We introduce new q-tangent numbers based on the Carlitz q-analog of the Eulerian polynomial and the so-called doubloon combinatorial set-up. Those new q-tangent numbers are polynomials with positive integral coefficients. They are divisible by products of binomials of the form 1+ q, the quotients being q-analogs of the reduced tangent numbers having an explicit combinatorial interpretation.

Automorphic distributions arise in connection with boundary values of modular forms and Maass forms. In most cases, these distributions have antiderivatives that are continuous functions. We shall look at the result of graphing the real vs. imaginary parts of these functions. Because of the automorphic properties of the distributions we consider, the graphs of their antiderivatives are curves w...

In this paper we approach the topology of smooth manifolds using differential tools, as opposed to algebraic ones such as homology or the fundamental group. The main result is the Poincaré-Hopf index theorem, which states the sum of the indices of a vector field with finitely many zeros on a smooth compact oriented boundaryless manifold is equal to the Euler characteristic of the manifold. To l...

For an integral homology 3-sphere embedded asymptotically flatly in an Euclidean space, we find a natural framing extending the standard trivialization on the asymptotically flat part. Suppose M is a 3-dimensional closed smooth manifold which has the same integral homology groups as the 3-sphere S. x0 is a fixed point in M . Embed M in a Euclidean space R such that x0 is the infinite point of t...

We provide an example of two closed sets S1, S2 ⊂ IR such that S1∩S2 = {0}. Yet, at the origin, a Boltyanskii tangent cone C1 to S1 and the Clarke tangent cone C2 to S2 are strongly transversal. This settles a question originally proposed by H. Sussmann. 1 Introduction Let t 7→ x∗(t) and t 7→ u∗(t) be respectively a trajectory and an optimal control for the Mayer problem    minimize φ ( x(...

A number of recent papers concerned various non-degeneracy conditions on embedding and immersions of smooth manifolds in affine and projective spaces defined in terms of mutual positions of the tangent spaces at distinct points, see [1, 2, 3, 4, 6, 7, 9]. Following Ghomi [1], a C-embedded manifold M ⊂ R is called T -embedded if the tangent spaces to M at distinct points do not intersect. For ex...

Levin’s MLS projection operator allows defining a surface from a set of points and represents a versatile procedure to generate points on this surface. Practical problems of MLS surfaces are a complicated non-linear optimization to compute a tangent frame and the (commonly overlooked) fact that the normal to this tangent frame is not the surface normal. An alternative definition of Point Set Su...

We deal with different properties of a smooth and strictly convex body that depend on the behavior of the planar sections of the body parallel to and close to a given tangent plane. The first topic is boundary points where any given convex domain in the tangent plane can be approximated by a sequence of suitably rescaled planar sections (so-called p-universal points). In the second topic, the g...