Monatshefte für Mathematik最新文献

On the circle of radius R centred at the origin, consider a “thin” sector about the fixed line (y = alpha x) with edges given by the lines (y = (alpha pm epsilon ) x), where (epsilon = epsilon _R rightarrow 0) as ( R rightarrow infty ). We establish an asymptotic count for (S_{alpha }(epsilon ,R)), the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of (epsilon ) and on the rationality/irrationality type of (alpha ). In particular, we demonstrate that if (alpha ) is Diophantine, then (S_{alpha }(epsilon ,R)) is asymptotic to the area of the sector, so long as (epsilon R^{t} rightarrow infty ) for some ( t<2 ).

Anosov diffeomorphisms are an important class of dynamical systems with many peculiar properties. Ever since they were introduced in the sixties, it has been an open question which manifolds can admit such diffeomorphisms, where tori of dimension greater than or equal to two are the typical examples. It is conjectured that the only manifolds supporting an Anosov diffeomorphism are finitely covered by a nilmanifold, a type of manifold closely related to rational nilpotent Lie algebras. In this paper, we study the existence of Anosov diffeomorphisms for a large class of these nilpotent Lie algebras, namely the ones that can be realized as a rational form in a Lie algebra associated to a graph. From a given simple undirected graph, one can construct a complex c-step nilpotent Lie algebra, which in general contains different non-isomorphic rational forms, as described by the authors in previous work. We determine precisely which forms correspond to a nilmanifold admitting an Anosov diffeomorphism, leading to the first class of complex nilpotent Lie algebras having several non-isomorphic rational forms and for which all the ones that are Anosov are described. In doing so, we put a new perspective on certain classifications in low dimensions and correct a false result in the literature.

We consider the following class of quasilinear Schrödinger equations introduced in plasma physics and nonlinear optics with Stein–Weiss convolution parts

where (kappa in mathbb {R}backslash {0}) is a parameter, (beta >0), (0<mu <2) with (0<2beta +mu <2) and H is the primitive of h that fulfills the critical exponential growth in the Trudinger–Moser sense. For (kappa <0): (i) via using a change of variable argument and the mountain-pass theorem, we investigate the existence of ground state solutions only assuming that (Vin C^0(mathbb {R}^2,mathbb {R}^+)) and (inf _{x in mathbb {R}^2}V(x)>0), which complements and generalizes the problems proposed in our recent work in Alves and Shen (J Differ Equ 344:352–404, 2023); (ii) by developing a new type of Trudinger–Moser inequality, we establish a Pohoz̆aev type ground solution by the constraint minimization approach when (Vequiv 1). Moreover, if (kappa >0) is small, combining the mountain-pass theorem and Nash–Moser iteration procedure, we obtain the existence of nontrivial solutions, where the asymptotical behavior is also considered when (kappa rightarrow 0^+). It seems that the results presented above are even new for the case (kappa =0).

In this research, we adopt a comprehensive approach to address the multifractal and fractal analysis problem. We introduce a novel definition for the general Hausdorff and packing measures by considering sums involving certain functions and variables. Specifically, we explore the sums of the form

where (mu ) represents a Borel probability measure on (mathbb R^d), and q and t are real numbers. The functions h and g are predetermined and play a significant role in our proposed intrinsic definition. Our observation reveals that estimating Hausdorff and packing pre-measures is significantly easier than estimating the exact Hausdorff and packing measures. Therefore, it is natural and essential to explore the relationships between the Hausdorff and packing pre-measures and their corresponding measures. This investigation constitutes the primary objective of this paper. Additionally, the secondary aim is to establish that, in the case of finite pre-measures, they possess a form of outer regularity in a metric space X that is not limited to a specific context or framework.

In the present article, we are focused to study the sharp estimates of the pre-Schwarzian and Schwarzian norms for subclasses of univalent functions. We will generalize the results of Carrasco and Hernández (Anal Math Phys 13(2):22, 2023) to the case of Janowski convex mappings in terms of the value (h^{prime prime }(0)). We will also derive the sharp bound of pre-Schwarzian norm for a subclass of harmonic mappings whose fixed analytic part is a convex function of order (alpha (0 le alpha <1)).