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We present and discuss seven different open problems in applied combinatorics. Additionally, the application areas relevant to this compilation include quantum computing, algorithmic differentiation, topological data analysis, iterative methods, hypergraph cut algorithms, and power systems.
Here we explore the potential of nanocrystals (a term used equivalently to nanoparticles) as building blocks for nanomaterials, and the current advances and open challenges for fundamental science developments and applications. Nanocrystal assemblies are inherently multiscale, and the generation of revolutionary material properties requires a precise understanding of the relationship between structure and function, the former being determined by classical effects and the latter often by quantum effects. With an emphasis on theory and computation, we discuss challenges that hamper current assembly strategies and to what extent nanocrystal assemblies represent thermodynamic equilibrium or kinetically trapped metastable states. We also examine dynamic effects and optimization of assembly protocols. Finally, we discuss promising material functions and examples of their realization with nanocrystal assemblies.
The purpose of this review article is to present some recent results on the modeling and control of large systems of agents. We focus on particular applications where the agents are capable of independent actions instead of simply reacting to external forces. In the literature, such agents were referred to as autonomous, intelligent, self-propelled, greedy, and others. The main applications we have in mind are social systems (as opinion dynamics), pedestrians’ movements (also called crowd dynamics), animal groups, and vehicular traffic. We note that the last three examples include physical constraints; however, the agents are able to inject energy into the system, thus preventing the typical conservation of momentum and energy. In addition, the control problems posed by such systems are new and require innovative methods. We illustrate some ideas developed recently, including the use of sparse controls, limiting the total variation of controls, and defining new control problems for measures. After reviewing various approaches, we discuss some future research directions of potential interest. The latter encompasses both new types of equations and new types of limiting procedures to connect several scales at which a system can be represented. We conclude by illustrating a recent real-life experiment using autonomous vehicles on an open highway to smooth traffic waves. This opens the door to a new era of interventions to control real-time multi-agent systems and to increase the societal impact of such interventions guided by control research.
Abstract We consider the solution of nonlinear inverse problems where the forward problem is a discretization of a partial differential equation. Such problems are notoriously difficult to solve in practice and require minimizing a combination of a data-fit term and a regularization term. The main computational bottleneck of typical algorithms is the direct estimation of the data misfit. Therefore, likelihood-free approaches have become appealing alternatives. Nonetheless, difficulties in generalization and limitations in accuracy have hindered their broader utility and applicability. In this work, we use a paired autoencoder framework as a likelihood-free estimator (LFE) for inverse problems. We show that the use of such an architecture allows us to construct a solution efficiently and to overcome some known open problems when using LFEs. In particular, our framework can assess the quality of the solution and improve on it if needed. We demonstrate the viability of our approach using examples from full waveform inversion and inverse electromagnetic imaging.
Real-world networks evolve over time via the addition or removal of vertices and edges. In current network evolution models, vertex degree varies or grows arbitrarily. A recently introduced degree-preserving network growth (DPG) family of models preserves vertex degree, resulting in structures significantly different from and more diverse than previous models. Despite its degree preserving property, the DPG model is able to replicate the output of several well-known real-world network growth models. Simulations showed that many real-world networks can also be constructed from small seed graphs via the DPG process. Here, we start the development of a rigorous mathematical theory underlying the DPG family of network growth models. We prove that the degree sequence of the output of some of the well-known, real-world network growth models can be reconstructed via the DPG process, using proper parametrization. We also show that the general problem of deciding whether a simple graph can be obtained via the DPG process from a small seed (DPG feasibility) is, however, NP-complete. In conclusion, it is an intriguing open problem to uncover whether there is a structural reason behind the DPG-constructability of real-world networks.
The emergence of hadron mass represents one of the most challenging and still open problems in contemporary hadron physics. The results on the nucleon resonance electroexcitation amplitudes available from the CLAS data on πN and π + π – p electroproduction analyzed within the continuum Schwinger method open up a new avenue for gaining insight into the strong interaction dynamics that are responsible for the generation of the dominant part of hadron mass. Future prospects of these studies in experiments of the 12 GeV era with CLAS12 and after a potential increase of the CEBAF energy up to 22 GeV will offer a unique opportunity to explore the full range of distances where the dominant part of hadron mass and N* structure emerge from QCD.
Abstract Scientists aim to discover meaningful formulae that accurately describe experimental data. Mathematical models of natural phenomena can be manually created from domain knowledge and fitted to data, or, in contrast, created automatically from large datasets with machine-learning algorithms. The problem of incorporating prior knowledge expressed as constraints on the functional form of a learned model has been studied before, while finding models that are consistent with prior knowledge expressed via general logical axioms is an open problem. We develop a method to enable principled derivations of models of natural phenomena from axiomatic knowledge and experimental data by combining logical reasoning with symbolic regression. We demonstrate these concepts for Kepler’s third law of planetary motion, Einstein’s relativistic time-dilation law, and Langmuir’s theory of adsorption. We show we can discover governing laws from few data points when logical reasoning is used to distinguish between candidate formulae having similar error on the data.
Abstract We review some recent developments in mathematical aspects of relativistic fluids. The goal is to provide a quick entry point to some research topics of current interest that is accessible to graduate students and researchers from adjacent fields, as well as to researches working on broader aspects of relativistic fluid dynamics interested in its mathematical formalism. Instead of complete proofs, which can be found in the published literature, here we focus on the proofs’ main ideas and key concepts. After an introduction to the relativistic Euler equations, we cover the following topics: a new wave-transport formulation of the relativistic Euler equations tailored to applications; the problem of shock formation for relativistic Euler; rough (i.e., low-regularity) solutions to the relativistic Euler equations; the relativistic Euler equations with a physical vacuum boundary; relativistic fluids with viscosity. We finish with a discussion of open problems and future directions of research.
A key contributor to high capital costs and schedule overruns for new nuclear power plants is lack of standardization, driven by site-specific customization and construction of multiple designs by competing vendors rather than commitment to a single standardized program. While advanced reactor vendors typically individually target repeat construction of standardized units, the many competing designs could exacerbate the problem. “Open Architecture”, the open specification of requirements and interfaces for structures, systems and components (SSCs), has been proposed as a means of promoting standardization, by facilitating existing non-nuclear suppliers to enter the industry and/or allowing SSCs to be configured for more than one reactor within the same technology type. Contracting mechanisms that facilitate information sharing and alignment of incentives between stakeholders may complement such an approach. A preliminary scheme is presented for selection of SSCs for which such strategies could be adopted, based on a vendor make/buy decision model and stakeholder interviews. SSCs are categorized according to number of suppliers and their contribution to the reactor’s competitive edge. SSCs with many potential suppliers and a high contribution to competitive edge may be attractive for widening the supply chain via open specification of system requirements and interfaces, e.g., SSCs in the power island. SSCs with few suppliers and low contribution to competitive edge may be potential avenues for common system specification between vendors, e.g., some of the auxiliary SSCs. Potential cost reductions from such strategies will depend upon the size of the build program and the reactor type.
A central problem in modern condensed matter physics is the understanding of materials with strong electron correlations. Despite extensive work, the essential physics of many of these systems is not understood and there is very little ability to make predictions in this class of materials. In this manuscript we share our personal views on the major open problems in the field of correlated electron systems. We discuss some possible routes to make progress in this rich and fascinating field. This manuscript is the result of the vigorous discussions and deliberations that took place at Johns Hopkins University during a three-day workshop January 27, 28, and 29, 2020 that brought together six senior scientists and 46 more junior scientists. Our hope, is that the topics we have presented will provide inspiration for others working in this field and motivation for the idea that significant progress can be made on very hard problems if we focus our collective energies.
The use of benchmark datasets has become an important engine of progress in machine learning (ML) over the past 15 years. Recently there has been growing interest in utilizing machine learning to drive advances in research-level mathematics. However, off-the-shelf solutions often fail to deliver the types of insights required by mathematicians. This suggests the need for new ML methods specifically designed with mathematics in mind. The question then is: what benchmarks should the community use to evaluate these? On the one hand, toy problems such as learning the multiplicative structure of small finite groups have become popular in the mechanistic interpretability community whose perspective on explainability aligns well with the needs of mathematicians. While toy datasets are a useful benchmark for initial work, they lack the scale, complexity, and sophistication of many of the principal objects of study in modern mathematics. To address this, we introduce a new collection of benchmark datasets, Algebraic Combinatorics Benchmarks (ACBench), representing either classic or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. After describing the datasets, we discuss the challenges involved in constructing “good” mathematics benchmarks, describe baseline model performance, and discuss some of the insights these datasets can provide that may be of interest even to those who are not interested in mathematics research itself.
The combination of machine learning (ML) models with chemistry-related tasks requires the description of molecular structures in a machine-readable way. The nature of these so-called molecular descriptors has a direct and major impact on the performance of ML models and remains an open problem in the field. Structural descriptors like SMILES strings or molecular graphs lack size-independence and can be memory intensive. Machine-learned descriptors can be of low dimensionality and constant size but lack physical significance and human interpretability. Sigma profiles, which are unnormalized histograms of the surface charge distributions of solvated molecules, combine physical significance with low dimensionality and size-independence, making them a suitable candidate for a universal molecular descriptor. However, their widespread adoption in ML applications requires open access to sigma profile generation, which is currently not available. This work details the development of OpenSPGen – an open-source tool for generating sigma profiles. Also presented are studies on the effect of different settings on the efficacy of the generated sigma profiles at predicting thermophysical material properties when used as inputs to a Gaussian process as a simple surrogate ML model. We find that a higher level of theory does not translate to more accurate results. We also provide further recommendations for sigma profile calculation and use in ML models.
Drone delivery, once thought of as fictitious, is becoming a reality with the efforts of both forward-looking enterprises and supportive government policies. This emerging mode of e-commerce delivery raises many concerns. One important concern is the energy efficiency of direct delivery drones compared with conventional delivery trucks at a regional systems level. Here, in this study, we develop and apply methods to quantify the regional energy impacts of drone delivery, then we assess these impacts and compare them with the impacts of truck delivery. To study this problem, we develop an optimization model that determines an optimal set of fulfillment centers (FCs) with variable service capacities that allow drones to make direct e-commerce deliveries. We adopt two drone delivery energy estimation models from the literature and use them as inputs to demonstrate the potential range of energy needs. We also develop another optimization model to account for the energy consumption of diesel trucks (DTs) and battery electric vehicles (BEVs). We test the models using validated simulation data for the Chicago metropolitan area in the U.S. to quantify the energy implications of these three delivery modes. For drone delivery, we further extend our analyses by considering the impact of wind speed and flight patterns. Our results show that direct delivery drones require 15.8% more energy than BEVs on an average windy day, and they need 15% more energy than DTs on a very windy day. We provide essential parameter values for reproducibility and list relevant open problems.
Incorporating physics information into the deep reinforcement learning (DRL) process is a promising approach for addressing the challenges faced in learning-based control design problems for physical systems. Power grid dynamics, being a physical system, adheres to specific physical laws, constraints, as well as operational and control rules. Therefore, consideration of such physics-based law improves the learning process drastically. In general, traditional grid control schemes rely on rule-based mechanisms that cannot adapt to changing operating conditions. To improve the adaptability and computation time, recent research has seen a surge of DRL-based applications in power grid control. A generic DRL-based control design imposes the system performance requirements through the design of reward functions. In some cases, some of the important physics information is injected through this reward function. However, due to the complex dynamics and large state-action space, learning an optimal DRL policy often becomes challenging. Inspired by the latest developments in general machine learning (ML) research, power system researchers have been investigating more direct ways of incorporating physics knowledge into DRL training. This chapter specifically focuses on these aspects of physics-informed DRL designs in grid control. It discusses the significance, applications, research gaps, and open problems that need to be addressed in future research.
The heavy jet mass event shape has large perturbative logarithms near the leading order kinematic threshold at ρ = $\frac{1}{3}$. Catani and Webber named these logarithms Sudakov shoulders and resummed them at double-logarithmic level. A resummation to next-to-leading logarithmic level was achieved recently. Here, we extend the resummation using an effective field theory framework to next-to-next-to-leading logarithmic order and show how to combine it with the resummation of dijet logarithms. We also solve the open problem of an unphysical singularity in the resummed momentum space distribution, in a way similar to how it is resolved in the Drell-Yan q T spectrum: through a careful analysis of the kinematics and scale-setting in position space. The heavy jet mass Sudakov shoulder is the first observable that does not involve transverse momentum for which position space resummation is critical. These advances may lead to a more precise extraction of the strong coupling constant from e + e – data.
Single-molecule photoluminescence (PL) spectroscopy of semiconductor nanocrystals (NCs) reveals the nature of exciton–phonon interactions in NCs. Understanding the homogeneous spectral line shapes and their temperature dependence remains an open problem. Here, we develop an atomistic model to describe the PL spectrum of NCs, accounting for excitonic effects, phonon dispersion relations, and exciton–phonon couplings. We validate our model using single-NC measurements on CdSe/CdS NCs from T = 4 to 290 K, and we find that the slightly asymmetric main peak at low temperatures is comprised of a narrow zero-phonon line (ZPL) and acoustic phonon sidebands. Furthermore, we identify the specific phonon modes that give rise to the optical phonon sidebands. At temperatures above 200 K, the spectral line width shows a stronger dependence upon the temperature, which we demonstrate to be correlated with higher order exciton–phonon couplings. We also identify the line width dependence upon reorganization energy, NC core sizes, and shell thicknesses.
Recently realized high-mobility semiconducting epitaxial graphene on silicon carbide (SiC) [Zhao, J. Nature 2024, 625 (7993), 60−65, 10.1038/s41586-023-06811-0] provided an important step toward integration of the graphene-based system into active components in postsilicon micro- and nanoelectronics. However, the exact atomic-scale structure and complex bonding configurations of the first epitaxial graphene carbon layer (C buffer ) remain an open problem. Our recent report [Kolmer, M. Communications Physics 2024, 7 (1), 16, 10.1038/s42005-023-01515-3] has shed new light on understanding this interface, where the external transverse electric field-dependent dynamic switching behavior of the C buffer –SiC bonds was observed. Here, using scanning tunneling microscopy and spectroscopy (STM and STS), we present direct evidence of silicon (Si) vacancies at the interface and provide their distribution at the topmost reconstructed SiC(0001) layer. Bias voltage and epitaxial graphene thickness-dependent characterization of the collective C buffer –SiC interface showed that “Si” vacancy sites beneath C buffer are stable under STM electric fields. Moreover, the vacancies introduce localized electronic states below the Fermi level, thereby enhancing the charge-transfer phenomenon across the interface.
Abstract The mysterious Planckian metal state, showing perfect T -linear resistivity associated with universal scattering rate, 1/ τ = α k B T /ℏ with α ~ 1, has been observed in the normal state of various strongly correlated superconductors close to a quantum critical point. However, its microscopic origin and link to quantum criticality remains an outstanding open problem. Here, we observe quantum-critical T / B -scaling of the Planckian metal state in resistivity and heat capacity of heavy-electron superconductor Ce 1− x Nd x CoIn 5 in magnetic fields near the edge of antiferromagnetism at the critical doping x c ~ 0.03. We present clear experimental evidences of Kondo hybridization being quantum critical at x c . We provide a generic microscopic mechanism to qualitatively account for this quantum critical Planckian state within the quasi-two dimensional Kondo-Heisenberg lattice model near Kondo breakdown transition. We find α is a non-universal constant and depends inversely on the square of Kondo hybridization strength.