Small-scale two-phase flows are subject to intense capillary accelerations that must be treated with care in order to avoid artifacts often associated with the numerical methodologies used, such as excessive fragmentation of structures. This analysis proposes a formulation of capillary actions for compressible viscous two-phase flows within the framework of discrete mechanics, where the concept of mass is abandoned in favor of a law of motion that describes the conservation of accelerations, one related to inertia and the other to external actions. With the introduction of the capillary term, the sum of a capillary potential gradient and the dual curl of a vector potential is consistent with the other terms of the law of motion, a formal Helmholtz–Hodge decomposition. This fully compressible formulation reproduces the capillary waves generated by the source terms and the contact and shock discontinuities in the two immiscible fluids. This methodology completely eliminates parasitic currents due mainly to the presence of residual curl in the capillary source terms. Several classic examples demonstrate the validity of this approach.

The differentiation potency of cells is governed by dynamic changes in gene expression, which can be inferred from single-cell RNA sequencing (scRNA-seq) data. While velocity-based approaches have been used to analyze cell state changes as vector fields, extracting acceleration (change of change) information remains challenging because of the sparsity and high-dimensionality of the data. Here, we develop ddHodge, a framework based on Hodge decomposition for precise vector-field reconstruction. ddHodge accurately recovers all basic components of the vector field, namely, the gradient, curl, and divergence, including the acceleration of the cell state, as second-order derivatives, even from biased and sparse samples. Furthermore, we extend the method to approximate high-dimensional gene expression dynamics on lower-dimensional data manifolds. By applying ddHodge to scRNA-seq data from mouse embryogenesis, we reveal that the gene expression dynamics during development follow a gradient system shaped by potential landscapes, which has not previously been validated with real data. Furthermore, we quantify differentiation potency as cell state stability on the basis of the divergence and identify key genes that drive potency. Our general computational framework for analyzing complex biological systems can elucidate cell fate decisions in developmental processes.

Abstract We extend a CDGA V with a perfect pairing of degree n on cohomology to a CDGA $$\hat{V}$$ V ^ with a pairing of degree n on chain level such that $$\hat{V}$$ V ^ admits a Hodge decomposition and retracts onto V preserving the pairing on cohomology; here we suppose that V is either 1-connected, or that V is connected, of finite type, and n is odd. We show that a Hodge decomposition of $$\hat{V}$$ V ^ induces a differential Poincaré duality model of V in a natural way. Assuming that $$\textrm{H}(V)$$ H ( V ) is 1-connected, we apply our extension to a Sullivan model of V in the proof of the existence and “uniqueness” of a 1-connected differential Poincaré duality model of V by Lambrechts & Stanley; we eliminate their extra assumptions in the uniqueness statement, including $$\textrm{H}^2(V)=0$$ H 2 ( V ) = 0 if n is odd.

Abstract We show that, under mild assumptions, the Fano surfaces of lines on smooth cubic threefolds are the only smooth subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. This in particular leads to a considerable strengthening of our previous work on the Shafarevich conjecture. A key idea is to control the Hodge decomposition on cohomology by a cocharacter of the Tannaka group of Hodge modules, and to play this off against an improvement of the Hodge number estimates for irregular varieties by Lazarsfeld–Popa and Lombardi.

Abstract Taking place naturally in a gas subject to a given wall temperature distribution, the “ghost effect” exhibits a rare kinetic effect beyond the prediction of classical fluid theory and Fourier law in such a classical problem in physics. As the Knudsen number goes to zero, the finite variation of temperature in the bulk is determined by an infinitesimal, ghost‐like velocity field, created by a given finite variation of the tangential wall temperature as predicted by Maxwell's slip boundary condition. Mathematically, such a finite variation leads to the presence of a severe singularity and a Knudsen layer approximation in the fundamental energy estimate. Neither difficulty is within the reach of any existing PDE theory on the steady Boltzmann equation in a general 3D bounded domain. Consequently, in spite of the discovery of such a ghost effect from temperature variation in as early as 1960s, its mathematical validity has been a challenging and intriguing open question, causing confusion and suspicion. We settle this open question in affirmative if the temperature variation is small but finite, by developing a new framework with four major innovations as follows: (1) a key ‐Hodge decomposition and its corresponding local ‐conservation law eliminate the severe bulk singularity, leading to a reduced energy estimate; (2) a surprising gain in via momentum conservation and a dual Stokes solution; (3) the ‐conservation, energy conservation, and a coupled dual Stokes–Poisson solution reduces to an boundary singularity; (4) a crucial construction of ‐cutoff boundary layer eliminates such boundary singularity via new Hardy's and BV estimates.

ABSTRACTSynthetic wind fields generated for wind turbine simulations do not satisfy incompressibility condition, thus, are not divergence‐free. This results in spurious pressure fluctuations when input as a boundary condition to, for example, incompressible large eddy simulations (LES). This study investigates the impact of divergence‐free correction on synthetic wind fields and their influence on wind turbine loads. Although divergence‐free correction methods exist, they often modify the wind field energy spectrum and unsteady characteristics. Ongoing research addresses these challenges, but the acceptability of such changes and their impact on wind turbine loads has not been adequately studied. This work enforced incompressibility using the Helmholtz–Hodge decomposition, solved through spectral and spatial methods. An efficient Fourier‐based spectral method was implemented, validated, and tested against the traditional finite difference method used for the spatial approach. Synthetic wind fields based on three coherence models were analyzed under three turbine operating conditions. An aeroelastic analysis of the IEA MW wind turbine was performed in the wind fields before and after divergence correction. Spectral analysis revealed a reduction in energy at specific frequencies after the correction for incompressibility. Additionally, the standard deviations of the wind velocities changed (despite similar means), consequently affecting the aeroelastic turbine response. A new iterative correction method is proposed to mitigate these effects, which preserves first‐ and second‐order statistics while enforcing a divergence‐free condition. This method is recursively applied, maintaining RMSE changes to the wind field within user‐specified bounds. Key findings show that the iterative method yields an excellent match in the longitudinal wind field energy spectrum and a closer match in wind field standard deviation across the rotor, reducing discrepancies in turbine response. Some discrepancies in the lateral and vertical velocity components' higher order statistics were observed. Standard divergence correction (without RMSE constraints) led to a decrease of up to 20% in the tower fore‐aft moment, while the proposed method reduces this change to −10%. The tower top side‐side moment was found to increase by % by using the former approach, while the proposed correction reduced this increase to %. Blade root flap‐wise bending moment was less affected (up to 5% reduction). Divergence‐free wind fields, even with similar statistical properties, influence aeroelastic loads. The proposed method aims to achieve physically consistent and more comparable wind field analyses and resulting wind loads.

Abstract We develop a finite element method for an elliptic Maxwell boundary value problem on polyhedral domains in ℝ 3 {\mathbb{R}^{3}} with a general topology. Our method is based on a Hodge decomposition approach that leads to standard scalar elliptic problems and elliptic saddle point problems for vector potentials that have previously been investigated in the study of fluid flow problems. We carry out an error analysis that does not involve assumed regularity of the solution and present corroborating numerical results.

We propose to analyze dynamically changing brain networks by decomposing them into three orthogonal components through the Hodge decomposition. We propose to quantify the magnitude and relative strength of each components. We performed extensive simulation studies with the known ground truth. The Hodge decomposition is then applied to the dynamically changing human brain networks obtained from the resting state functional magnetic resonance imaging study. Our study indicates that the components of the Hodge decomposition contain biologically interpretable topological features that provide statistically significant results that are difficult to obtain with the traditional methods.

We introduce a novel framework that integrates Hodge decomposition with Filtered Average Short-Term (FAST) functional connectivity to analyze dynamic functional connectivity (DFC) in EEG signals. This method leverages graph-based topology and simplicial analysis to explore transient connectivity patterns at multiple scales, addressing noise, sparsity, and computational efficiency. The temporal EEG data are first sparsified by keeping only the most globally important connections, instantaneous connectivity at these connections is then filtered by global long-term stable correlations. This tensor is then decomposed into three orthogonal components to study signal flows over higher-order structures such as triangle and loop structures. Our analysis of Alzheimer-related MCI patients show significant temporal differences related to higher-order interactions that a pairwise analysis on its own does not implicate. This allows us to capture higher-dimensional interactions at high temporal resolution in noisy EEG signal recordings.

Abstract In the quest to uncovering the fundamental structures that underlie some of the asymptotic Swampland conjectures the authors initiate the general study of asymptotic period vectors of Calabi–Yau manifolds. The strategy is to exploit the constraints imposed by completeness, symmetry, and positivity, which are formalized in asymptotic Hodge theory. The general principles are used to study the periods near any boundary in complex structure moduli space and explain that near most boundaries, leading exponentially suppressed corrections must be present for consistency. The only exception are period vectors near the well‐studied large complex structure point. Together with the classification of possible boundaries, the procedure makes it possible to construct general models for these asymptotic periods. The starting point for this construction is the ‐data classifying the boundary, which is used to construct the asymptotic Hodge decomposition known as the nilpotent orbit. The authors then use the latter to determine the asymptotic period vector. This program has been explicitly carried out for all possible one‐ and two‐moduli boundaries in Calabi–Yau threefolds, and general models for their asymptotic periods have been written down.

So as to assess systemic economic value and risk, it is of crucial interest to develop methods to detect interwoven production chains in large-scale economies. Commodity transactions between firms induce a complex network in which it is challenging to identify production chains, not only due to the size of the underlying network but also because of its inherent cyclical connections and loops. We present a novel method, Restricted Gradient Extraction (RGE), which is based on Hodge decomposition, which is capable of extracting the gradient flow of a production network. The RGE method, being of relatively low computational complexity, is demonstrated to both a synthetic and real country-sized production network. Application of RGE on the syntactic data set shows that the resulting gradient flow is a directed acyclic graph, a weighted subgraph of the original network, and the gradient flow is retained. Application to the economic production network of the Netherlands shows that production chains can be readily detected and described. The method is applicable to weighted directed networks in general and is not limited to economic production networks.

Hodge theory is pivotal in studying algebraic varieties' intricate geometry and topology: it provides essential insights into their structure. The Hodge decomposition theorem establishes a profound link between the geometry of varieties and their cohomology groups, helping to understand their underlying properties. Moreover, Hodge theory was crucial at the inception of the field of mirror symmetry, revealing deep connections among seemingly disparate algebraic varieties. It also sheds light on studying algebraic cycles and motives, crucial objects in algebraic geometry. This article explores Hodge polynomials and their properties, specifically focusing on non-Kähler complex manifolds. We investigate a diverse range of such manifolds, including (quasi-)Hopf, (quasi-)Calabi-Eckmann, and LVM manifolds, alongside a class of definable complex manifolds encompassing both algebraic varieties and the aforementioned special cases. Our research establishes the preservation of the motivic nature of Hodge polynomials inside this broader context. Through explicit calculations and thorough analyses, this work contributes to a deeper understanding of complex manifold geometry beyond the realm of algebraic varieties. The outcomes of this study have potential applications in various areas of mathematics and physics where complex manifolds play a significant role.

The inference of rankings plays a central role in the theory of social choice, which seeks to establish preferences from collectively generated data, such as pairwise comparisons. Examples include political elections, ranking athletes based on competition results, ordering web pages in search engines using hyperlink networks, and generating recommendations in online stores based on user behavior. Various methods have been developed to infer rankings from incomplete or conflicting data. One such method, HodgeRank, introduced by Jiang etal. [Math. Program. 127, 203 (2011) 10.1007/s10107-010-0419-x], utilizes Hodge decomposition of cochains in Higher-Order Networks to disentangle gradient and cyclical components contributing to rating scores, enabling a parsimonious inference of ratings and rankings for lists of items. This paper presents a systematic study of HodgeRank's performance under the influence of quenched disorder and across networks with complex topologies generated by four different network models. The results reveal a transition from a regime of perfect retrieval of true rankings to one of imperfect retrieval as the strength of the quenched disorder increases. A range of observables is analyzed, and their scaling behavior with respect to the network model parameters is characterized. This work advances the understanding of social choice theory and the inference of ratings and rankings within complex network structures.

The lattice Boltzmann method has become a popular tool for simulating complex flows, including incompressible turbulent flows; however, as an artificial compressibility method, it can generate spurious pressure oscillations whose impact on the statistics of incompressible turbulence has not been systematically examined. In this work, we propose a theoretical approach to analyse the origin of compressibility-induced oscillations (CIOs) and explore ways to suppress or remove them. We begin by decomposing the velocity field and pressure field each into the solenoidal component and the compressive component, and then study the evolution of these two components analytically and numerically. The analysis yields an evolution equation of the mean-square pressure fluctuation which reveals several coupling effects of the two components. The evolution equation suggests that increasing the bulk-to-shear viscosity ratio can suppress CIOs, which is confirmed by numerical simulations. Furthermore, based on the derived evolution equation and data from the simulation, a model is developed to predict the long-term behaviours of the mean-square pressure fluctuations. In the case of decaying turbulence in a periodic domain, we show that the Helmholtz–Hodge decomposition can be used to obtain the solenoidal components reflecting the true evolution of incompressible turbulent flow, from the mesoscopic artificial compressibility approach. The study provides general theoretical guidelines to understand, suppress and even remove CIOs in other related pseudo-compressibility methods.

In this article, the Hodge decomposition for any degree of differential forms is investigated on the whole space ℝ n and the half-space ℝ + n on different scales of function spaces namely the homogeneous and inhomogeneous Besov and Sobolev spaces, H ˙ s,p , B ˙ p,q s , H s,p and B p,q s , for p∈(1,+∞), s∈(-1+1 p,1 p). The bounded holomorphic functional calculus, and other functional analytic properties, of Hodge Laplacians is also investigated in the half-space, and yields similar results for Hodge–Stokes and other related operators via the proven Hodge decomposition. As consequences, the homogeneous operator and interpolation theory revisited by Danchin, Hieber, Mucha and Tolksdorf is applied to homogeneous function spaces subject to boundary conditions and leads to various maximal regularity results with global-in-time estimates that could be of use in fluid dynamics. Moreover, the bond between the Hodge Laplacian and the Hodge decomposition will even enable us to state the Hodge decomposition for higher order Sobolev and Besov spaces with additional compatibility conditions, for regularity index s∈(-1+1 p,2+1 p). In order to make sense of all those properties in desired function spaces, we also give appropriate meaning of partial traces on the boundary in the appendix.“La raison d’être” of this paper lies in the fact that the chosen realization of homogeneous function spaces is suitable for non-linear and boundary value problems, but requires a careful approach to reprove results that are already morally known.

The ecient numerical method of Maxwells equation needs to satisfy the interface condition of electromagnetic eld, so the nite element method of the electromagnetic eld problem generally uses the edge nite element space. Compared with the traditional nodal element, the disadvantage of the edge element is that it has many degrees of freedom and the condition number of the linear system is poor. In this paper, a method based on Hodge decomposition is used to convert Maxwells equation into a standard elliptic boundary value problem, then use node element to solve the ellipse problem and then get the numerical solution of Maxwells equation. Because Hodge decomposition is used, non-physical numerical solutions are avoided in numerical solutions. This paper uses Superior Capsular Reconstruction (SCR) and Polynomial Preserving Recovery (PPR) techniques to post-process the nite element numerical solution, which eectively improves the accuracy of the numerical solution, and establishes a reliable posterior error indicator and adaptive nite element method. Finally, four examples are given to verify the eectiveness and accuracy of the method.

We give a new CR invariant treatment of the bigraded Rumin complex and related cohomology groups via differential forms. A key benefit is the identification of balanced A ∞ A_\infty -structures on the Rumin and bigraded Rumin complexes. We also prove related Hodge decomposition theorems. Among many applications, we give a sharp upper bound on the dimension of the Kohn–Rossi groups H 0 , q ( M 2 n + 1 ) H^{0,q}(M^{2n+1}) , 1 ≤ q ≤ n − 1 1\leq q\leq n-1 , of a closed strictly pseudoconvex manifold with a contact form of nonnegative pseudohermitian Ricci curvature; we prove a sharp CR analogue of the Frölicher inequalities in terms of the second page of a natural spectral sequence; we give new proofs of selected topological properties of closed Sasakian manifolds; and we generalize the Lee class L ∈ H 1 ( M ; P ) \mathcal {L}\in H^1(M;\mathscr {P}) — whose vanishing is necessary and sufficient for the existence of a pseudo-Einstein contact form — to all nondegenerate orientable CR manifolds.

This proposal introduces a novel decision-making framework to advance safe economic activities in cyberspace. We focus on identifying anomalies within crypto-asset trading, recognized as potential sources of criminal activity, severely undermining the credibility of such assets. Detecting and mitigating such anomalies holds significant societal implications, particularly in fostering trust within blockchain networks. We aim to bolster the “social trust” inherent to blockchain technology by facilitating informed economic activities in cyberspace. To achieve this, we propose integrating two artificial intelligence (AI) systems into a blockchain-based decentralized autonomous organization (DAO). The first AI application involves amalgamating various anomaly indicators, spanning from cluster coefficient, entropy, triangular motif analysis, correlation tensor analysis, loop component by Hodge decomposition, loop causality detection, network classification using graph Laplacian, and persistent homology analysis, into a comprehensive indicator using a Boltzmann machine. The second AI application entails deploying conversational AI to guide and support traders, aiding them in making informed trading decisions. This system is designed to alert DAO members to anomalies based on the integrated indicators, especially during massive price fluctuations. We operate under the assumption of close collaboration between governments, experts, traders, system developers, and operators to effectively organize DAOs. The primary technical challenge in our proposal lies in developing a wallet assisted by an intelligent software agent capable of safe interactions with traders within a unified DAO. With this organization, we envision fostering a global economic ecosystem where physical and cyber worlds converge, allowing democratic economic participation.

Computational trajectory analysis is a key computational task for inferring differentiation trees from this single-cell data. An open challenge is the prediction of complex and multi-branching trees from multimodal data. To address these challenges, we present PHLOWER (decomposition of the Hodge Laplacian for inferring trajectories from flows of cell differentiation), which leverages the harmonic component of the Hodge decomposition on simplicial complexes to infer trajectory embeddings from single-cell multimodal data. These natural representations of cell differentiation facilitate the estimation of their underlying differentiation trees. We evaluate PHLOWER through benchmarking with multi-branching differentiation trees and using kidney organoid multimodal and spatial single-cell data. These demonstrate the power of PHLOWER in both the inference of complex trees and the identification of transcription factors regulating off-target cells in kidney organoids. Thus, PHLOWER enables inference of complex branching trajectories and prediction of transcriptional regulators by leveraging multimodal data.

We study neighbourhoods of submanifolds in generalized complex geometry. Our first main result provides sufficient criteria for such a submanifold to admit a neighbourhood on which the generalized complex structure is B -field equivalent to a holomorphic Poisson structure. This is intimately tied with our second main result, which is a rigidity theorem for generalized complex deformations of holomorphic Poisson structures. Specifically, on a compact manifold with boundary we provide explicit conditions under which any generalized complex perturbation of a holomorphic Poisson structure is B -field equivalent to another holomorphic Poisson structure. The proofs of these results require two analytical tools: Hodge decompositions on almost complex manifolds with boundary, and the Nash–Moser algorithm. As a concrete application of these results, we show that on a four-dimensional generalized complex submanifold which is generically symplectic, a neighbourhood of the entire complex locus is B -field equivalent to a holomorphic Poisson structure. Furthermore, we use the neighbourhood theorem to develop the theory of blowing down submanifolds in generalized complex geometry.