Weak Form Research Articles

A stable localized weak strong form radial basis function method for modelling variably saturated groundwater flow induced by pumping and injection

The unsaturated zone profoundly affects groundwater (GW) flow induced by pumping and injection due to the capillary forces. However, current radial basis function (RBF) numerical models for GW pumping and injection mostly ignore the unsaturated zone. To bridge this gap, we developed a new three-dimensional weak strong form RBF model in this study, called CCHE3D-GW-RBF. Flow processes were modelled by the mixed-form Richards equation which was iteratively solved by the modified Picard iteration. Soil-water characteristic curves were represented by the widely applicable formulas, the van Genuchten (1980) model. Differential operators were approximated by the localized Gaussian RBF, and the weighted singular value decomposition method was used to construct stable bases. The injection/pumping wells and the flux boundaries were handled by the weak formulation using Meshless Local Petrov Galerkin method, and the strong-form equation using the collocation RBF method was enforced on the other points. Good agreement was found between the simulation results from our numerical model and the well-accepted solutions in all three verification cases, demonstrating the accuracy and applicability of this model. In addition, a smaller RBF shape parameter was found to produce a more accurate modelling resulting, indicating the necessity of implementing stable bases for RBF models.

Symmetric and Dual PRFs from Standard Assumptions: A Generic Validation of a Prevailing Assumption

A two-input function is a dual PRF if it is a PRF when keyed by either of its inputs. Dual PRFs are assumed in the design and analysis of numerous primitives and protocols including HMAC, AMAC, TLS 1.3 and MLS. But, not only do we not know whether particular functions on which the assumption is made really are dual PRFs; we do not know if dual PRFs even exist. What if the goal is impossible? This paper addresses this with a foundational treatment of dual PRFs, giving constructions based on standard assumptions. This provides what we call a generic validation of the dual PRF assumption. Our approach is to introduce and construct symmetric PRFs, which imply dual PRFs and may be of independent interest. We give a general construction of a symmetric PRF based on a function having a weak form of collision resistance coupled with a leakage hardcore function, a strengthening of the usual notion of hardcore functions we introduce. We instantiate this general construction in two ways to obtain two specific symmetric and dual PRFs, the first assuming any collision-resistant hash function and the second assuming any one-way permutation. A construction based on any one-way function evades us and is left as an intriguing open problem.

Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations

Abstract In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.

Numerical study of the multi-dimensional Galilei invariant fractional advection–diffusion equation using direct mesh-less local Petrov–Galerkin method

This article presents a local mesh-less procedure for simulating the Galilei invariant fractional advection–diffusion (GI-FAD) equations in one, two, and three-dimensional spaces. The proposed method combines a second-order Crank–Nicolson scheme for time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula. This time discretization scheme ensures unconditional stability and convergence with an order of O(τ2). In the spatial domain, a mesh-less weak form is employed based on the direct mesh-less local Petrov–Galerkin (DMLPG) method. The DMLPG method employs the generalized moving least-square (GMLS) approximation in conjunction with the local weak form of the equation. By utilizing simple polynomials as shape functions in the GMLS approximation, the necessity for complex shape function construction in the MLS approximation is eliminated. To validate and demonstrate the effectiveness of the proposed algorithm, a variety of problems in one, two, and three dimensions are investigated on both regular and irregular computational domains. The numerical results obtained from these investigations confirm the accuracy and reliability of the developed approach in simulating GI-FAD equations.

Initial Value and Initial Current Discrepancy in Simulating Lithium-Ion Battery Packs - Resolution from Dr. Ralph White’s Analytical Approach

Battery models are useful tools that provide insight into electrochemical processes within an electrochemical device those can typically not be measured, either due to a lack of experimental methods or due to the transient nature of the phenomena in question. Irrespective of the complexity of the models involved, a general view is that a good model, once established for a single cell in terms of complexity of the physics, choice of simulation algorithm, value for the parameters, and mechanisms for fade, can be easily extended for pack simulations.1–3 This type of extension is critical to the virtual engineering efforts currently underway at OEMs which rely on accurate and representative battery models. In this work, pitfalls during scaling up approximate and detailed physics-based models developed for single-cell to pack-level simulations are highlighted using representative examples. Interesting mathematical nuances were found depending on the numerical simulation approach used. A discrepancy at initial times (t = 0) during the pack-level simulations was highlighted and resolved using the Laplace transform approach to get an analytical solution for the simplest model. A few thoughts on numerical challenges and the index of differential-algebraic equations (DAEs)4,5 while using the strong form and the weak form formulation of DAEs are also provided in the paper.6

Critical Raw Materials, Economic Statecraft and Europe’s Dependence on China

ABSTRACT The People’s Republic of China (PRC) holds a dominant position along wide swaths of critical mineral and technology value chains that are necessary for achieving Europe’s decarbonisation policy goals. Many fear that the PRC will pursue increasingly coercive measures to leverage its supply chain advantage in pursuit of broader geopolitical objectives. Examining three mineral groups in which Beijing has enhanced supply chain restrictions in 2023, namely rare earth elements, gallium and graphite, highlights that the PRC’s priority has long been the pursuit of economic and industrial policy goals rather than a broader diplomatic agenda, though this could change. Arguably, leveraging raw materials is ultimately a weak form of economic statecraft, eliciting effective diversification strategies that in turn weaken a supplier state’s sources of leverage. Europe’s response, while slow to emerge, has been a strategy to reinvest in its own mineral extraction, refining and recycling capacities, on the one hand, and diversify global supply chains by expanding its strategic partnerships, on the other.

Deep finite volume method for partial differential equations

In this paper, we introduce the Deep Finite Volume Method (DFVM), an innovative deep learning framework tailored for solving high-order (order ≥2) partial differential equations (PDEs). Our approach centers on a novel loss function crafted from local conservation laws derived from the original PDE, distinguishing DFVM from traditional deep learning methods. By formulating DFVM in the weak form of the PDE rather than the strong form, we enhance accuracy, particularly beneficial for PDEs with less smooth solutions compared to strong-form-based methods like Physics-Informed Neural Networks (PINNs). A key technique of DFVM lies in its transformation of all second-order or higher derivatives of neural networks into first-order derivatives which can be computed directly using Automatic Differentiation (AD). This adaptation significantly reduces computational overhead, particularly advantageous for solving high-dimensional PDEs. Numerical experiments demonstrate that DFVM achieves equal or superior solution accuracy compared to existing deep learning methods such as PINN, Deep Ritz Method (DRM), and Weak Adversarial Networks (WAN), while drastically reducing computational costs. Notably, for PDEs with nonsmooth solutions, DFVM yields approximate solutions with relative errors up to two orders of magnitude lower than those obtained by PINN. The implementation of DFVM is available on GitHub at https://github.com/Sysuzqs/DFVM.

Моделирование течения воздуха в упруго-деформируемой пористой среде, аппроксимирующей легкие человека: структура модели, ее основные уравнения и разрешающие соотношения

This paper is devoted to refining the mathematical model of the human respiratory system designed to predict the occurrence of respiratory pathologies caused by the negative effects of atmospheric air pollution. It is difficult to identify individual air channels in the human lungs, consisting of small respiratory tracts, called alveoli, through which oxygen from the air passes into blood. Therefore, the human lungs are modeled as a two-phase elastically deformable saturated porous medium, in which one phase is the deformable skeleton of the medium (lung tissue) described by the model of deformable solid body, and the other phase is gas filling the porous space. We construct the model of air flow in the human lungs, which consists of the sub-model of elastic deformation of the porous lung medium, experiencing large displacement gradients, and the sub-model of air filtration through the deformable porous medium. The model takes into account the relationships between sub-models of the respiratory system, i.e. we consider the coupled problems of deformation and filtration in differential form. Since at present the analytical solution of the stated essentially nonlinear problem is unfeasible, the problem is formulated in a generalized weak form to allow subsequent application of numerical methods using the Galerkin method. For the numerical solution of the nonlinear problem of deformation of the two-phase lung medium, it is suggested to use the finite element method, for the filtration problem - the finite volume method. The resolving relations needed to solve the formulated sub-problems are derived. To solve the nonlinear problem, we developed algorithms and a set of programs. The description of these computational tools, as well as analysis of the results obtained will be presented in our the next paper.

Role of Artificial Intelligence and Big Data in Sustainable Entrepreneurship

There is a pressing necessity to shift our economy, society, and culture to systems and actions that promote ecological sustainability. This radical transformation necessitates an equally radical transformation of resource utilization and decision-making strategies. Sustainable entrepreneurship (SE) is frequently touted as the solution to the triple-bottom-line challenges that businesses encounter; however, there are tangible constraints on its potential. SE is currently in the first phase of implementing technological frontier tools that provide empirical guidance throughout the entrepreneurial decision-making process. The potential for artificial intelligence (AI) to inform decision-making is advanced by Big Data (BD), which also establishes pathways to attain desired outcomes. The interactions between AI, BD, and SE have been generally under-studied thus far. The absence of work that consolidates and synthesizes this literature is the primary focus of this conceptual paper. We propose that AI and BD are capable of rapidly contributing to the continued sustainable development of the weak form, but they also hold significant potential for attaining the strong sustainability ideal. We present two proposals for the integration of AI and BD to inform and facilitate SE. Finally, we outline potential areas for future research. The core of human cosmology and ethics has always been the definition of his uniqueness. He ceased to be the species situated at the center of the universe, accompanied by the sun and stars, with the arrival of Copernicus and Galileo. He ceased to be the species that was created and specially endowed by God with soul and reason with the arrival of Darwin. With Freud, he ceased to be the species whose behavior could potentially be regulated by the rational mind. He has ceased to be the species that is uniquely capable of complex, intelligent manipulation of his environment as we begin to produce mechanisms that think and learn.

Existence of Weak Solution for Black-Scholes Partial Differential Equation and Application of Energy Estimate Theorem in Sobolev Space

Purpose:This paper aims to solve the BS second-order parabolic equation in Sobolev spaces to obtain weak solutions for financial applications, extending previous work in this field.Methodology:This paper constructs a set of functions that transforms the Black-Scholes partial differential equation into weak formulations. This study focuses on the Mathematics of finance, particularly the evolution of option pricing. An option's underlying asset involves agreements to buy or sell at a future strike price. The Black-Scholes (BS) equation, widely used in financial applications, models this.Findings:The analytical solutions, existence, uniqueness and other estimates were also obtained in weak form using boundary conditions to establish the effects of their financial implications in Sobolev spaces.Implication:The problem's regularity conditions were considered, and the coefficients and boundary of the domain are all smooth functions. To this end, this paper illustrates the definitions and assumptions that led to valuable assertions.

CHEER and markets’ integration: evidence from selected East Asian economies

PurposeThis paper aims to investigate markets’ integration using the capital enhanced equilibrium exchange rate (CHEER) model for seven, highly competitive, East Asian countries.Design/methodology/approachThe sample consists of monthly observations, whereas unit root and cointegration techniques with structural shifts have been used.FindingsThe evidence shows that the weak form of the CHEER approach holds for Malaysia and Thailand. For China, Japan, Korea, Singapore and Taipei, only the uncovered interest parity condition is validated, implying capital markets integration. In contrast, for these five countries, the results indicate absence of goods’ markets integration. This outcome can be attributed to the impact of quite high non-tariff barriers and the Balassa–Samuelson effect.Originality/valueTo the best of the author’s knowledge, this is the first study that investigate markets’ integration in several East Asian economies, using the CHEER approach and more accurate price indices.

Harmonic weak Maass forms and periods II

Harmonic weak Maass forms and periods II

Physics‐informed neural operator solver and super‐resolution for solid mechanics

AbstractPhysics‐Informed Neural Networks (PINNs) have solved numerous mechanics problems by training to minimize the loss functions of governing partial differential equations (PDEs). Despite successful development of PINNs in various systems, computational efficiency and fidelity prediction have remained profound challenges. To fill such gaps, this study proposed a Physics‐Informed Neural Operator Solver (PINOS) to achieve accurate and fast simulations without any required data set. The training of PINOS adopts a weak form based on the principle of least work for static simulations and a strong form for dynamic systems in solid mechanics. Results from numerical examples indicated that PINOS is capable of approximating solutions notably faster than the benchmarks of PINNs in both static an dynamic systems. The comparisons also showed that PINOS reached a convergence speed of over 20 times faster than finite element software in two‐dimensional and three‐dimensional static problems. Furthermore, this study examined the zero‐shot super‐resolution capability by developing Super‐Resolution PINOS (SR‐PINOS) that was trained on a coarse mesh and validated on fine mesh. The numerical results demonstrate the great performance of the model to obtain accurate solutions with a speed up, suggesting effectiveness in increasing sampling points and scaling a simulation. This study also discusses the differentiation methods of PINOS and SR‐PINOS and suggests potential implementations related to forward applications for promising machine learning methods for structural designs and optimization.

Mutual visibility of luminous robots despite angular inaccuracy

We initiate the study of the Mutual Visibility problem using N opaque luminous point robots that have inaccurate movements. Each robot operates in Look-Compute-Move cycles and has a persistent light attached to it to have a weak form of communication between robots using a constant number of colors. The inaccuracy for a robot r is an angular deviation from its target point T to a point T′ such that the angle ∠TrT′<90∘. The problem becomes unsolvable if this angle is ≥90∘. From any initial configuration of the robots on the Euclidean plane, the problem aims to arrange the robots in a configuration such that any two robots are visible to each other. We assume that the robots agree on one coordinate axis. We present two collision-free algorithms, a 2 color algorithm (which is optimal in the number of colors used) for semi-synchronous setting and a 3 color algorithm for asynchronous setting, both of which run in O(N) epochs. We also study the problem in the presence of mobile faulty robots. A robot can exhibit both mobility failure and angular inaccuracies in its movement. We present a fault-tolerant algorithm that aims to bring the robots in a configuration where no three non-faulty robots are collinear, and no faulty robot lies between two non-faulty robots. This algorithm uses 10 colors and takes O(N) epochs under asynchronous settings.

Modeling the mechanosensitive collective migration of cells on the surface and the interior of morphing soft tissues.

Cellular contractility, migration, and extracellular matrix (ECM) mechanics are critical for a wide range of biological processes including embryonic development, wound healing, tissue morphogenesis, and regeneration. Even though the distinct response of cells near the tissue periphery has been previously observed in cell-laden microtissues, including faster kinetics and more prominent cell-ECM interactions, there are currently no models that can fully combine coupled surface and bulk mechanics and kinetics to recapitulate the morphogenic response of these constructs. Mailand et al. (Biophys J 117(5):975-986, 2019) had shown the importance of active elastocapillarity in cell-laden microtissues, but modeling the distinct mechanosensitive migration of cells on the periphery and the interior of highly deforming tissues has not been possible thus far, especially in the presence of active elastocapillary effects. This paper presents a framework for understanding the interplay between cellular contractility, migration, and ECM mechanics in dynamically morphing soft tissues accounting for distinct cellular responses in the bulk and the surface of tissues. The major novelty of this approach is that it enables modeling the distinct migratory and contractile response of cells residing on the tissue surface and the bulk, where concurrently the morphing soft tissues undergo large deformations driven by cell contractility. Additionally, the simulation results capture the changes in shape and cell concentration for wounded and intact microtissues, enabling the interpretation of experimental data. The numerical procedure that accounts for mechanosensitive stress generation, large deformations, diffusive migration in the bulk and a distinct mechanism for diffusive migration on deforming surfaces is inspired from recent work on bulk and surface poroelasticity of hydrogels involving elastocapillary effects, but in this work, a two-field weak form is proposed and is able to alleviate numerical instabilities that were observed in the original method that utilized a three-field mixed finite element formulation.

Inoculation theory

ABSTRACT Inoculation theory is a theory of resistance to influence that builds on analogous relationships with medical vaccination. In short: Exposure to weakened forms of challenges motivates resistance to stronger challenges encountered later. Throughout its 60 + year lifespan, scholars have applied inoculation theory as a messaging strategy across issues and contexts, including politics, health, and commerce. Additionally, inoculation theory scholarship has clarified how inoculation messaging confers resistance, moving beyond the traditional components (threat and refutational preemption) to consider other factors, including emotions. This article reviews key findings in inoculation theory research and then proposes particularly promising areas for future research.

Numerical integration in the virtual element method with the scaled boundary cubature scheme

AbstractThe virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

Determinants of stock return in 10 biggest market capitalization on the Indonesian stock exchange

Several research results on the capital market in Indonesia fall into the category of market efficiency in the weak form, so the results of previous research do not reflect the theory of capital market efficiency in the strong form. The purpose of this research is to analyze and answer the existence of inconsistencies in the results of previous research as well as the phenomenon of stock returns which are not as described in capital market theory in strong form. This is what prompted researchers to do it again using different time series and cross-sectional data. This type of research is quantitative descriptive with a panel data multiple regression analysis method using 10 sample companies that are members of 50 large capitalization companies for 7 years. This research formula is to maximize the Stock Return value through leverage as an intervening variable using research objects of companies on the Indonesia Stock Exchange. Two research models are integrated into one and each goes through a model selection test stage, namely the Chow Test, Hausman Test, and Lagrange Multiplier Test using the eviews12 application. The results of this research explain that increasing ROA can have an impact on reducing Leverage (DER) and through DER it can explain the impact on Stock Returns, but ROA cannot directly explain Stock Returns. It is hoped that these results can help as a guide for investors to get maximum Stock Returns.

Indirect Tax Reform And Market Efficiency: An Empirical Study of Indian Stock Market

The biggest tax reform of the Indian Government in this present decade is the Goods and Services Tax (GST), which was passed in parliament on 8th September 2016 and was enforced from 1st July 2017. Eight years of data from the Bombay Stock Exchange Sensex and the National Stock Exchange Nifty50 index is studied. The data is divided into two sub-periods before GST was passed from parliament period - I and after GST was passed from the parliament of India period - II. The study intends to test the market efficiency in its weak form. To check the time series data for normality, stationarity, random walk, and autocorrelation function, Kolmogorov-Smirnov (KS) test, Augmented Dickey-Fuller (ADF) test, Phillips Perron (PP) test is used for the analyse of the time-series data. The study found that returns are not normally distributed and the market is inefficient in a weak form for both periods under study.

Evolving Efficiency of Cryptocurrency Market: Evidence from Leading Cryptocurrencies

The article investigated the efficiency of the cryptocurrency market, with a focus on Bitcoin (BTC), Ethereum (ETH), Tether (USDT), and Binance Coin (BNB-USD), based on their substantial market capitalization. For this purpose, we observed the daily closing prices spanning from January 2018 to December 2023 and employed a set of robust tests, including the generalized spectral test, automatic portmanteau test, and automatic variance ratio test. The findings of the study reveal a random pattern in price fluctuations, indicating weak form efficiency. Furthermore, we adopted the rolling window approach to investigate whether market efficiency is dynamic or static over time. The empirical result illustrated that the crypto market efficiency remains static over time, except for USDT. In conclusion, the overall empirical results support the notion of the random walk hypothesis, indicating that past price movements offer no predictive insight into future prices. These findings have significant implications for investors, emphasizing the lack of predictive insight from past price movements. Policymakers are urged to establish a robust framework for market integrity and reliable price discovery.