Two-dimensional nonmonotonic conservative maps are recognized to be of relevance in modeling a number of nonlinear systems, for instance, laser acceleration of charged particles @1–6#, and the nonlinear flow of magnetic field lines in fusion machines such as tokamaks and others @7,8#. As opposed to more traditional monotonic versions, nonmonotonic maps are characterized by frequency curves that...

A connection is a device that defines the concept of parallel transport on bundle, identifies fibers over nearby points. Fiber bundles form are natural mathematical framework for gauge filed theories. Also affine most elementary type connection, means transfer tangent vectors manifold from one point to another. In any with positive dimension there an infinite number connection; junctions among ...

A theorem proved by C. C. Hsiung in a recent paper [l]1 may be stated as follows : In a linear space Sn of n ( ̂ 3) dimensions let Nx be a conjugate net and w be a fixed hyperplane ; then the points M, M of intersection of the fixed hyperplane w and the two tangents at a point x of the net Nx describe two conjugate nets Nm, Nm in the hyperplane ir, respectively, and one of the two nets Nm, N¡¡ i...

The aim of this paper is to solve some interesting interpolation problems using rational B-spline curve. If a sequence of planar points and vectors are given then a free-form curve can calculated which interpolates the points and has the given tangent vectors in these points. Our method gives a fast interpolation of these data using extra control points. Then we provide a method which allows to...

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group G, we compute a certain subgroup IO′(G) of the representation ring RO(G). This allows us to prove that a finite perfect group G has a smooth 2–proper ac...

Let F be a linear system on a curve C. In Part I we described a method for studying the secant varieties KJ locally. The varieties V¡ are contained in the rf-fold symmetric product C(</'. In this paper (Part II) we apply the methods from Part I. We give a formula for local tangent space dimensions of the varieties V\ valid in all characteristics (Theorem 2.4). Assume rk V = n+1 and criara = 0 ....

Then, for example, taking (r, s) = (1, 3) gives us (x, y, z) = (3, 4, 5). (2) Geometric solution: Solve X2 + Y 2 = 1 in rationals, where X = xz and Y = y z . Therefore, we are looking for rational points on the circle. Finding real points is easy, just take X = sin θ, Y = cos θ, however this doesn’t help us very much. Instead, fix the point at (−1, 0) and look at lines from that point that inte...

Let L be a line intersecting the sidelines of triangle ABC at X , Y , Z respectively. The lines joining these intercepts to the centroid give rise to six more intercepts on the sidelines which lie on a conic Q(L , G). We show that this conic (i) degenerates in a pair of lines if L is tangent to the Steiner inellipse, (ii) is a parabola if L is tangent to the ellipse containing the trisection po...

1. TANGENT SPHERES. Any four mutually tangent spheres determine six points of tangency. We say that a pair of tangencies {ti , t j } is opposite if the two spheres determining ti are distinct from the two spheres determining t j . Thus the six tangencies are naturally grouped into three opposite pairs, corresponding to the three ways of partitioning the four spheres into two pairs. Altshiller-C...

In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to optimize ground-state approximations, implement real-time evolution and describe elementary ...