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Object navigation in multi-floor environments presents a formidable challenge in robotics, requiring sophisticated spatial reasoning and adaptive exploration strategies. Traditional approaches have primarily focused on single-floor scenarios, overlooking the complexities introduced by multi-floor structures. To address these challenges, we first propose a Multi-floor Navigation Policy (MFNP) and implement it in Zero-Shot object navigation tasks. Our framework comprises three key components: (i) Multi-floor Navigation Policy, which enables an agent to explore across multiple floors; (ii) Multi-modal Large Language Models (MLLMs) for reasoning in the navigation process; and (iii) Inter-Floor Navigation, ensuring efficient floor transitions. We evaluate MFNP on the Habitat-Matterport 3D (HM3D) and Matterport 3D (MP3D) datasets, both include multi-floor scenes. Our experiment results demonstrate that MFNP significantly outperforms all the existing methods in Zero-Shot object navigation, achieving higher success rates and improved exploration efficiency. Ablation studies further highlight the effectiveness of each component in addressing the unique challenges of multi-floor navigation. Meanwhile, we conducted real-world experiments to evaluate the feasibility of our policy. Upon deployment of MFNP, the Unitree quadruped robot demonstrated successful multi-floor navigation and found the target object in a completely unseen environment. By introducing MFNP, we offer a new paradigm for tackling complex, multi-floor environments in object navigation tasks, opening avenues for future research in visual-based navigation in realistic, multi-floor settings.

Indefinite quadratic programs (QPs) are known to be very difficult to be solved to global optimality, so are linear programs with linear complementarity constraints. Treating the former as a subclass of the latter, this paper presents a progressive mixed integer linear programming method for solving a general linear program with linear complementarity constraints (LPCC). Instead of solving the LPCC with a full set of integer variables expressing the complementarity conditions, the presented method solves a finite number of mixed integer subprograms by starting with a small fraction of integer variables and progressively increasing this fraction. After describing the PIP (for progressive integer programming) method and its various implementations, we demonstrate, via an extensive set of computational experiments, the superior performance of the progressive approach over the direct solution of the full-integer formulation of the LPCCs. It is also shown that the solution obtained at the termination of the PIP method is a local minimizer of the LPCC, a property that cannot be claimed by any known non-enumerative method for solving this nonconvex program. In all the experiments, the PIP method is initiated at a feasible solution of the LPCC obtained from a nonlinear programming solver, and with high likelihood, can successfully improve it. Thus, the PIP method can improve a stationary solution of an indefinite QP, something that is not likely to be achievable by a nonlinear programming method. Finally, some analysis is presented that provides a better understanding of the roles of the LPCC suboptimal solutions in the local optimality of the indefinite QP.

Geometric frustration is known to completely damage kinetic processes of some of the orbitals (and their associated quantum coherence) as to produce flat bands in the non-interacting systems. The impact of introducing additional interaction to the system in such frustrated systems is, however, a highly controversial issue. On the one hand, numerical studies on geometrically frustrated systems of hard-core boson (equivalent to a spin-1/2 systems) typically lead to glass or solid phases containing only local many-body coherence, indicating the persistence of the damage in quantum coherence. On the other, there continues to be noticeable claims of development of superfluidity that implies kinetic flow of particles. To resolve this apparent contradiction of great significance, we present a rigorous proof showing that density-density interaction is incapable of defeating the geometric frustration to allow propagation of those immobile particles, let alone sustaining a superfluidity. Instead, the frustrated systems develop many $\textit{compact}$ many-body localized states as "many-body scars" that do not thermalize, making them good candidates for storing $\textit{robust}$ and $\textit{manipulable}$ quantum information., Comment: 6 pages, 4 figures

In great contrast to the numerous discoveries of superconductivity in layer-stacked graphene systems, the absence of superconductivity in the simplest and cleanest monolayer graphene remains a big puzzle. Here, through realistic computation of electronic structure, we identify a systematic trend that superconductivity appears to emerge only upon alteration of the low-energy electronic lattice from the underlying honeycomb atomic structure. We then demonstrate that this inhibition can result from from geometric frustration of the bond lattice that disables quantum phase coherence of the order parameter residing on it. In comparison, upon deviating from the honeycomb lattice, relief of geometric frustration allows robust superfluidity with non-trivial spatial structure. For the specific examples of bilayer and trilayer graphene under an external electric field, such bond centered order parameter would develop superfluidity with staggered flux that breaks the time-reversal symmetry. Our study also suggests the possible realization of the long-sought superconductivity in single-layer graphene via the application of uni-directional strain.

Robotic agents must master common sense and long-term sequential decisions to solve daily tasks through natural language instruction. The developments in Large Language Models (LLMs) in natural language processing have inspired efforts to use LLMs in complex robot planning. Despite LLMs' great generalization and comprehension of instruction tasks, LLMs-generated task plans sometimes lack feasibility and correctness. To address the problem, we propose a RoboGPT agent\footnote{our code and dataset will be released soon} for making embodied long-term decisions for daily tasks, with two modules: 1) LLMs-based planning with re-plan to break the task into multiple sub-goals; 2) RoboSkill individually designed for sub-goals to learn better navigation and manipulation skills. The LLMs-based planning is enhanced with a new robotic dataset and re-plan, called RoboGPT. The new robotic dataset of 67k daily instruction tasks is gathered for fine-tuning the Llama model and obtaining RoboGPT. RoboGPT planner with strong generalization can plan hundreds of daily instruction tasks. Additionally, a low-computational Re-Plan module is designed to allow plans to flexibly adapt to the environment, thereby addressing the nomenclature diversity challenge. The proposed RoboGPT agent outperforms SOTA methods on the ALFRED daily tasks. Moreover, RoboGPT planner exceeds SOTA LLM-based planners like ChatGPT in task-planning rationality for hundreds of unseen daily tasks, and even other domain tasks, while keeping the large model's original broad application and generality.

The modularity of elliptic curves always intrigues number theorists. Recently, Thorne had proved a marvelous result that for a prime $ p $, every elliptic curve defined over a $ p $-cyclotomic extension of $ \mathbb{Q} $ is modular. The method is to use some automorphy lifting theorems and study non-cusp points on some specific elliptic curves by Iwasawa theory for elliptic curves. Since the modularity of elliptic curves over real quadratic was proved, one may ask whether it is possible to replace $ \mathbb{Q} $ with a real quadratic field $ K $. Following Thorne's idea, we give some assumptions first and prove the modularity of elliptic curves over the $\mathbb{Z}_p$-extension of some real quadratic fields., Comment: 7 pages. Some typos are corrected