Exact Differential Forms Research Articles

Linking integrals on rank-1-symmetric spaces

In symmetric spaces of rank one we obtain a right inverse of the Cartan differential on exact differential forms as an integral operator. As a corollary we extend Gauss’ linking integral beyond space forms.

Cohomology of annuli, duality and L∞-differential forms on Heisenberg groups

In the last few years the authors proved Poincaré and Sobolev type inequalities in Heisenberg groups Hn for differential forms in the Rumin's complex. The need to substitute the usual de Rham complex of differential forms for Euclidean spaces with the Rumin's complex is due to the different stratification of the Lie algebra of Heisenberg groups. The crucial feature of Rumin's complex is that dc is a differential operator of order 1 or 2 according to the degree of the form.Roughly speaking, Poincaré and Sobolev type inequalities are quantitative formulations of the well known topological problem whether a closed form is exact. More precisely, for suitable p and q, we mean that every exact differential form ω in Lp admits a primitive ϕ in Lq such that ‖ϕ‖Lq≤C‖ω‖Lp. The cases of the norm Lp, p≥1 and q<∞ have been already studied in a series of papers by the authors. In the present paper we deal with the limiting case where q=∞: it is remarkable that, unlike in the scalar case, when the degree of the forms ω is at least 2, we can take q=∞ in the left-hand side of the inequality. The corresponding inequality in the Euclidean setting RN (p=N and q=∞) was proven by Bourgain & Brezis.

On the BMY inequality on surfaces

In this paper, we are concerned with the relation between the ordinarity of surfaces of general type and the failure of the BMY inequality in positive characteristic. We consider semistable fibrations where S is a smooth projective surface and C is a smooth projective curve. Using the exact sequence relating the locally exact differential forms on S, C, and S/C, we prove an inequality relating and c 2 for ordinary surfaces which admit generically ordinary semistable fibrations. This inequality differs from the BMY inequality by a correcting term which vanishes if the fibration is ordinary.

Evaluation of the 4-D singular and near singular potential integrals via the Stokes' theorem

A closed-form solution for singular and near-singular double surface integrals arising in boundary integral equations of electrostatics is given for the case of arbitrary coplanar polygonal surfaces. To date, a limited number of closed-form solutions for these integrals were published only for the case of coincident triangular surfaces. The second result of the paper is the evaluation method itself, which is extendable to non-coplanar and non-polygonal surfaces. The main idea of the evaluation method is to construct exact differential forms to perform integration via the Stokes' theorem. Free of coordinates, the differential forms approach has several important advantages over the traditional algebraic singularity subtraction and singularity cancellation methods. Numerical tests on triangles show that the proposed method maintains accuracy even as the aspect ratio of a triangle tends to infinity while the existing methods fail in that limit. Thus, the results of this paper are expected to greatly improve the accuracy and efficiency of the computational electrostatics codes. We also show that the proposed method is extendable to other kernels, e.g., Helmholtz, and therefore has the potential to greatly speed up the matrix build of the computational electromagnetics and acoustics codes.

Introduction to Econophysics

The future contains terms (V) that are valid at all times, and terms (U) that are presently unknown. In economics, (V) and (U) correspond to ex ante and ex post, in physics to conservative and not conservative, in calculus, to exact and not exact differential forms or to Riemann and Stokes integrals, and to linear or non-linear equations, in statistics to real and probable terms. Apparently, the authors may represent the (V) and (U) structure in economics and physics by calculus, probability theory, by non-linearity, and chaos theory. The present paper applies Stokes integrals to double entry accounting. The resulting laws replace neoclassical theory and correspond to the first and second laws of thermodynamics. Economics and physics have the same structure, leading to the name econophysics. Production is a two level cycle with cheap production, expensive sales, corresponding to the Carnot cycle of a motor with cold air and hot exhaust. In a running motor, efficiency, the difference between hot and cold, becomes always higher. In an economy the gap between rich and poor always rises.

Elliptic functions and maximal unitarity

Scattering amplitudes at loop level can be reduced to a basis of linearly independent Feynman integrals. The integral coefficients are extracted from generalized unitarity cuts which define algebraic varieties. The topology of an algebraic variety characterizes the difficulty of applying maximal cuts. In this work, we analyze a novel class of integrals whose maximal cuts give rise to an algebraic variety with irrational irreducible components. As a phenomenologically relevant example we examine the two-loop planar double-box contribution with internal massive lines. We derive unique projectors for all four master integrals in terms of multivariate residues along with Weierstrass' elliptic functions. We also show how to generate the leading-topology part of otherwise infeasible integration-by-parts identities analytically from exact meromorphic differential forms.

On a canonical class of Green currents for the unit sections of abelian schemes

We show that on any abelian scheme over a complex quasi-projective smooth variety, there is a Green current for the zero-section, which is axiomatically determined up to \partial and \bar\partial -exact differential forms. On an elliptic curve, this current specialises to a Siegel function. We prove generalisations of classical properties of Siegel functions, like distribution relations and reciprocity laws. Furthermore, as an application of a refined version of the arithmetic Riemann-Roch theorem, we show that the above current, when restricted to a torsion section, is the realisation in analytic Deligne cohomology of an element of the (Quillen) K_1 group of the base, the corresponding denominator being given by the denominator of a Bernoulli number. This generalises the second Kronecker limit formula and the denominator 12 computed by Kubert, Lang and Robert in the case of Siegel units. Finally, we prove an analog in Arakelov theory of a Chern class formula of Bloch and Beauville, where the canonical current plays a key role.

Application of local coordinates rectification in linearization of selected parameters of dynamic nonlinear systems

Purpose – The purpose of this paper is to aim to an application of element of the theory of differential geometry for building the state space transformation, linearizing nonlinear dynamic system into a linear form. Design/methodology/approach – It is assumed that the description of nonlinear electric circuits with concentrated parameters or electromechanical systems is given by nonlinear system of differential equations of first order (state equations). The goal is to find transformation which leads nonlinear state equation (written in one coordinate system) to the linear in the other – sought coordinate system. Findings – The necessary conditions fulfilled by nonlinear system undergoing linearization process are presented. Numerical solutions of the nonlinear equations of state together with linearized system obtained from direct transformation of the state space are included (transformation input – the state of the nonlinear system). Originality/value – Application of first order exact differential forms for determining the transformation linearizing the nonlinear state equation. Simple linear models obtained with the use of the linearizing transformation are very useful (mainly because of the known and well-mastered theory of linear systems) in solving of various practical technical problems.

Topology and Geometry of Smectic Order on Compact Curved Substrates

Smectic order on arbitrary curved substrate can be described by a differential form of rank one (1-form), whose geometric meaning is the differential of the local phase field of the density modulation. The exterior derivative of 1-form is the local dislocation density. Elastic deformations are described by superposition of exact differential forms. We use the formalism of differential forms to systematically classify and characterize all low energy smectic states on torus as well as on sphere. A two dimensional smectic order confined on either manifold exhibits many topologically distinct low energy states. Different states are not accessible from each other by local fluctuations. The total number of low energy states scales as the square root of the system area. We also address the energetics of 2D smectic on a curved substrate and calculate the mean field phase diagram of smectic on a thin torus. Finally, we discuss the motion of disclinations for spherical smectics as low energy excitations, and illustrate the interesting connection between spherical smectic and the theory of elliptic functions.

Topology of Smectic Order on Compact Substrates

Smectic orders on curved substrates can be described by differential forms of rank one (1-forms), whose geometric meaning is the differential of the local phase field of density modulation. The exterior derivative of the 1-form is the local dislocation density. Elastic deformations are described by superposition of exact differential forms. Applying this formalism to study smectic order on a torus as well as on a sphere, we find that both systems exhibit many topologically distinct low energy states that can be characterized by two integer topological charges. The total number of low energy states scales as the square root of the substrate area. For a smectic on a sphere, we also explore the motion of disclinations as possible low energy excitations, as well as its topological implications.

Stabilizability conditions in terms of critical Hamiltonians and symbols

Symmetric functions of critical Hamiltonians, called symbols, were used in such problems of nonlinear control as the characterization of symmetries and feedback invariants. We derive here a stabilizability condition in the class of almost continuous feedback controls based on symbols. The methodology proposed consists of defining a selector of the multivalued covector field satisfying an extra degree condition on a sphere close to the origin. Then the above selector is extended to some neighborhood in order to get an exact differential form. Integration of that form gives us a control Lyapunov function candidate (Theorem 3.1). The condition proposed is applied to the analysis of a planar control-affine system with polynomial homogeneous vector fields. Unlike known results in the literature, we consider the case when each vector field vanishes at the origin and the control is bounded. We give stabilizability conditions explicitly in terms of the control system parameters (Theorem 4.1).

Growth of a primitive of a differential form

For an exact differential form on a Riemannian manifold to have a primitive bounded by a given function f, by Stokes it has to satisfy some weighted isoperimetric inequality. We show the converse up to some constants if M has bounded geometry. For a volume form, it suffices to have the inequality (|Ω|≤∫ ∂Ω fdσ for every compact domain Ω⊂M). This implies in particular the “well-known” result that if M is the universal covering of a compact Riemannian manifold with non-amenable fundamental group, then the volume form has a bounded primitive. Thanks to a recent theorem of A.Z ˙uk, we also obtain that if the fundamental group is infinite, the volume form always has a primitive with linear growth.

Note on the nonequilibrium partition function and generalized potentials

In this paper, we present a nonperturbative method of calculating the nonequilibrium partition function appearing in the nonequilibrium ensemble method for dilute gases. A set of first-order partial differential equations is derived for the nonequilibrium partition function. The set can be put into an exact differential form and its integral gives the nonequilibrium partition function with which various macroscopic variables can be expressed. In particular, on solving the aforementioned partial differential equations for the derivatives, the nonequilibrium constitutive relations are directly obtained for the generalized potentials of the substance which are conjugate to the nonconserved variables such as stresses, heat fluxes, diffusion fluxes, and so on. The results for a couple of special cases are presented without proof.

On the Onsager Variation Principle and its Generalization for Nonlinear Irreversible Thermodynamics

The Onsager variation principle is examined from the viewpoint of the thermodynamic analogue of the D'Alembert principle in mechanics when the irreversible processes are linear and thus the system is near equilibrium. The thermodynamic D'Alembert principle is shown to be a precursor to the Onsager variation principle. The thermodynamic D'Alembert principle is then generalised to the cases of nonlinear irreversible processes occurring removed from equilibrium and a generalised form of the Onsager variation principle is obtained under some restricting conditions. The restricted variation principle so deduced has an accompanying exact differential form generalising the Clausius entropy differential (equilibrium Gibbs relations) and contains in it the essence of the thermodynamics of irreversible processes in systems where non-linear transport processes occur. An example is given for the nonlinear dissipation function in the variation functional. The evolution equations for fluxes are shown to yield those known in the literature.

A linear algebraic framework for dynamic feedback linearization

To any accessible nonlinear system we associate a so-called infinitesimal Brunovsky form. This gives an algebraic criterion for strong accessibility as well as a generalization of Kronecker controllability indices. An output function which defines a right-invertible system without zero dynamics is shown to exist if and only if the basis of the Brunovsky form can be transformed into a system of exact differential forms. This is equivalent to the system being differentially flat and hence constitutes a necessary and sufficient condition for dynamic feedback linearizability.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The principle of virtual work and a symplectic reduction of non-local continuum mechanics

We study a symplectic formulation of continuum mechanics on the Fréchet manifold of (smooth) embeddings J : M → N. From a virtual work principle we derive a generalized Hamiltonian dynamics, which allows to include the description non-hyperelastic media as well as a non-linear and non-local material behaviour. Dividing out the (rigid) translation group, the deformation gradient and the stress tensor appear—via Marsden-Weinstein reduction—as natural geometric objects. The balance law on the reduced phase space becomes a weak equation for exact differential forms, which relates the dynamics of the deformation gradient to the stress. This equation is subjected to an interesting gauge freedom and yields the classical continuum dynamics as well as boundary conditions for the stress.

An exact differential form to check analytical theories

As a result of a Von Zeipel's elimination of periodic arguments, the linear differential form ∑W i dv i , where theW i 's are the constant actions and thev i 's the mean motions of the averaged angles, is exact, because, as we prove here, it is the differential of the principal function averaged over the new angles. We propose to use this exact form to check the analytical expressions developed from an elimination of periodic arguments, and we take as an example the main problem of artificial satellites.

Logarithmically exact differential forms

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Betti numbers and exact differential forms.

Betti numbers and exact differential forms.

On the relation between homological structure of Riemannian spaces and exact differential forms which are invariant under holonomy groups, I

On the relation between homological structure of Riemannian spaces and exact differential forms which are invariant under holonomy groups, I