Differential Forms Research Articles

On the Moduli of Lipschitz Homology Classes

Abstract We define a type of modulus $$\operatorname {dMod}_p$$ dMod p for Lipschitz surfaces based on $$L^p$$ L p -integrable measurable differential forms, generalizing the vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a homological duality theorem, where for Hölder conjugate exponents $$p, q \in (1, \infty )$$ p , q ∈ ( 1 , ∞ ) , every relative Lipschitz k-homology class c has a unique dual Lipschitz $$(n-k)$$ ( n - k ) -homology class $$c'$$ c ′ such that $$\operatorname {dMod}_p^{1/p}(c) \operatorname {dMod}_q^{1/q}(c') = 1$$ dMod p 1 / p ( c ) dMod q 1 / q ( c ′ ) = 1 and the Poincaré dual of c maps $$c'$$ c ′ to 1. As $$\operatorname {dMod}_p$$ dMod p is larger than the classical surface modulus $$\operatorname {Mod}_p$$ Mod p , we immediately recover a more general version of the estimate $$\operatorname {Mod}_p^{1/p}(c) \operatorname {Mod}_q^{1/q}(c') \le 1$$ Mod p 1 / p ( c ) Mod q 1 / q ( c ′ ) ≤ 1 , which appears in works by Freedman and He and by Lohvansuu. Our theory is formulated in the general setting of Lipschitz Riemannian manifolds, though our results appear new in the smooth setting as well. We also provide a characterization of closed and exact Sobolev forms on Lipschitz manifolds based on integration over Lipschitz k-chains.

Adaptive Construction of Freeform Surface to Integrable Ray Mapping Using Implicit Fixed-Point Iteration

Constructing a freeform surface that accurately satisfies both integrable condition and Snell’s law under a given invariant source–target map is challenging for freeform design. Here, we propose a fixed-point iteration method to address this problem. This process involves solving a set of balanced gradient equations in the form of fixed-point iterations that are derived from equivalent integrability conditions and Snell’s law. By using the convergence theorem of fixed-point iteration, a unique solution for the balanced gradient equations exists, which is determined by the natural geometric properties of the freeform surface and is independent of the mapping. The gradient operators on the left-hand side of the equations are converted into a differential matrix form via a finite difference scheme. In one iteration, differential operations are forward-performed on the right-hand side of the equations, and the system of linear equations is solved on the left-hand side of the equation. The constructed freeform surfaces work well in both the paraxial and nonparaxial regions, and convergence in the nonparaxial region is faster than that in the paraxial region. The robustness and high efficiency of the proposed method are demonstrated with several design examples.

Modeling the growth and dynamics of uropathogenic Escherichia coli in sugarcane juice for shelf life predictions.

In this study, we assessed the effects of temperature and dilution on uropathogenic Escherichia coli (UPEC) growth in sugarcane juice and modeled the kinetics for shelf life simulation. Diluted and undiluted sugarcane juice samples inoculated with a four-strain UPEC cocktail were stored at 4, 10, 15, 20, 30, and 40°C to evaluate their growth during storage. Changes in UPEC growth were fitted using three primary models (Baranyi, Huang, and reparameterized Gompertz models), and two secondary models (Huang square-root and Ratkowsky square-root models) were selected to evaluate the effect of temperature on specific growth rates. The best-fitted models (reparameterized Gompertz and Huang square-root models) were validated by an additional experiment under isothermal storage (35°C). Combining differential forms of the Baranyi model, the prediction performances of these models were further validated under storage at dynamic temperature profiles (15 to 37°C in 4-h cycles). Results revealed that UPEC growth was observed in diluted and undiluted juice samples at storage temperatures of 15 to 40°C. No growth of UPEC was found in either type of juice at storage temperatures below 10°C. The estimated minimum growth temperatures (Tmin) of UPEC in sugarcane juice and diluted juice were 10.7 and 11.2°C, respectively. Validation results showed that the model perfectly estimated UPEC growth under isothermal conditions in both types of sugarcane juice (proportion of prediction error (pPE)=1.00). Predictions of dynamic conditions in sugarcane juice (pPE = 0.73) and diluted juice (pPE = 0.91) were acceptable, at pPE > 0.7. Using the model, shelf life charts were derived based on the time to reach a certain microbial concentration. In conclusion, these results together with shelf life charts provide valuable information for the food industry to enhance sugarcane juice safety.

Continuous primitives for higher degree differential forms in euclidean spaces, heisenberg groups and applications

Continuous primitives for higher degree differential forms in euclidean spaces, heisenberg groups and applications

The Port-Hamiltonian Structure of Continuum Mechanics

In this paper, we present a novel approach to the geometric formulation of solid and fluid mechanics within the port-Hamiltonian framework, which extends the standard Hamiltonian formulation to non-conservative and open dynamical systems. Leveraging Dirac structures, instead of symplectic or Poisson structures, this formalism allows the incorporation of energy exchange within the spatial domain or through its boundary, which allows for a more comprehensive description of continuum mechanics. Building upon our recent work in describing nonlinear elasticity using exterior calculus and bundle-valued differential forms, this paper focuses on the systematic derivation of port-Hamiltonian models for solid and fluid mechanics in the material, spatial, and convective representations using Hamiltonian reduction theory. This paper also discusses constitutive relations for stress within this framework including hyper-elasticity, for both finite and infinitesimal strains, as well as viscous fluid flow governed by the Navier–Stokes equations.

Predefined-time adaptive robust simultaneous stabilisation control for two nonlinear singular time-delay systems

This paper mainly studies the simultaneous stabilisation control problem of a class of nonlinear singular systems with predefined time. We first develop two singular system in differential algebraic forms via an appropriate state feedback. And then the two systems are extended to one system for further research. We have solved the difficulty of constructing Lyapunov functions by constructing appropriate Hamiltonian functions for our research. In addition, when considering nonlinear singular systems with unknown disturbances, parameter uncertainties and time delays, a control strategy is developed to achieve simultaneous stabilisation within a predefined time. Finally, the effectiveness of the proposed controller was verified through simulation.

The Bigraded Rumin Complex via Differential Forms

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.

Covariant interacting fractons

We show that there exists a natural analog of the Yang-Mills equations using the Frölicher-Nijenhuis bracket between vector-valued differential forms. The gauge field is a rank-2 tensor, and when one constrains it to be symmetric, then the system exhibits fractonic behaviors. In the linearized limit, the constrained equations of motion reduce to those of the covariant fracton model [., ., and Notes from the bulk, Ph.D. thesis, Università di Genova, 2024.]. Published by the American Physical Society 2025

Effect of Severe Salt Stress on Respiratory and Biochemical Parameters in Legumes With Differential Nodulation Form

ABSTRACTLegumes are among the most utilised agronomic plant species due to their symbiotic association with N2‐fixing bacteria. Since N2 fixation entails high ATP cost, salt stress disrupts N2 fixation in the symbiont, but increases the production of osmolytes and antioxidant systems in the host plant. This results in competition for C allocation between osmoprotection in the host and continued supply to the symbiont for N acquisition, which may result in different plant responses to salinity. Two‐nodule types of plant species with contrasting carbon requirements for organic N2 fixation can be found within legume species; determinate and indeterminate. In this study, we tested responses of respiratory carbon metabolism, nitrogen assimilation and antioxidant machinery in leaves and roots of Phaseolus vulgaris (determinate nodules) and Pisum sativum (indeterminate nodules) 24 and 72 h after salt treatment (300 mM of NaCl). In P. sativum, we observed that nitrogenase activity was maintained at 24 h, but showed a strong decrease at 72 h together with cytochrome activity. On contrast, in P. vulgaris, respiration rates were maintained by an enhanced antioxidant activity under salinity although at the expense of nodule metabolism. Despite of the severity of the salt stress for N2 fixation, both species showed similar mechanisms to cope with salinity, like the maintenance of alternative respiration and increased antioxidant defence, that are worthy to be tested in the long term under field conditions.

Maxwell’s and Stokes’s operators associated with elliptic differential complexes

We propose a regular method for generating consistent systems of partial differential equations (PDEs) that describe a wide class of models in natural sciences. Such systems appear within typical constructions of the homological algebra as complexes of differential operators describing compatibility conditions for overdetermined PDEs. Additional assumptions on the ellipticity/parameter-dependent ellipticity of the differential complexes provide a wide range of elliptic, parabolic, and hyperbolic operators. In particular, most equations related to modern mathematical physics are generated by the de Rham complex of differentials on exterior differential forms. These include the equations based on elliptic Laplace and Lamé type operators; the parabolic heat and mass transfer equations; the Euler type and Navier-Stokes type equations in hydrodynamics; the hyperbolic wave equation and the Maxwell equations in Electrodynamics; the Klein–Gordon equation in relativistic quantum mechanics; and so on. The advantages of our approach are that (1) the proposed method of PDE generation covers a broad class of generated systems, especially in high dimensions, due to different underlying algebraic structures and (2) it enables a direct constructing of the fundamental solutions/parametrices for the generated systems.

Cadabra and python algorithms in general relativity and cosmology I: Generalities

The aim of this work is to present a series of concrete examples which illustrate how the computer algebra system Cadabra can be used to manipulate expressions appearing in General Relativity and other gravitational theories. We highlight the way in which Cadabra's philosophy differs from other systems with related functionality. The use of various new built-in packages is discussed, and we show how such packages can also be created by end-users directly using the notebook interface. The current paper focuses on fairly generic applications in gravitational theories, including the use of differential forms, the derivation of field equations and the construction of their solutions.

Analytical approach to control the irreversibilities in a chemically reactive nanofluid flow over a Darcy–Forchheimer medium with nonuniform heat source/sink

AbstractIn real‐world processes, such as heat transfer, fluid flow, and chemical reactions, irreversibility occurs frequently, which induces an augment in entropy. The optimization of the entropy generation in a mechanical system is one of the fundamental areas of research to channel the energy of the system for utilization. This research exhibits an entropy generation in a nanofluid flow drenched in a fluid‐saturated non‐Darcy porous medium toward a stagnation point. The flow line also experiences the existence of nonuniform heat generation/absorption following the first‐order chemical reaction, which enhances the applicability of this model in different domains of science and engineering. A special form of the Lie group approach is applied to revert the governing partial differential system into its self‐similar ordinary differential form. The solutions of the transformed nonlinear ODEs are acquired analytically through the DTM ‐Padé as well as numerically by the Runge–Kutta–Fehlberg (RKF‐45) method coupled with the shooting process. The fallouts are vividly sketched graphically. One of the upshots reveals that the escalation of space and temperature‐reliant heat absorption parameters can optimize the thermal irreversibility of the system and so as the total entropy generation. Meanwhile, by incrementing the Darcy parameter from 0 to 0.8, the wall shear stress decays by 23.7%, whereas with the same hike in the Forchheimer resistance factor, declines the rate of heat and mass transfer approximately by 2.6% and 3.6%, respectively. Another upshot reveals that with the escalation of the space‐dependent heat source parameter from 0.1 to 0.5, the Nusselt number significantly decreased by 23.9%. Meanwhile boosting the temperature‐reliant heat sink parameter from ‐0.1 to ‐0.5 causes an inclination in the rate of heat transportation by 10.5%. Moreover, it is found that mass transport irreversibility can be controlled by enhancing the constructive chemical reaction rate. We hope that this study will be beneficial for many engineering and industrial processes, particularly in geothermal energy extraction, nuclear waste disposal management, groundwater filtration machinery and food processing equipment.

On Locally Projectively Flat Finsler Space of a Special Exponential Metric with Constant Flag Curvature

From the point of view of Hilbert's fourth problem, Finsler metrics on an open subset of $\mathbb{R}^n$ with positive geodesics that are straight lines are known as locally projectively flat Finsler metrics. In this article, we have studied such projectively flat $(\alpha,\beta)$-metrics in the form of the special exponential Finsler metric, where $\alpha$ is a Riemannian metric and $\beta$ is a differential 1-form. We found that the special exponential metric is locally projectively flat if and only if $\alpha$ is locally projectively flat and $\beta$ is parallel with respect to $\alpha$. Furthermore, we obtained the flag curvature and proved that the special exponential metric is locally Minkowskian.

ON IDENTIFICATION OF PARAMETERS OF A NONLINEAR CONSOLIDATION MODEL OF SANDY SOIL

Previously, the authors of this article formulated a physically and geometrically nonlinear formulation of the problem of porous fluid-saturated medium deformation during fluid filtration (consolidation problem) in terms of the rate of solid phase displacement and the change in pore pressure in differential and variational forms. The developed consolidation model takes into account changes in the porosity and permeability of the medium during deformation. Deformation-type constitutive relations are used in the model. The developed consolidation model can be used to simulate non-stationary processes in the soil, for example, the formation of ruts and unevenness of dirt roads, as well as to calculate the uneven settlement of engineering structures. This work is devoted to the experimental determination of the deformation and strength properties of water-saturated sandy soils, which is the next stage in the consolidation process simulation. The results of the experimental determination of the bulk and shear properties of sandy soil using the ASIS automated complex (OOO NPP “Geotek”) are presented. The studies were carried out on three quartz sands of various grain sizes. To determine the volumetric moduli of dry and water-saturated sandy soils, compression tests were carried out under a continuously growing vertical load at a constant strain rate. The experiments were carried out for various strain rates in the range from 3·10–6 to 3·10–3 s–1. According to the experimental results, the bulk properties do not depend on the strain rate in the specified range. Deformation and strength shear characteristics of sandy soils were determined by the method of multiplanar shear, approximating a simple shear. The tests were carried out under a kinematically applied shear load with a given constant strain rate according to the scheme of consolidated-drained shear. The dependences of the deformation and strength properties of coarse and fine quartz sands on the shear strain rate in the range from 2·10–4 tо 4·10–3 s–1 were studied. Increasing, decreasing and nonmonotonic dependences of the internal friction angle on the shear strain rate were obtained for dry and water-saturated sands of various grain sizes. For water-saturated sands, the maximum spread in the values of the internal friction angle for different strain rates does not exceed 7 %. A technique has been developed for recalculating the obtained properties and experimental dependences into the parameters of the proposed sandy soil consolidation model.

Analysis And Spectral Theory Of Neck-Stretching Problems

We study the mapping properties of a large class of elliptic operators PT in gluing problems where two non-compact manifolds with asymptotically cylindrical geometry are glued along a neck of length 2T. In the limit where T→∞, we reduce the question of constructing approximate solutions of PTu=f to a finite-dimensional linear system, and provide a geometric interpretation of the obstructions to solving this system. Under some assumptions on the real roots of the model operator P0 on the cylinder, we construct a Fredholm inverse for PT with good control on the growth of its norm. As applications of our method, we study the decay rate and density of the low eigenvalues of the Laplacian acting on differential forms, and give improved estimates for compact G2-manifolds constructed by twisted connected sum. We relate our results to the swampland distance conjectures in physics.

Dynamical Mechanism of Memristor and its Phase Transition

Abstract A device is defined as the memristor if it exhibits a pinched hysteresis loop in the current-voltage plane, and the loop area shrinks with the increasing of driven frequency until it gets a single-valued curve. However, it is still limited to explain the underlying mechanism for these fingerprints. In this letter, we propose the differential form of the memristor function, and we disclose the dynamical mechanism of the memristor according to the differential form. The symmetry of the curve is only determined by the driven signal, and the shrinking loop area results from the shrinking area enclosed by driven signal and the time coordinate axis. Significantly, we find the condition for the phase transition of a memristor, and the resistance switches between the positive resistance, local zero resistance, and local negative resistance. This phase transition is confirmed in HP memristor. These results advance the understanding of the dynamics mechanism and phase transition of memristor.

Locally-Verifiable Sufficient Conditions for Exactness of the Hierarchical B-spline Discrete de Rham Complex in $$\mathbb {R}^n$$

AbstractGiven a domain $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on $$\Omega $$ Ω , and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When $$\Omega $$ Ω is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such $$\Omega $$ Ω , we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements—these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them.

Manipulator Differential Kinematics: Part 2: Acceleration and Advanced Applications

This is the second and final article on the tutorial on manipulator differential kinematics. In Part 1, we described a method of modelling kinematics using the elementary transform sequence (ETS), before formulating forward kinematics and the manipulator Jacobian. We then described some basic applications of the manipulator Jacobian including resolved-rate motion control (RRMC), inverse kinematics (IK), and some manipulator performance measures. In this article, we formulate the second-order differential kinematics, leading to a definition of manipulator Hessian. We then describe the differential kinematics' analytical forms, which are essential to dynamics applications. Subsequently, we provide a general formula for higher-order derivatives. The first application we consider is advanced velocity control. In this section, we extend resolved-rate motion control to perform sub-tasks while still achieving the goal before redefining the algorithm as a quadratic program to enable greater flexibility and additional constraints. We then take another look at numerical inverse kinematics with an emphasis on adding constraints. Finally, we analyse how the manipulator Hessian can help to escape singularities. We have provided Jupyter Notebooks to accompany each section within this tutorial. The Notebooks are written in Python code and use the Robotics Toolbox for Python, and the Swift Simulator to provide examples and implementations of algorithms. While not absolutely essential, for the most engaging and informative experience, we recommend working through the Jupyter Notebooks while reading this article. The Notebooks and setup instructions can be accessed at https://github.com/jhavl/dkt.

Topological, Differential Geometry Methods and Modified Variational Approach for Calculation of the Propagation Time of a Signal, Emitted by a GPS-Satellite and Depending on the Full Set of 6 Kepler Parameters

Abstract In preceding publications a mathematical approach has been developed for calculation of the propagation time of a signal, emitted by a moving along an elliptical orbit satellite and accounting also for the General Relativity Theory (GRT) Effects. So far, the formalism has been restricted to one dynamical parameter (the true anomaly or the eccentric anomaly angle). In this paper the aim is to extend the formalism to the case, when also the other five Kepler parameters will be changing and thus, the following important problem can be stated: if two satellites move on two space-distributed orbits and they exchange signals, how can the propagation time be calculated? This paper requires the implementation of differential geometry and topological methods. In this approach, the action functional for the propagation time is represented in the form of a quadratic functional in the differentials of the Kepler elements, consequently the problem is related to the first and second quadratic forms from differential geometry. Such an approach has a clear advantage, because if the functional is written in terms of Cartesian coordinates X, Y, Z, the extremum value after the application of the variational principle is shown to be the straight line - a result, known from differential geometry, but not applicable to the current problem of signal exchange between satellites on different orbits. So the known mapping from celestial mechanics is used, when by means of a transformation the 6 Kepler parameters are mapped into the cartesian coordinates X, Y, Z. This is in fact a submersion of a manifold of 6 parameters into a manifold of 3 parameters. If a variational approach is applied with respect to a differential form in terms of the differentials of the Kepler parameters, the second variation will be different from zero and the Stokes theorem can be applied, provided that the second partial derivatives of the Cartesian coordinates with respect to the Kepler parameters are assumed to be different from zero. ‘From topology this requirement is equivalent to the existence of the s.c. Morse functions (non-degenerate at the critical points). In the given case it has been shown that Morse function cannot exist with respect to each one of the Kepler parameters- Morse function cannot be defined with respect to the omega angle.

Variational discretizations of ideal magnetohydrodynamics in smooth regime using structure-preserving finite elements

We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approaches developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincaré principle, and to further exploit the geometrical structure of the problem, vector fields are represented by their action as Lie derivatives on differential forms of any degree. The resulting semi-discrete approximations are shown to conserve the total mass, entropy and energy of the solutions for a wide class of finite element approximations. In addition, the divergence-free nature of the magnetic field is preserved in a pointwise sense and a time discretization is proposed, preserving those invariants and giving a reversible scheme at the fully discrete level. Numerical simulations are conducted using spline elements to verify the accuracy of our approach and its ability to preserve the invariants for several test problems.