The analytic index of the operator d+δ on a compact orientable Riemannian manifold M is the Euler characteristic of M, χ (M) . This paper extends this result to a class of complete noncompact manifolds, those with finitely generated rational homology and finitely many quasi-periodic ends. The latter term means that there is a neighborhood of each end which is quasi-isometric to a neighborhood o...

Aiming at optimizing the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points M\"obius energy. The gradients are computed with respect Sobolev inner products similar $W^{3/2,2}$-inner product. This leads optimization methods that significantly more efficient and robust than standard techniques based on $L^2$-gradien...

In this paper, we first introduce the notion of $c$-affine functions for $c> 0$. Then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. Moreover, a Hyers–-Ulam stability result for strongly convex functions is shown.

about electronic signatures there are more or less some writings concerning their legal aspect. but in those writings there has been a little attention to credibility and evidential value of the electronic signatures. some questions and ambiguities despite being set before has not been answered at all or answered unsatisfactorily. meanwhile, some authors have made mistakes and some have set tri...

in this paper we consider contact cr-warped product submanifolds of the type $m = n_ttimes_f n_perp$, of a nearly kenmotsu generalized sasakian space form $bar m(f_1‎, ‎f_2‎, ‎f_3)$ and by use of hopf's lemma we show that $m$ is simply contact cr-product under certain condition‎. ‎finally‎, ‎we establish a sharp inequality for squared norm of the second fundamental form and equality case i...

A manifold M is like a curve in 2D and a surface in 3D. It is a set of points with topology (or neighbor structures), and locally looks like R. Example: our earth. Locally it is planar. Rigidly, for a patch U ⊂ M , we have a local coordinate system xU : U 7→ R as i = 1 · n local coordinates. Typically M can be covered by several such patches. (eg, A sphere can be covered by 2 patches but not on...