Related papers: A new formula for rotation number

For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of…

We study the relationship between the smoothness of a plane curve and that of its evolute, especially in the cases where the parent curve is no more two or three times continuously differentiable, and exhibit the same kind of apparent…

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from…

The main result is a wall crossing formula for central projections defined on submanifolds of a real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the…

For a continuous map on a topological graph containing a unique loop S, it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and…

In this article, firstly, some simple and smoothness properties of the weighted numerical radius and the weighted Crawford number functions are investigated. Then, some generalization formulas for lower and upper bounds of the weighted…

The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…

Using a general relativistic exact model for spherical structures in a cosmological background, we have put forward an algorithm to calculate the test particle geodesics within such cosmological structures in order to obtain the velocity…

To determine the global root number of an elliptic curve defined over a number field, one needs to understand all the local root numbers. These have been classified except at places above 2, and in this paper we attempt to complete the…

In computational 3D geometric problems involving rotations, it is often that people have to convert back and forth between a rotational matrix and a rotation described by an axis and a corresponding angle. For this purpose, Rodrigues'…

In this paper we consider the normal map of a closed plane curve as a vector field on the cylinder. We interpret the critical points geometrically and study their Poincar\'{e} index, including the points at infinity. After projecting the…

The winding number is a concept in complex analysis which has, in the presence of chiral symmetry, a physics interpretation as the topological index belonging to gapped phases of fermions. We study statistical properties of this topological…

We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as…

In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover,…

The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits…

We derive a simple, fast one-dimensional integral for the equatorial rotation curve of a thin disk with surface density Sigma(R) modeled as a spheroid with axis ratio q. The result is simpler than standard expressions even in the limit of…

A formula expressing a point of order 8 on an elliptic curve, in terms of the roots of the associated cubic polynomial, is given. Doubling such a point yields a point of order 4 distinct from the well-known points of order 4 given in…

We establish several contraction formulas for Kirchhoff index. We relate Kirchhoff index with some other metrized graph invariants. By applying our contraction formulas successively when the graph is a tree, we derive new formulas for…

We consider curves which go around Whitney umbrella. Then we consider the geodesic and the normal curvatures, ruled surfaces generated by the normal vector and normal developable surfaces with respect to the tangent and bi-tangent vectors…

Generalized product formulas and index transforms, involving products of Whittaker's functions of different indices are established and investigated. The corresponding inversion formulas are found. Particular cases cover index transforms…