim to determine the emission height of Herbig disks and compare the resulting vertical extent of both groups. ALMA Band 6 observations of 12CO emission lines at sufficient velocity and spatial resolution of eight Herbig disks (four group I and four group II sources) are used to determine the emission heights from the channel maps via geometrical methods developed in other works. We find that all group I disks are vertically extended with a height to radius ratio of at least 0.25, and for three of the disks the gas emission profile can be traced out to 200-500 au. The group II disks are divided between MWC 480 and HD 163296 which have similar emission height profiles as the group I disks, and AK Sco and HD 142666 which are very flat (not exceeding a height of 10 au) and more compact (<200 au in size). The brightness temperatures show no differences between the disks when the luminosity of the host star is accounted for. Our findings agree with previo
Astronomers have released the largest gravitational wave catalog ever, revealing 161 new black hole collisions and pushing the total number of detections to 390。 Among the highlights are the clearest gravitational wave signal ever recorded, the most accurate location of a black hole merger, and growing evidence that some black holes are the product
If Homo floresiensis wasn't a fire-using hunter, its origins could be different than we thought
Digital Services Act may force Meta to make big changes on its platforms
An unusual gravitational wave signal has renewed hopes that primordial black holes, long considered purely theoretical, may finally be within reach of discovery。 If confirmed, they could solve one of astronomy's greatest mysteries by explaining the nature of dark matter
What if consciousness isn’t limited to brains like ours。 Philosophers Eric Schwitzgebel and Jeremy Pober argue that consciousness could arise in many different forms of life, even in beings built from radically different materials than those found on Earth。 Drawing on the vastness of the universe and the likely existence of countless alien civiliza
Researchers have created an AI-based simulation that makes it much faster to model how neutron star mergers produce many of the universe's heaviest elements。 The new tool could improve predictions of these powerful explosions while helping scientists better connect observations in space with experiments on Earth