Let be the modified Bessel function of the first kind of order . Motivated by a conjecture on the convexity of the ratio for , using the monotonicity rules for a ratio of two power series and an elementary technique, we present fully the convexity of the functions , and for on in different value ranges of , which give an answer to the conjecture and extend known results. As consequences, some monotonicity results and new functional inequalities for are established. As applications, an open problem and a conjectures are settled. Finally, a conjecture on the complete monotonicity of for is proposed.

Assume that is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph . We consider the combinatorial multivariable Poincaré series associated with and its counting functions, which encode rich topological information. Using the 'periodic constant' of the series (with reduced variables associated with an arbitrary subset of the set of vertices) we prove surgery formulae for the normalized Seiberg-Witten invariants: the periodic constant associated with appears as the difference of the Seiberg-Witten invariants of and for any .

In the present paper, we consider the nonparametric regression model with random design based on a -valued strictly stationary and ergodic continuous time process, where the regression function is given by , for a measurable function . We focus on the estimation of the location (mode) of a unique maximum of by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve . Within this context, we obtain the consistency with rate and the asymptotic normality results for under mild local smoothness assumptions on and the design density of . Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under the mixing condition. The usefulness of our results will be illustrated in the construction of confidence regions.

In this paper we present an elementary proof of a pointwise radial monotonicity property of heat kernels that is shared by the Euclidean spaces, spheres and hyperbolic spaces. The main result was discovered by Cheeger and Yau in 1981 and rediscovered in special cases during the last few years. It deals with the monotonicity of the heat kernel from special points on revolution hypersurfaces. Our proof hinges on a non straightforward but elementary application of the parabolic maximum principle. As a consequence of the monotonicity property, we derive new inequalities involving classical special functions.

We continue the study of the space  of functions with bounded fractional variation in  of order introduced in our previous work (Comi and Stefani in J Funct Anal 277(10):3373-3435, 2019). After some technical improvements of certain results of Comi and Stefani (2019) which may be of some separated insterest, we deal with the asymptotic behavior of the fractional operators involved as . We prove that the -gradient of a -function converges in to the gradient for all as . Moreover, we prove that the fractional -variation converges to the standard De Giorgi's variation both pointwise and in the -limit sense as . Finally, we prove that the fractional -variation converges to the fractional -variation both pointwise and in the -limit sense as for any given .

Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D'Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D'Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D'Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension.