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We study the geometry of the algebraic set of tuples of composable matrices which multiply to a fixed matrix, using tools from the theory of quiver representations. In particular, we determine its codimension $C$ and the number $\theta$ of its top-dimensional irreducible components. Our solution is presented in three forms: a Poincar\'e series in equivariant cohomology, a quadratic integer program, and an explicit formula. In the course of the proof, we establish a surprising property: $C$ and $\theta$ are invariant under arbitrary permutations of the dimension vector. We also show that the real log-canonical threshold of the function taking a tuple to the square Frobenius norm of its product is $C/2$. These results are motivated by the study of deep linear neural networks in machine learning and Bayesian statistics (singular learning theory) and show that deep linear networks are in a certain sense ``mildly singular"., Comment: 28 pages, 2 figures. Comments welcome!

We investigate the charge density wave phase in the strongly correlated Hubbard model without any other broken symmetry phase. Starting from the atomic Hamiltonian with no hopping, we generate quasiparticle operators corresponding to holons and doublons in the strongly correlated limit of the repulsive Hubbard model. We develop a real space composite operator formalism using the equation of motion technique to include the intersite hopping perturbatively. Our fully self-consistent calculation stabilizes multiple unidirectional translation symmetry broken states within the doping range $\delta=0.07$ to $0.2$. The charge-ordered states become increasingly unfavorable with hole-doping. The unidirectional density waves manifest as periodic modulations of half-filled Mott regions separated by hole-rich regions. Notably, density wave solutions with periods of $3$ to $8$ lattice spacing remain energetically higher than those with larger periods. Quenched disorder on the charge-ordered states induces the merging of the Mott regions and, consequently, forms short-ranged charge modulations. The density of states shows signatures of strongly correlated Mott regions, potentially relevant to the physics of underdoped cuprates., Comment: 13 Pages, 6 Figures

We use the Bogoliubov-de Gennes (BDG) formalism to undertake a microscopic investigation of a vortex lattice in a strongly correlated, type-II, d-wave superconductor (SC) treating strong correlation within Gutzwiller formalism. We demonstrate that in the underdoped region, the vortex core changes from metallic-type to insulating-type in the presence of subdominant charge and bond order, in contrast to Mott-type, when these orders are absent. We have investigated that such subdominant order changes the structure and spectrum of the d-wave vortex in the underdoped region. We have demonstrated the formation of charge and bond modulation at the vortex center by decreasing the doping and reaching an underdoped zone.

Let $F$ be a finite unramified extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_F$, and let $\mathbf{G}$ denote a split, connected reductive group over $\mathcal{O}_F$. We fix a Borel subgroup $\mathbf{B} = \mathbf{T}\mathbf{U}$ with maximal torus $\mathbf{T}$ and unipotent radical $\mathbf{U}$, and let $L(\lambda)$ denote an irreducible representation of $G_0 := \mathbf{G}(\mathcal{O}_F)$ with coefficients in a sufficiently large field of characteristic $p$. Set $G := \mathbf{G}(F)$, etc. Assuming $\lambda$ is a $p$-small and sufficiently regular character and that $p - 1$ is greater than the Coxeter number of $\mathbf{G}$, we show that the complex $L(U,\textrm{c-ind}_{G_0}^{G}(L(\lambda)))$ splits as the orthogonal direct sum of its cohomology objects in the derived category of smooth $T$-representations in characteristic $p$. (Here $L(U, -)$ denotes Heyer's left adjoint of parabolic induction, from the derived category of smooth $G$-representations to the derived category of smooth $T$-representations.) Consequently, this gives rise to a collection of morphisms of graded spherical Hecke algebras $$\displaystyle{\bigoplus_{i \in \mathbb{Z}}\textrm{Ext}_{G}^{i}\left(\textrm{c-ind}_{G_0}^{G}(L(\lambda)),~\textrm{c-ind}_{G_0}^{G}(L(\lambda))\right) \longrightarrow \bigoplus_{i \in \mathbb{Z}}\textrm{Ext}_{T}^{i}\left(\textrm{c-ind}_{T_0}^{T}(L^n(U_0,L(\lambda))),~\textrm{c-ind}_{T_0}^{T}(L^n(U_0,L(\lambda)))\right)}$$ indexed by $n=-[F:\mathbb{Q}_p]\dim(\mathbf{U}), \ldots, 0$, which we refer to as derived Satake morphisms. For $\lambda=0$ and $n=0$, this recovers the graded mod $p$ Satake homomorphism constructed by Ronchetti. We also give some partial results for general standard parabolic subgroups $\mathbf{P} = \mathbf{M}\mathbf{N} \subset \mathbf{G}$., Comment: 40 pages. Comments welcome!

Quantum materials featuring both itinerant and localized degrees of freedom exhibit numerous exotic phases and transitions that deviate from the Ginzburg-Landau paradigm. This work uses the composite operator formalism to examine two-orbital strongly correlated Hubbard models. We observe the spontaneous breaking of orbital symmetry, where the electron density in one of the orbitals reaches half-filling, resulting in an orbitally selective Mott phase (OSMP). This broken symmetry phase becomes unstable at a critical average electronic density away from half-filling. Furthermore, significant orbital differentiation persists up to a moderate inter-orbital hopping, beyond which the system abruptly transitions to an orbitally uniform phase. In the OSMP phase, the electrons in the two orbitals are weakly hybridized, resulting in a small Fermi surface. The volume of the Fermi surface jumps at the transition from the OSMP to the orbitally uniform phase. We also discuss the physical mechanisms leading to the collapse of the OSMP phase under different perturbations.

The recent surge in the capabilities of Large Language Models (LLMs) has led to claims that they are approaching a level of creativity akin to human capabilities. This idea has sparked a blend of excitement and apprehension. However, a critical piece that has been missing in this discourse is a systematic evaluation of LLM creativity, particularly in comparison to human divergent thinking. To bridge this gap, we leverage recent advances in creativity science to build a framework for in-depth analysis of divergent creativity in both state-of-the-art LLMs and a substantial dataset of 100,000 humans. We found evidence suggesting that LLMs can indeed surpass human capabilities in specific creative tasks such as divergent association and creative writing. Our quantitative benchmarking framework opens up new paths for the development of more creative LLMs, but it also encourages more granular inquiries into the distinctive elements that constitute human inventive thought processes, compared to those that can be artificially generated., Comment: First two and last listed authors are corresponding authors. The first two listed authors contributed equally to this work

The recent advances in deep learning (DL) have been accelerated by access to large-scale data and compute. These large-scale resources have been used to train progressively larger models which are resource intensive in terms of compute, data, energy, and carbon emissions. These costs are becoming a new type of entry barrier to researchers and practitioners with limited access to resources at such scale, particularly in the Global South. In this work, we take a comprehensive look at the landscape of existing DL models for medical image analysis tasks and demonstrate their usefulness in settings where resources are limited. To account for the resource consumption of DL models, we introduce a novel measure to estimate the performance per resource unit, which we call the PePR score. Using a diverse family of 131 unique DL architectures (spanning 1M to 130M trainable parameters) and three medical image datasets, we capture trends about the performance-resource trade-offs. In applications like medical image analysis, we argue that small-scale, specialized models are better than striving for large-scale models. Furthermore, we show that using existing pretrained models that are fine-tuned on new data can significantly reduce the computational resources and data required compared to training models from scratch. We hope this work will encourage the community to focus on improving AI equity by developing methods and models with smaller resource footprints., Comment: Accepted to be published at the Northern Lights Deep Learning Conference (NLDL), 2025. Source code available at https://github.com/saintslab/PePR

Version Control Systems, such as Git and Mercurial, manage the history of a project as a Directed Acyclic Graph encoding the various divergences and synchronizations happening in its life cycle. A popular workflow in the industry, called the feature branch workflow, constrains these graphs to be of a particular shape: a unique main branch, and non-interfering feature branches. Here we focus on the uniform random generation of those graphs with n vertices, including k on the main branch, for which we provide three algorithms, for three different use-cases. The first, based on rejection, is efficient when aiming for small values of k (more precisely whenever k = O($\sqrt$ n)). The second takes as input any number k of commits in the main branch, but requires costly precalculation. The last one is a Boltzmann generator and enables us to generate very large graphs while targeting a constant k/n ratio. All these algorithms are linear in the size of their outputs.

Wildlife disease surveillance programs and research studies track infection and identify risk factors for wild populations, humans, and agriculture. Often, several types of samples are collected from individuals to provide more complete information about an animal's infection history. Methods that jointly analyze multiple data streams to study disease emergence and drivers of infection via epidemiological process models remain underdeveloped. Joint-analysis methods can more thoroughly analyze all available data, more precisely quantifying epidemic processes, outbreak status, and risks. We contribute a paired data modeling approach that analyzes multiple samples from individuals. We use "characterization maps" to link paired data to epidemiological processes through a hierarchical statistical observation model. Our approach can provide both Bayesian and frequentist estimates of epidemiological parameters and state. We motivate our approach through the need to use paired pathogen and antibody detection tests to estimate parameters and infection trajectories for the widely applicable susceptible, infectious, recovered (SIR) model. We contribute general formulas to link characterization maps to arbitrary process models and datasets and an extended SIR model that better accommodates paired data. We find via simulation that paired data can more efficiently estimate SIR parameters than unpaired data, requiring samples from 5-10 times fewer individuals. We then study SARS-CoV-2 in wild White-tailed deer (Odocoileus virginianus) from three counties in the United States. Estimates for average infectious times corroborate captive animal studies. Our methods use general statistical theory to let applications extend beyond the SIR model we consider, and to more complicated examples of paired data., Comment: 22 pages, 5 figures, 1 table

We consider the most energetically favorable symmetry-allowed spin-singlet and spin-triplet superconducting pairing symmetries in twisted bilayer graphene at the magic angle, whose normal state physics is described by a six-band effective tight-binding model. We compute the Chern number as a function of the superconducting order parameter strength and the chemical potential and we find a topological phase transition only for the chiral $p+ip'$ superconducting state. Different from the regular graphene systems for which this happens at the van Hove singularity, for TBG the topological phase transition arises at the point where the Fermi surface becomes tangent to the boundary of the first Brillouin zone. For each pairing symmetry we study the formation of subgap impurity states for both scalar and magnetic impurities. We analyze the number of subgap states as well as their spin polarized density of states that we find to exhibit peculiar properties that allows one to distinguish between spin-singlet and triplet pairing. Thus only triplet-paired states may exhibit opposite-energy impurity states with the same spin, same as for regular graphene systems\, moreover we find that this spin may flip at the twist-induced van Hove singularity.

Let $p$ be an odd prime. Let $F$ be a non-archimedean local field of residue characteristic $p$, and let $\mathbb{F}_q$ be its residue field. Let $\mathcal{H}^{(1)}_{\mathbb{F}_q}$ be the pro-$p$-Iwahori-Hecke algebra of the $p$-adic group ${\textrm GL_2}(F)$ with coefficients in $\mathbb{F}_q$, and let $Z(\mathcal{H}^{(1)}_{\mathbb{F}_q})$ be its center. We define a scheme $X(q)_{\mathbb{F}_q}$ whose geometric points parametrize the semisimple two-dimensional Galois representations of ${\textrm Gal}(\overline{F}/F)$ over $\overline{\mathbb{F}}_q$. Then we construct a morphism from the spectrum of $Z(\mathcal{H}^{(1)}_{\mathbb{F}_q})$ to $X(q)_{\mathbb{F}_q}$ generalizing the morphism appearing in \cite{PS2} for $F=\mathbb{Q}_p$. In the case $F/\mathbb{Q}_p$, we show that the induced map from Hecke modules to Galois representations, when restricted to supersingular modules, coincides with Grosse-Kl\"onne's bijection \cite{GK18}. For this, we determine the Lubin-Tate $(\varphi,\Gamma)$-modules associated to absolutely irreducible Galois representations., Comment: The note arXiv:1911.11820 has been included as an Appendix of the present preprint

We use the composite operator method (COM) to analyze the strongly correlated repulsive Hubbard model, investigating the effect of nearest-neighbor hoppings up to fourth order on a square lattice. We consider two sets of self-consistent equations, one enforcing the Pauli principle and the other imposing charge-charge, spin-spin, and pair-pair correlations using a decoupling scheme developed by L. Roth. We extract three distinct solutions from these equations: COM1 and COM2 by imposing the Pauli principle and one from Roth decoupling. An overview of the method studying the validity of particle-hole symmetry and the Luttinger theorem for each solution is presented. Additionally, we extend the initial basis to study superconductivity, concluding that it is induced by the Van Hove singularity. Finally, we include higher-order hoppings using realistic estimates for tight binding parameters and compare our results with ARPES measurements on cuprates.

We compute transport and thermodynamic properties of a two-band spin-fermion model describing itinerant fermions in two dimensions interacting via $Z_2$ antiferromagnetic quantum critical fluctuations by means of a sign-problem-free quantum Monte-Carlo approach. We show that the phase diagram of this model indeed contains a $d$-wave superconducting phase at low enough temperatures. However, a crucial question that arises is whether a non-Fermi-liquid metallic regime exists above $T_c$ exhibiting hallmark strange-metal transport phenomenology. Interestingly, we find that this version of the model describes a non-Fermi-liquid metallic regime that displays an approximately $T$-linear resistivity above $T_c$ for a strong fermion-boson interaction. Using Nernst-Einstein relation, our QMC results also show that this strange metal phase exhibits a crossover from being characterized by a charge compressibility given approximately by $\chi_{c}\sim 1/T$ at high temperatures to being described by a charge diffusivity consistent with the scaling $D_{c}\sim 1/T$ at low temperatures. Therefore, our work adds support to the view that the $Z_2$ antiferromagnetic spin-fermion model at strong coupling can be considered a minimal model that describes both unconventional superconductivity and strange metallicity, which are fundamentally interconnected in many important strongly-correlated quantum materials., Comment: 12 pages, 9 figures

Network pruning is a widely used technique to reduce computation cost and model size for deep neural networks. However, the typical three-stage pipeline significantly increases the overall training time. In this paper, we develop a systematic weight-pruning optimization approach based on Surrogate Lagrangian relaxation, which is tailored to overcome difficulties caused by the discrete nature of the weight-pruning problem. We prove that our method ensures fast convergence of the model compression problem, and the convergence of the SLR is accelerated by using quadratic penalties. Model parameters obtained by SLR during the training phase are much closer to their optimal values as compared to those obtained by other state-of-the-art methods. We evaluate our method on image classification tasks using CIFAR-10 and ImageNet with state-of-the-art MLP-Mixer, Swin Transformer, and VGG-16, ResNet-18, ResNet-50 and ResNet-110, MobileNetV2. We also evaluate object detection and segmentation tasks on COCO, KITTI benchmark, and TuSimple lane detection dataset using a variety of models. Experimental results demonstrate that our SLR-based weight-pruning optimization approach achieves a higher compression rate than state-of-the-art methods under the same accuracy requirement and also can achieve higher accuracy under the same compression rate requirement. Under classification tasks, our SLR approach converges to the desired accuracy $3\times$ faster on both of the datasets. Under object detection and segmentation tasks, SLR also converges $2\times$ faster to the desired accuracy. Further, our SLR achieves high model accuracy even at the hard-pruning stage without retraining, which reduces the traditional three-stage pruning into a two-stage process. Given a limited budget of retraining epochs, our approach quickly recovers the model's accuracy., Comment: arXiv admin note: text overlap with arXiv:2012.10079

Directed acyclic graphs (DAGs) are directed graphs in which there is no path from a vertex to itself. DAGs are an omnipresent data structure in computer science and the problem of counting the DAGs of given number of vertices and to sample them uniformly at random has been solved respectively in the 70's and the 00's. In this paper, we propose to explore a new variation of this model where DAGs are endowed with an independent ordering of the out-edges of each vertex, thus allowing to model a wide range of existing data structures. We provide efficient algorithms for sampling objects of this new class, both with or without control on the number of edges, and obtain an asymptotic equivalent of their number. We also show the applicability of our method by providing an effective algorithm for the random generation of classical labelled DAGs with a prescribed number of vertices and edges, based on a similar approach. This is the first known algorithm for sampling labelled DAGs with full control on the number of edges, and it meets a need in terms of applications, that had already been acknowledged in the literature., Comment: 32 pages, 12 figures. For the implementation of the algorithms, see https://github.com/Kerl13/randdag

Computing accurate splines of degree greater than three is still a challenging task in today's applications. In this type of interpolation, high-order derivatives are needed on the given mesh. As these derivatives are rarely known and are often not easy to approximate accurately, high-degree splines are difficult to obtain using standard approaches. In Beaudoin (1998), Beaudoin and Beauchemin (2003), and Pepin et al. (2019), a new method to compute spline approximations of low or high degree from equidistant interpolation nodes based on the discrete Fourier transform is analyzed. The accuracy of this method greatly depends on the accuracy of the boundary conditions. An algorithm for the computation of the boundary conditions can be found in Beaudoin (1998), and Beaudoin and Beauchemin (2003). However, this algorithm lacks robustness since the approximation of the boundary conditions is strongly dependant on the choice of $\theta$ arbitrary parameters, $\theta$ being the degree of the spline. The goal of this paper is therefore to propose two new robust algorithms, independent of arbitrary parameters, for the computation of the boundary conditions in order to obtain accurate splines of any degree. Numerical results will be presented to show the efficiency of these new approaches.

Non-local quasiparticles in correlated quantum materials can exhibit the proximity effect. For instance, in metal superconductor hybrid systems, the leaking of cooper pairs to the metallic region induces superconducting correlations in a standard metal. This paper explores the proximity effects of charge density wave (CDW) on metal using the attractive Hubbard model, which harbors CDW state at half-filling. Our fully self-consistent calculations demonstrate that periodic charge modulations develop in a metal due to the tunneling of finite momentum particle-hole pairs from the CDW region. Upon doping the normal region, the commensurate CDW changes to an incommensurate one by incorporating regular phase shifts. Furthermore, the induced CDW produces a soft gap in the density of states and thus can be detected in tunneling experiments. We discuss our results in light of recent reports of such proximity-induced charge order in different two-dimensional heterostructures., Comment: 8 Pages, 6 Figures

The recent advancements in wireless technology enable connected autonomous vehicles (CAVs) to gather data via vehicle-to-vehicle (V2V) communication, such as processed LIDAR and camera data from other vehicles. In this work, we design an integrated information sharing and safe multi-agent reinforcement learning (MARL) framework for CAVs, to take advantage of the extra information when making decisions to improve traffic efficiency and safety. We first use weight pruned convolutional neural networks (CNN) to process the raw image and point cloud LIDAR data locally at each autonomous vehicle, and share CNN-output data with neighboring CAVs. We then design a safe actor-critic algorithm that utilizes both a vehicle's local observation and the information received via V2V communication to explore an efficient behavior planning policy with safety guarantees. Using the CARLA simulator for experiments, we show that our approach improves the CAV system's efficiency in terms of average velocity and comfort under different CAV ratios and different traffic densities. We also show that our approach avoids the execution of unsafe actions and always maintains a safe distance from other vehicles. We construct an obstacle-at-corner scenario to show that the shared vision can help CAVs to observe obstacles earlier and take action to avoid traffic jams., Comment: This paper gets the Best Paper Award in the DCAA workshop of AAAI 2023

We revisit a phenomenological model of fermions coupled to fluctuating bosons that emerges from finite-momentum particle-particle pairs for describing the strange metal phase in the cuprates. The incoherent bosons dominate the transport properties for the resistivity and optical conductivity in the non-Fermi liquid phase. Within the Kubo formalism, the resistivity is approximately linear in temperature with a Drude form for the optical conductivity, such that the Drude lifetime is inversely proportional to the temperature. Additionally, when the bosons emerge from electron pairs with spin-triplet symmetry, the magnetoresistance has a linear magnetic field dependence. Here, we compute the transport properties of such bosons within the memory-matrix approach that successfully captures the hydrodynamic regime. This technique emerges as the appropriate framework for describing the transport coefficients of the strange metal phase. Our analysis confirms the $T$-linear resistivity due to the Umklapp scattering that we obtained for this effective model. Finally, we provide new predictions regarding the variation of the thermal conductivity with temperature and examine the validity of the Wiedemann-Franz law.

We consider the most energetically favorable symmetry-allowed spin-singlet and spin-triplet superconducting pairing symmetries in monolayer and few-layer graphene, and for each calculate the energy spectrum in the presence of a scalar or magnetic impurity. We find that two doubly degenerate subgap states exist for scalar impurities for all types of pairing, except for the spin-singlet $s$-wave state. For magnetic impurities, two or four subgap states may form depending on the order parameter symmetry. We find that the spin polarization of these states allows one to distinguish between spin-singlet and triplet pairing, for example, only the spin-triplet states show opposite-energy subgap states with the same spin. We also calculate the quasi-particle interference patterns associated with the subgap states and find that they exhibit features that could distinguish between different types of pairing symmetries, especially a breaking of rotational symmetry for nodal states, stronger for the spin-singlet $d_{xy}$ and $d_{x^2-y^2}$ than for the spin-triplet $p_x$ and $p_y$ states.

We construct new six-functor formalisms capturing cohomological invariants of varieties with potentials. Starting from any six-functor formalism $C$, encoded as a coefficient system, we associate a new six-functor formalism $C_{\text{exp}}$. This requires in particular constructing the convolution product symmetric monoidal structure at the $\infty$-categorical level. We study $C_{\text{exp}}$ and how it relates to $C$. We also define motives in $C_{\text{exp}}$ attached to varieties with potential and study their properties., Comment: 50 pages; preliminary version subject to change

Electric vehicles (EVs) are being rapidly adopted due to their economic and societal benefits. Autonomous mobility-on-demand (AMoD) systems also embrace this trend. However, the long charging time and high recharging frequency of EVs pose challenges to efficiently managing EV AMoD systems. The complicated dynamic charging and mobility process of EV AMoD systems makes the demand and supply uncertainties significant when designing vehicle balancing algorithms. In this work, we design a data-driven distributionally robust optimization (DRO) approach to balance EVs for both the mobility service and the charging process. The optimization goal is to minimize the worst-case expected cost under both passenger mobility demand uncertainties and EV supply uncertainties. We then propose a novel distributional uncertainty sets construction algorithm that guarantees the produced parameters are contained in desired confidence regions with a given probability. To solve the proposed DRO AMoD EV balancing problem, we derive an equivalent computationally tractable convex optimization problem. Based on real-world EV data of a taxi system, we show that with our solution the average total balancing cost is reduced by 14.49%, and the average mobility fairness and charging fairness are improved by 15.78% and 34.51%, respectively, compared to solutions that do not consider uncertainties., Comment: 16 pages

We study the emergence of electronic edge states in superconducting (SC) monolayer, bilayer, and trilayer graphene for both spin-singlet and spin-triplet SC order parameters. We focus mostly on the gapped chiral $p+ip'$- and $d+id'$-wave SC states that show a non-zero Chern number and a corresponding number of edge states. For the $p+ip'$-wave state, we observe a rich Chern phase diagram when tuning the chemical potential and the SC order parameter amplitudes, which depends strongly on the number of layers and their stacking, and is also modified by trigonal warping. At small parameter values we observe a region whose Chern number is unique to rhombohedrally stacked graphene, and is independent of the number of layers. Our results can be understood in relation not only to the SC order parameter winding as expected, but also to the normal state band structure. This observation establishes the importance of the normal state characteristics for understanding the topology in SC graphene systems.

We review all symmetry-allowed spin-singlet and spin-triplet superconducting (SC) order parameters in graphene ($s$-wave, $d$-wave, $p$-wave, and $f$-wave) generated by generic onsite, nearest-neighbor (NN), and next-to-nearest-neighbor (NNN) pairing interactions in a tight-binding model. For each pairing channel, we calculate both the band structure and the dependence of the density of states on energy, chemical potential, and on the pairing strength. In particular, we distinguish between nodal superconducting states and fully gapped states and study the dependence of gap closing points on the chemical potential and the superconducting pairing strength. We further investigate the difference between mono-, bi-, and tri-layer ABC and ABA graphene, including accounting for the effects of trigonal warping.

As electric vehicle (EV) technologies become mature, EV has been rapidly adopted in modern transportation systems, and is expected to provide future autonomous mobility-on-demand (AMoD) service with economic and societal benefits. However, EVs require frequent recharges due to their limited and unpredictable cruising ranges, and they have to be managed efficiently given the dynamic charging process. It is urgent and challenging to investigate a computationally efficient algorithm that provide EV AMoD system performance guarantees under model uncertainties, instead of using heuristic demand or charging models. To accomplish this goal, this work designs a data-driven distributionally robust optimization approach for vehicle supply-demand ratio and charging station utilization balancing, while minimizing the worst-case expected cost considering both passenger mobility demand uncertainties and EV supply uncertainties. We then derive an equivalent computationally tractable form for solving the distributionally robust problem in a computationally efficient way under ellipsoid uncertainty sets constructed from data. Based on E-taxi system data of Shenzhen city, we show that the average total balancing cost is reduced by 14.49%, the average unfairness of supply-demand ratio and utilization is reduced by 15.78% and 34.51% respectively with the distributionally robust vehicle balancing method, compared with solutions which do not consider model uncertainties., Comment: This paper has been published in IROS2020

In this paper, we present our studies of the phase diagram of the cuprate superconductors performed in recent years. We describe how a few field-theoretical concepts can be used to account for the puzzling properties of these compounds. Starting with a short exposition of recent experimental developments, we then introduce the concept of an emergent SU(2) symmetry, which rotates between the superconducting and charge order parameters. We describe a solvable model for which this symmetry turns out to be exact and derive the corresponding effective field theory in terms of a non-linear {\sigma} model. We then turn to the experimental consequences of our model with the presence of skyrmions inside the vortex core and an elegant account of the B-T phase diagram. Next, we move on to the second concept, which is the one of an emergent U(1) gauge field in the theory. Within this more general framework, the pseudogap transition can be viewed as a confining/deconfining transition. Experimental consequences for a few spectroscopic probes like inelastic neutron scattering, Raman spectroscopy, phonon line in x-ray scattering and angle-resolved photoemission experiments are studied. This theory suggests the presence of incoherent bosons at moderate temperatures in the phase diagram of the cuprates, which would undergo a jamming transition at optimal doping [1]. Consequences of the presence of such bosons on the transport properties of these materials are examined., Comment: 16 pages, 13 figures