For a continuous nonvanishing complex-valued function g on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of det G. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

The paper defines and studies new semi-Riemannian generalized Fisher metrics Fisher-like metrics, associated with entropies divergences. Examples of seven such families are provided, based on exponential PDFs. particular case when the basic entropy is a ϕ-deformed one, in sense Naudts, investigated detail, emphasis variation emergent scalar curvatures. Moreover, highlights impact these geometri...

we establish a sharp maximal function estimate for some vector-valued multilinear singular integral operators. as an application, we obtain the $(l^p, l^q)$-norm inequality for vector-valued multilinear operators.

For a continuous complex-valued function g on the real line without zeros, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of detG. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

In this paper, Szabos algorithm is used as the main tool to find locally symmetric left invariant Riemannian metrics on some $4$-dimensional Lie groups. Locally groups constitute an important subclass of with zero-divergence Weyl-tensor so-called C-manifolds. Some properties curvature operator these 4-dimensional C-manifolds are studied.

We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rkG− rkH ≤ 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement. Herzlich gave an optimal upper bound for the lowest eigenvalue of the Dirac operator on spheres with arbitrary Riemannian metrics in [9] using a method developed by Vafa and Wi...