Pfaffian Equations Research Articles

∞-structures in the integration of involutive distributions

For a system of ordinary differential equations (ODEs) or, more generally, an involutive distribution of vector fields, the problem of its integration is considered. Among the many approaches to this problem, solvable structures provide a systematic procedure of integration via Pfaffian equations that are integrable by quadratures. In this paper structures more general than solvable structures (named -structures) are considered. The symmetry condition in the concept of solvable structure is weakened for -structures by requiring their vector fields be just -symmetries. For -structures there is also an integration procedure, but the corresponding Pfaffian equations, although completely integrable, are not necessarily integrable by quadratures. The well-known result on the relationship between integrating factors and Lie point symmetries for first-order ODEs is generalized for -structures and involutive distributions of arbitrary corank by introducing symmetrizing factors. The role of these symmetrizing factors on the integrability by quadratures of the Pfaffian equations associated with the -structure is also established. Some examples that show how these objects and results can be applied in practice are also presented.

On the Existence of the Conditionally Linear Integral in Conservative Holonomic Systems with Two Degrees of Freedom

For the Lagrange equations of the 2nd kind the problem on the existence of the conditionally linear in the velocities integral, which possesses the property that its total time derivative is identically proportional to the integral itself, is considered. The Lagrange function is assumed to be given in arbitrary generalized coordinates. The conditions for the existence of such integral are reduced to the study of the compatibility of two equations in partial derivatives of the 2nd order for one unknown function of two independent arguments. These equations are written in the invariant form in an arbitrary system of generalized coordinates and the problem is transformed to the investigation of the set of Pfaffian equations.

On the Existence of the Conditionally Linear Integral in Conservative Holonomic Systems with Two Degrees of Freedom

For the Lagrange equations of the 2nd kind the problem of the existence of the conditionally linear in the velocities integral, which possesses the property that its total time derivative is identically proportional to the integral itself, is considered. The Lagrange function is assumed to be given in arbitrary generalized coordinates. The conditions for the existence of such integral are reduced to the study of the compatibility of two equations in partial derivatives of the 2nd order for one unknown function of two independent arguments. These equations are written in the invariant form in an arbitrary system of generalized coordinates and the problem is transformed into the investigation of the set of Pfaffian equations.

Formula omitted]-symmetries of distributions and integrability

An extension of the notion of solvable structure for involutive distributions of vector fields is introduced. It is based on a generalization of the concept of symmetry of a distribution of vector fields, inspired in the extension of Lie point symmetries to C∞-symmetries for ODEs developed in the recent years. The new structures, named C∞-structures, play a fundamental role in the integrability of the distribution: the knowledge of a C∞-structure for a corank k involutive distribution allows to find its integral manifolds by solving k successive completely integrable Pfaffian equations. These results have important consequences for the integrability of differential equations. In particular, we derive a new procedure to integrate an mth-order ordinary differential equation by splitting the problem into m completely integrable Pfaffian equations. This step-by-step integration procedure is applied to fully integrate several equations that cannot be solved by standard procedures.

Macaulay matrix for Feynman integrals: linear relations and intersection numbers

We elaborate on the connection between Gel’fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations for Feynman Integrals. We propose a novel, more efficient algorithm to compute Macaulay matrices, which are used to derive Pfaffian systems of differential equations. The Pfaffian matrices are then employed to obtain linear relations for mathcal{A} -hypergeometric (Euler) integrals and Feynman integrals, through recurrence relations and through projections by intersection numbers.

Solution of matching equations of IDA-PBC by Pfaffian differential equations

Finding the general solution of partial differential equations (PDEs) is essential for controller design in newly developed methods. Interconnection and damping assignment passivity-based control (IDA-PBC) is one of such methods in which the solution to corresponding PDEs which are called matching equations is needed to apply it in practice. In this paper, these matching equations are transformed to corresponding Pfaffian differential equations. Furthermore, it is shown that upon satisfaction of the integrability condition, the solution to the corresponding third-order Pfaffian differential equation may be obtained quite easily. The method is applied to the PDEs of IDA-PBC in some benchmark systems such as Magnetic levitation system, Pendubot, and underactuated cable-driven robot to verify its applicability.

Constantin Carathéodory axiomatic approach and Grigory Perelman thermodynamics for geometric flows and cosmological solitonic solutions

We elaborate on statistical thermodynamics models of relativistic geometric flows as generalisations of G. Perelman and R. Hamilton theory centred around C. Carath\'{e}odory axiomatic approach to thermodynamics with Pfaffian differential equations. The anholonomic frame deformation method, AFDM, for constructing generic off--diagonal and locally anisotropic cosmological solitonic solutions in the theory of relativistic geometric flows and general relativity is developed. We conclude that such solutions can not be described in terms of the Hawking--Bekenstein thermodynamics for hypersurface, holographic, (anti) de Sitter and similar configurations. The geometric thermodynamic values are defined and computed for nonholonomic Ricci flows, (modified) Einstein equations, and new classes of locally anisotropic cosmological solutions encoding solitonic hierarchies.

{mathfrak {S}}_5-equivariant syzygies for the Del Pezzo surface of degree 5

The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in {mathbb {P}}^5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group {mathfrak {S}}_5. We give canonical explicit {mathfrak {S}}_5-invariant Pfaffian equations through a 6times 6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of {mathfrak {S}}_5. Finally, we give {mathfrak {S}}_5-invariant equations for the embedding of Y inside ({mathbb {P}}^1)^5, and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.

Pfaffian Equations and Contiguity Relations of the Hypergeometric Function of Type (<i>k</i>+1, <i>k</i>+<i>n</i>+2) and Their Applications

We study the structures of Pfaffian equations and contiguity relations of the hypergeometric function of type $(k+1,k+n+2)$ by using twisted cohomology groups and the intersection form on them. We apply our results to algebraic statistics; numerical evaluation of the normalizing constants of two way contingency tables with fixed marginal sums.

On the exact maximum likelihood inference of Fisher\u2013Bingham distributions using an adjusted holonomic gradient method

Holonomic function theory has been successfully implemented in a series of recent papers to efficiently calculate the normalizing constant and perform likelihood estimation for the Fisher–Bingham distributions. A key ingredient for establishing the standard holonomic gradient algorithms is the calculation of the Pfaffian equations. So far, these papers either calculate these symbolically or apply certain methods to simplify this process. Here we show the explicit form of the Pfaffian equations using the expressions from Laplace inversion methods. This improves on the implementation of the holonomic algorithms for these problems and enables their adjustments for the degenerate cases. As a result, an exact and more dimensionally efficient ODE is implemented for likelihood inference.

Systems of Pfaffian equations and controlled systems

The relation between the classical theory of Pfaffian systems and the modern theory of controlled systems is discussed. It is shown that this relation helps solve classification problems and terminal control problems for controlled systems.

SOLVING NOT COMPLETELY INTEGRABLE QUANTILE PFAFFIAN DIFFERENTIAL EQUATIONS

The present work deals with quantile Pfaffian differential equations which are constructed using two-dimensional conditional quantiles of multidimensional probability distributions. As it was shown in [3] in case when the initial probability distributions have reproducible conditional quantiles this kind of Pfaan equations is completely integrable and the integral manifold is the conditional quantile of maximum dimension. In this paper we discuss properties of integral manifolds of maximum possible dimension for quantile Pfaffian equations which are not completely integrable. Manifolds of this type are described in terms of conditional quantiles of intermediate dimensions.

Structure of magnetic field lines

In this paper the Hamiltonian structure of magnetic lines is studied in many ways. First it is used vector analysis for defining the Poisson bracket and Casimir variable for this system. Second it is derived Pfaffian equations for magnetic field lines. Third, Lie derivative and derivative of Poisson bracket is used to show structure of this system. Finally, it is shown Nambu structure of the magnetic field lines.

Geometric transitions between Calabi–Yau threefolds related to Kustin–Miller unprojections

We study Kustin–Miller unprojections between Calabi–Yau threefolds, or more precisely the geometric transitions they induce. We use them to connect many families of Calabi–Yau threefolds with Picard number one to the web of Calabi–Yau complete intersections. This result enables us to find explicit description of a few known families of Calabi–Yau threefolds in terms of equations. Moreover, we find two new examples of Calabi–Yau threefolds with Picard group of rank one, which are described by Pfaffian equations in weighted projective spaces.

CANONICAL TRANSFORMATIONS IN THREE-DIMENSIONAL PHASE-SPACE

Canonical transformation in a three-dimensional phase-space endowed with Nambu bracket is discussed in a general framework. Definition of the canonical transformations is constructed based on canonoid transformations. It is shown that generating functions, transformed Hamilton functions and the transformation itself for given generating functions can be determined by solving Pfaffian differential equations corresponding to that quantities. Types of the generating functions are introduced and all of them are listed. Infinitesimal canonical transformations are also discussed. Finally, we show that the decomposition of canonical transformations is also possible in three-dimensional phase space as in the usual two-dimensional one.

Fractional Nambu Mechanics

The fractional generalization of Nambu mechanics is constructed by using the differential forms and exterior derivatives of fractional orders. The generalized Pfaffian equations are obtained and one example is investigated in details.

Residues of codimension one singular holomorphic distributions

For a codimension one locally-free singular holomorphic distribution, we give a residue formula in terms of the conormal sheaf given by Pfaffian equations. We also prove a Baum-Bott type residue formula for singular distributions.

Differential geometry of Lagrange-like webs

1. LAGRANGE-LIKE WEBS LLWt(n, r) ON Xnr We consider a d-web W (d, n, r) of codimension r on a differentiable manifold Xnr formed by d foliations Fu, u = 1, . . . , d, which are in general position onXnr and located onXnr discretely. Because for d ≤ n such a web is parallelizable, in what follows we will assume that d ≥ n+ 1. In this paper, we will consider (n+ 1)-websW (n+ 1, n, r). The foliations Fu defining the web W (n+ 1, n, r) can be given on Xnr by completely integrable systems of Pfaffian equations

Pfaffian equations satisfied by differential modular forms

The ring of (ordinary) isogeny covariant differential modular forms introduced in (3) was shown in (1) to be described by two basic forms introduced in (3) and (1) respectively. We prove that these two forms satisfy a simple triangular system of Pfaffian equations (in characteristic zero). The equation giving the form in (1) is integrable by quadratures which gives a closed form expression for this form; the equation giving the form in (3) is shown not to be integrable by quadratures.

Bounded decomposition in the Brieskorn lattice and Pfaffian Picard–Fuchs systems for Abelian integrals

We suggest an algorithm for derivation of the Picard–Puchs system of Pfaffian equations for Abelian integrals corresponding to semiquasihomogeneous Hamiltonians. It is based on an effective decomposition of polynomial forms in the Brieskorn lattice. The construction allows for an explicit upper bound on the norms of the polynomial coefficients, an important ingredient in studying zeros of these integrals.