Related papers: Application of complex-valued functions in plane d…
We study complex analytic (possibly singular) projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the…
We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained…
A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.
In this note we compute some enumerative invariants of real and complex projective spaces by means of some enriched graphs called floor diagrams.
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
Geometric programming problem is a powerful tool for solving some special type non-linear programming problems. It has a wide range of applications in optimization and engineering for solving some complex optimization problems. Many…
We review and extend a technique for recovering a smooth function from its averages over a wide class of curves in a general region of Euclidean space. The method is based on complexification of the underlying vector fields defining the…
A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic…
Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal…
Eliminating the arbitrary coefficients in the equation of a generic plane curve of order $n$ by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to…
We address the question of identifying non-smooth points in affine real algebraic varieties. A simple algebraic criterion will be formulated and proven. As an application we can answer several questions about the configuration spaces of…
It is well known that the Gauss map for a complex plane curve is birational, whereas the Gauss map in positive characteristic is not always birational. Let $q$ be a power of a prime integer. We study a certain plane curve of degree…
Let $\mathcal{C}$ be a plane curve given by an equation $f(x,y)=0$ with $f\in K[x][y]$ a monic squarefree polynomial. We study the problem of computing an integral basis of the algebraic function field $K(\mathcal{C})$ and give new…
The embeddings of complex plane projective curves in the plane are a cornerstone of the topological study of algebraic varieties. In this work, we deal with the local and global aspects of these embeddings, with a special attention to its…
The efficacy of using complex numbers for understanding geometric questions related to polar equations and general cycloids is demonstrated.
We give examples of smooth plane quartics over $\mathbb{Q}$ with complex multiplication over $\overline{\mathbb{Q}}$ by a maximal order with primitive CM type. We describe the required algorithms as we go, these involve the reduction of…
We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most…
In this article we present pictorially the foundation of differential geometry which is a crucial tool for multiple areas of physics, notably general and special relativity, but also mechanics, thermodynamics and solving differential…
Using the method of moving frames we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is…
The correct computation of orbits of discrete dynamical systems on the interval is considered. Therefore, an arbitrary-precision floating-point approach based on automatic error analysis is chosen and a general algorithm is presented. The…