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We introduce the bilayer construction, as a specific purification scheme for a general mixed state, where each mixed state has a one-to-one correspondence with a bilayer pure state with two constraints: non-negativity of the bilayer wavefunction; and the presence of an anti-unitary layer-exchange symmetry T. Different from the Choi-Jamio{\l}kowski isomorphism, any mixed state can be realized as the monolayer reduced density matrix of a bilayer pure state, and its physical properties can be experimentally realized and detected in non-magnetic bilayer 2D materials with a layer-exchange mirror symmetry. We study a variety of mixed state phenomena in the bilayer construction: (1) strong and weak symmetries, their explicit and spontaneous breakings in mixed states can be understood as usual Landau-type symmetry breakings in the bilayer pure state, and their criteria can be derived accordingly; (2) decoherence of a pure state by local errors can be mapped to quantum quench dynamics of the bilayer pure states; (3) mixed symmetry protected topological (SPT) states and mixed state topological orders can be classified, characterized and realized as pure state SPTs and topological orders in the bilayer. We further study examples of strong-to-weak spontaneous symmetry breaking (SWSSB) and their critical scalings at the SWSSB transition in the bilayer construction., Comment: 15 pages, 1 figure

Topological magnons give rise to possibilities for engineering novel spintronics devices with critical applications in quantum information and computation, due to its symmetry-protected robustness and low dissipation. However, to make reliable and systematic predictions about material realization of topological magnons has been a major challenge, due to the lack of neutron scattering data for most materials. In this work, we significantly advance the symmetry-based approach for identifying topological magnons through developing a fully automated algorithm, utilizing the theory of symmetry indicators, that enables a highly efficient and large-scale search for candidate materials hosting field-induced topological magnons. This progress not only streamlines the discovery process but also expands the scope of materials exploration beyond previous manual or traditional methods, offering a powerful tool for uncovering novel topological phases in magnetic systems. Performing a large-scale search over all 1649 magnetic materials in Bilbao Crystallographic Server with a commensurate magnetic order, we discover 387 candidate materials for topological magnons, significantly expanding the pool of topological magnon materials. We further discuss examples and experimental accessibility of the candidate materials, shedding light on future experimental realizations of topological magnons in magnetic materials., Comment: Main text: 30 pages with 2 tables + 12 figures. Supplementary materials: 945 pages with 900+ tables and 460+ figures

Floquet Topological Insulators (FTIs) in two spatial dimensions can exhibit anomalous chiral edge modes despite a fully localized bulk, captured by a new topological invariant other than the Chern number. In this work, we focus on (2 + 1)D FTIs in symmetry class A with a spacetime symmetry known as n-fold time screw symmetry. We show that the bulk invariant can be partially determined by symmetry indicators associated with n = 2, 3, 4, 6 time screw symmetry, which only depend on the time evolution in high symmetry momenta. We derive simple formula that relates symmetry indicators to the bulk invariant, and demonstrate its validity in examples of FTI models. Our results can simplify the engineering of FTIs in real materials, and especially shed light on systems driven by circularly polarized light., Comment: 6 pages, 2 figures

Motivated by the abundance of symmetry breaking states in magic-angle twisted bilayer graphene and other two-dimensional materials, we study superconducting (SC) and charge orders in two-dimensional topological flat bands in the strong correlation regime. By relating the half-filled 2D topological flat bands to the surface states of 3D topological insulators in symmetry class AIII, we reveal the topological obstruction to the formation of gapped SC and inter-valley charge orders without intrinsic topological orders, in the presence of the anti-unitary particle-hole symmetry at half filling. This is a generalization of the Li-Haldane arguments for nodal superconductivity to strongly interacting electrons. In contrast to the $\mathbb{Z}$-valued obstruction derived from the non-interacting band topology, the topological obstruction of interacting electrons in half-filled flat bands has a $\mathbb{Z}_{8}$ classification, depending on the charge (valley) Chern number of the superconducting (inter-valley charge) orders. This is demonstrated by an interacting Hamiltonian for half-filled flat bands with a net Chern number $C=4$, where superconductivity and $\mathbb{Z}_2$ topological order coexist in a gapped ground state with particle-hole symmetry., Comment: 5+9 pages, 2+1 figures; We added a short discussion about materials in v2

Vision and Language Navigation (VLN) is a challenging task that requires agents to understand instructions and navigate to the destination in a visual environment.One of the key challenges in outdoor VLN is keeping track of which part of the instruction was completed. To alleviate this problem, previous works mainly focus on grounding the natural language to the visual input, but neglecting the crucial role of the agent's spatial position information in the grounding process. In this work, we first explore the substantial effect of spatial position locating on the grounding of outdoor VLN, drawing inspiration from human navigation. In real-world navigation scenarios, before planning a path to the destination, humans typically need to figure out their current location. This observation underscores the pivotal role of spatial localization in the navigation process. In this work, we introduce a novel framework, Locating be for Planning (Loc4Plan), designed to incorporate spatial perception for action planning in outdoor VLN tasks. The main idea behind Loc4Plan is to perform the spatial localization before planning a decision action based on corresponding guidance, which comprises a block-aware spatial locating (BAL) module and a spatial-aware action planning (SAP) module. Specifically, to help the agent perceive its spatial location in the environment, we propose to learn a position predictor that measures how far the agent is from the next intersection for reflecting its position, which is achieved by the BAL module. After the locating process, we propose the SAP module to incorporate spatial information to ground the corresponding guidance and enhance the precision of action planning. Extensive experiments on the Touchdown and map2seq datasets show that the proposed Loc4Plan outperforms the SOTA methods., Comment: arXiv admin note: text overlap with arXiv:2203.13838 by other authors

We present a biased atomic qubit, universally implementable across all atomic platforms, encoded as a `spin-cat' within ground state Zeeman levels. The key characteristic of our configuration is the coupling of the ground state spin manifold of size $F_g \gg 1$ to an excited Zeeman spin manifold of size $F_e = F_g - 1$ using light. This coupling results in eigenstates of the driven atom that include exactly two dark states in the ground state manifold, which are decoupled from light and immune to spontaneous emission from the excited states. These dark states constitute the `spin-cat', leading to the designation `dark spin-cat'. We demonstrate that under strong Rabi drive and for large $F_g$, the `dark spin-cat' is autonomously stabilized against common noise sources and encodes a qubit with significantly biased noise. Specifically, the bit-flip error rate decreases exponentially with $F_g$ relative to the dephasing rate. We provide an analysis of dark spin-cats, their robustness to noise, and discuss bias-preserving single qubit and entangling gates, exemplified on a Rydberg tweezer platform.

This paper provides a unified treatment to the recovery of structured signals living in a star-shaped set from general quantized measurements $\mathcal{Q}(\mathbf{A}\mathbf{x}-\mathbf{\tau})$, where $\mathbf{A}$ is a sensing matrix, $\mathbf{\tau}$ is a vector of (possibly random) quantization thresholds, and $\mathcal{Q}$ denotes an $L$-level quantizer. The ideal estimator with consistent quantized measurements is optimal in some important instances but typically infeasible to compute. To this end, we study the projected gradient descent (PGD) algorithm with respect to the one-sided $\ell_1$-loss and identify the conditions under which PGD achieves the same error rate, up to logarithmic factors. For multi-bit case, these conditions only ensure local convergence, and we further develop a complementary approach based on product embedding. When applied to popular models such as 1-bit compressed sensing with Gaussian $\mathbf{A}$ and zero $\mathbf{\tau}$ and the dithered 1-bit/multi-bit models with sub-Gaussian $\mathbf{A}$ and uniform dither $\mathbf{\tau}$, our unified treatment yields error rates that improve on or match the sharpest results in all instances. Particularly, PGD achieves the information-theoretic optimal rate $\tilde{O}(\frac{k}{mL})$ for recovering $k$-sparse signals, and the rate $\tilde{O}((\frac{k}{mL})^{1/3})$ for effectively sparse signals. For 1-bit compressed sensing of sparse signals, our result recovers the optimality of normalized binary iterative hard thresholding (NBIHT) that was proved very recently.

Thoroughly testing autonomy systems is crucial in the pursuit of safe autonomous driving vehicles. It necessitates creating safety-critical scenarios that go beyond what can be safely collected from real-world data, as many of these scenarios occur infrequently on public roads. However, the evaluation of most existing NVS methods relies on sporadic sampling of image frames from the training data, comparing the rendered images with ground truth images using metrics. Unfortunately, this evaluation protocol falls short of meeting the actual requirements in closed-loop simulations. Specifically, the true application demands the capability to render novel views that extend beyond the original trajectory (such as cross-lane views), which are challenging to capture in the real world. To address this, this paper presents a novel driving view synthesis dataset and benchmark specifically designed for autonomous driving simulations. This dataset is unique as it includes testing images captured by deviating from the training trajectory by 1-4 meters. It comprises six sequences encompassing various time and weather conditions. Each sequence contains 450 training images, 150 testing images, and their corresponding camera poses and intrinsic parameters. Leveraging this novel dataset, we establish the first realistic benchmark for evaluating existing NVS approaches under front-only and multi-camera settings. The experimental findings underscore the significant gap that exists in current approaches, revealing their inadequate ability to fulfill the demanding prerequisites of cross-lane or closed-loop simulation. Our dataset is released publicly at the project page: https://3d-aigc.github.io/XLD/., Comment: project page: https://3d-aigc.github.io/XLD/

We derive selection rules in optical absorption and Raman scattering spectra, that can determine the parity of pairing order parameters under inversion symmetry in two classes of \emph{clean} superconductors: (i) chiral superconductors with strong spin-orbit couplings, (ii) singlet superconductors with negligible spin-orbit couplings. Experimentally, the inversion parity of pair wave functions can be determined by comparing the "optical gap" $\Delta_\text{op}$ in Raman and optical spectroscopy and the "thermodynamic gap" $2\Delta$ in specific heat measurements, and the selection rules apply when $\Delta_\text{op}>2\Delta$. We demonstrate the selection rules in superconductivity in models of (i) doped Weyl semimetals and (ii) doped graphene. Our derivation is based on the relation between pairing symmetry and fermion projective symmetry group of a superconductor. We further derive similar selection rules for two-dimensional superconductors with 2-fold rotational symmetry, and discuss how they apply to the superconducting state in magic-angle twisted bilayer graphene., Comment: 16 pages, 12 figures

We present EgoExo-Fitness, a new full-body action understanding dataset, featuring fitness sequence videos recorded from synchronized egocentric and fixed exocentric (third-person) cameras. Compared with existing full-body action understanding datasets, EgoExo-Fitness not only contains videos from first-person perspectives, but also provides rich annotations. Specifically, two-level temporal boundaries are provided to localize single action videos along with sub-steps of each action. More importantly, EgoExo-Fitness introduces innovative annotations for interpretable action judgement--including technical keypoint verification, natural language comments on action execution, and action quality scores. Combining all of these, EgoExo-Fitness provides new resources to study egocentric and exocentric full-body action understanding across dimensions of "what", "when", and "how well". To facilitate research on egocentric and exocentric full-body action understanding, we construct benchmarks on a suite of tasks (i.e., action classification, action localization, cross-view sequence verification, cross-view skill determination, and a newly proposed task of guidance-based execution verification), together with detailed analysis. Code and data will be available at https://github.com/iSEE-Laboratory/EgoExo-Fitness/tree/main., Comment: Accepted by ECCV2024

Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity of such data provides vast new opportunities for modeling and analysis, but efficiently extracting information content from them, both statistically and computationally, presents unique and fundamental challenges. Addressing these challenges requires an interdisciplinary approach that brings together tools and insights from statistics, optimization and numerical linear algebra among other fields. Despite these hurdles, significant progress has been made in the last decade. This review seeks to examine some of the key advancements and identify common threads among them, under eight different statistical settings.

In this paper, we study the sample complexity and develop efficient optimal algorithms for 1-bit phase retrieval: recovering a signal $\mathbf{x}\in\mathbb{R}^n$ from $m$ phaseless bits $\{\mathrm{sign}(|\mathbf{a}_i^\top\mathbf{x}|-\tau)\}_{i=1}^m$ generated by standard Gaussian $\mathbf{a}_i$s. By investigating a phaseless version of random hyperplane tessellation, we show that (constrained) hamming distance minimization uniformly recovers all unstructured signals with Euclidean norm bounded away from zero and infinity to the error $\mathcal{O}((n/m)\log(m/n))$, and $\mathcal{O}((k/m)\log(mn/k^2))$ when restricting to $k$-sparse signals. Both error rates are shown to be information-theoretically optimal, up to a logarithmic factor. Intriguingly, the optimal rate for sparse recovery matches that of 1-bit compressed sensing, suggesting that the phase information is non-essential for 1-bit compressed sensing. We also develop efficient algorithms for 1-bit (sparse) phase retrieval that can achieve these error rates. Specifically, we prove that (thresholded) gradient descent with respect to the one-sided $\ell_1$-loss, when initialized via spectral methods, converges linearly and attains the near optimal reconstruction error, with sample complexity $\mathcal{O}(n)$ for unstructured signals and $\mathcal{O}(k^2\log(n)\log^2(m/k))$ for $k$-sparse signals. Our proof is based upon the observation that a certain local (restricted) approximate invertibility condition is respected by Gaussian measurements., Comment: substantially cleaner proofs provided

This paper presents a topological analytics approach within the 5-level Cyber-Physical Systems (CPS) architecture for the Stream-of-Quality assessment in smart manufacturing. The proposed methodology not only enables real-time quality monitoring and predictive analytics but also discovers the hidden relationships between quality features and process parameters across different manufacturing processes. A case study in additive manufacturing was used to demonstrate the feasibility of the proposed methodology to maintain high product quality and adapt to product quality variations. This paper demonstrates how topological graph visualization can be effectively used for the real-time identification of new representative data through the Stream-of-Quality assessment., Comment: The paper has been submitted to Manufacturing Letters (Under Review)

Quantum entanglement is one of the primary features which distinguishes quantum computers from classical computers. In gate-based quantum computing, the creation of entangled states or the distribution of entanglement across a quantum processor often requires circuit depths which grow with the number of entangled qubits. However, in teleportation-based quantum computing, one can deterministically generate entangled states with a circuit depth that is constant in the number of qubits, provided that one has access to an entangled resource state, the ability to perform mid-circuit measurements, and can rapidly transmit classical information. In this work, aided by fast classical FPGA-based control hardware with a feedback latency of only 150 ns, we explore the utility of teleportation-based protocols for generating non-local, multi-partite entanglement between superconducting qubits. First, we demonstrate well-known protocols for generating Greenberger-Horne-Zeilinger (GHZ) states and non-local CNOT gates in constant depth. Next, we utilize both protocols for implementing an unbounded fan-out (i.e., controlled-NOT-NOT) gate in constant depth between three non-local qubits. Finally, we demonstrate deterministic state teleportation and entanglement swapping between qubits on opposite side of our quantum processor.

Modern particle physics experiments usually rely on highly complex and large-scale spectrometer devices. In high energy physics experiments, visualization helps detector design, data quality monitoring, offline data processing, and has great potential for improving physics analysis. In addition to the traditional physics data analysis based on statistical methods, visualization provides unique intuitive advantages in searching for rare signal events and reducing background noises. By applying the event display tool to several physics analyses in the BESIII experiment, we demonstrate that visualization can benefit potential physics discovery and improve the signal significance. With the development of modern visualization techniques, it is expected to play a more important role in future data processing and physics analysis of particle physics experiments., Comment: 19 pages, 7 figures

This paper studies the principal components (PC) estimator for high dimensional approximate factor models with weak factors in that the factor loading ($\boldsymbol{\Lambda}^0$) scales sublinearly in the number $N$ of cross-section units, i.e., $\boldsymbol{\Lambda}^{0\top} \boldsymbol{\Lambda}^0 / N^\alpha$ is positive definite in the limit for some $\alpha \in (0,1)$. While the consistency and asymptotic normality of these estimates are by now well known when the factors are strong, i.e., $\alpha=1$, the statistical properties for weak factors remain less explored. Here, we show that the PC estimator maintains consistency and asymptotical normality for any $\alpha\in(0,1)$, provided suitable conditions regarding the dependence structure in the noise are met. This complements earlier result by Onatski (2012) that the PC estimator is inconsistent when $\alpha=0$, and the more recent work by Bai and Ng (2023) who established the asymptotic normality of the PC estimator when $\alpha \in (1/2,1)$. Our proof strategy integrates the traditional eigendecomposition-based approach for factor models with leave-one-out analysis similar in spirit to those used in matrix completion and other settings. This combination allows us to deal with factors weaker than the former and at the same time relax the incoherence and independence assumptions often associated with the later.

ENN Science and Technology Development Co., Ltd. (ENN) is committed to generating fusion energy in an environmentally friendly and cost-effective manner, which requires abundant aneutronic fuel. Proton-boron ( p-$^{11}$B or p-B) fusion is considered an ideal choice for this purpose. Recent studies have suggested that p-B fusion, although challenging, is feasible based on new cross-section data, provided that a hot ion mode and high wall reflection can be achieved to reduce electron radiation loss. The high beta and good confinement of the spherical torus (ST) make it an ideal candidate for p-B fusion. By utilizing the new spherical torus energy confinement scaling law, a reactor with a major radius $R_0=4$ m, central magnetic field $B_0=6$ T, central temperature $T_{i0}=150$ keV, plasma current $I_p=30$ MA, and hot ion mode $T_i/T_e=4$ can yield p-B fusion with $Q>10$. A roadmap for p-B fusion has been developed, with the next-generation device named EHL-2. EHL stands for ENN He-Long, which literally means ``peaceful Chinese Loong". The main target parameters include $R_0\simeq1.05$ m, $A\simeq1.85$, $B_0\simeq3$ T, $T_{i0}\simeq30$ keV, $I_p\simeq3$ MA, and $T_i/T_e\geq2$. The existing ST device EXL-50 was simultaneously upgraded to provide experimental support for the new roadmap, involving the installation and upgrading of the central solenoid, vacuum chamber, and magnetic systems. The construction of the upgraded ST fusion device, EXL-50U, was completed at the end of 2023, and it achieved its first plasma in January 2024. The construction of EHL-2 is estimated to be completed by 2026., Comment: 16 pages, 8 figures

The Ginzburg-Landau (GL) theory is very successful in describing the pairing symmetry, a fundamental characterization of the broken symmetries in a paired superfluid or superconductor. However, GL theory does not describe fermionic excitations such as Bogoliubov quasiparticles or Andreev bound states that are directly related to topological properties of the superconductor. In this work, we show that the symmetries of the fermionic excitations are captured by a Projective Symmetry Group (PSG), which is a group extension of the bosonic symmetry group in the superconducting state. We further establish a correspondence between the pairing symmetry and the fermion PSG. When the normal and superconducting states share the same spin rotational symmetry, there is a simpler correspondence between the pairing symmetry and the fermion PSG, which we enumerate for all 32 crystalline point groups. We also discuss the general framework for computing PSGs when the spin rotational symmetry is spontaneously broken in the superconducting state. This PSG formalism leads to experimental consequences, and as an example, we show how a given pairing symmetry dictates the classification of topological superconductivity., Comment: 21 pages, 8 tables

Higher-order topological insulators in two spatial dimensions display fractional corner charges. While fractional charges in one dimension are known to be captured by a many-body bulk invariant, computed by the Resta formula, a many-body bulk invariant for higher-order topology and the corresponding fractional corner charges remains elusive despite several attempts. Inspired by recent work by Tada and Oshikawa, we propose a well-defined many-body bulk invariant for $C_n$ symmetric higher-order topological insulators, which is valid for both non-interacting and interacting systems. Instead of relating them to the bulk quadrupole moment as was previously done, we show that in the presence of $C_n$ rotational symmetry, this bulk invariant can be directly identified with quantized fractional corner charges. In particular, we prove that the corner charge is quantized as $e/n$ with $C_n$ symmetry, leading to a $\mathbb{Z}_n$ classification for higher-order topological insulators in two dimensions., Comment: 12 pages, 7 figures, references updated

Resonators with weak single-photon self-Kerr nonlinearities can theoretically be used to prepare Fock states in the presence of a loss much larger than their nonlinearities. Two necessary ingredients are large displacements and a two-photon (parametric) drive. Here, we find that these systems can be controlled to achieve any desired gate operation in a finite dimensional subspace (whose dimensionality can be chosen at will). Moreover, we show that the two-photon driving requirement can be relaxed and that full controllability is achievable with only 1-photon (linear) drives. We make use of both Trotter-Suzuki decompositions and gradient-based optimization to find control pulses for a desired gate, which reduces the computational overhead by using a small blockaded subspace. We also discuss the infidelity arising from input power limitations in realistic settings, as well as from corrections to the rotating-wave approximation. Our universal control protocol opens the possibility for quantum information processing using a wide range of lossy systems with weak nonlinearities., Comment: 7 pages of main text with 3 figures. 15 pages of supplemental material with 4 figures

PtTe$_2$ and PdTe$_2$ are among the first transition metal dichalcogenides that were predicted to host type-II Dirac fermions, exotic particles prohibited in free space. These materials are layered and air-stable, which makes them top candidates for technological applications that take advantage of their anisotropic magnetotransport properties. Here, we provide a detailed characterization of the electronic structure of PtTe$_2$ and PdTe$_2$ using Angle Resolved Photoemission Spectroscopy (ARPES) and Density Functional Theory (DFT) calculations, unveiling a new three-dimensional dispersion in these materials. Through the use of circularly polarized light, we report a different behavior of such dispersion in PdTe$_2$ compared to PtTe$_2$, that we relate to a symmetry analysis of the dipole matrix element. Such analysis reveals a link between the observed circular dichroism and the different momentum-dependent terms in the dispersion of these two compounds, despite their close similarity in crystal structure. Additionally, our data shows a clear difference in the circular dichroic signal for the type-II Dirac cones characteristic of these materials, compared to their topologically protected surface states. Our work provides a useful reference for the ARPES characterization of other transition metal dichalcogenides with topological properties and illustrates the use of circular dichroism as a guide to identify the topological character of two otherwise equivalent band dispersions, and to recognize different attributes in the band structure of similar materials., Comment: 14 pages, 9 figures

Many important tasks of large-scale recommender systems can be naturally cast as testing multiple linear forms for noisy matrix completion. These problems, however, present unique challenges because of the subtle bias-and-variance tradeoff of and an intricate dependence among the estimated entries induced by the low-rank structure. In this paper, we develop a general approach to overcome these difficulties by introducing new statistics for individual tests with sharp asymptotics both marginally and jointly, and utilizing them to control the false discovery rate (FDR) via a data splitting and symmetric aggregation scheme. We show that valid FDR control can be achieved with guaranteed power under nearly optimal sample size requirements using the proposed methodology. Extensive numerical simulations and real data examples are also presented to further illustrate its practical merits.

Action Quality Assessment (AQA) is a task that tries to answer how well an action is carried out. While remarkable progress has been achieved, existing works on AQA assume that all the training data are visible for training at one time, but do not enable continual learning on assessing new technical actions. In this work, we address such a Continual Learning problem in AQA (Continual-AQA), which urges a unified model to learn AQA tasks sequentially without forgetting. Our idea for modeling Continual-AQA is to sequentially learn a task-consistent score-discriminative feature distribution, in which the latent features express a strong correlation with the score labels regardless of the task or action types.From this perspective, we aim to mitigate the forgetting in Continual-AQA from two aspects. Firstly, to fuse the features of new and previous data into a score-discriminative distribution, a novel Feature-Score Correlation-Aware Rehearsal is proposed to store and reuse data from previous tasks with limited memory size. Secondly, an Action General-Specific Graph is developed to learn and decouple the action-general and action-specific knowledge so that the task-consistent score-discriminative features can be better extracted across various tasks. Extensive experiments are conducted to evaluate the contributions of proposed components. The comparisons with the existing continual learning methods additionally verify the effectiveness and versatility of our approach. Data and code are available at https://github.com/iSEE-Laboratory/Continual-AQA., Comment: 16 pages, 8 figures

Advancements in quantum system lifetimes and control have enabled the creation of increasingly complex quantum states, such as those on multiple bosonic cavity modes. When characterizing these states, traditional tomography scales exponentially in both computational and experimental measurement requirement, which becomes prohibitive as the state size increases. Here, we implement a state reconstruction method whose sampling requirement instead scales polynomially with subspace size, and thus mode number, for states that can be expressed within such a subspace. We demonstrate this improved scaling with Wigner tomography of multimode entangled W states of up to 4 modes on a 3D circuit quantum electrodynamics (cQED) system. This approach performs similarly in efficiency to existing matrix inversion methods for 2 modes, and demonstrates a noticeable improvement for 3 and 4 modes, with even greater theoretical gains at higher mode numbers., Comment: 6 pages of main text with 4 figures. 10 pages of supplementary information with 6 figures