Additional Constraints Research Articles

Magmatic water content in HIMU basalts from the Cook-Austral Islands: constraints on degassing processes and source composition from clinopyroxene phenocrysts

The amount of water recycled during subduction is unclear. Water/Ce estimates in HIMU (high-238U/204Pb) basalts are variable, ranging from < 100 to > 250 in glasses and melt inclusions. Because clinopyroxene (cpx) is a common early liquidus phase and the cpx/melt partitioning of water is well constrained, cpx phenocrysts provide an additional constraint on magmatic water contents. We present water and trace element concentrations in HIMU-basalt-hosted cpx phenocrysts from the Austral Islands. Calculated melt [H2O] (up to 4.3 wt.%) and H2O/Ce ratios (22–825) are higher than in olivine-hosted inclusions, as are estimated equilibration pressures. Correlation between estimated melt [H2O] and crystallization or entrapment pressure suggests significant water loss during magma ascent. Most ocean island basalts (OIBs) span a limited range in H2O (~ 1–1.5 wt.%), and low (< 100) H2O/Ce ratios are primarily observed in melts with unusually high [Ce] (up to 350 ppm). Additionally, [H2O] and H2O/Ce in some suites correlate with entrapment pressure despite having quench pressures high enough to prevent significant water loss from open- or closed-system degassing (< 100 MPa). Polybaric “sparging”, whereby low-P melts re-equilibrate with CO2-rich fluids exsolved at higher pressure, may result in water loss at pressures less than CO2 saturation. This may more accurately describe OIB degassing processes than open or closed system degassing. After correcting for degassing, primary Australs melts likely have H2O/Ce of ~ 300–600. If applicable to OIB sources in general, this limits the total water budget of the mantle, including the mantle transition zone, to < 2.4 ocean masses.

MAC: Maximal Cliques for 3D Registration.

This paper presents a 3D registration method with maximal cliques (MAC) for 3D point cloud registration (PCR). The key insight is to loosen the previous maximum clique constraint and mine more local consensus information in a graph for accurate pose hypotheses generation: 1) A compatibility graph is constructed to render the affinity relationship between initial correspondences. 2) We search for maximal cliques in the graph, each representing a consensus set. 3) Transformation hypotheses are computed for the selected cliques by the SVD algorithm and the best hypothesis is used to perform registration. In addition, we present a variant of MAC if given overlap prior, called MAC-OP. Overlap prior further enhances MAC from many technical aspects, such as graph construction with re-weighted nodes, hypotheses generation from cliques with additional constraints, and hypothesis evaluation with overlap-aware weights. Extensive experiments demonstrate that both MAC and MAC-OP effectively increase registration recall, outperform various state-of-the-art methods, and boost the performance of deep-learned methods. For instance, MAC combined with GeoTransformer achieves a state-of-the-art registration recall of 95.7% / 78.9% on 3DMatch / 3DLoMatch. We perform synthetic experiments on 3DMatch-LIR / 3DLoMatch-LIR, a dataset with extremely low inlier ratios for 3D registration in ultra-challenging cases. Code will be available at: https://github.com/zhangxy0517/3D-Registration-with-Maximal-Cliques.

Frequency Shift in Binary Microlensing

Abstract Gravitational microlensing with binary lensing is one of the channels for detecting exoplanets. Due to the degeneracy of the lens parameters for the binary microlensing, additional features such as parallax and finite-size effects need to identify the lens parameters. The frequency-shift effect as the relativistic analogy of the gravity assist for the photons, is an extra observation that provides additional constraint between the lens parameters . In this work, we extend the application of the frequency shift effect to binary microlensing and derive the frequency shift during the lensing and caustic crossing. The frequency shift for the binary lens is of the order of Δν/ν ∼ 10−12. We also investigate the feasibility of detecting this effect by employing Cross-Correlation methods .

Systematic Elastostatic Stiffness Model of Over-Constrained Parallel Manipulators Without Additional Constraint Equations

The establishment of an elastostatic stiffness model for over constrained parallel manipulators (PMs), particularly those with over constrained subclosed loops, poses a challenge while ensuring numerical stability. This study addresses this issue by proposing a systematic elastostatic stiffness model based on matrix structural analysis (MSA) and independent displacement coordinates (IDCs) extraction techniques. To begin, the closed-loop PM is transformed into an open-loop PM by eliminating constraints. A subassembly element is then introduced, which considers the flexibility of both rods and joints. This approach helps circumvent the numerical instability typically encountered with traditional constraint equations. The IDCs and analytical constraint equations of nodes constrained by various joints are summarized in the appendix, utilizing multipoint constraint theory and singularity analysis, all unified within a single coordinate frame. Subsequently, the open-loop mechanism is efficiently closed by referencing the constraint equations presented in the appendix, alongside its elastostatic model. The proposed method proves to be both modeling and computationally efficient due to the comprehensive summary of the constraint equations in the Appendix, eliminating the need for additional equations. An example utilizing an over constrained subclosed loops demonstrate the application of the proposed method. In conclusion, the model proposed in this study enriches the theory of elastostatic stiffness modeling of PMs and provides an effective solution for stiffness modeling challenges they present.

Evaluation of a building envelope Heat Transfer Coefficient in use: Bayesian approach to improve the inclusion of solar gains

Accurately measuring the actual building performance would be a significant step toward performance contracting. Methods under occupancy have operational advantages but come with additional constraints due to the variability of weather and occupancy conditions, which cannot be overlooked. Especially in recent, well-insulated buildings, since solar gains are a larger contribution to the global heat balance. This work aims to enhance the accuracy of the Heat Transfer Coefficient estimation in an occupied building by taking better account of solar gains in multilinear regression models. The approach adopted here considers solar gains from both the incident solar heat flux through glazing and from the opaque elements of a building by calculating an equivalent outdoor temperature. This model feeds a multilinear regression model whose parameters are estimated by Bayesian inference. Our model for estimating HTC is validated against a set of virtual data from a simulated detached house under different weather conditions.

A Tandem Photovoltaic-Electrochemical Photothermal Process for CO2 Conversion to Butene

Decarbonizing the chemicals industry requires net carbon negative technologies capable of displacing current greenhouse gas emitting processes. In particular, the generation of valuable monomers, such as butene, are currently produced by significant greenhouse gas emitting processes, including ethylene oligomerization and crude oil refining, which require high temperatures and pressures. Integrated, tandem solar fuels devices for the conversion of CO2 to ethylene, using solar-driven CO2 reduction (CO2R), combined with a photothermal reactor for ethylene oligomerization, offers a potential path to producing clean butene from CO2, using solar energy as the only driving force. A key challenge for the successful operation of a tandem photo-driven electrochemical-photothermal system is the co-design of the two processes, which are typically optimized under different operating conditions. Specifically, photo-driven electrochemical CO2R (pCO2R) reactors operate under ambient temperature and pressure conditions while thermally-driven ethylene oligomerization reactors prefer elevated temperatures (80-140 °C) and pressures (30-150 atm) to promote high activity as well as pure ethylene in the inlet stream. Operation of the tandem process under solar-driven conditions imposes additional constraints on the maximum power output of the pCO2R reactor and the maximum temperature which can be obtained in the photothermal reactor. Herein, we present an unassisted solar-driven tandem system coupling pCO2R with photothermal ethylene oligomerization for the conversion of CO2 to butene in a single pass. This work presents a critical advancement in the development of solar fuels technologies for larger hydrocarbon generation and identifies the co-design principles which must be considered for implementation of tandem, solar-driven electrochemical-thermal processes.

On accounting for the effects of crust and uppermost mantle structure in global scale full-waveform inversion

Summary Fundamental mode surface wave data have often been used to construct global shear velocity models of the upper mantle under the so-called “path average approximation”, an efficient approach from the computational point of view. With the advent of full-waveform inversion and numerical wavefield computations, such as afforded by the spectral element method, accounting for the effects of the crust accurately becomes challenging. Here, we assess the merits of accounting for crustal and uppermost mantle effects on surface and body waveforms using fundamental mode dispersion data and a smooth representation of the shallow structure. For this we take as reference a model obtained by full waveform inversion and wavefield computations using the spectral element method, model SEMUCB-WM1 (French and Romanowicz, 2014) and compare the waveform fits of synthetics to different parts of three component observed teleseismic records, in the period band 32-300 s for body waves and 40-300 s for surface waves and their overtones for three different models. The latter are: a dispersion-only based model (model Disp_20s_iter5), and two models modified from SEMUCB-WM1 by successively replacing the top 200 km (model Merged _200km) and top 80 km (model Merged _80km), respectively, by a model constrained solely by fundamental mode surface wave dispersion data between periods of 20 and 150 s. The crustal part of these 3 models (resp. SEMUCB-WM1) is constrained from dispersion data in the period range 20-60 s (resp. 25-60 s), using the concept of homogenization (e.g., Backus 1962, Capdeville &amp; Marigo 2007) which is tailored to simplify complex geological features, enhancing the computational efficiency of our seismic modeling. We evaluate the fits to observed waveforms provided by these 3 models compared to those of SEMUCB-WM1 by computing three component synthetics using the spectral element method for 5 globally distributed events recorded at 200+ stations, using several measures of misfit. While fits to waveforms for model 3 are similar to those for SEMUCB-WM1, the other two models provide increasingly poorer fits as the distance travelled by the corresponding seismic wave increases and/or as it samples deeper in the mantle. In particular, models 1 and 2 are biased towards fast shear velocities, on average. Our results suggest that, given a comparable frequency band, models constructed using fundamental mode surface wave data alone and the path average approximation, fail to provide acceptable fits to the corresponding waveforms. However, the shallow part of such a 3D radially anisotropic model can be a good starting model for further full waveform inversion using numerical wavefield computations. Moreover, the shallow part of such a model, including its smooth crustal model, and down to a maximum depth that depends on the frequency band considered, can be fixed in FWI iterations for deeper structure. This can save significant computational time when higher resolution is sought in the deeper mantle. In the future, additional constraints for the construction of the homogenized model of the crust can be implemented from independent short period studies, either globally or regionally.

Macroscopic state‐based reactive voltage control of virtual synchronous generator in AC microgrid

AbstractIn building a smarter and more flexible low‐carbon smart grid system, alternating current (AC) microgrids using virtual synchronous generator (VSG) technology are viewed as a key link in integrating distributed renewable energy access into the main grid. Given that renewable energy sources (such as solar, hydroenergy, and wind) do not have sufficient capacity for reactive power when not available, AC microgrids face challenges in maintaining stable operation. In order to overcome this difficulty, it is hoped that digging deeper and applying more system information can significantly improve the overall performance of the microgrid. This paper proposes a novel method based on macroscopic state dynamic modeling. This method expands the understanding of the inherent rational control mechanism within the microgrid, enabling the overall control objective of the microgrid to be expressed in a more abstract and direct manner. Additionally, by implementing additional convergence constraint conditions on the macroscopic state dynamics, such as based on some optimality criteria, a set of macroscopic state controllers can be obtained to meet specific performance indicators. Theoretical analysis combined with simulation validation demonstrate the effectiveness of this macroscopic state based control strategy. It proves that when meeting the predefined design requirements, the designed controller can enhance the transient response of microgrids in practical applications, thus supporting higher rate of renewable energy access and promoting the development of the smart grid.

The Effects of the Introduction of Volume-Based Liquidity Constraints in Portfolio Optimization with Alternative Investments

Recently, liquidity issues in financial markets and portfolio asset management have attracted much attention among investors and scholars, fuelling a stream of research devoted to exploring the role of liquidity in investment decisions. In this paper, we aim to investigate the effects of introducing liquidity in portfolio optimization problems. For this purpose, first we consider three volume-based liquidity measures proposed in the literature and we build a new one particularly suited to portfolio optimization. Secondly, we formulate an extended version of the Markowitz portfolio selection problem, named mean–variance–liquidity, wherein the goal is to minimize the portfolio variance subject to the usual constraint on the expected portfolio return and an additional constraint on the portfolio liquidity. Thirdly, we consider a sensitivity analysis, with the aim to assess the trade-offs between liquidity and return, on the one hand, and between liquidity and risk, on the other hand. In the second part of the paper, the portfolio optimization framework is applied to a dataset of US ETFs comprising both standard and alternative, often illiquid, investments. The analysis is carried out with all the liquidity measures considered, allowing us to shed light on the relationships among risk, return and liquidity. Finally, we study the effects of the introduction of a Bitcoin ETF, as an asset with an extremely high expected return and risk.

Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables.

Chance constrained optimization problems allow to model problems where constraints involving stochastic components should only be violated with a small probability. Evolutionary algorithms have been applied to this scenario and shown to achieve high quality results. With this paper, we contribute to the theoretical understanding of evolutionary algorithms for chance constrained optimization. We study the scenario of stochastic components that are independent and normally distributed. Considering the simple single-objective (1+1) EA, we show that imposing an additional uniform constraint already leads to local optima for very restricted scenarios and an exponential optimization time. We therefore introduce a multi-objective formulation of the problem which trades off the expected cost and its variance. We show that multi-objective evolutionary algorithms are highly effective when using this formulation and obtain a set of solutions that contains an optimal solution for any possible confidence level imposed on the constraint. Furthermore, we prove that this approach can also be used to compute a set of optimal solutions for the chance constrained minimum spanning tree problem. In order to deal with potentially exponentially many trade-offs in the multi-objective formulation, we propose and analyze improved convex multi-objective approaches. Experimental investigations on instances of the NP-hard stochastic minimum weight dominating set problem confirm the benefit of the multi-objective and the improved convex multi-objective approach in practice.

Effects of Resistivity on the Reconstructed Plasma Fields Revealed by a Three‐Dimensional Empirical Reconstruction Model

AbstractWe extend the previous three‐dimensional (3D) empirical reconstruction (ER) model for a set of ideal magnetohydrodynamics (MHD) constraints into a resistive MHD 3D ER model that includes additional resistive MHD constraints and additional measurements from NASA's Magnetospheric Multiscale (MMS) mission. The same form of a stochastic optimization algorithm is used as in the previous ideal MHD 3D ER model to directly minimize the loss function that includes a few more highly nonlinear terms characterizing the model‐measurement differences and the model departures from physical constraints. The resistive MHD 3D ER model is applied to three regions of MMS measurements that correspond to direct sampling of an electron diffusion region (EDR), a region adjacent to the EDR, and one far away from the EDR. The reconstructed plasma and electromagnetic fields are of high quality in all three regions as measured by model‐measurement difference indices and physics‐based quality indicators. The reconstructed fields in the EDR provide us with a good view of the spatial configuration of the reconnection site. We specifically examine the effect of resistivity on energy exchange in the vicinity of the EDR. It was discovered that in the EDR, the energy exchange shows an exclusive and systematic one‐channel process between the plasma thermal energy and electromagnetic energy with the conversion rate highly correlated with the strength of the turbulent electromagnetic fields. In the other two regions away from the EDR, the energy exchange between the electromagnetic energy and the plasma thermal and kinetic energies shows rapidly‐varying and random characteristics.

Learning Oriented Object Detection via Naive Geometric Computing.

Detecting oriented objects along with estimating their rotation information is one crucial step for image analysis, especially for remote sensing images. Despite that many methods proposed recently have achieved remarkable performance, most of them directly learn to predict object directions under the supervision of only one (e.g., the rotation angle) or a few (e.g., several coordinates) groundtruth (GT) values individually. Oriented object detection would be more accurate and robust if extra constraints, with respect to proposal and rotation information regression, are adopted for joint supervision during training. To this end, we propose a mechanism that simultaneously learns the regression of horizontal proposals, oriented proposals, and rotation angles of objects in a consistent manner, via naive geometric computing, as one additional steady constraint. An oriented center prior guided label assignment strategy is proposed for further enhancing the quality of proposals, yielding better performance. Extensive experiments on six datasets demonstrate the model equipped with our idea significantly outperforms the baseline by a large margin and several new state-of-the-art results are achieved without any extra computational burden during inference. Our proposed idea is simple and intuitive that can be readily implemented. Source codes are publicly available at: https://github.com/wangWilson/CGCDet.git.

Urban sources of methane characterised by long-term eddy covariance observations in central Europe

Multiyear eddy covariance measurements of methane and carbon dioxide were performed in Innsbruck, Austria, a mid-sized European city in the Alps. Annual mean methane and carbon dioxide fluxes were on the order of 21.6 nmol/m2/s and 11.7 μmol/m2/s respectively. Methane fluxes increased significantly with decreasing temperature and showed a negligible weekend to weekday effect. Accompanying non-methane volatile organic compound (NMVOC) measurements provided additional constraints on urban methane sources during an intensive 6-week campaign in 2021. Positive matrix factorization analysis identified a unique and dominant methane source. Contributions from biomass burning and vehicle emissions were found to be small. No evidence was found for a large unknown biogenic source. A second intensive campaign in 2023 revealed that the measured ethane to methane enhancement in turbulent plumes showed a typical ratio of 5.2 ± 0.6%, close to the unburned ratio of natural gas (5%) in Austria. The analysis suggests that urban methane fluxes are likely to be dominated by end-user releases rather than natural gas leaks in the pipeline system. Long-term eddy flux observations also allowed identifying the existence of urban super-emitters, which brought area weighted methane fluxes up to 39 nmol/m2/s in 2022. On average, the 100 y global warming potential of methane emitted by natural gas combustion activities is 6% relative to the total urban CO2 flux, and ∼12% relative to CO2 emissions from the residential, commercial and public sectors in Innsbruck.

FARN: Fetal Anatomy Reasoning Network for Detection With Global Context Semantic and Local Topology Relationship.

Accurate recognition of fetal anatomical structure is a pivotal task in ultrasound (US) image analysis. Sonographers naturally apply anatomical knowledge and clinical expertise to recognizing key anatomical structures in complex US images. However, mainstream object detection approaches usually treat each structure recognition separately, overlooking anatomical correlations between different structures in fetal US planes. In this work, we propose a Fetal Anatomy Reasoning Network (FARN) that incorporates two kinds of relationship forms: a global context semantic block summarized with visual similarity and a local topology relationship block depicting structural pair constraints. Specifically, by designing the Adaptive Relation Graph Reasoning (ARGR) module, anatomical structures are treated as nodes, with two kinds of relationships between nodes modeled as edges. The flexibility of the model is enhanced by constructing the adaptive relationship graph in a data-driven way, enabling adaptation to various data samples without the need for predefined additional constraints. The feature representation is further refined by aggregating the outputs of the ARGR module. Comprehensive experimental results demonstrate that FARN achieves promising performance in detecting 37 anatomical structures across key US planes in tertiary obstetric screening. FARN effectively utilizes key relationships to improve detection performance, demonstrates robustness to small-scale, similar, and indistinct structures, and avoids some detection errors that deviate from anatomical norms. Overall, our study serves as a resource for developing efficient and concise approaches to model inter-anatomy relationships.

Robust Hamiltonian Engineering for Interacting Qudit Systems

Dynamical decoupling and Hamiltonian engineering are well-established techniques that have been used to control qubit systems. However, designing the corresponding methods for qudit systems has been challenging due to the lack of a Bloch sphere representation, more complex interactions, and additional control constraints. By identifying several general structures associated with such problems, we develop a formalism for the robust dynamical decoupling and Hamiltonian engineering of strongly interacting qudit systems. Our formalism significantly simplifies qudit pulse-sequence design while naturally incorporating robustness conditions necessary for experimental practicality. We experimentally demonstrate these techniques in a strongly interacting, disordered ensemble of spin-1 nitrogen-vacancy centers, achieving more than an order-of-magnitude improvement in coherence time over existing pulse sequences. We further describe how our techniques enable the engineering of exotic many-body phenomena such as quantum many-body scars, and open up new opportunities for quantum metrology with enhanced sensitivities. These results enable wide-reaching new applications for dynamical decoupling and Hamiltonian engineering in many-body physics and quantum metrology. Published by the American Physical Society 2024

Investigation of a new method for direct chemistry integration in Conditional Source-term Estimation

This paper examines a new Conditional Source-term Estimation (CSE) implementation in which an integral inversion is performed for species mass fractions without any chemistry tabulation. To tackle the numerical stiffness due to chemistry, a constant-pressure chemical reactor is considered in conditional space. The objective of this work is to investigate this new method in CSE by comparing its numerical performance (accuracy of predictions and computational efficiency) with previous CSE implementations using tabulated chemistry. For the first time, the constraints for mass and mixture fraction conservation in conditional space are included in the inversion process. The investigation is performed in a Reynolds Averaged Navier-Stokes (RANS) framework using a well-documented turbulent flame with detailed measurements in conditional and physical spaces for many reacting scalars. A chemical mechanism including 16 species and 35 reactions is incorporated. CSE predictions of conditional and Favre averaged temperature and different species mass fractions are analysed. CSE with full chemistry performs as well as CSE with tabulated chemistry, if not slightly better. Numerical errors are discussed. The new CSE implementation is made possible by the proposed numerical treatment of stiffness due to chemistry and first-order Tikhonov regularisation technique with additional physical constraints. The new approach is found to be more CPU intensive than CSE with a pre-tabulation technique, but as a preliminary analysis, the computational cost of CSE with direct integration of a chemical mechanism appears to be much smaller than the run time associated with CMC for a similar set-up. Further investigation is required to refine this initial conclusion. These results open new opportunities towards CSE simulations that involve fuel blends and more complex operating conditions for example with spray and multi-stream inlets with non-uniform compositions for which accurate chemistry tabulations are difficult to generate.

Search Me Knot, Render Me Knot: Embedding Search and Differentiable Rendering of Knots in 3D

AbstractWe introduce the problem of knot‐based inverse perceptual art. Given multiple target images and their corresponding viewing configurations, the objective is to find a 3D knot‐based tubular structure whose appearance resembles the target images when viewed from the specified viewing configurations. To solve this problem, we first design a differentiable rendering algorithm for rendering tubular knots embedded in 3D for arbitrary perspective camera configurations. Utilizing this differentiable rendering algorithm, we search over the space of knot configurations to find the ideal knot embedding. We represent the knot embeddings via homeomorphisms of the desired template knot, where the weights of an invertible neural network parametrize the homeomorphisms. Our approach is fully differentiable, making it possible to find the ideal 3D tubular structure for the desired perceptual art using gradient‐based optimization. We propose several loss functions that impose additional physical constraints, enforcing that the tube is free of self‐intersection, lies within a predefined region in space, satisfies the physical bending limits of the tube material, and the material cost is within a specified budget. We demonstrate through results that our knot representation is highly expressive and gives impressive results even for challenging target images in both single‐view and multiple‐view constraints. Through extensive ablation study, we show that each proposed loss function effectively ensures physical realizability. We construct a real‐world 3D‐printed object to demonstrate the practical utility of our approach.

Branch spiral beam harvester for uni-directional ultra-low frequency excitations

In recent years, there has been a growing interest in piezoelectric energy harvesting systems, particularly for their potential to recharge or replace batteries in energy-efficient electronic devices and wireless sensor networks. Nonetheless, the conventional linear piezoelectric energy harvesters (PEH) face limitations in ultra-low frequency vibrations (1–10 Hz) due to their narrow operating bandwidth and higher resonance frequencies. To address this, researchers explored compact shaped geometries, with spiral PEH being one such design to lower resonance frequencies by reducing structural stiffness. While trying to achieve this lower resonance frequency, spiral designs overlooked that they were spreading the stress across the structure. This was a significant drawback because it reduced the structure's ability to stress the piezoelectric transducer. The issue remains unaddressed, limiting the power generation of spiral beam harvesters. Furthermore, spiral structures also fail to broaden the operating bandwidth, posing additional constraints on their effectiveness. This study introduces a novel solution – the “branch spiral beam harvester,” combining the benefits of both spiral and branch beam designs. The integration of the branch beam concept into the spiral structure aimed to broaden the effective frequency range and establish a concentrated stress area for the placement of the piezoelectric transducer. Finite Element Analysis (FEA) was employed to assess operating bandwidth and stress distribution, while experimental studies evaluated voltage and power generation. Once the workability was confirmed, a statistical optimisation method was introduced to tailor the harvester for specific frequencies in the ultra-low frequency range. Results indicated that the branch spiral beam harvester exhibits a wider operating bandwidth with six natural frequencies in the ultra-low frequency range. It harnessed significantly higher output voltages and power compared to conventional linear PEH. This innovation presents a promising advancement in piezoelectric energy harvesting, offering improved performance without the need for proof masses or additional accessories.

Global solutions to a chemotaxis system with singular density-suppressed motility and superlinear consumption

In this paper, we consider a migration-consumption taxis system involving singular density-suppressed motility and superlinear consumption(⋆){ut=Δ(uϕ(v)),vt=Δv−uvm in a bounded smooth domain Ω⊂Rn(n≥2) with no-flux boundary conditions. Here, m≥1 is a constant. The motility function ϕ generalizes the singular prototype given by ϕ(ξ)=ξ−α(ξ>0) with α>0. In Tao and Winkler (2023) [32], Tao and Winkler studied the system (⋆) with m=1 and obtained respectively the global existence of very weak-strong and weak-strong solutions. We study the system (⋆) with m≥1 and prove that a so-called very weak-strong solution can be established for arbitrarily large initial data in W1,∞(Ω). Moreover, under the additional constraints of(⋆⋆)(6−n)α+2m(n+2)−(3n+2)>0, and with an additional assumption on ϕ′, the model (⋆) possesses a global weak-strong solution. In particular, when 2≤n≤5 and m=1, (⋆⋆) is equivalent to α>n−26−n, which consistent with the result of Tao and Winkler. However, our result contains the case n≥6. When n=6, (⋆⋆) is equivalent to m>54 without any requirement for α>0. When n>6, (⋆⋆) is equivalent to m>3n+22n+4 and 0<α<2m(n+2)−(3n+2)n−6.

M-Distance-regular graphs and their relation to multivariate P-polynomial association schemes

An association scheme is P-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate P-polynomial association schemes of type (α,β) were introduced by Bernard et al., and multivariate P-polynomial association schemes were later defined by Bannai et al. In this paper, the notion of m-distance-regular graph is defined and shown to give a graph interpretation of the multivariate P-polynomial association schemes. Various examples are provided. Refined structures and additional constraints for multivariate P-polynomial association schemes and m-distance-regular graphs are also considered. In particular, bivariate P-polynomial schemes of type (α,β) are discussed, and their connection to 2-distance-regular graphs is established.