Differential Forms Research Articles

Religion in the Thought of the Young Hegel

Religion is one of the central themes of the young Hegel. This is where intellectual problems arise, the treatment of which led him to discover his speculative way of thinking. Starting in the footsteps of Kant’s ethicotheology, Hegel quickly realized that religion cannot be a vehicle for introducing autonomous morality. Under titles such as love and life, he then develops a kind of Spinozist thinking of unification of everything, including the finite and the infinite. However, it turns out that the quasi-divine performance of unity cannot be thought of as such, since thinking is bound to discursive forms of reflection that are always mediated through differences. As soon as the religious performance of unity is to be thought of, it slips away from the form of reflection. This problem can be solved if, on the one hand, the differential form of thinking is brought into a self-application and, on the other hand, difference itself is put into the performance of unity, even if it is named as absolute. The former becomes the nucleus of the figure of negation characteristic for Hegel’s speculative thinking; the latter leads to an understanding of the absolute as spirit, which, according to its self-being, which encompasses difference, is always for the other and is known by the other. Religion brings this to mind in the form of imagination, according to Hegel’s later concept of religion.

The Lavrentiev phenomenon in calculus of variations with differential forms

In this article we study convex non-autonomous variational problems with differential forms and corresponding function spaces. We introduce a general framework for constructing counterexamples to the Lavrentiev gap, which we apply to several models, including the double phase, borderline case of double phase potential, and variable exponent. The results for the borderline case of double phase potential provide new insights even for the scalar case, i.e., variational problems with 0-forms.

Noncommutative Riemannian geometry of Kronecker algebras

We study aspects of noncommutative Riemannian geometry of the path algebra arising from the Kronecker quiver with N arrows. To start with, the framework of derivation based differential calculi is recalled together with a discussion on metrics and bimodule connections compatible with the ⁎-structure of the algebra. As an illustration, these concepts are applied to the noncommutative torus where examples of torsion free and metric (Levi-Civita) connections are given.In the main part of the paper, noncommutative geometric aspects of (generalized) Kronecker algebras are considered. The structure of derivations and differential calculi is explored, and torsion free bimodule connections are studied together with their compatibility with hermitian forms, playing the role of metrics on the module of differential forms. Moreover, for several different choices of Lie algebras of derivations, non-trivial Levi-Civita connections are constructed.

Tristan Needham: Visual differential geometry and forms. A mathematical drama in five acts

Tristan Needham: Visual differential geometry and forms. A mathematical drama in five acts

The Hodge-Dirac operator and Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary

Dabrowski et al. [Spectral metric and Einstein functionals for Hodge–Dirac operator, preprint (2023), arXiv:2307.14877] gave spectral Einstein bilinear functionals of differential forms for the Hodge–Dirac operator [Formula: see text] on an oriented even-dimensional Riemannian manifold. In this paper, we generalize the results of Dabrowski et al. to the cases of 4-dimensional oriented Riemannian manifolds with boundary. Furthermore, we give the proof of Dabrowski–Sitarz–Zalecki-type theorems associated with the Hodge–Dirac operator for manifolds with boundary.

Dynamics of accelerating and decelerating flows: A boundary layer examination with non-similar approach and convective boundary condition

Moisture/air-flow over the aircraft wing and bonnet of a moving car in aerodynamical systems exhibit dynamic behavior characterized by accelerating and decelerating flows. These analysis is essential and applicable in various fields, including physics, engineering, and fluid mechanics. The present investigation aim to explore the dynamics of accelerating and decelerating flows within the framework of boundary layer examination. To achieve these objectives, a non-similar group of transformations is employed to convert the governing dimensional two-variable dynamical system into a dimensionless differential form, with η and ξ as the independent variables. To gain physical understanding of the mathematical equations governing the flow, an efficient bivariate spectral collocation method (BSCM) is employed for analyzing the flow dynamics. This method allows for a thorough investigation of the behavior and characteristics of the accelerating and decelerating Casson flows. Subject to the convergence and residual error analysis of the numerical results, the effects of pertinent parameters on the flow variables were investigated. The findings of this study show that the effects of flow parameters on decelerating and accelerating flows are not completely opposite. Additionally, the Biot parameter positively impacts the velocity and temperature profiles of the two dynamical flows. However, an increase in the Casson fluid parameter β has an opposing effect on the temperature and concentration profiles of both flows.

Practical Aspects for Applying Picard Iterations to the SIR Model Using Actual Data

The updated version of the Picard method for solving systems of differential equations is employed to solve the SIR system. A local performance of the Picard iteration algorithm combined with the Gauss-Seidel approach is applied to the SIR model. The integral form of the SIR model, in addition to the use of Gauss-Seidel philosophy (using the most recent calculated values), achieved more accuracy in the computational work than those obtained using the differential forms. Documented data regarding the spread of corona virus 19 in the Kingdom of Saudi Arabia region from April to the end of December 2020 were used to calculate the corresponding actual values for the parameters and the initial conditions. Due to efficient management and to obtain representable behaviours, we restricted the size of the study to only 1% of the population. The global characteristics of the integral formulation have affected the calculations accurately. The initial conditions and the model’s parameters are established depending on the documented data. The results illustrate the superiority of the updated Picard formulation over the classical Picard within their domain of convergence. The results of this study illuminate and validate the importance of mathematical modeling. These findings can provide valuable insights into mathematical modeling for those involved in environmental health research, especially those responsible for devising strategic plans.

Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms

In this paper we introduce the magnetic Hodge Laplacian, which is a generalization of the magnetic Laplacian on functions to differential forms. We consider various spectral results, which are known for the magnetic Laplacian on functions or for the Hodge Laplacian on differential forms, and discuss similarities and differences of this new “magnetic-type” operator.

Novel Lax pair and many conservation laws to a [formula omitted]-dimensional generalized combined Calogero–Bogoyavlenskii–Schiff-type equation in biohydrodynamics

Under investigation in this paper is a (2+1)-dimensional generalized combined Calogero–Bogoyavlenskii–Schiff-type equation for certain nonlinear phenomena in biohydrodynamics, fluids and plasmas. A novel differential form Lax pair with certain conditions is constructed with an arbitrary function H(y,t), while an iterative operator different from those in the existing literatures is derived and fully verified to satisfy the above Lax pair. Based on the iterative operator, n-fold DT and gDT are derived to obtain the higher-order solutions with n being a positive integer. The nth-order solutions are influenced by the form of H(y,t) in the Lax pair, such as the dark solitons with H(y,t) being a log function whereas the bright solitons with H(y,t) being a completely quadratic function. Additionally, the solution’s format may be impacted by the alteration in the Lax pair’s form caused by H(y,t)=0 under specific circumstances. Two types of the many conservation laws are generated separately by discovering two identity relations, which are independent of the form of H(y,t). It is beneficial in providing a range of physical information for subsequent studies, such as modeling of neural network models for certain fluids as described in this equation.

Numerical computation of joule heating and chemical reaction for the radiated Casson fluid on the slender parabolic surface

AbstractThis work dealing to solve the numerical results of magnetohydrodynamic flow material (Casson fluid) moving over the slender parabolic surface with temperature dependent viscosity. To make the enhancement in the thermal and solutal fields, we have determined the impact of thermal radiation, viscous dissipation, joule heating, and chemical reaction on the considered flow material. Due to huge applications in coating processes and aerodynamic industries, the suction/Injection is assumed at the surface. With the help of assumed assumptions, the governing equations are obtained in partial differential equation forms, and these equations are translated into ordinary differential equations form by using the Levy‐Lee similarity transformations. The translated forms of ODE's are then numerically solved by using the Adams‐Milne (Predictor–Corrector) technique. The graphical and numerical findings are determined by producing the emerging parameters. The decrement in velocity is noted due to fluid parameter and suction coefficient whereas the enhancement happens due to natural convection. The numerical determination of physical quantities including, Nusselt number, Skin friction and Sherwood number are also examined here and compared the findings with two numerical techniques (Adam‐Milnes and Bvp4c method).

Investigation on the thermo‐piezo‐flexoelectric energy harvesting performance of self‐powered microbeam devices considering strain gradient and dual‐phase‐lag effects

AbstractAchieving renewable power for self‐powered electronic devices such as micro/nano‐electro‐mechanical systems is attractive in the fields of energy resources and environment, in this regard, dielectric material structures with excellent energy storage and electromechanical responses have stimulated a surge of research interest. However, in the micro/nanoscale filed, the existing thermo‐electromechanical continuum theories are not capable of fully and effectively characterizing the arising effects, for example, size‐dependent effect and heat lagging effect, etc. To further develop the existing theories, a theoretical prototype for piezo‐flexoelectricity by incorporating strain gradient effect and dual‐phase‐lag heat conduction model is proposed for the first time in this study, and then this proposed model is applied to investigating the energy harvesting performances of a piezo‐flexoelectric micro‐beam under piezoelectric‐thermoelastic coupling. In terms of the differential form of two‐phase elasticity and Hamilton's principle, the governing equations and the boundary conditions for the transverse vibration of a micro‐beam based on the Euler–Bernoulli beam model are derived. By means of the assumed mode method, the influences of the flexoelectricity, the load resistance and the strain gradient on voltage and output power are examined. From the obtained results, it can be aware of that this study not only presents the piezoelectric‐thermoelastic coupling effect of piezo‐flexoelectric configuration in microscale field, but also presents that a piezo‐flexoelectric system at microscale can generate higher power than that reported in previous works. It is hoped that this study could provide some guidelines in designing and optimizing dielectric energy harvesters of micro/nano electronic devices.

A Pseudodifferential Analytic Perspective on Getzler's Rescaling

Inspired by Gilkey's invariance theory, Getzler's rescaling method and Scott's approach to the index via Wodzicki residues, we give a localisation formula for the $\mathbb Z_2$-graded Wodzicki residue of the logarithm of a class of differential operators acting on sections of a spinor bundle over an even-dimensional manifold. This formula is expressed in terms of another local density built from the symbol of the logarithm of a limit of rescaled differential operators acting on differential forms. When applied to complex powers of the square of a Dirac operator, it amounts to expressing the index of a Dirac operator in terms of a local density involving the logarithm of the Getzler rescaled limit of its square.

Efficient passive GDQLL scrutinization of an advanced steady EMHD mixed convective nanofluid flow problem via Wakif–Buongiorno approach and generalized transport laws

Owing to the practical importance of nanofluids and their adjustable thermal capabilities, this study intends to develop a robust generalized differential quadrature local linearization (GDQLL) algorithm for examining realistically the heat and mass aspects of electrically conducting nanofluids during their non-Darcian laminar motion nearby a convectively heating vertical surface of an active electromagnetic actuator (i.e., Riga plate). By invoking Wakif–Buongiorno model and Oberbeck–Boussinesq approximations along with other generalized transport laws (i.e., Cattaneo–Christov and non-Fick’s laws) and Grinberg’s concept, a set of gigantic partial differential equations is stated appropriately in the sense of the boundary layer approximations for describing exhaustively the present EMHD mixed convective nonhomogeneous flow under the passive control strategy of nanoparticles within the nanofluidic medium. Operationally, the dimensionless differential forms of the governing boundary equations are derived properly by introducing reasonable mathematical adjustments into the preliminary formulation. In this case, the differential complexity of the leading differential structure is reduced to a nonlinear coupled system of ordinary differential equations, whose discrete numerical solutions are computed perfectly via a well-structured GDQLL algorithm. As foremost outcomes, it is demonstrated that the nanofluid motion and its surface thermal enhancement rate can be reinforced significantly through the thermal strengthening in the convective heating and mixed convective process as well as via the electromagnetic improvement in the driven aspect of Lorentz’s forces.

Homogeneous Sasakian and 3-Sasakian structures from the spinorial viewpoint

We give a spinorial construction of Sasakian and 3-Sasakian structures in arbitrary dimension, generalizing previously known results in dimensions 5 and 7. Furthermore, we obtain a complete description of the space of invariant spinors on a homogeneous 3-Sasakian space, and show that it is spanned by the Clifford products of invariant differential forms with a certain invariant Killing spinor. Finally, we give a basis for the space of invariant Riemannian Killing spinors on a homogeneous 3-Sasakian space, and determine which of these induce the homogeneous 3-Sasakian structure.

Anisotropic mesh adaptation for high-order finite elements spaces with the log-simplex method. Application to discontinuous Galerkin methods

In this article, a high-order solution-based mesh adaptation method is investigated. This later, which is called the log-simplex method, relies on the approximation of high-order differential form of the solution by a dedicated metric-field. The method is a natural extension of Hessian-based methods to the high-order case.

Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology

We show how the tangent functor extends from ordinary smooth maps to “microformal morphisms” (also called “thick morphisms”) of supermanifolds. Microformal morphisms generalize ordinary maps and correspond to formal canonical relations between the cotangent bundles specified by generating functions depending on position variables on the source manifold and momentum variables on the target manifold (as formal power expansions), regarded as part of the structure. Microformal morphisms act on functions by non-linear (in general) pullbacks. We obtain here non-linear pullbacks of (pseudo)differential forms and show that they respect the de Rham differentials as “non-linear chain maps” that can induce non-linear transformations of cohomology.

Port-Hamiltonian formulations of the incompressible Euler equations with a free surface

In this paper, we present port-Hamiltonian formulations of the incompressible Euler equations with a free surface governed by surface tension and gravity forces, modelling e.g. capillary and gravity waves and the evolution of droplets in air. Three sets of variables are considered, namely (v,Σ), (η,ϕ∂,Σ) and (ω,ϕ∂,Σ), with v the velocity, η the solenoidal velocity, ϕ∂ a potential, ω the vorticity, and Σ the free surface, resulting in the incompressible Euler equations in primitive variables and the vorticity equation. First, the Hamiltonian formulation for the incompressible Euler equations in a domain with a free surface combined with a fixed boundary surface with a homogeneous boundary condition will be derived in the proper Sobolev spaces of differential forms. Next, these results will be extended to port-Hamiltonian formulations allowing inhomogeneous boundary conditions and a non-zero energy flow through the boundaries. Our main results are the construction and proof of Dirac structures in suitable Sobolev spaces of differential forms for each variable set, which provides the core of any port-Hamiltonian formulation. Finally, it is proven that the state dependent Dirac structures are related to Poisson brackets that are linear, skew-symmetric and satisfy the Jacobi identity.

A new improved fractional Tikhonov regularization method for moving force identification

To address the problem of poor recognition accuracy of integer-order Tikhonov regularization method (Tik) in identifying bridge moving loads, a bridge moving load recognition method based on Improved fractional Tikhonov (IF-Tik) regularization method is proposed in this paper. The bridge moving load identification model is established according to the theory of time domain method, and the process of driving a two-axle vehicle on the bridge is simulated. The moving load is represented as the differential form of the kernel function of bending moment response and acceleration response. The differential equation is transformed into a linear system of equations by the discretization method, and solved by the improved fractional-order Tikhonov regularization method. The results show that the IF-Tik method has certain advantages over the F-Tik and Tik methods in terms of recognition accuracy and noise immunity, and the IF-Tik method has a higher accuracy of load recognition, and the recognition error is only 15% of that of the F-Tik method; the recognition results are less affected by the moment response, and the proposed method has good robustness, and is more suitable for the bridge moving load identification.

Toric differential forms and periods of complete intersections

Let n be an even natural number. We compute the periods of any n2-dimensional complete intersection algebraic cycle inside an n-dimensional non-degenerated intersection of a projective simplicial toric variety. Using this information we determine the cycle class of such algebraic cycles. As part of the proof we develop a toric generalization of a classical theorem of Macaulay about complete intersection Artin Gorenstein rings, and we generalize an algebraic cup formula for residue forms due to Carlson and Griffiths to the toric setting.

Interface vanishing of $ d-\delta $ systems

<abstract><p>We introduce $ d-\delta $ systems on differential forms in Eucliean spaces and show the interface vanishing of the solution. This result generalizes previous theorems on stationary and non-stationary Maxwell's equation. Other applications are also given.</p></abstract>