arXiv research

The paper uses complex-valued functions to simplify plane differential geometry and kinematics.

In this work, the Z$_3$-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algeb…

Hopf algebra structure on the differential algebra of the extended $q$-plane is defined. An algebra of forms which is obtained from the generators of the extended $q$-plane is introduced and its Hopf algebra structure is given.

Geometric structures on surfaces relate to 2-plane distributions in 5D.

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator $x$ is invertible and furthermore working polynomials in $\ln x$ instead of polynomials in $x$. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual H…

For a pair of points in a smooth locally convex surface in 3-space, its mid-plane is the plane containing its mid-point and the intersection line of the corresponding pair of tangent planes. In this paper we show that the limit of mid-planes when one point tends to the other along a direction is the Transon plane of th…

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomor…

Mathematical analysis of Prytz planimeter using sub-Riemannian geometry.

In this paper, by the method of moving planes, we prove the symmetry result which says that classical solutions of Monge-Ampere system in the whole plane are symmetric about some point. Our system under consideration comes from the differential geometry problem.

We present a complete set of criteria for determining A-types of plane-to-plane map-germs of corank one with A-codimension <7, which provides a new insight into the A-classification theory from the viewpoint of recognition problem. As an application to generic differential geometry, we discuss about projections of smoo…

The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of a real plane curve as the Hessian curve of a smooth function. The plane curves …

A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. In this paper, compositions of…

The study finds bounds on metrics with constant curvature in the plane.

In-plane drill rotations are impossible for smooth shells.

Study rectifying curves in 3D multiplicative Euclidean space.

Investigates the vertex curve of smooth surfaces in 3D space, connecting geometry and image analysis.

In this paper we study the general affine geometry of curves in affine space $A^2$. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…

Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.

In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the notion of Birkhoff orthogonality from Banach space theory, we study all curvature types of curves in gauge planes, thus g…

The dimensions of the spaces of $k$-homogeneous $\mathrm{Spin}(9)$-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular inte…

The study identifies unique fluid flow patterns.

Research on refined algebraic domains respecting differential geometry.

The paper explores geometric properties of interception curves on planes and spheres.

The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.

Study of curves in dual space with constant curvature and torsion.

The paper proves global invertibility for certain local diffeomorphisms and biholomorphisms in higher dimensions.

This paper explores 4-planes in Spin(7) manifolds with symplectic structures.

The paper explores how the Gauss curvature of Riemannian surfaces can be represented as the divergence of a vector field.

New framework reveals limits of flexible, periodic thin surfaces.

Model financial dynamics using 2-manifold geometries, revealing the torus as best for cyclical data.

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…

Study k-folding map-germs to understand surface geometry.

We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes through 0 such that all sections of $B$ by these planes are linearly equivalent. Then…

Let $(Σ,p)$ be a pointed Riemann surface of genus $g\geq 1$. For any integer $k\geq 1$, we parametrize the space of meromorphic quadratic differentials on $Σ$ with a pole of order $(k+2)$ at $p$, having a connected critical graph and an induced metric composed of $k$ Euclidean half-planes. The parameters form a finite-…

We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…

Differentiable cutting-plane layers solve parametric mixed-integer linear optimization problems.

The study extends inscription problems to non-Euclidean geometries.

An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature $k$-symplectic formalism is used to provide the optimal control problems associated to some families of partial dif…

Lecture notes introduce Abelian differentials and their flat surfaces, focusing on families and Teichmüller dynamics.

The study explores conformal planes with finite areas.

Linear ODEs are solved by geodesics in hyperbolic geometry.

Study on isoperimetric problem in Randers planes achieving maximum area.

For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well known singularities of generic vector fields on the plane and the singularit…

We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincaré characteristic zero) in ${\bold R}^3$ of constant mean curvature which meet planes $Π_1$ and $Π_2$ in constant contact angles $γ_1$ and $γ_2$ and bound, together with those planes, a…

We construct a two-parameter covariant differential calculus on the quantum $h$-exterior plane. We also give a deformation of the two-dimensional fermionic phase space.

The paper proves a conjecture about the shape of floating bodies.

Only vertical planes are asymptotic to other planes in 3D space.

Harmonic maps from complex plane to hyperbolic space constructed using heat flow.