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15 pages, 1914 KB

Open AccessArticle

Nonequilibrium Steady States in Active Systems: A Helmholtz–Hodge Perspective

by Horst-Holger Boltz and Thomas Ihle

Entropy 2025, 27(5), 525; https://doi.org/10.3390/e27050525 - 14 May 2025

Viewed by 642

We revisit the question of the existence of a potential function, the Cole–Hopf transform of the stationary measure, for nonequilibrium steady states, in particular those found in active matter systems. This has been the subject of ongoing research for more than fifty years, [...] Read more.

We revisit the question of the existence of a potential function, the Cole–Hopf transform of the stationary measure, for nonequilibrium steady states, in particular those found in active matter systems. This has been the subject of ongoing research for more than fifty years, but continues to be relevant. In particular, we want to make a connection to some recent work on the theory of Helmholtz–Hodge decompositions and address the recently suggested notion of typical trajectories in such systems. Full article

(This article belongs to the Collection Foundations of Statistical Mechanics)

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15 pages, 280 KB

Open AccessFeature PaperArticle

An Interior Regularity Property for the Solution to a Linear Elliptic System with Singular Coefficients in the Lower-Order Term

by Teresa Radice

Mathematics 2025, 13(3), 489; https://doi.org/10.3390/math13030489 - 31 Jan 2025

Viewed by 650

Abstract

This paper deals with the interior higher differentiability of the solution u to the Dirichlet problem related to system

- div (A (x) D u) + B (x, u) = f

on a bounded Lipschitz domain [...] Read more.

This paper deals with the interior higher differentiability of the solution u to the Dirichlet problem related to system

- div (A (x) D u) + B (x, u) = f

on a bounded Lipschitz domain

Ω

in

R^{n}

. The matrix

A (x)

lies in the John and Nirenberg space

B M O

. The lower-order term

B (x, u)

is controlled with respect to the spatial variable by a function

b (x)

belonging to the Marcinkiewicz space

L^{n, \infty}

. The novelty here is the presence of a singular coefficient in the lower-order term. Full article

17 pages, 426 KB

Open AccessArticle

A Simplex Model of Long Pathways in the Brain Related to the Minimalist Program in Linguistics

by Atsuhide Mori

Symmetry 2025, 17(2), 207; https://doi.org/10.3390/sym17020207 - 29 Jan 2025

Viewed by 618

Abstract

Marcolli, Chomsky, and Berwick described the minimalist program, proposed by Chomsky in generative linguistics, as an algebra of binary trees in an analogy of quantum physics on Feynman diagrams. In this paper, we proposed another model of the minimalist program based on simplicial [...] Read more.

Marcolli, Chomsky, and Berwick described the minimalist program, proposed by Chomsky in generative linguistics, as an algebra of binary trees in an analogy of quantum physics on Feynman diagrams. In this paper, we proposed another model of the minimalist program based on simplicial Hodge theory by taking the relevant brain neural network into account. We focused on a long directed pathway connecting distant areas in the brain, and took the (abstract) simplex spanning the locations on the terminal area, which the signals going through the pathway can reach. The identity of each signal is represented by the symmetry of the corresponding face, consisting of locations receiving the signal simultaneously. Then, we showed that this model fits the minimalist program. Further, we calculated the spectrum and eigenspaces of the Hodge Laplacian in important cases and found their surprising rationality. According to this rationality, we could draw pictures of syntactic relations based only on the calculation without using linguistic knowledge. In addition, though word order depends on what language is used, and thus has nothing to do with the minimalist program, planar word arrangements are still possible and within the scope of our model. Full article

(This article belongs to the Section Life Sciences)

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15 pages, 744 KB

Open AccessArticle

Causal Hierarchy in the Financial Market Network—Uncovered by the Helmholtz–Hodge–Kodaira Decomposition

by Tobias Wand, Oliver Kamps and Hiroshi Iyetomi

Entropy 2024, 26(10), 858; https://doi.org/10.3390/e26100858 - 11 Oct 2024

Cited by 1 | Viewed by 1575

Abstract

Granger causality can uncover the cause-and-effect relationships in financial networks. However, such networks can be convoluted and difficult to interpret, but the Helmholtz–Hodge–Kodaira decomposition can split them into rotational and gradient components which reveal the hierarchy of the Granger causality flow. Using Kenneth [...] Read more.

Granger causality can uncover the cause-and-effect relationships in financial networks. However, such networks can be convoluted and difficult to interpret, but the Helmholtz–Hodge–Kodaira decomposition can split them into rotational and gradient components which reveal the hierarchy of the Granger causality flow. Using Kenneth French’s business sector return time series, it is revealed that during the COVID crisis, precious metals and pharmaceutical products were causal drivers of the financial network. Moreover, the estimated Granger causality network shows a high connectivity during the crisis, which means that the research presented here can be especially useful for understanding crises in the market better by revealing the dominant drivers of crisis dynamics. Full article

(This article belongs to the Special Issue Complexity in Financial Networks)

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15 pages, 281 KB

Open AccessArticle

Some Properties of the Potential Field of an Almost Ricci Soliton

by Adara M. Blaga and Sharief Deshmukh

Mathematics 2024, 12(19), 3049; https://doi.org/10.3390/math12193049 - 28 Sep 2024

Cited by 1 | Viewed by 1055

Abstract

In this article, we are interested in finding necessary and sufficient conditions for a compact almost Ricci soliton to be a trivial Ricci soliton. As a first result, we show that positive Ricci curvature and, for a nonzero constant c, the integral [...] Read more.

In this article, we are interested in finding necessary and sufficient conditions for a compact almost Ricci soliton to be a trivial Ricci soliton. As a first result, we show that positive Ricci curvature and, for a nonzero constant c, the integral of

Ric (c ξ, c ξ)

satisfying a generic inequality on an n-dimensional compact and connected almost Ricci soliton

(M^{n}, g, ξ, σ)

are necessary and sufficient conditions for it to be isometric to the n-sphere

S^{n} (c)

. As another result, we show that, if the affinity tensor of the soliton vector field

ξ

vanishes and the scalar curvature

τ

of an n-dimensional compact almost Ricci soliton

(M^{n}, g, ξ, σ)

satisfies

τ (n σ - τ) \geq 0

, then

(M^{n}, g, ξ, σ)

is a trivial Ricci soliton. Finally, on an n-dimensional compact almost Ricci soliton

(M^{n}, g, ξ, σ)

, we consider the Hodge decomposition

ξ = \bar{ξ} + \nabla h

, where

div \bar{ξ} = 0

, and we use the bound on the integral of

Ric (\bar{ξ}, \bar{ξ})

and an integral inequality involving the scalar curvature to find another characterization of the n-sphere. Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

14 pages, 271 KB

Open AccessArticle

Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

by Hanan Alohali, Sharief Deshmukh, Bang-Yen Chen and Hemangi Madhusudan Shah

Mathematics 2024, 12(17), 2628; https://doi.org/10.3390/math12172628 - 24 Aug 2024

Cited by 1 | Viewed by 1091

Abstract

For a compact Riemannian m-manifold

(M^{m}, g), m > 1,

endowed with a nontrivial conformal vector field

ζ

with a conformal factor

σ

, there is an associated skew-symmetric tensor

φ

called the associated tensor, and [...] Read more.

For a compact Riemannian m-manifold

(M^{m}, g), m > 1,

endowed with a nontrivial conformal vector field

ζ

with a conformal factor

σ

, there is an associated skew-symmetric tensor

φ

called the associated tensor, and also,

ζ

admits the Hodge decomposition

ζ = \bar{ζ} + \nabla ρ

, where

\bar{ζ}

satisfies

div \bar{ζ} = 0

, which is called the Hodge vector, and

ρ

is the Hodge potential of

ζ

. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on

M^{m}

. The first result of this article states that a compact Riemannian m-manifold

M^{m}

is an m-sphere

S^{m} (c)

if and only if (1) for a nonzero constant c, the function

- σ / c

is a solution of the Poisson equation

Δ ρ = m σ

, and (2) the Ricci curvature satisfies

R i c (\bar{ζ}, \bar{ζ}) \geq {∥φ∥}^{2}

. The second result states that if

M^{m}

has constant scalar curvature

τ = m (m - 1) c > 0

, then it is an

S^{m} (c)

if and only if the Ricci curvature satisfies

R i c (\bar{ζ}, \bar{ζ}) \geq {∥φ∥}^{2}

and the Hodge potential

ρ

satisfies a certain static perfect fluid equation. The third result provides another new characterization of

S^{m} (c)

using the affinity tensor of the Hodge vector

\bar{ζ}

of a conformal vector field

ζ

on a compact Riemannian manifold

M^{m}

with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold

M^{m}

,

m > 2

, is a Euclidean m-space if and only if it admits a nontrivial conformal vector field

ζ

whose affinity tensor vanishes identically and

ζ

annihilates its associated tensor

φ

. Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

19 pages, 357 KB

Open AccessArticle

Arakelov Inequalities for a Family of Surfaces Fibered by Curves

by Mohammad Reza Rahmati

Mathematics 2024, 12(13), 1963; https://doi.org/10.3390/math12131963 - 25 Jun 2024

Viewed by 1214

Abstract

The numerical invariants of a variation of Hodge structure over a smooth quasi-projective variety are a measure of complexity for the global twisting of the limit-mixed Hodge structure when it degenerates. These invariants appear in formulas that may have correction terms called Arakelov [...] Read more.

The numerical invariants of a variation of Hodge structure over a smooth quasi-projective variety are a measure of complexity for the global twisting of the limit-mixed Hodge structure when it degenerates. These invariants appear in formulas that may have correction terms called Arakelov inequalities. We investigate numerical Arakelov-type equalities for a family of surfaces fibered by curves. Our method uses Arakelov identities in weight-one and weight-two variations of Hodge structure in a commutative triangle of two-step fibrations. Our results also involve the Fujita decomposition of Hodge bundles in these fibrations. We prove various identities and relationships between Hodge numbers and degrees of the Hodge bundles in a two-step fibration of surfaces by curves. Full article

(This article belongs to the Section B: Geometry and Topology)

13 pages, 292 KB

Open AccessArticle

Q-Form Field on a p-Brane with Codimension Two

by Ziqi Chen, Chun’e Fu, Xiaoyu Zhang, Chen Yang and Li Zhao

Symmetry 2023, 15(10), 1819; https://doi.org/10.3390/sym15101819 - 25 Sep 2023

Viewed by 1210

Abstract

This paper investigates gauge invariance in a bulk massless q-form field on a p-brane with codimension two, utilizing a general Kaluza–Klein (KK) decomposition. The KK decomposition analysis reveals four distinct KK modes: the conventional q-form, two

(q - 1)

-forms [...] Read more.

This paper investigates gauge invariance in a bulk massless q-form field on a p-brane with codimension two, utilizing a general Kaluza–Klein (KK) decomposition. The KK decomposition analysis reveals four distinct KK modes: the conventional q-form, two

(q - 1)

-forms and one

(q - 2)

-form. These diverse modes are essential for maintaining gauge invariance. We also find eight Schrödinger-like equations for the four modes due to the two extra dimensions, and their mass spectra are closely related. The KK decomposition process gives rise to four dualities on the p-brane, originating from the inherent Hodge duality present in the bulk. Notably, these dual symmetries play a significant role in maintaining the equivalence of bulk dual fields during dimensional reduction. Full article

(This article belongs to the Section Physics)

24 pages, 5225 KB

Open AccessArticle

Reconstruction of the Instantaneous Images Distorted by Surface Waves via Helmholtz–Hodge Decomposition

by Bijian Jian, Chunbo Ma, Yixiao Sun, Dejian Zhu, Xu Tian and Jun Ao

J. Mar. Sci. Eng. 2023, 11(1), 164; https://doi.org/10.3390/jmse11010164 - 9 Jan 2023

Cited by 3 | Viewed by 2597

Abstract

Imaging through water waves will cause complex geometric distortions and motion blur, which seriously affect the correct identification of an airborne scene. The current methods main rely on high-resolution video streams or a template image, which limits their applicability in real-time observation scenarios. [...] Read more.

Imaging through water waves will cause complex geometric distortions and motion blur, which seriously affect the correct identification of an airborne scene. The current methods main rely on high-resolution video streams or a template image, which limits their applicability in real-time observation scenarios. In this paper, a novel recovery method for the instantaneous images distorted by surface waves is proposed. The method first actively projects an adaptive and adjustable structured light pattern onto the water surface for which random fluctuation will cause the image to degrade. Then, the displacement field of the feature points in the structured light image is used to estimate the motion vector field of the corresponding sampling points in the scene image. Finally, from the perspective of fluid mechanics, the distortion-free scene image is reconstructed based on the Helmholtz-Hodge Decomposition (HHD) theory. Experimental results show that our method not only effectively reduces the distortion to the image, but also significantly outperforms state-of-the-art methods in terms of computational efficiency. Moreover, we tested the real-scene sequences of a certain length to verify the stability of the algorithm. Full article

(This article belongs to the Special Issue Underwater Engineering and Image Processing)

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19 pages, 4762 KB

Open AccessArticle

The Role of Inertia in the Onset of Turbulence in a Vortex Filament

by Jean-Paul Caltagirone

Fluids 2023, 8(1), 16; https://doi.org/10.3390/fluids8010016 - 2 Jan 2023

Cited by 2 | Viewed by 2335

Abstract

The decay of the kinetic energy of a turbulent flow with time is not necessarily monotonic. This is revealed by simulations performed in the framework of discrete mechanics, where the kinetic energy can be transformed into pressure energy or vice versa; this persistent [...] Read more.

The decay of the kinetic energy of a turbulent flow with time is not necessarily monotonic. This is revealed by simulations performed in the framework of discrete mechanics, where the kinetic energy can be transformed into pressure energy or vice versa; this persistent phenomenon is also observed for inviscid fluids. Different types of viscous vortex filaments generated by initial velocity conditions show that vortex stretching phenomena precede an abrupt onset of vortex bursting in high-shear regions. In all cases, the kinetic energy starts to grow by borrowing energy from the pressure before the transfer phase to the small turbulent structures. The result observed on the vortex filament is also found for the Taylor–Green vortex, which significantly differs from the previous results on this same case simulated from the Navier–Stokes equations. This disagreement is attributed to the physical model used, that of discrete mechanics, where the formulation is based on the conservation of acceleration. The reasons for this divergence are analyzed in depth; however, a spectral analysis allows finding the established laws on the decay of kinetic energy as a function of the wave number. Full article

(This article belongs to the Special Issue Recent Advances in Fluid Mechanics: Feature Papers, 2022)

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13 pages, 273 KB

Open AccessArticle

2D Discrete Hodge–Dirac Operator on the Torus

by Volodymyr Sushch

Symmetry 2022, 14(8), 1556; https://doi.org/10.3390/sym14081556 - 28 Jul 2022

Cited by 1 | Viewed by 1844

Abstract

We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide [...] Read more.

We discuss a discretization of the de Rham–Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge–Dirac and Laplace operators in which key geometric aspects of the continuum counterpart are captured. We provide and prove a discrete version of the Hodge decomposition theorem. The goal of this work is to develop a satisfactory discrete model of the de Rham–Hodge theory on manifolds that are homeomorphic to the torus. Special attention has been paid to discrete models on a combinatorial torus. In this particular case, we also define and calculate the cohomology groups. Full article

(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)

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15 pages, 303 KB

Open AccessArticle

The Injectivity Theorem on a Non-Compact Kähler Manifold

by Jingcao Wu

Symmetry 2021, 13(11), 2222; https://doi.org/10.3390/sym13112222 - 20 Nov 2021

Viewed by 1824

Abstract

In this paper, we establish an injectivity theorem on a weakly pseudoconvex Kähler manifold X with negative sectional curvature. For this purpose, we develop the harmonic theory in this circumstance. The negative sectional curvature condition is usually satisfied by the manifolds with hyperbolicity, [...] Read more.

In this paper, we establish an injectivity theorem on a weakly pseudoconvex Kähler manifold X with negative sectional curvature. For this purpose, we develop the harmonic theory in this circumstance. The negative sectional curvature condition is usually satisfied by the manifolds with hyperbolicity, such as symmetric spaces, bounded symmetric domains in

C^{n}

, hyperconvex bounded domains, and so on. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

13 pages, 1055 KB

Open AccessArticle

A Monolithic Approach of Fluid–Structure Interaction by Discrete Mechanics

by Stéphane Vincent and Jean-Paul Caltagirone

Fluids 2021, 6(3), 95; https://doi.org/10.3390/fluids6030095 - 1 Mar 2021

Cited by 1 | Viewed by 2308

Abstract

The unification of the laws of fluid and solid mechanics is achieved on the basis of the concepts of discrete mechanics and the principles of equivalence and relativity, but also the Helmholtz–Hodge decomposition where a vector is written as the sum of divergence-free [...] Read more.

The unification of the laws of fluid and solid mechanics is achieved on the basis of the concepts of discrete mechanics and the principles of equivalence and relativity, but also the Helmholtz–Hodge decomposition where a vector is written as the sum of divergence-free and curl-free components. The derived equation of motion translates the conservation of acceleration over a segment, that of the intrinsic acceleration of the material medium and the sum of the accelerations applied to it. The scalar and vector potentials of the acceleration, which are the compression and shear energies, give the discrete equation of motion the role of conservation law for total mechanical energy. Velocity and displacement are obtained using an incremental time process from acceleration. After a description of the main stages of the derivation of the equation of motion, unique for the fluid and the solid, the cases of couplings in simple shear and uniaxial compression of two media, fluid and solid, make it possible to show the role of discrete operators and to find the theoretical results. The application of the formulation is then extended to a classical validation case in fluid–structure interaction. Full article

(This article belongs to the Special Issue Fluid Structure Interaction: Methods and Applications)

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18 pages, 357 KB

Open AccessArticle

The Weitzenböck Type Curvature Operator for Singular Distributions

by Paul Popescu, Vladimir Rovenski and Sergey Stepanov

Mathematics 2020, 8(3), 365; https://doi.org/10.3390/math8030365 - 6 Mar 2020

Cited by 1 | Viewed by 2215

Abstract

We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior [...] Read more.

We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their

L^{2}

adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones. Full article

(This article belongs to the Special Issue Geometric Structures and Interdisciplinary Applications)

14 pages, 527 KB

Open AccessArticle

Stochastic Thermodynamics of Oscillators’ Networks

by Simone Borlenghi and Anna Delin

Entropy 2018, 20(12), 992; https://doi.org/10.3390/e20120992 - 19 Dec 2018

Cited by 1 | Viewed by 3915

Abstract

We apply the stochastic thermodynamics formalism to describe the dynamics of systems of complex Langevin and Fokker-Planck equations. We provide in particular a simple and general recipe to calculate thermodynamical currents, dissipated and propagating heat for networks of nonlinear oscillators. By using the [...] Read more.

We apply the stochastic thermodynamics formalism to describe the dynamics of systems of complex Langevin and Fokker-Planck equations. We provide in particular a simple and general recipe to calculate thermodynamical currents, dissipated and propagating heat for networks of nonlinear oscillators. By using the Hodge decomposition of thermodynamical forces and fluxes, we derive a formula for entropy production that generalises the notion of non-potential forces and makes transparent the breaking of detailed balance and of time reversal symmetry for states arbitrarily far from equilibrium. Our formalism is then applied to describe the off-equilibrium thermodynamics of a few examples, notably a continuum ferromagnet, a network of classical spin-oscillators and the Frenkel-Kontorova model of nano friction. Full article

(This article belongs to the Section Statistical Physics)

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