W. K. Schief
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View article: Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation
Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation Open
View article: Canonical reductions of the TED equation: integrable deformations of heavenly-type equations
Canonical reductions of the TED equation: integrable deformations of heavenly-type equations Open
Natural classes of integrability-preserving reductions of a 4+4-dimensional generalisation (TED equation) of the general heavenly equation are recorded. In particular, these reductions lead to integrable ‘deformations’ of various other ava…
View article: Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation
Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation Open
It is shown that a canonical geometric setting of the integrable TED equation is a Kahlerian tangent bundle of an affine manifold. The remarkable multi-dimensional consistency of this 4+4-dimensional dispersionless partial differential equ…
View article: Log-aesthetic curves: Similarity geometry, integrable discretization and variational principles
Log-aesthetic curves: Similarity geometry, integrable discretization and variational principles Open
In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. In the framework of similarity geometry, those curves are characterized as invariant…
View article: Relativistic perfect fluids: integrability of the classical Taub system
Relativistic perfect fluids: integrability of the classical Taub system Open
It is established that the classical Taub system governing isentropic planar motions of perfect fluids in special relativity may be formulated entirely in soliton-theoretic terms. The well-established tools of integrable systems theory may…
View article: A two-dimensional magnetohydrodynamic system: geometric decomposition and canonical reduction
A two-dimensional magnetohydrodynamic system: geometric decomposition and canonical reduction Open
View article: On 2+1-dimensional fibre-reinforced fluid motions: a magnetohydrodynamics nexus
On 2+1-dimensional fibre-reinforced fluid motions: a magnetohydrodynamics nexus Open
An intrinsic geometric decomposition is applied to provide via a canonical third-order nonlinear equation a connection between 2+1-dimensional fibre-reinforced motions of a fluid and a magnetohydrodynamic system. A Lagrange-type parametris…
View article: On non-steady planar motions of fibre-reinforced fluids. Geometry and integrable structure
On non-steady planar motions of fibre-reinforced fluids. Geometry and integrable structure Open
It is shown that the kinematic system describing planar non-steady motions of ideal fibre-reinforced fluids may be reduced to a single two-dimensional third-order partial differential equation in which time enters parametrically. A procedu…
View article: On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability
On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability Open
It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential e…
View article: On Discrete Conjugate Semi-Geodesic Nets
On Discrete Conjugate Semi-Geodesic Nets Open
We introduce two canonical discretizations of nets on generic surfaces, which consist of an one-parameter family of geodesics and its associated family of conjugate lines. The two types of “discrete conjugate semi-geodesic nets” constitute…
View article: On relativistic gasdynamics: invariance under a class of reciprocal-type transformations and integrable Heisenberg spin connections
On relativistic gasdynamics: invariance under a class of reciprocal-type transformations and integrable Heisenberg spin connections Open
A classical system of conservation laws descriptive of relativistic gasdynamics is examined. In the two-dimensional stationary case, the system is shown to be invariant under a novel multi-parameter class of reciprocal transformations. The…
View article: On non-steady planar motions of fibre-reinforced fluids. Geometry and integrable structure
On non-steady planar motions of fibre-reinforced fluids. Geometry and integrable structure Open
It is shown that the kinematic system describing planar non-steady motions of ideal fibre-reinforced fluids may be reduced to a single two-dimensional third-order partial differential equation in which time enters parametrically. A procedu…
View article: On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs
On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs Open
View article: On modulated coupled systems. Canonical reduction via reciprocal transformations
On modulated coupled systems. Canonical reduction via reciprocal transformations Open
View article: On an integrable multi-dimensionally consistent 2 <i>n</i> + 2 <i>n</i> -dimensional heavenly-type equation
On an integrable multi-dimensionally consistent 2 <i>n</i> + 2 <i>n</i> -dimensional heavenly-type equation Open
Based on the commutativity of scalar vector fields, an algebraic scheme is developed which leads to a privileged multi-dimensionally consistent 2 n + 2 n -dimensional integrable partial differential equation with the associated eigenfuncti…
View article: Checkerboard incircular nets: Laguerre geometry and parametrisation
Checkerboard incircular nets: Laguerre geometry and parametrisation Open
View article: On an integrable multi-dimensionally consistent 2n+2n-dimensional heavenly-type equation
On an integrable multi-dimensionally consistent 2n+2n-dimensional heavenly-type equation Open
Based on the commutativity of scalar vector fields, an algebraic scheme is developed which leads to a privileged multi-dimensionally consistent 2n+2n-dimensional integrable partial differential equation with the associated eigenfunction co…
View article: Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principles
Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principles Open
In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. We consider these curves in the framework of the similarity geometry and characteriz…
View article: Discrete projective minimal surfaces
Discrete projective minimal surfaces Open
View article: Log-aesthetic curves as similarity geometric analogue of Euler's elasticae
Log-aesthetic curves as similarity geometric analogue of Euler's elasticae Open
View article: Discrete projective minimal surfaces
Discrete projective minimal surfaces Open
We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models…
View article: Log-aesthetic Curves as Similarity Geometric Analogue of Euler's Elasticae
Log-aesthetic Curves as Similarity Geometric Analogue of Euler's Elasticae Open
In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by…
View article: Random orthogonal matrix simulation with exact means, covariances, and multivariate skewness
Random orthogonal matrix simulation with exact means, covariances, and multivariate skewness Open
View article: Gaussian and mean curvatures for discrete asymptotic nets
Gaussian and mean curvatures for discrete asymptotic nets Open
View article: On the Combinatorics of Demoulin Transforms and (Discrete) Projective Minimal Surfaces
On the Combinatorics of Demoulin Transforms and (Discrete) Projective Minimal Surfaces Open
View article: Circle Complexes and the Discrete CKP Equation
Circle Complexes and the Discrete CKP Equation Open
In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated P…
View article: Rom Simulation with Exact Means, Covariances, and Multivariate Skewness
Rom Simulation with Exact Means, Covariances, and Multivariate Skewness Open
View article: On a discretization of confocal quadrics. I. An integrable systems\n approach
On a discretization of confocal quadrics. I. An integrable systems\n approach Open
Confocal quadrics lie at the heart of the system of confocal coordinates\n(also called elliptic coordinates, after Jacobi). We suggest a discretization\nwhich respects two crucial properties of confocal coordinates: separability and\nall t…
View article: On a discretization of confocal quadrics. I. An integrable systems approach
On a discretization of confocal quadrics. I. An integrable systems approach Open
Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-…
View article: Critical points, Lauricella functions and Whitham-type equations
Critical points, Lauricella functions and Whitham-type equations Open
A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted\nas the equations governing families of critical points of functions obeying the\nclassical linear Darboux equations for conjugate nets.The distinguished role of…